Devices and methods employing vibration damping in fluids

Abstract

A physical property of a fluid is determined by vibrating a vibration transducer element in the fluid at a vibration frequency wherein the vibration transducer element comprises an elongate member contacting the fluid characterized by a width, a half width equal to half the width, and a length greater than the width, the half width is smaller than the propagation depth of shear waves in fluid under the vibration frequency; measuring vibrations of the vibration transducer element in the fluid at the vibration frequency; and determining a physical property of the fluid based on the measurement of vibration, such as viscosity, viscoelasticity, density, fluid stiffness, loss tangent, storage modulus, loss modulus, or yield stress.

Description

Technical Field

[0001] The present disclosure relates to vibration damping within fluids, including the use of damping to obtain measurements of physical and rheological properties of materials, such as measurements of viscosity. Background Art

[0002] The physical and rheological properties of a fluid can be measured by applying an oscillating stimulus to the fluid and observing the mechanical response of the fluid. Based on the observed mechanical response of the fluid (e.g., the degree of damping and / or stiffness, and / or resonant frequency), measurements of fluid properties such as viscosity, density, storage modulus, loss modulus, and loss tangent can be obtained.

[0003] As examples, the degree of damping may be determined based on the amplitude or change in amplitude of the vibration, the resonant frequency or change in resonant frequency, the rate of decay of the vibration, or the quality (Q) factor or loss factor, where the loss factor is the inverse of the quality factor. Summary of the invention

[0004] According to the technology of the present disclosure, geometric damping is used to provide damping of vibrations of a body vibrating in a fluid and / or determine properties of the fluid, such as viscosity, viscoelasticity, density, fluid stiffness, loss tangent, storage modulus, loss modulus, and yield stress. As discussed in more detail below, geometric damping can be achieved using a slender member vibrating in a fluid, the slender member having a half-width that is relatively smaller than the viscoelastic propagation depth of a shear wave in the fluid at the vibration frequency. The slender member can be cylindrical (in which case the half-width is equal to the radius of the cylinder), but does not have to be cylindrical.

[0005] By adopting geometric damping, improved linearity can be obtained. In any measurement scenario, if the measurement does not require any compensation for nonlinearity, such as an algorithm that may need to modify the output of the measurement to compensate for the nonlinear deviation of the output relative to the measured variable, the quality and integrity of the measurement are improved. In addition to simplicity, linearity can provide improved accuracy by not requiring nonlinear correction, which may introduce errors due to imperfect compensation algorithms. In addition, by adopting geometric damping, greater independence from secondary variables can be obtained. For example, damping-related measurements can depend on viscosity rather than density, elasticity or frequency. Therefore, it may be expected that any changes in density, elasticity or frequency that occur in real-world measurement situations will not affect the measurement of viscosity. When determining, for example, viscosity, it may not be necessary to measure all these factors separately and compensate for their changes.

[0006] According to a first aspect, a method for determining physical properties of a fluid is provided, the method comprising: causing a vibration transducer element to vibrate in the fluid at a vibration frequency, wherein the vibration transducer element comprises a slender member contacting the fluid, the slender member being characterized by a width, a half-width equal to half of the width, and a length greater than the width, wherein the half-width is less than a propagation depth of a shear wave in the fluid at the vibration frequency; measuring the vibration of the vibration transducer element in the fluid at the vibration frequency; and determining the physical properties of the fluid based on the measurement of the vibration.

[0007] In some embodiments, determining a physical property of the fluid based on the measurement of the vibrations includes determining one or more of: viscosity, viscoelasticity, density, fluid stiffness, loss tangent, storage modulus, loss modulus, and yield stress.

[0008] In some embodiments, measuring vibrations includes determining a first quantity indicative of a degree of damping of a vibration transducer element in a fluid at a vibration frequency, wherein the method further includes vibrating the vibration transducer element in the fluid at another vibration frequency and determining a second quantity indicative of a degree of damping of the vibration transducer element in the fluid at the other vibration frequency, wherein determining a physical property of the fluid based on the measurement of the vibrations includes determining the viscoelasticity of the fluid based on the quantities indicative of the degree of damping at the vibration frequency and at another vibration frequency.

[0009] In some embodiments, measuring the vibrations includes determining a resonant frequency of a vibrating transducer element in the fluid, wherein determining a physical property of the fluid based on the measurement of the vibrations includes determining a density of the fluid based on the resonant frequency.

[0010] In some embodiments, the propagation depth is the distance over which the amplitude of a shear wave propagating in a fluid at the vibration frequency decreases by 1 / e, where e is the base of the natural logarithm.

[0011] In some embodiments, the propagation depth of a shear wave propagating in a fluid at a vibration frequency is given by the following expression: where μ is the viscosity of the fluid, ρ is the density of the fluid, ω is the angular frequency of vibration, and Δ varies between 0 and π / 2 (in radians) and is defined by the loss tangent tanΔ, and where tanΔ is equal to the following expression, where G' is the storage modulus of the fluid:

[0012] In some embodiments, the half-width of the slender member is less than 75% of the propagation depth, optionally less than 60% of the propagation depth, optionally less than 50% of the propagation depth, optionally less than 40% of the propagation depth, optionally less than 25% of the propagation depth, optionally less than 10% of the propagation depth, optionally less than 5% of the propagation depth, optionally less than 2% of the propagation depth, optionally less than 1% of the propagation depth, or optionally less than 0.5% of the propagation depth.

[0013] In some embodiments, the elongated member has a substantially or completely circular cross-section along 50%, 70%, 90% or 100% of its length. Optionally, along 50%, 70%, 90% or 100% of the length of the elongated member, the cross-section of the elongated member has a circularity in the range of 0.75 to 1, optionally in the range of 0.8 to 1, optionally in the range of 0.85 to 1, optionally in the range of 0.9 to 1, optionally in the range of 0.95 to 1, more preferably in the range of 0.9 to 1, optionally in the range of 0.95 to 1, wherein the circularity of the cross-sectional shape is given by 4πA / p 2 Calculate , where A is the convex area of ​​the cross-sectional shape and p is the convex perimeter of the cross-sectional shape.

[0014] In some embodiments, the half-width of the slender member calculated at a point along the length of the slender member is based on the convex perimeter or convex area of ​​the cross-section of the slender member at that point along its length. Optionally, the half-width is calculated based on the convex perimeter of the cross-sectional shape by the expression p / 2π. Alternatively, the half-width can also be calculated based on the convex area of ​​the cross-sectional shape by the expression √(A / π). If the slender member has a circular cross-section, these two expressions produce the radius of the circle, and therefore the half-width of the circular cross-section is the radius of the circle.

[0015] In some embodiments, the elongated member has a constant cross-section along more than 50%, more than 60%, more than 70%, more than 80%, more than 90%, or 100% of its length.

[0016] In some embodiments, the elongated member has a constant cross-section along only less than 50%, less than 40%, less than 30%, less than 20%, less than 10% of its length, or the cross-section varies continuously along its length.

[0017] In some embodiments, the cross-sectional area increases or decreases monotonically along the length of the elongated member.

[0018] In some embodiments, the elongated member is straight.

[0019] In some embodiments, the elongated member is axially symmetric along its length.

[0020] In some embodiments, the elongated member is non-straight. For example, the elongated member may include a closed loop.

[0021] In some embodiments, the elongated member comprises one of: a cylinder, a cone, a frustum of a cone, a torus, and an arcuate portion of a torus.

[0022] In some embodiments, the half-width of the elongated member that is less than the propagation depth is the maximum half-width along the length of the elongated member.

[0023] In some embodiments, the half width of the elongated member that is less than the propagation depth is the average half width along the length of the elongated member. Optionally, the average half width is calculated as the arithmetic mean of the half widths along the length of the elongated member, or is calculated as twice the volume of the elongated member divided by the average half width of the surface area of ​​the elongated member.

[0024] In some embodiments, the width of the elongated member is greater than 0.5 mm, and / or greater than 1 mm, and / or greater than 2 mm, and / or greater than 5 mm, and / or greater than 10 mm, and / or greater than 20 mm, and / or greater than 50 mm.

[0025] In some embodiments, the length of the elongated member is greater than a multiple of half the width of the elongated member (half the width is half the width of the elongated member), and wherein the multiple is one of: 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 25, 30, 35, 40, 45, or 50. In other words, the length of the elongated member can be greater than a multiple of the width of the elongated member, and wherein the multiple is one of: 1, 1.5, 2, 2.5, 3, 3.5, 4, 4.5, 5, 5.5, 6, 6.5, 7, 7.5, 8, 8.5, 9, 9.5, 10, 12.5, 15, 17.5, 20, 22.5, and 25.

[0026] In some embodiments, the viscosity of the fluid is greater than 100 Pa·s, and / or greater than 200 Pa·s, and / or greater than 500 Pa·s.

[0027] In some embodiments, the density of the fluid is 500 kg / m 3 With 1500kg / m 3 Between, and / or above 700kg / m 3 With 1300kg / m 3 Between, and / or above 900kg / m 3 With 1100kg / m 3 between.

[0028] In some embodiments, the vibration frequency is less than 10 kHz, and / or less than 7 kHz, and / or less than 5 kHz, and / or less than 3 kHz, and / or less than 2 kHz, and / or less than 1 kHz, and / or less than 500 Hz.

[0029] In some embodiments, the width of the elongated member is between 1nm and 500nm. Such embodiments can be described as "nanoscale" or nanometer-scale embodiments. In some other embodiments, the width of the elongated member is between 500nm and 500μm. Such embodiments can be described as "micrometer-scale" or micrometer-scale embodiments. Appropriately sized devices (which can be nanometer-scale or micrometer-scale devices) can vibrate at low frequencies to advantageously measure the fluid properties of fluids with low viscosity (such as less than 1mPa·s), or can vibrate at high frequencies to advantageously measure the fluid properties of fluids with low viscosity (such as less than 1mPa·s), because at such small scales, the width of the elongated member can be relatively small relative to the propagation depth at such high frequencies.

[0030] In some embodiments, the viscosity of the fluid can be less than 100 Pa·s, less than 10 Pa·s, less than 1 Pa·s, less than 100 mPa·s, less than 10 mPa·s, or less than 1 mPa·s. At any such viscosity, the vibration frequency can be greater than 500 Hz, greater than 1 kHz, greater than 2 kHz, greater than 3 kHz, greater than 4 kHz, greater than 5 kHz, greater than 7 kHz, and / or greater than 10 kHz.

[0031] In some embodiments, the fluid is a Newtonian fluid. In other embodiments, the fluid is a non-Newtonian fluid, such as a viscoelastic fluid or a yield stress fluid.

[0032] In some embodiments, the vibration transducer element comprises a shaft having a longitudinal axis, wherein the elongated member is connected to the shaft, and wherein the elongated member is not co-linear with the longitudinal axis of the shaft. Optionally, during vibration of the vibration transducer element at the vibration frequency, the flow of the fluid around the elongated member is laminar flow. Alternatively or additionally, the Reynolds number Re of the fluid flow around the slender member is less than 1, wherein the Reynolds number is equal to 2Rvρ / μ, wherein μ is the viscosity of the fluid, ρ is the density of the fluid, R is the half-width of the slender member, and v is the maximum (vibration) velocity of the slender member relative to the fluid during vibration of the vibration transducer, wherein, optionally, the Reynolds number is less than 1000, or less than 300, or less than 100, or less than 30, or less than 10, or less than 3, or less than 1, or less than 0.9, or less than 0.8, or less than 0.75, or less than 0.7, or less than 0.6, or less than 0.5, or less than 0.4, or less than 0.3, or less than 0.25, or less than 0.2, or less than 0.1. Alternatively or additionally, the elongated member may have a first end and a second end, wherein one or both of the first end and the second end are spaced apart from the longitudinal axis of the shaft by an offset distance greater than half the width of the elongated member, wherein, optionally, the offset distance is greater than a multiple of the width, and the multiple is 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 20, 30, or 50. Alternatively or additionally, the vibration transducer element may include a plurality of elongated members connected to the shaft, each elongated member not being co-linear with the axis of the shaft, each elongated member having a half-width less than a propagation depth of a shear wave in a fluid at a vibration frequency, and wherein, optionally, the half-width of a first elongated member of the plurality of elongated members is different from the half-width of a second elongated member of the plurality of elongated members, and wherein, optionally, two, three, four, five, or more of the plurality of elongated members may have uniquely different half-widths. Alternatively or additionally, the elongated member may include a first end and a second end, wherein the elongated member is connected to the shaft at the first end and optionally also connected to the shaft at the second end.Alternatively or additionally, the shaft may include a bob, and the elongated member may be connected to the shaft at the bob.

[0033] In some embodiments, vibrating the vibration transducer element comprises vibrating the vibration transducer element in an oscillatory rotational motion.Alternatively, the elongate member may be straight and vibrating the transducer element may comprise vibrating the elongate member in an oscillatory rotational motion about an axis along the length of the elongate member.

[0034] In some embodiments, vibrating the vibration transducer may include vibrating the vibration transducer in an oscillating linear or curvilinear motion. Alternatively, where the vibration transducer includes a shaft having a longitudinal axis and the elongated member is connected to the shaft but not co-linear with the longitudinal axis of the shaft, vibrating the vibration transducer in an oscillating linear or curvilinear motion may include vibrating the shaft in an oscillating rotational, linear or curvilinear motion, which causes the elongated member to move in an oscillating linear or curvilinear motion.

[0035] According to another aspect, a method for determining a property of a fluid is provided, the method comprising: vibrating a vibration transducer element in a fluid at a vibration frequency, first with a first vibration amplitude and then with a second vibration amplitude, wherein the vibration transducer element comprises an elongated member contacting the fluid, the elongated member being characterized by a width, a half-width equal to half of the width, and a length greater than the width; determining a first quantity indicating a degree of damping based on the vibration of the vibration transducer element in the fluid at the first amplitude; determining a second quantity indicating a degree of damping based on the vibration of the vibration transducer element in the fluid at the second amplitude; and determining the property of the fluid based on a difference between the first quantity and the second quantity.

[0036] In some embodiments, determining the property of the fluid includes determining a dipole of a wave field around the vibrating transducer at the vibration frequency based on a difference between the first quantity and the second quantity. Optionally, the method may also include determining, based on the determined dipole, the extent to which the half-width of the elongated member is less than the propagation depth of the shear wave in the fluid at the vibration frequency and / or determining whether the half-width of the elongated member is less than the propagation depth of the shear wave in the fluid at the vibration frequency or less than a predetermined fraction of the propagation depth, such as less than 50% of the propagation depth.

[0037] In some embodiments, a quantity indicative of the dipole nature of the wave field surrounding the vibrating transducer at the vibration frequency is not explicitly determined, and a difference between a first quantity indicative of the degree of damping and a second quantity can be used to determine whether the half-width of the slender member is less than the propagation depth of the shear wave in the fluid at the vibration frequency (or some predetermined fraction of the propagation depth, such as less than 50% of the propagation depth) and / or the extent to which the half-width of the slender member is less than the propagation depth of the shear wave in the fluid at the vibration frequency.

[0038] In some embodiments, determining the property of the fluid includes determining a Reynolds number of the fluid based on a difference between the first quantity and the second quantity.

[0039] In some embodiments, determining the property of the fluid includes determining a vibration velocity relative to the fluid, a viscosity of the fluid, or a density of the fluid based on a difference between the first quantity and the second quantity.

[0040] In some embodiments, determining the first and second quantities indicative of the degree of damping includes determining first and second Q factors. In other embodiments, determining the first and second quantities indicative of the degree of damping includes determining first and second loss factors, wherein the loss factor is the inverse of the Q factor.

[0041] In some embodiments, the determined property of the fluid is a property of the flow of the fluid due to vibration of a vibrating transducer element in the fluid at a vibration frequency of one or both of a first amplitude and a second amplitude.

[0042] In some embodiments, the change in Reynolds number can be considered to be proportional to the ratio between the change in Q factor and the change in vibration amplitude. If the ratio is zero or negligible, the Reynolds number is low and the vibrating flow can be estimated as laminar. If the ratio is non-zero or greater than a threshold, the fluid flow may not be completely laminar and the Reynolds number may have increased. The extent to which the Reynolds number increases may depend on the value of the ratio.

[0043] In some embodiments, the method further comprises performing any of the above methods for determining the physical properties of the fluid. For example, performing multiple vibration tests at different amplitudes may indicate whether secondary damping or geometric damping is present, and therefore whether geometric damping is expected when determining the physical properties of the fluid. In secondary damping, the damping force varies quadratically with the square of the velocity. As will be discussed in more detail later, in geometric damping, the damping force varies linearly with velocity.

[0044] According to another aspect, an apparatus is provided, comprising: a shaft configured to vibrate at a vibration frequency, the shaft having a longitudinal axis; and an elongated member connected to the shaft but not colinear with the longitudinal axis of the shaft, the elongated member being characterized by a width, a half-width equal to half the width, and a length greater than the width. The apparatus can be configured to vibrate the shaft in a fluid such as a liquid. The fluid can be a Newtonian fluid or a non-Newtonian fluid.

[0045] In some embodiments, at least a portion of the elongated member is offset from the longitudinal axis by an offset distance greater than half the width of the elongated member. Optionally, the offset distance is greater than a multiple of half the width of the elongated member, and the multiple is 2, 3, 4, 5, 10, 15, 20, 30, or 50. At least a portion of the elongated member that is offset from the longitudinal axis by such an offset distance may be a portion along the longitudinal axis that extends through the center of the elongated member along the length of the elongated member.

[0046] In some embodiments, the elongated member has a first end and a second end, wherein one or both of the first end and the second end are spaced apart from the longitudinal axis of the shaft by an offset distance greater than half the width. Optionally, the offset distance is greater than a multiple of half the width of the elongated member, and the multiple is 2, 3, 4, 5, 10, 15, 20, 30, or 50.

[0047] In some embodiments, the elongated member has a substantially or completely circular cross-section along 50%, 70%, 90% or 100% of its length. Optionally, along 50%, 70%, 90% or 100% of the length of the elongated member, the circularity of the cross-section of the elongated member is in the range of 0.75 to 1, optionally in the range of 0.8 to 1, optionally in the range of 0.85 to 1, optionally in the range of 0.9 to 1, optionally in the range of 0.95 to 1, optionally in the range of 0.98 to 1, optionally in the range of 0.99 to 1, wherein the circularity of the cross-sectional shape is given by 4πA / p 2 Calculate , where A is the convex area of ​​the cross-sectional shape and p is the convex perimeter of the cross-sectional shape.

[0048] In some embodiments, the half-width of the elongated member is calculated at a point along the length of the elongated member based on the convex perimeter or convex area of ​​the cross-section of the elongated member at that point along its length. Alternatively, the half-width is calculated based on the convex perimeter of the cross-sectional shape by the expression p / 2π. Alternatively, the half-width can also be calculated based on the convex area of ​​the cross-sectional shape by the expression √(A / π). If the elongated member has a circular cross-section, these two expressions produce the radius of the circle, and therefore the half-width of the circular cross-section is the radius of the circle.

[0049] In some embodiments, the elongated member has a constant cross-section along more than 50%, more than 60%, more than 70%, more than 80%, more than 90%, or 100% of its length.

[0050] In some embodiments, the elongated member has a constant cross-section along only less than 50%, less than 40%, less than 30%, less than 20%, less than 10% of its length, or the cross-section varies continuously along its length.

[0051] In some embodiments, the cross-sectional area increases or decreases monotonically along the length of the elongated member.

[0052] In some embodiments, the elongated member is straight.

[0053] In some embodiments, the elongated member is axially symmetric along its length.

[0054] In some embodiments, the elongated member is non-straight. For example, the elongated member may include a closed loop.

[0055] In some embodiments, the elongated member comprises one of: a cylinder, a cone, a frustum of a cone, a torus, and an arcuate portion of a torus.

[0056] In some embodiments, the half-width of the elongated member that is less than the propagation depth is the maximum half-width along the length of the elongated member.

[0057] In some embodiments, the half width of the elongated member that is less than the propagation depth is the average half width along the length of the elongated member. Optionally, the average half width is calculated as the arithmetic mean of the half widths along the length of the elongated member, or is calculated as the average half width of twice the volume of the elongated member divided by the surface area of ​​the elongated member.

[0058] In some embodiments, the width of the elongated member is greater than 0.5 mm, and / or greater than 1 mm, and / or greater than 2 mm, and / or greater than 5 mm, and / or greater than 10 mm, and / or greater than 20 mm, and / or greater than 50 mm.

[0059] In some embodiments, the length of the slender member is greater than a multiple of the half-width of the slender member, and wherein the multiple is one of the following: 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 25, 30, 35, 40, 45 or 50.

[0060] In some embodiments, the device comprises a plurality of elongated members connected to the shaft, each elongated member having a corresponding width, half-width and length. Optionally, the half-width of a first elongated member in the plurality of elongated members is different from the half-width of a second elongated member in the plurality of elongated members, and wherein, optionally, two, three, four, five or more elongated members in the plurality of elongated members may have uniquely different half-widths.

[0061] In some embodiments, the device is configured to vibrate the shaft torsionally about its longitudinal axis and / or longitudinally along its longitudinal axis and / or transversely to its longitudinal axis.

[0062] In some embodiments, the device comprises a boss. Optionally, at least one elongate member is connected to the shaft at the boss.

[0063] In some embodiments, the device is a device for determining a physical property of a fluid, the device being configured to vibrate a shaft in a fluid when the elongated member is in contact with the fluid, the device being configured to: measure the vibration of the shaft in the fluid; and determine the physical property of the fluid based on the measurement of the vibration. The determined physical property may be the viscosity of the fluid or the density of the fluid or the storage modulus of the fluid or the loss modulus of the fluid or the loss tangent of the fluid. Such a device may be used to implement any of the above methods.

[0064] According to another aspect, a system is provided comprising a fluid and any of the above-described devices, wherein at least a portion of the elongated member and the shaft are in contact with the fluid, and the shaft is configured to vibrate at a vibration frequency, wherein the half-width of the elongated member is less than the propagation depth of the shear wave in the fluid at the vibration frequency. In some embodiments, the system may include a fluid captured in a container. In some embodiments, the fluid may flow in a system, such as in an open or closed loop conduit system, with the fluid flowing through the elongated member and the shaft. In some embodiments, the device may be a damper, wherein the elongated member is configured to damp vibrations (such as torsional vibrations) in the shaft at the vibration frequency. In this manner, the shaft may be coupled or integrated with a vibration source in a machine or device and provide damping to such vibrations.

[0065] According to another aspect, a method of controlling damping behavior of a shaft configured to vibrate in a fluid is provided, the method comprising: providing a shaft configured to vibrate at a vibration frequency, the shaft having a longitudinal axis; and providing an elongated member connected to the shaft but not colinear with the longitudinal axis of the shaft, the elongated member being characterized by a width, a half-width equal to half of the width, and a length greater than the width, the elongated member and at least a portion of the shaft being configured to vibrate in the fluid at the vibration frequency, wherein the half-width of the elongated member is less than a propagation depth of a shear wave in the fluid at the vibration frequency. The propagation depth may be a distance over which the amplitude of a shear wave propagating in the fluid at the vibration frequency decreases by 1 / e, wherein e is the base of the natural logarithm, and may be a viscoelastic propagation depth as described herein.

[0066] In some embodiments, the elongated member has a first end and a second end, wherein one or both of the first end and the second end are spaced apart from the longitudinal axis of the shaft by an offset distance that is greater than half the width of the elongated member.

[0067] In some embodiments, the elongated member is configured such that during vibration of the shaft at the vibration frequency, flow of the fluid around the elongated member is laminar.

[0068] In some embodiments, the elongated member is configured such that during vibration of the shaft in the fluid at the vibration frequency, the Reynolds number Re of the fluid flowing around the elongated member is less than 1000, or less than 300, or less than 100, or less than 30, or less than 10, or more preferably less than 3, or less than 1, or less than 0.9, or less than 0.8, or less than 0.75, or less than 0.7, or less than 0.6, or less than 0.5, or less than 0.4, or less than 0.3, or less than 0.25, or less than 0.2, or less than 0.1, wherein the Reynolds number is given by: Where μ is the viscosity of the fluid, ρ is the density of the fluid, R is the half-width of the elongated member, and v is the maximum velocity of the elongated member relative to the fluid during vibration of the shaft.

[0069] In some embodiments, the shaft is provided with a plurality of elongated members connected to the shaft, each elongated member is not colinear with the longitudinal axis of the shaft, each elongated member having a half-width less than the propagation depth of the shear wave in the fluid at the vibration frequency. Optionally, the half-width of a first elongated member of the plurality of elongated members may be different from the half-width of a second elongated member of the plurality of elongated members.

[0070] In some embodiments, one or more elongated members provided to the shaft are all straight, all non-straight, or a mixture of straight and non-straight elongated members.

[0071] According to another aspect, a device is provided, which includes components for performing any of the above methods. For each step of the method, the device may include components for performing the step. The components for performing any step of the method may be components for performing multiple steps of the method.

[0072] According to another aspect, a computer-readable medium having instructions stored thereon is provided that, when executed by a processor of a vibration transducer including an elongated member, such as an elongated member according to the techniques described herein, causes an apparatus to perform any of the above methods. BRIEF DESCRIPTION OF THE DRAWINGS

[0073] The present invention will now be described in more detail, by way of example only, with reference to the accompanying drawings, in which:

[0074] Figure 1 shows the propagation of shear waves from a flat plane surface in a viscous fluid;

[0075] Figure 2 shows a shear wave propagating radially from a curved surface in a viscoelastic fluid;

[0076] Figure 3 shows shear waves propagating radially from the curved surface of a cylinder, where one of the cylinders is displaced from the axis of rotation;

[0077] Figure 4 shows a graph of measured damping factors illustrating application of the techniques of the present disclosure;

[0078] Figure 5 A cylindrical element is shown under transverse vibration causing a dipole wave field;

[0079] Figure 6 shows laminar flow around a cylindrical element under lateral vibration;

[0080] Figures 7 to 31 A vibration transducer or portion thereof according to the techniques of this disclosure or for use with a process according to the techniques of this disclosure is shown;

[0081] Fig.32 schematically illustrates an example apparatus for analyzing a fluid using one or more techniques of the present disclosure;

[0082] Fig.33 Another example apparatus for analyzing a fluid using one or more techniques of the present disclosure is schematically illustrated;

[0083] Fig.34 a flowchart showing a method according to the techniques of the present disclosure; and

[0084] Fig.35 A flow chart showing another method in accordance with the techniques of this disclosure is shown. DETAILED DESCRIPTION

[0085] Figure 1 shows that a shear wave in a viscous fluid moves from a surface velocity V 0 Propagation of an oscillating flat surface. The depth at which the amplitude or velocity (shear velocity) of the wave decreases to 1 / e of its value at the surface due to viscous damping is the propagation depth δ μ , which is given by the expression: where μ′ is the dynamic viscosity, ρ is the density of the fluid, and ω is the angular frequency.

[0086] The reduction in amplitude of the wave caused by viscosity generates a shear stress τ at the vibrating surface Sμ , the shear stress is the rate of change of velocity at the surface (i.e., the shear rate ) and the fluid viscosity μ′, which is given by the following expression:

[0087] Shear rate at the oscillating surface caused by wave decay This can be determined by differentiating the expression for the wave velocity with respect to the distance from the surface and evaluating the expression at the surface, yielding the following expression: Where V 0 is the shear velocity at the oscillating surface.

[0088] The shear rate due to viscous attenuation is proportional to the square root of the frequency and to the square root of the density and to the square root of the inverse of the viscosity.

[0089] Therefore, the shear stress at the surface (which is the product of the viscosity and the shear rate at the surface) is nonlinear:

[0090] In addition to viscous effects, fluids can also exhibit elastic behavior, which depends on the storage modulus G'. The presence of G' reduces the loss tangent tanΔ, which is defined by the following expression:

[0091] For a purely viscous fluid, tanΔ = ∞. As elastic behavior increases, the fluid becomes less lossy, allowing the wave to propagate further into the fluid. Taking into account the elasticity, the propagation depth is given by the following expression: where δ μ,G′ is the viscoelastic propagation depth, δ μ is the purely viscous propagation depth, and 1 / (sin(Δ / 2)√(2sinΔ)) is the amount by which the propagation depth is scaled due to the elastic behavior of the fluid. This amount is also equal to 1 / √(sinΔ(1-cosΔ)).

[0092] For a purely viscous fluid, Δ is equal to 90°, and is therefore given by (or equivalently 1 / √(sinΔ(1-cosΔ))) gives a scaling equal to 1, and a purely viscous propagation depth δ μ is recovered. Therefore, it is appropriate to use this expression to refer to the viscoelastic propagation depth even for fluids that exhibit little or no viscoelasticity. For values ​​of Δ less than 90°, the scaling amount is greater than 1, so the propagation depth increases relative to the purely viscous propagation depth.

[0093] The shear rate at the oscillating surface due to both viscosity and elasticity is given by the expression:

[0094] The shear stress at the oscillating surface due to both viscosity and elasticity is given by the following expression:

[0095] Shear stress is a nonlinear function of fluid viscosity, fluid density, frequency, and storage modulus (via loss tangent). As elasticity G' increases, tanΔ decreases, Δ decreases from a maximum value of π / 2, and both sinΔ and sinΔ / 2 decrease, so damping shear stress decreases with increasing elasticity G'. This explains why viscoelastic fluids show reduced damping compared to Newtonian fluids of "similar" viscosity.

[0096] Figure 2Figure 1 shows a shear wave propagating radially from a curved surface with radius R in a viscoelastic fluid. Although viscoelastic fluids are relatively lossless over short distances, Figure 2 The amplitude decreases as the radial distance increases, because the potential energy of each crest must be maintained, and the energy is spread over an increasing circumferential length (2πr). The line 10 of constant potential energy is Figure 2 The distribution of energy over the increasing circumferential length results in a reduction in energy per unit volume and therefore a smaller peak height. This amplitude reduction due to geometric considerations looks similar to damping, although it does not dissipate energy itself.

[0097] The change in height causes a decrease in velocity proportional to 1 / r, and this change in velocity produces a shear rate This shear rate combines with the viscosity to produce the shear stress τ RAD , the shear stress will have a component in phase with the surface velocity, which results in energy dissipation. This effect is referred to herein as "geometric damping".

[0098] If instead of Figure 1 If a plane surface oscillates in the air, a cylindrical surface of radius R oscillates, then an expression for the velocity of the radial shear wave at position r from the central axis of the cylinder can be obtained and differentiated with respect to r to obtain the radial shear rate: Where V r is the shear velocity at the surface, and φ is given by tan -1 The phase adjustment angle is given by (β / α) and is equal to π / 2-Δ / 2, where β is the wave number of the propagating wave (ie, 2π / λ, where λ is the wavelength of the propagating wave).

[0099] The shear stress at the cylindrical surface is then given by the following expression, where r = R: The 1 / R term is the in-phase shear gradient. For any degree of viscoelasticity, the shear rate of this component is in phase with the velocity. μG The term is the out-of-phase shear gradient. Depending on the degree of viscoelasticity, the shear rate of this component is out of phase by φ. The phase shift angle represents the angle between the shear stress and the velocity. A shear stress that is in phase with the velocity dissipates energy.

[0100] For much smaller than δ μG′For values ​​of R, the in-phase part dominates and the shear rate becomes less dependent on or even independent of viscosity, density, frequency and storage modulus (via tanΔ / 2, which is a function of G″). If 1 / δ is compared to 1 / R, μG′ is negligible, then the shear stress at the cylindrical surface is given by the expression:

[0101] For much larger than δ μG′ The heterogeneous part dominates and the shear rate becomes a nonlinear function of viscosity, density, frequency and storage modulus. μG′ In comparison, 1 / R is negligible, and the shear stress at the cylindrical surface is given by the expression:

[0102] The critical value of R is R onset =δ μG′ , as this represents the crossover point for the R value where the shear stress due to the 1 / R term becomes greater than that due to 1 / δ μG′ According to the techniques described in this paper, this can be considered as the onset of geometric damping.

[0103] When R<δ μG′ The dependence of shear stress on the nonlinear functions of viscosity, density, frequency and storage modulus is further reduced when the cylinder radius is less than half the viscoelastic propagation depth, that is, where R <R onset / 2. In other words, R geo =R onset / 2, where R geo can be understood as the cylinder radius, which defines the regime in which geometric damping can be assumed to dominate and define the damping behavior.

[0104] To measure the physical properties of the fluid, parameters can be selected to provide a fluid loading factor (such as a fluid damping factor C F , stiffness load factor K F and inertia load factor J F ) has improved linearity. If R = δ μG′ , then the expression for the fluid viscosity at which geometric damping begins to occur is given by: Where ω = 2πf.

[0105] For example, given a cylindrical element of radius R of 2 mm vibrating at a frequency of 5 kHz in a purely viscous fluid (sin(Δ / 2)√(2sinΔ)=1), the fluid has a mass of 1000 kg / m 3 The density ρ of the fluid is given by the following formula, whose unit is Pa·s, where R = R onset : μ′ onset =2 2 π 5000 1000 = 62

[0106] For the geometric damping to begin to dominate (i.e., where R geo =R onset / 2), the required viscosity is four times higher, that is: μ′ geo =(2·2) 2 ·π·5000·1000=250.

[0107] Similarly, the radius and / or vibration frequency of the vibrating element may be selected to exploit geometric damping for a given operating range of viscosity and density according to operational requirements.

[0108] Figure 4 Graph showing the damping factor of heavy mineral oil measured at a frequency of 5 kHz using a vibrating cylinder with a radius of 2 mm. The fluid has a mass of 1000 kg / m 3 The damping factor is measured over a range of viscosity values ​​(obtained by applying heat to the heavy mineral oil). The measured damping factor is given by Figure 4 The solid line indicated by 'A' in the figure also shows that if the damping is described by non-geometric damping (i.e., if with 1 / δ μG′ 1 / R is negligible in comparison). Figure 4 The graph also shows that if the damping is described by geometric damping (i.e., if 1 / δ is compared to 1 / R μG′ is negligible). The line is Figure 4 Indicated by 'C'.

[0109] Figure 4 It is shown that, although the viscosity is relatively low, i.e., below about 60 Pa·s, the radius of the vibrating element is larger than the viscous penetration depth and the shear rate is a nonlinear function of the viscosity. At viscosities above about 60 Pa·s, the shear wave attenuation starts to be described by geometric damping, and the damping factor appears to become increasingly linear.

[0110] Figure 4Three regions are identified. The first region, indicated by reference numeral 70, is the nonlinear region and covers the region below μ onset The second region indicated by reference numeral 80 is a linear transition region and covers μ onset and μ geo The third region, indicated by reference numeral 90, is a completely linear region and covers viscosity values ​​above μ geo Compared to operating in the first region 70, operating in the second region 80 provides improved linearity. Compared to operating in the second region 80, operating in the third region 90 provides improved linearity.

[0111] For a vibrating cylinder of sufficiently wide radius where geometric damping is negligible, the fluid damping factor C F , stiffness load factor K F and inertia load factor J F The expression for is given by the following expression:

[0112] In these expressions, each of the factors has a nonlinear dependence on μ′, G′, or ρ due to the presence of these terms (or quantities that are functions of these terms, such as Δ and its dependence on G′) within the brackets.

[0113] If the vibrating cylinder has a sufficiently small radius so that non-geometric damping is negligible, then C F , K F and J F The expression of is given by:

[0114] From these expressions, where non-geometric damping is negligible, it can be seen that the fluid damping factor C F , stiffness load factor K F and inertia load factor J F There is no longer any nonlinear dependence on μ′, G′ or ρ. F , K F and J F are directly proportional to μ′, G′ and ρ, respectively, and the proportionality constant depends only on the geometric parameters.

[0115] Improved linearity of these fluid load factors may be advantageous. A mechanical system may have damping C, stiffness K, and inertia J. These determine the vibration frequency ω and Q factor of the system by:

[0116] When the system vibrates in air or vacuum, these mechanical factors can be stated as C 0 , K 0 and J 0 When the system vibrates in the fluid, the physical properties of the fluid are respectively F , K F and J F "Load" these factors.

[0117] The overall values ​​of the damping factor, stiffness factor, and inertia factor of the system taking into account the fluid load can be expressed as: C=C 0 +C F (Formula 24) K=K 0 +K F (Formula 25) J=J 0 +J F (Formula 26)

[0118] The overall values ​​of C, K, and J are related to the frequency and Q factor above and are easily measured. F , K F and J F The contribution of the vibration behavior to determine the physical properties of the fluid. As discussed below, the technology of the present disclosure can provide C F , K F and J F Simple linear relationships between ρ and the physical properties of interest such as density ρ, viscosity μ′, and storage modulus G′.

[0119] In these expressions, A is the fluid contact surface area of ​​the cylindrical element, and R G is the "radius of gyration" of the element and is equal to the radius R of the cylindrical element when it vibrates torsionally about its axis.

[0120] When considering the effect of moment of inertia on the rotational motion of a body, the radius of gyration is the radial distance to a point that, if the total mass of the body were concentrated there, would have the same moment of inertia as the actual mass distribution of the body. The term "radius of gyration" in this disclosure is a generalization of this concept to consider torsional factors in addition to moment of inertia.

[0121] The damping factor C F In the case of , the radius of gyration represents the radial distance to a point at which, if the damping were concentrated, the point would have the same damping effect as the actual damping effect of the body.

[0122] In the stiffness load factor K F In the case of a body, the radius of gyration represents the radial distance to a point at which, if the stiffness load is concentrated, the point will have the same stiffness load effect as the actual stiffness load effect of the body.

[0123] Inertia load factor K F In the case of a body, the radius of gyration represents the radial distance to a point where, if the inertial load is concentrated at that point, the point will have the same inertial load effect as the actual inertial load effect of the body.

[0124] Therefore, the radius of gyration, as more generally defined in this disclosure, represents a convenient measure of the radial effect of these load factors. The radius of gyration will be defined by the specific geometry of the cylindrical element, but is generally assumed to have upper and lower boundaries defined by the maximum and minimum radial extents of the cylindrical element from the axis of rotation, and when the cylindrical element is vibrating torsionally about its axis, the radius of gyration is equal to the radius R of the cylindrical element, since all surface loads occur at the cylindrical surface, at a distance R from the axis.

[0125] Figure 3 A first cylinder 20 of radius R is shown, which is twisted about an axis 25 running longitudinally through the center of the cylinder. The cylinder has a length l of rotation about its axis, wherein the length is long enough so that the area of ​​the end (πR 2 ) is smaller than the area of ​​the curved side (2πRl), and R<<δ μG′ , and R = R G , damping factor C F , stiffness load factor K F and inertia load factor J F can be written as: C F =AR μ′=(2πR 2 l)μ′ (Equation 27) K F =ARG′=(2πR 2 l)G′ (Formula 28) J F =AR 3 ρ=(2πR 4 l)ρ. (Equation 29)

[0126] Although geometric damping of radial wave propagation brings advantages by relying on geometric parameters rather than fluid properties, the condition of geometric damping encourages the use of cylindrical elements with small radius, which leads to small effective surface area. Small R in damping factor, stiffness load factor and inertia load factor 2 The term indicates that the damping factor, elastic load factor, or inertial load factor from torsional vibration is small, even at high viscosities.

[0127] Figure 3 Also shown is a second cylinder 30 having the same dimensions as the first cylinder 20, wherein the second cylinder 30 is vertically displaced from the axis 25 of the first cylinder 20 by an offset radius R greater than R. O The second cylinder 30 also vibrates torsionally about the axis 25 of the first cylinder 20. This has the effect of reducing the radius of gyration R G Change from R (the distance from the cylindrical surface to the axis 25) to R O (The effect of the radial offset of the entire cylinder).

[0128] If the geometric damping (i.e., negligible non-geometric damping) is used, the above damping factor C F , stiffness load factor K F and inertia load factor J F The provided equation holds true, then the equation can be expressed as:

[0129] If Ro is greater than R, the load factor is amplified by the square of the ratio between Ro and R, i.e., by causing the vibration of the cylindrical element to be offset from the axis. O / R) 2 In J F In the case of O / R) 4 , that is, the fourth power of the ratio between Ro and R.

[0130] However, these equations only hold true if the cylindrical element is geometrically damped. By displacing the cylindrical element, it no longer undergoes pure torsional oscillations, but instead vibrates transversely at the displacement distance. These load factor equations do not automatically apply because, under transverse vibrations, the cylindrical element may form a dipole wave field instead of a monopole wave field.

[0131] Figure 5A cylindrical element is shown under lateral vibration causing a dipole wave field. Under the dipole wave field, there is a 180° phase difference between the wave fields on each side of the cylindrical element. The formation of a dipole wave field is problematic because the resulting waves are pressure (P) waves rather than shear (S) waves. The fluid mechanics of pressure waves are different from shear waves, and the relationships previously defined for shear waves no longer apply.

[0132] For example, the damping factor is defined differently for shear waves than for pressure waves. For shear waves, the shear rate and the stress generated by the shear rate result in a relatively well-defined and controllable damped wave. In contrast, pressure waves follow what is known as "quadratic damping," where the damping force is proportional to the square of the velocity. This results in a damping factor of the following form: C quadratic = b·v (Formula 33) Where b is a constant and v is the velocity. In other words, the damping factor varies unfavorably with the vibration velocity.

[0133] However, if the Reynolds number is kept low, the monopole wave field can be maintained. The Reynolds number represents the ratio between inertial and viscous forces. As the Reynolds number decreases, the viscous forces become larger relative to the inertial forces.

[0134] Figure 6 Laminar flow around a cylinder vibrating in a left-right direction perpendicular to the axis of the cylinder is shown. In the case of laminar flow, the forces across either side of the cylinder become shear, and so shear waves propagate due to the lateral vibrations. Without wishing to be bound by theory, it is believed that the laminar flow results in inertial forces being sufficiently low relative to viscous forces that the wave field is largely, or at least partially, defined by shear waves arising from the upper and lower portions of the cross section of the cylinder, i.e., perpendicular to the axis and the direction of vibration. As a result, as the Reynolds number decreases, the extent to which the wave field has a dipole form decreases, and the extent to which the wave field has a monopole form increases. The low Reynolds number has recovered the portion of the shear wave field that is in phase in the space in which it propagates. The benefits of geometric damping are retained, but with the advantage of an increased load factor gain due to the offset of the element from the axis.

[0135] Advantageously, a cylindrical element displaced from the axis can achieve a relatively high degree of amplification with little increase in size and weight, since relatively low Reynolds numbers are readily achievable at small length scales. In some embodiments of the disclosed technology, fluid load factors equivalent to those of larger and heavier vibrating elements can be achieved.

[0136] It is further recognized that relatively low Reynolds numbers can be readily achieved with micro- and nano-scale devices for nearly all fluids of interest, regardless of their physical properties. Submicron needle structures on vibrating substrates can form the same radially displaced elements described above and achieve the same benefits of geometric damping. Some embodiments can feature multiple cylindrical elements or cylindrical elements (such as pins and spikes) formed by micro- or nano-fabrication processes, which can allow micro-surfaces to exhibit high fluid loading factors.

[0137] It is also recognized that low Reynolds numbers may result in wave fields that are only partially defined by shear waves, so the expressions for the load factors under geometric damping conditions may differ from the above expressions, but may be proportional to those expressions, with the proportionality constant depending on the extent to which the wave field is defined by shear waves. In the following expressions, a proportionality constant h is introduced, where if the wave field is completely defined by shear waves, then h will have a value of 1, and if the wave field is 50% defined by shear waves, then h will have a value of 0.5, which may be a reasonable assumption in practice: C F =h(2 π l R O 2 )μ′ (Equation 34) K F =h(2 π l R O 2 )G′ (Formula 35) J F =h(2 π l R O 4 )ρ. (Equation 36)

[0138] If h is assumed to be 0.5, the above expression simplifies to: C F =(π l R o 2 )μ′ (Equation 37) K F =(π l R o 2 )G′ (Formula 38) J F =(π l R O 4 )ρ. (Equation 39)

[0139] It is also recognised that the techniques of the present disclosure may have particular application in the measurement of properties of yield stress fluids following the techniques described in WO 2018 / 197902 A1 (“The Measurement Of Properties Of Flowing Yield Stress Fluids”) to Hydramotion Ltd and WO 2018 / 197900 A1 (“The Measurement Of Properties Of Vibrated Yield Stress Fluids”) to Hydramotion Ltd, the entire contents of both documents being hereby incorporated by reference into the present disclosure.

[0140] In the context of the present disclosure, a "low" Reynolds number means that the Reynolds number is low enough so that laminar flow is obtained, and the flow due to vibration can be characterized to some extent by a shear wave field, and provides the advantages of geometric damping at least to some extent. It has been recognized that the transition from laminar to turbulent flow occurs within a range of Reynolds numbers, and the precise range in which this transition occurs depends on the geometry. Compared with higher Reynolds numbers, lower Reynolds numbers are more likely to result in flow behavior that causes a partial shear wave field. Without wishing to be bound by theory, it is believed that the extent to which the shear wave field develops and therefore some of the advantages of the technology of the present disclosure are obtained depends on the Reynolds number. For example, a Reynolds number of 1000 can exhibit a certain degree of laminar flow and cause a shear wave field to a certain extent. A Reynolds number of 100 can exhibit an even greater degree of laminar flow and cause a shear wave field to an even greater extent. A Reynolds number of 10 can exhibit an even greater degree of laminar flow and cause a shear wave field to an even greater extent. A Reynolds number of 1 can exhibit an even greater degree of laminar flow and cause a shear wave field to an even greater extent. A Reynolds number of 1 can exhibit an even greater degree of laminar flow and cause a shear wave field to an even greater extent. Generally, lower Reynolds numbers may be preferred, but the reader will recognize that achieving the lowest possible Reynolds number must be balanced with other technical considerations.

[0141] In these disclosures, a description of the viscoplastic boundary region of a yield stress fluid and techniques for determining physical properties of the fluid by transmitting shear waves of varying propagation depths, some of which extend only into the viscoplastic boundary region and some of which extend through the liquid boundary region are presented. The techniques of the present disclosure allow for the wave propagation depth to be determined by the geometry of the cylindrical element, and specifically by its radius, provided that geometric damping conditions are met. By using multiple cylindrical elements including at least two different radii, the cylindrical elements will generate waves that propagate a known distance into the yield stress fluid, thereby allowing for straightforward design and specification of measurement systems to probe the liquid boundary layer and beyond a yield stress fluid, such as a flowing yield stress fluid.

[0142] Such techniques may include a method for estimating the yield stress of a flowing yield stress fluid using one or more vibration transducers, the one or more vibration transducers having a vibration surface in contact with the yield stress fluid, the method comprising: vibrating the vibration surface of the vibration transducer to transmit waves from the vibration surface into a viscoplastic boundary layer of the flowing yield stress fluid; making one or more measurements of the degree of damping of the vibrations using the vibrations of the vibration transducer; and estimating the yield stress of the flowing yield stress fluid based on the one or more measurements of the degree of damping of the vibrations, wherein the one or more vibration transducers include a plurality of cylindrical elements as described herein, optionally offset from the axis of the vibration transducer, either on a single vibration transducer or distributed across a plurality of vibration transducers, wherein the criterion of geometric damping is satisfied.

[0143] For example, a first measurement of the degree of damping of vibrations may be performed when the vibration surface of the vibration transducer vibrates at a first vibration frequency to transmit a wave propagating a first distance into the viscoplastic boundary layer of the flowing yield stress fluid, and a second measurement of the degree of damping of vibrations may be performed when the vibration surface of the vibration transducer vibrates at a second vibration frequency different from the first frequency to transmit a wave propagating a second distance less than the first distance into the viscoplastic boundary layer of the flowing yield stress fluid, and the yield stress of the flowing yield stress fluid may be estimated based on a linear combination of the first measurement of the degree of damping of vibrations and the second measurement. The method may also include correcting one or both of the first measurement of the degree of damping of vibrations and the second measurement based on the first vibration frequency and the second vibration frequency and the power law exponent of the yield stress fluid. The estimated value of the yield stress of the flowing yield stress fluid is proportional to: Wherein V1 is a first measure of the degree of damping of the vibration, V2 is a second measure of the degree of damping of the vibration, ω1 is an angular frequency of the first vibration frequency, ω2 is an angular frequency of the second vibration frequency, and n is a power law index. The method may also include: i) taking a third measure of the degree of damping of the vibration using a vibrating surface of a vibration transducer, the vibration transducer vibrating at a third vibration frequency different from the first vibration frequency and the second vibration frequency to transmit a wave that propagates a third distance less than the first distance into a viscoplastic boundary layer of a flowing yield stress fluid; and ii) estimating a power law index of the flowing yield stress fluid based on the third measure of the degree of damping of the vibration and the third vibration frequency, and further based on one of: the first measure of the degree of damping of the vibration and the first vibration frequency; and the second measure of the degree of damping of the vibration and the second vibration frequency. Therefore, the generation of shear waves of known propagation depth by the technology of the present disclosure is easily and advantageously applied.

[0144] The techniques of this disclosure also allow for determination of whether a cylindrical element radiates waves in a monopole-like or dipole-like pattern, and for inferences to be made based on such determination.

[0145] The damping of a dipole does not follow the shear wave theory because the dipole generates a P wave, under which the magnitude of the damping force is determined by av 2 is given by where a is a constant and v is the fluid velocity. In other words, the damping force is proportional to the square of the velocity. This results in a damping factor given by (Eq. 33). In other words, the damping factor varies with velocity.

[0146] However, the geometric damping described herein is independent of vibration velocity, as explained above in conjunction with, for example, (Equation 30), (Equation 31), and (Equation 32).

[0147] The vibration velocity is a function of the vibration amplitude, and changing the vibration amplitude will change the vibration velocity. By changing the vibration amplitude and measuring the vibrations within the fluid at different vibration amplitudes (i.e., without changing the frequency), the degree of dipole behavior or "dipoleness" of the wave field around the cylindrical element can be detected. Dipoleness can be thought of as a quantity that indicates the degree to which the wave field has a dipole rather than a monopole form. In particular, a change in the Q factor will indicate a change in the damping factor, since the Q factor is inversely proportional to the damping factor.

[0148] Therefore, the change in Q factor is proportional to the degree of dipole property, namely: ΔQ∝ dipole (Equation 40)

[0149] Furthermore, the dipole is a function of the Reynolds number, which is given by Re = 2Rvρ / μ′. Therefore: ΔQ∝Re=2Rvρ / μ′. (Formula 41)

[0150] Therefore, changes in Q by modulating the velocity (and in turn by modulating the amplitude of the vibration) can be used to indicate changes in the Reynolds number, which is useful in determining whether there is geometrically damped monopole-like behavior, or as an alternative means of determining a parameter (such as velocity, viscosity or density) for determining the Reynolds number.

[0151] The change from a vibrating element in a fluid causing a wave field with high dipole, where the waves generated are acoustic (i.e., pressure) waves, to causing a monopole or near-monopole wave field, where the waves generated are only or mostly shear waves, requires that the disturbed fluid around the vibrating element is laminar as the element vibrates and moves through the fluid. Thus, the condition for laminar flow is a low Reynolds number under small amplitude vibration conditions, such as a Reynolds number of less than 1000, less than 100, less than 10, or less than 1, where a lower Reynolds number indicates a greater degree of laminar flow.

[0152] Parameters that determine the Reynolds number, such as viscosity in particular, may be advantageous because the length scale and density can be assumed to be fixed in the system. The variation in the Reynolds number may be determined entirely or primarily by the ratio between velocity and viscosity. At very high viscosities, the Reynolds number will be small even for a wide range of vibration velocities. At low viscosities, a change in velocity may modulate the Reynolds number to a higher value, resulting in a loss of laminar flow conditions and the generation of acoustic waves. Acoustic waves follow secondary damping, which varies with the square of the velocity, and therefore a change in the vibration amplitude changes the local velocity, resulting in a detectable change in damping. Therefore, an estimate of the Reynolds number may be obtained from the difference in the Q factor.

[0153] In some examples according to the presently disclosed techniques, the change in Reynolds number may be considered to be proportional to the ratio between the change in Q factor and the change in vibration amplitude: ΔQ / ΔA∝ΔRe, (Equation 42) Where ΔA represents the change in amplitude and ΔRe represents the change in Reynolds number. If ΔQ / ΔA is zero or negligible, the Reynolds number is low and the vibrating flow can be estimated as laminar. If ΔQ / ΔA is non-zero or greater than a threshold, the fluid flow is not completely laminar and the Reynolds number has increased. The extent of the increase in the Reynolds number depends on the value of ΔQ / ΔA.

[0154] In the case of determining the Reynolds number or determining whether a monopole-like behavior of geometric damping exists (or the extent to which it exists), the technique represents determining a property of the fluid, wherein the determined property of the fluid is a property of the flow of the fluid due to the vibration of a vibrating transducer element in the fluid at a vibration frequency of one or both of the first amplitude and the second amplitude. Determining the velocity, viscosity or density based on the determined Reynolds number also represents determining a property of the fluid, which is a property of the flow of the fluid due to the vibration of a vibrating transducer element in the fluid at a vibration frequency of one or both of the first amplitude and the second amplitude.

[0155] These techniques that utilize the change in Q factor between vibrations of different amplitudes to determine fluid properties (such as the properties of the flow of the fluid around the vibrating element) or the properties of the fluid itself (such as viscosity) can be applied to fluids with a yield stress. These techniques can also be applied to fluids with zero yield stress because they do not rely on the fluid having a yield stress.

[0156] Figures 7 to 31 A vibration transducer or portion thereof according to the technology of the present disclosure or for use with a vibration transducer or portion thereof for use with a process according to the technology of the present disclosure is shown. Although these figures show vibrations relative to an axis in a specific direction such as torsion, lateral, or longitudinal, it will be appreciated that the technology of the present disclosure is not limited to any specific vibration direction. A vibration transducer with a vibration element according to the technology of the present disclosure can be configured to vibrate in multiple vibration directions, including a combination of torsion and lateral vibrations or torsion and longitudinal vibrations. In addition, the foregoing discussion has treated the vibration element as a cylinder. However, while this can simplify the analysis, the vibration element does not need to be a cylinder or even have a perfectly circular cross-section to achieve at least some of the advantages identified herein. Therefore, at least some vibration transducers or portions thereof are discussed using the more general expression "elongated member".

[0157] Figures 7 to 9 A vibration transducer for use with the techniques of the present disclosure is shown. Figures 7 to 9 In each of the embodiments, the vibration transducer has a cylindrical vibration element 110 in contact with the fluid 100. The cylindrical vibration element has a radius R around an axis 112 along the length of the cylindrical vibration element. According to the technology of the present disclosure, geometric damping is obtained when the radius R is less than the viscoelastic propagation depth of the shear wave in the fluid 100, which is a function of the angular frequency ω of the vibration and the physical properties of the fluid 100.

[0158] exist Figure 7 In FIG. 1 , a cylindrical vibrating element 110 is configured to torsionally vibrate about an axis 112 at an angular frequency ω.

[0159] exist Figure 8In FIG. 1 , a cylindrical vibrating element 110 is configured to vibrate longitudinally along an axis 112 at an angular frequency ω.

[0160] exist Fig. 9 In the embodiment, the cylindrical vibration element 110 is configured to vibrate laterally at an angular frequency ω in a direction perpendicular to the axis 112.

[0161] Fig.10 and Fig.11 A vibration transducer for use with the techniques of the present disclosure is shown. Fig.10 and Fig.11 In each of the embodiments, the vibration transducer has an elongated member in contact with the fluid, the elongated member having a circular cross-section along the length of a major axis passing through the elongated member, and the elongated member is configured to vibrate torsionally about the major axis at an angular frequency ω.

[0162] exist Fig.10 In the embodiment, at the end of the elongated member 120 , the radius of the circular cross section decreases linearly with the position along the axis 122 , and one end of the elongated member 120 forms a conical shape.

[0163] exist Fig.11 , the radius of the circular cross section along the elongated member 130 decreases discontinuously with position along the axis 132 in a piecewise constant manner, from a maximum radius R max Step to minimum radius R min , the elongated member 130 has an average radius R along the length of the axis 132 ave .

[0164] If a portion of the elongated member has a radius that is less than the viscoelastic propagation depth in the fluid 100, geometric damping will occur along at least a portion of the elongated member. According to the disclosed technique, when the average radius R of the elongated member along the length of the axis is ave A beneficial level of geometric damping is obtained when the maximum radius R is less than the viscoelastic propagation depth of the shear wave in the fluid 100. max Smaller than the viscoelastic propagation depth, higher levels of geometric damping can be achieved.

[0165] Fig.12A vibration transducer for use with the technology of the present disclosure is shown. The vibration transducer has a slender member 140 in contact with a fluid 100. The slender member 140 is not straight, but has a curve. An axis 142 extends longitudinally through the center of the slender member. The axis 142 itself is curved. The slender member is configured to vibrate in a linear direction that is roughly aligned with the axis 142. The slender member 140 has a circular cross-section (although there is a curvature) along its length and has a radius R around the axis 142. According to the technology of the present disclosure, geometric damping is obtained when the radius R is less than the viscoelastic propagation depth of the shear wave in the fluid 100, which is a function of the angular frequency ω of the vibration and the physical properties of the fluid 100.

[0166] Figures 13 to 20 Additional vibration transducers for use with the techniques of this disclosure are illustrated in which an elongated member is connected to a shaft having a longitudinal axis, the elongated member being offset from the longitudinal axis of the shaft.

[0167] Fig.13 A vibration transducer for use with the technology of the present disclosure is shown. The vibration transducer includes a shaft 210 and an elongated member 220 attached to the shaft. The shaft 210 is cylindrical and is configured to torsionally vibrate at an angular frequency of vibration ω about a longitudinal axis 212 passing through the center of the shaft 210. The elongated member 220 is also cylindrical and is connected to the shaft 210 at one end of the shaft 210. The elongated member 220 has a longitudinal axis 222 that is parallel to the axis 212 of the shaft 210 about which the shaft 210 vibrates, but the longitudinal axis 222 is radially offset from the axis 212 of the shaft 210 by an offset distance R O The elongated member 220 has a radius R. The shaft 210 and the elongated member 220 are in contact with the fluid 200. In accordance with the disclosed techniques, a beneficial level of geometric damping is obtained when the viscoelastic propagation distance of a shear wave in the fluid 200 at the angular frequency of vibration is greater than the radius R of the elongated member 220 and when the Reynolds number at which the fluid 200 flows around the elongated member 220 is low. The offset of the elongated member 220 from the axis 212 of the shaft 210 provides greater damping than would be obtained if the axis 222 of the elongated member 220 were in line with the axis 212 of the shaft 210.

[0168] Fig.14 A vibration transducer for use with the technology of the present disclosure is shown. The vibration transducer includes a shaft 210 and an elongated member 220 attached to the shaft. The shaft 210 is cylindrical and is configured to vibrate torsionally at an angular frequency ω of vibration about a longitudinal axis 212 passing through the center of the shaft 210. The elongated member 220 is also cylindrical and is connected to the shaft 210. Fig.13 Compared to the arrangement shown, Fig.14The elongated member 220 is connected to the curved side wall of the shaft 210 and has a longitudinal axis 222 perpendicular to the axis 212 of the shaft 210, and the shaft 210 vibrates around the axis 212. The elongated member 220 has a radius R. The proximal end of the elongated member 220 is offset from the axis 212 of the shaft 210 by a first offset distance R O,1 , the first offset distance is equal to the radius of the shaft 210. The distal end of the elongated member 220 is offset from the axis 212 of the shaft 210 by a second offset distance R O,2 , the second offset distance being equal to the sum of the radius of the shaft 210 and the length of the elongated member 220. The elongated member 220 is offset from the axis 212 of the shaft 210 along its entire length. The shaft 210 and the elongated member 220 are in contact with the fluid 200. According to the techniques of the present disclosure, a beneficial level of geometric damping is obtained when the viscoelastic propagation distance of a shear wave in the fluid 200 at the angular frequency of vibration is greater than the radius R of the elongated member 220 and when the Reynolds number of the fluid 200 flowing around the elongated member 220 is low. The offset of the elongated member 220 from the axis 212 of the shaft 210 provides greater damping than would be obtained if the axis 222 of the elongated member 220 were in line with the axis 212 of the shaft 210.

[0169] Fig.15 A vibration transducer for use with the technology of the present disclosure is shown. The vibration transducer includes a shaft 210 and an elongated member 220 attached to the shaft. The shaft 210 is cylindrical and is configured to vibrate torsionally at an angular frequency ω of vibration about a longitudinal axis 212 passing through the center of the shaft 210. The elongated member 220 is also cylindrical and is connected to the shaft 210. Fig.13 Compared to the arrangement shown, Fig.15 The elongated member 220 is connected to the curved side wall of the shaft 210. Fig.13 and Fig.14 Compared to the arrangement shown, Fig.15 The elongated member 220 extends in an inclined direction that is neither completely radial nor completely axial. The elongated member 220 has a longitudinal axis 222 that is inclined relative to the axis 212 of the shaft 210, and the shaft 210 vibrates around the axis 212. The elongated member 220 has a radius R. The proximal end of the elongated member 220 is offset from the axis 212 of the shaft 210 by a first offset distance R O,1 , the first offset distance is equal to the radius of the shaft 210. The distal end of the elongated member 220 is offset from the axis 212 of the shaft 210 by a second offset distance R O,2, the second offset distance being equal to the sum of the radial component of the radius of the shaft 210 and the length of the elongated member 220. The elongated member 220 is offset from the axis 212 of the shaft 210 along its entire length. The shaft 210 and the elongated member 220 are in contact with the fluid 200. According to the techniques of the present disclosure, a beneficial level of geometric damping is obtained when the viscoelastic propagation distance of a shear wave in the fluid 200 at the angular frequency of vibration is greater than the radius R of the elongated member 220 and when the Reynolds number of the fluid 200 flowing around the elongated member 220 is low. The offset of the elongated member 220 from the axis 212 of the shaft 210 provides greater damping than would be obtained if the axis 222 of the elongated member 220 were in line with the axis 212 of the shaft 210.

[0170] Fig.16 A vibration transducer for use with the technology of the present disclosure is shown. The vibration transducer includes a shaft 210 and an elongated member 230 attached to the shaft. The shaft 210 is cylindrical and is configured to vibrate torsionally at an angular frequency of vibration ω about a longitudinal axis 212 passing through the center of the shaft 210. The elongated member 230 is connected to the shaft 210 at one end thereof and has a longitudinal axis 232 extending in a direction parallel to the axis 212 of the shaft 210, but offset from the axis 212 of the shaft by an offset distance R O .and Fig.13 Compared to the arrangement shown in , although the slender member 230 has a circular cross-section along its length, the slender member 230 does not have a constant radius along its length. In contrast, the slender member 230 includes a first cylindrical portion having a first radius and a second cylindrical portion having a second radius different from the first radius, the first cylindrical portion is connected to the shaft 210, and the second cylindrical portion is connected to the first cylindrical portion. The first cylindrical portion and the second cylindrical portion of the slender member 230 share the same common axis 232. The first radius is the maximum radius R of the slender member 230. max , and the second radius is the minimum radius R of the elongated member 230 min The average radius of the elongated member 230 along its length is R ave The shaft 210 and the elongated member 230 are in contact with the fluid 200. According to the disclosed technology, when the viscoelastic propagation distance of the shear wave in the fluid 200 at the angular frequency of vibration is greater than R min (More preferably greater than R ave , more preferably greater than R max ) and a beneficial level of geometric damping is obtained when the Reynolds number of the fluid 200 flowing around the elongated member 230 is low. The offset of the elongated member 230 from the axis 212 of the shaft 210 provides greater damping than would be obtained if the axis 222 of the elongated member 230 was in line with the axis 212 of the shaft 210.

[0171] Fig.17 A vibration transducer for use with the technology of the present disclosure is shown. The vibration transducer includes a shaft 210 and an elongated member 240 attached to the shaft. The shaft 210 is cylindrical and is configured to torsionally vibrate at an angular frequency of vibration ω about a longitudinal axis 212 passing through the center of the shaft 210. The elongated member 240 is connected to the shaft 210 at one end thereof and has a longitudinal axis 242 extending in a direction parallel to the axis 212 of the shaft 210, but offset from the axis 212 of the shaft by an offset distance R O .and Fig.13 Compared to the arrangement shown in , although the elongated member 240 has a circular cross-section along its length, the elongated member 230 does not have a constant radius along its length. In contrast, the elongated member 240 is conical, having a maximum radius R where the elongated member 240 is connected to the shaft 210. max The average radius of the elongated member 240 along its length is R ave The shaft 210 and the elongated member 240 are in contact with the fluid 200. According to the disclosed technology, when the viscoelastic propagation distance of the shear wave in the fluid 200 at the angular frequency of vibration is greater than R ave (More preferably greater than R max ) and a beneficial level of geometric damping is obtained when the Reynolds number of the fluid 200 flowing around the elongated member 240 is low. The offset of the elongated member 240 from the axis 212 of the shaft 210 provides greater damping than would be obtained if the axis 222 of the elongated member 240 was in line with the axis 212 of the shaft 210.

[0172] Fig.18 A vibration transducer for use with the technology of the present disclosure is shown. The vibration transducer includes a shaft 210 and an elongated member 240 attached to the shaft. The shaft 210 is cylindrical and is configured to vibrate torsionally at an angular frequency of vibration ω about a longitudinal axis 212 passing through the center of the shaft 210. The elongated member 240 is connected to the shaft 210. The elongated member 240 is conical with a maximum radius R at the point where the elongated member 240 is connected to the shaft 210. max The average radius of the elongated member 240 along its length is R ave .and Fig.17 Compared to the arrangement shown, Fig.18 The elongated member 240 is connected to the curved side wall of the shaft 210 and has a longitudinal axis 242 perpendicular to the axis 212 of the shaft 210, and the shaft 210 vibrates around the axis 212. The proximal end of the elongated member 240 is offset from the axis 212 by a first offset distance R O,1 , the first offset distance being equal to the radius of the shaft 210. The distal end of the elongated member 240 is offset from the axis 212 by a second offset distance R O,2, the second offset distance is equal to the sum of the radius of the shaft 210 and the length of the elongated member 240. The entirety of the elongated member 240 (including both ends) is offset from the axis 212 of the shaft 210. The shaft 210 and the elongated member 240 are in contact with the fluid 200. According to the technology disclosed in the present invention, when the viscoelastic propagation distance of the shear wave in the fluid 200 at the angular frequency of vibration is greater than R ave (More preferably greater than R max ) and a beneficial level of geometric damping is obtained when the Reynolds number of the fluid 200 flowing around the elongated member 240 is low. The offset of the elongated member 240 from the axis 212 of the shaft 210 provides greater damping than would be obtained if the axis 242 of the elongated member 240 was in line with the axis 212 of the shaft 210.

[0173] Fig.19 A vibration transducer for use with the technology of the present disclosure is shown. The vibration transducer includes a shaft 210 and an elongated member 250 attached to the shaft. The shaft 210 is cylindrical and is configured to vibrate torsionally at an angular frequency ω of vibration around a longitudinal axis 212 passing through the center of the shaft 210. The elongated member 250 is connected to the shaft 210. The elongated member 250 is not straight, but has a curve. The axis 252 extends longitudinally through the center of the elongated member 250. The axis 252 itself is curved. The elongated member 250 has a circular cross-section (although with curvature) along its length and has a radius R around the axis 242. The elongated member 250 is connected to the shaft 210 at the end of the shaft, extending in a direction substantially consistent with the axis 210 of the shaft, but the elongated member 250 deviates from the axis 212 of the shaft 210. The proximal end of the elongated member 250 is offset from the axis 212 of the shaft 210 by a first offset distance R O,1 , and the distal end of the elongated member 250 is offset from the axis 212 of the shaft 210 by a second distance R O,2 . An axis 252 extending longitudinally through the center of the elongated member 250 is offset from the axis 212 of the shaft 210 along the entire length of the elongated member 250. The shaft 210 and the elongated member 250 are in contact with the fluid 200. In accordance with the techniques of the present disclosure, a beneficial level of geometric damping is obtained when the viscoelastic propagation distance of a shear wave in the fluid 200 at the angular frequency of vibration is greater than R and when the Reynolds number of the flow of the fluid 200 around the elongated member 250 is low. The offset of the elongated member 250 from the axis 212 of the shaft 210 provides greater damping than would be obtained if the axis 252 of the elongated member 250 were in line with the axis 212 of the shaft 210.

[0174] Fig. 20A vibration transducer for use with the technology of the present disclosure is shown. The vibration transducer includes a shaft 210 and a slender member 260 attached to the shaft. The shaft 210 is cylindrical and is configured to vibrate torsionally at an angular frequency of vibration ω about a longitudinal axis 212 passing through the center of the shaft 210. The slender member 260 is not straight, but has a curve. The slender member 260 has a first end and a second end. The first end and the second end of the slender member 260 are connected to the shaft 210 at positions spaced apart from each other longitudinally (in a direction aligned with the axis 212 of the shaft) along the curved side wall of the shaft 210, and the slender member 260 forms a closed loop with the shaft 210. The axis 262 extends longitudinally along a curved path through the center of the slender member 260. The slender member 260 has a circular cross-section along its length (although there is a curvature) and has a radius R about the axis 262. In Fig. 20 In the embodiment, the elongated member 260 has a semi-circular torus form. The first end and the second end of the elongated member 260 are both offset from the axis 212 of the shaft 210 by a first offset distance R O,1 and the second offset distance R O,2 , the first offset distance and the second offset distance are both equal to the radius of the shaft 210. The axis 262 of the elongated member 260 is offset from the axis 210 of the shaft 210 along the entire length of the elongated member 260. The average offset distance R of the elongated member 262 along the length of the elongated member 260 is O,ave greater than the offset distance R at either end of the elongated member 260 O,1 and R O,2 The shaft 210 and the elongated member 250 are in contact with the fluid 200. According to the techniques of the present disclosure, a beneficial level of geometric damping is obtained when the viscoelastic propagation distance of the shear wave in the fluid 200 at the angular frequency of vibration is greater than R and when the Reynolds number of the flow of the fluid 200 around the elongated member 260 is low. The offset of the elongated member 260 from the axis 212 of the shaft 210 provides greater damping than would be obtained if the axis 262 of the elongated member 260 were in line with the axis 212 of the shaft 210.

[0175] Figures 21 to 23 Shown are different from Figures 13 to 20 The vibration transducer shown in FIG. Figure 21 to Figure 23 The vibration transducers shown in include an elongated member connected to a shaft, but they differ in that each shaft includes a boss at the end of the shaft where the elongated member is connected to the shaft.

[0176] Fig.21A vibration transducer for use with the technology of the present disclosure is shown. The vibration transducer includes a shaft 210 and a boss 214 at the end of the shaft 210 and an elongated member 220 connected to the shaft 210 at the boss 214. The shaft 210 is cylindrical and is configured to vibrate torsionally at an angular frequency of vibration ω about a longitudinal axis 212 passing through the center of the shaft 210. The boss 214 represents the end of the shaft 210, which has a larger diameter than the rest of the shaft 210. The boss 214 has three axial sections axially aligned with the axis 212 of the shaft 210, including: i) a first axial section of a frustoconical shape, wherein the radius increases linearly from an initial radius equal to the shaft radius to a final radius over the axial section; ii) a second axial section, which is cylindrical, and whose radius is equal to the final radius of the first axial section; and iii) a third axial section of a conical shape, wherein the radius decreases linearly from the radius of the second axial section to zero. The elongated member 220 is cylindrical and has a radius R, and is connected to the third axial section of the boss 214. The elongated member 220 has a longitudinal axis 222 that is parallel to the axis 212 of the shaft 210 about which the shaft 210 vibrates, but is radially offset from the axis 212 of the shaft 210 by an offset distance R. O The shaft 210 (including the boss 214) and the elongated member 220 are in contact with the fluid 200. According to the disclosed techniques, a beneficial level of geometric damping is obtained when the viscoelastic propagation distance of the shear wave in the fluid 200 at the angular frequency of vibration is greater than the radius R of the elongated member 220 and when the Reynolds number at which the fluid 200 flows around the elongated member 220 is low. The offset of the elongated member 220 from the axis 212 of the shaft 210 provides greater damping than would be obtained if the axis 222 of the elongated member 220 were in line with the axis 212 of the shaft 210.

[0177] Fig. 22 A vibration transducer for use with the technology of the present disclosure is shown. The vibration transducer includes a shaft 210 and a boss 214 at the end of the shaft 210 and an elongated member 220 connected to the shaft 210 at the boss 214. The shaft 210 is cylindrical and is configured to vibrate torsionally at an angular frequency ω of vibration about a longitudinal axis 212 passing through the center of the shaft 210. Fig.21Like the vibration transducer of the present invention, the boss 214 has three axial sections axially aligned with the axis 212 of the shaft 210, including: i) a first axial section of a frustoconical shape, wherein the radius increases linearly from an initial radius equal to the shaft radius to a final radius over the axial section; ii) a second axial section, which is cylindrical, and whose radius is equal to the final radius of the first axial section; and iii) a third axial section of a conical shape, wherein the radius decreases linearly from the radius of the second axial section to zero. The slender member 220 is cylindrical and has a radius R, and is connected to the second axial section of the boss 214. The slender member 220 has a longitudinal axis 222, which is perpendicular to the axis 212 of the shaft 210, and the shaft 210 vibrates around the axis 212. The proximal end of the slender member 220 is offset from the axis 212 of the shaft 210 by a first offset distance R O,1 , the first offset distance is equal to the radius of the second axial section of the boss 214. The distal end of the elongated member 220 is offset from the axis 212 of the shaft 210 by a second offset distance R O,2 , the second offset distance is equal to the sum of the radius of the second axial section of the boss 214 and the length of the elongated member 220. The elongated member 220 is offset from the axis 212 of the shaft 210 along its entire length. The shaft 210 (including the boss 214) and the elongated member 220 are in contact with the fluid 200. According to the technology of the present disclosure, a beneficial level of geometric damping is obtained when the viscoelastic propagation distance of the shear wave in the fluid 200 at the angular frequency of vibration is greater than the radius R of the elongated member 220 and when the Reynolds number of the fluid 200 flowing around the elongated member 220 is low. The offset of the elongated member 220 from the axis 212 of the shaft 210 provides greater damping than would be obtained if the axis 222 of the elongated member 220 was in line with the axis 212 of the shaft 210.

[0178] Fig.23 A vibration transducer for use with the technology of the present disclosure is shown. The vibration transducer includes a shaft 210 and a boss 214 at the end of the shaft 210 and two elongated members 240a, 240b connected to the shaft 210 at the boss 214. The shaft 210 is cylindrical and is configured to vibrate torsionally at an angular frequency of vibration ω about a longitudinal axis 212 passing through the center of the shaft 210. Fig.21Like the vibration transducer of the present invention, the boss 214 has three axial sections axially aligned with the axis 212 of the shaft 210, including: i) a first axial section of frustoconical shape, wherein the radius increases linearly from an initial radius equal to the shaft radius to a final radius over the axial section; ii) a second axial section, which is cylindrical, having a radius equal to the final radius of the first axial section; and iii) a third axial section of conical shape, wherein the radius decreases linearly from the radius of the second axial section to zero. Each elongated member 240a, 240b is connected to the second axial section of the boss 214 on the radially opposite side of the second axial section of the boss 214. Each elongated member 240a, 240b is tapered around an axis 242, which is perpendicular to the axis of the shaft 210 about which the shaft 210 vibrates. Each elongated member 240a, 240b has a maximum radius R at its proximal end where it is connected to the boss 214. max The average radius of the elongated members 240a, 240b along their length is R ave , which is less than R max For each elongated member 240a, 240b, the proximal end is offset from the axis 212 by a first offset distance R O,1 , the first offset distance is equal to the radius of the second axial section of the boss 214, and the distal end is offset from the axis 212 by a second offset distance R O,2 , the second offset distance is equal to the sum of the radius of the second axial section of the boss 214 and the length of the elongated members 240a, 240b. The entirety of each elongated member 240a, 240b (including both ends) is offset from the axis 212 of the shaft 210. The shaft 210 (including the boss 214) and the two elongated members 240a, 240b are in contact with the fluid 200. According to the technology disclosed in the present invention, when the viscoelastic propagation distance of the shear wave in the fluid 200 at the angular frequency of vibration is greater than R ave (More preferably greater than r max ) and when the Reynolds number of the fluid 200 flowing around the elongated members 240a, 240b is low, a beneficial level of geometric damping is obtained. The offset of the elongated members 240a, 240b from the axis 212 of the shaft 210 provides greater damping than would be obtained if each elongated member 240a, 240b had its axis 242 aligned with the axis 212 of the shaft 210.

[0179] In another configuration of the technology according to the present disclosure, the vibration transducer includes a shaft and a plurality of elongated members. If the vibration transducer includes a boss, the elongated member can be connected to the vibration transducer at the boss. Alternatively or additionally, the vibration transducer can include an elongated member connected to the vibration transducer at the shaft. A plurality of elongated members are spaced apart around the circumference of the shaft or boss, extending outward from the shaft or boss in a completely radial direction or in a direction having a radial component and an axial component or in a completely axial direction (not colinear with the longitudinal axis of the shaft / boss). A plurality of elongated members can be evenly distributed around the circumference, which can alleviate or avoid any interference of the center of mass relative to the longitudinal axis, or can be unevenly distributed around the circumference. A plurality of elongated members can be connected to the shaft or boss at the same axial position along the length of the shaft or boss, or can be connected at different axial positions, for example, in a spiral pattern around the outer surface of the shaft or boss.

[0180] If a plurality of elongated members extend from a shaft or boss in all or part of the axial direction, the elongated members may include a support member spaced apart from the outer surface of the shaft or boss to provide a radial offset to the elongated members. Alternatively, a plurality of elongated members may extend from the ends of the shaft or boss, such as being distributed in a circle around the longitudinal axis and extending from the ends of the shaft or boss. The ends of the shaft or boss may be flat, curved, conical, or have some other profile.

[0181] The elongate member may have a width and a half-width such that geometric damping (monopole behavior) may occur around the elongate member during vibration of the vibration transducer.The vibration of the vibration transducer may be torsional about a longitudinal axis of the shaft.

[0182] The vibration transducer may include two or more elongated members, such as three, four, five, six, seven, eight, nine, ten or more elongated members. The elongated member may have a constant cross-section along its length (such as a cylindrical elongated member or an elongated member having a square / rectangular, rounded square / rectangular (e.g., in the form of a superellipse), triangular or elliptical cross-section), or may have a varying cross-sectional shape or size along its length, such as i) a cone with a linear decrease in cross-sectional area as the distance from the axis or boss increases, or ii) a stepwise change in size or shape (e.g., a stepwise decrease in size) as the distance from the axis or boss increases.

[0183] In a specific configuration, the vibration transducer includes a shaft configured for torsional vibration, the shaft having a proximal end and a distal end, and the vibration is driven at the proximal end. At or near the distal end of the shaft (e.g., closer to the distal end than the proximal end, or within the last quarter of the fluid contact length of the shaft, or within the last tenth of the fluid contact length of the shaft, or within the last twentieth of the fluid contact length of the shaft), a plurality of elongated members extend radially outward from the shaft at a common axial position along the length of the shaft. Eight elongated members are evenly distributed around the circumference of the shaft in increments of 45°. In another specific configuration, a boss is present at the distal end of the shaft, and eight elongated members extend radially outward from the boss. Other configurations include more or fewer elongated members, which are evenly distributed or unevenly distributed around the shaft / boss. For example, a configuration includes six elongated members evenly distributed around the circumference of the shaft.

[0184] In another specific configuration, the vibration transducer includes a shaft and a plurality of elongated members that are axially aligned with the longitudinal axis of the shaft but not colinear with the longitudinal axis of the shaft. There is a boss at the distal end of the shaft. The boss has the form of a cylinder that is coaxial with the longitudinal axis of the shaft but has a larger radius than the shaft. A plurality of elongated members extend axially outward from the end of the boss, each of which is connected to the end of the boss at the same radial offset as the longitudinal axis and is evenly distributed around the longitudinal axis. The plurality of elongated members includes eight elongated members that are evenly distributed around the longitudinal axis in increments of 45°. The vibration transducer is configured to vibrate torsionally around the longitudinal axis. Other configurations include more or fewer elongated members that are evenly distributed or unevenly distributed around the longitudinal axis. For example, a configuration includes six elongated members evenly distributed around the longitudinal axis.

[0185] Figures 24 to 31 Another vibration transducer for use with the disclosed techniques is shown, wherein a shaft 310 is configured to vibrate torsionally about a longitudinal axis 312, the shaft having a boss 314 having three axial sections axially aligned with the axis 312 of the shaft 310, including: i) a first axial section of frustoconical shape, wherein the radius increases linearly over the axial section from an initial radius equal to the shaft radius to a final radius; ii) a second axial section which is cylindrical with a radius equal to the final radius of the first axial section; and iii) a third axial section of conical shape, wherein the radius decreases linearly from the radius of the second axial section to zero. Figures 24 to 31 In each of the plurality of elongated members, a plurality of elongated members are connected to the shaft 310 at a boss 314 .

[0186] Fig.24A vibration transducer is shown in which two elongated members 320a, 320b are connected to the second axial section of the boss 314. The two elongated members 320a, 320b are cylindrical and have the same radius that is constant along their length. The first of the elongated members 320a is radially spaced apart from the other elongated member 320b around the boss 314. Each elongated member 320a, 320b extends outwardly from the boss 314 in a radial direction perpendicular to the axis 312 of the shaft 310.

[0187] Fig.25 A vibration transducer is shown in which two elongated members 320a, 320b are connected to the third axial section of the boss 314. Each elongated member 320a, 320b is aligned along a respective axis parallel to the axis 312 of the shaft 310 but radially offset from the axis 312 of the shaft 310. The two elongated members 320a, 320b are cylindrical and have the same radius that is constant along their length. The first of the elongated members 320a is radially spaced apart from the other elongated member 320b around the third axial section of the boss 314.

[0188] Fig.26 A vibration transducer is shown in which two elongated members 340a, 340b are connected to the second axial section of the boss 314. The two elongated members 340a, 340b are conical. The first of the elongated members 340a is radially spaced apart from the other elongated member 340b around the boss 314. Each elongated member 340a, 340b extends outward from the boss 314 in a radial direction perpendicular to the axis 312 of the shaft 310. Each elongated member 340a, 340b has a maximum radius at the proximal end, where the elongated members 340a, 340b are connected to the second axial section of the boss 314.

[0189] Fig. 27 A vibration transducer is shown in which four elongated members 340a, 340b, 340c, 340d are connected to the second axial section of the boss 314. The four elongated members 340a, 340b, 340c, 340d are conical. The first elongated member 340a is radially spaced from the second elongated member 340b around the boss 314, but is in the same axial position as the second elongated member 340b along the axis 312. The third elongated member 340c is radially spaced from the fourth elongated member 340d around the boss 314, but is in the same axial position as the fourth elongated member 340d along the axis 312. The first elongated member 320a and the second elongated member 320b are axially spaced from the third elongated member 340c and the fourth elongated member 340d respectively along the axis 312. Each elongated member 340a, 340b, 340c, 340d has a maximum radius at a proximal end where the elongated member connects to a second axial section of the boss 314.

[0190] Fig.28 A vibration transducer is shown in which two elongated members 360a, 360b are connected to the second axial section of the boss 314. Both elongated members 360a, 360b have a semicircular form. The first elongated member 360a is curved and connected to the boss 314 at two axially spaced locations and has a circular cross-section along the curved axis passing through the elongated member 360a. The second elongated member 360b is curved and connected to the boss 314 at two axially spaced locations, which are radially spaced around the boss and the location where the first elongated member 360a is connected to the boss 314. The second elongated member also has a circular cross-section along the curved axis passing through the elongated member 360b.

[0191] Fig.29 A vibration transducer is shown in which four elongated members 320a, 320b, 320c, 320d are connected to the second axial section of the boss 314. Each of the four elongated members 320a, 320b, 320c, 320d is cylindrical, has the same constant radius, and extends from the boss 314 in the radial direction from positions distributed around the circumference of the second axial section and distributed at 90° to the adjacent elongated members. Each distal end of the four elongated members 320a, 320b, 320c, 320d is connected to a fifth elongated member 324 that surrounds the boss 314 and has the form of a torus.

[0192] Fig.30 A vibration transducer is shown in which four elongated members 320a, 320b, 320c, 320d are connected to the second axial section of the boss 314. The four elongated members 320a, 320b, 320c, 320d are cylindrical, but have different radii. The first elongated member 320a has the largest radius. The radius of the second elongated member 320b is smaller than the radius of the first elongated member 320a. The radius of the third elongated member 320c is smaller than the radius of the second elongated member 320b. The radius of the fourth elongated member 320d is smaller than the radius of the third elongated member 320c. The first elongated member 320a and the second elongated member 320b both extend radially outward from the second axial section of the boss 314 from respective positions axially offset from each other, but otherwise extend radially outward from the same circumferential position. The third elongated member 320c and the second elongated member 320d each extend radially outward from respective positions axially offset from each other from the second axial section of the boss 314, but otherwise extend radially outward from the same circumferential position. The first elongated member 320a and the second elongated member 320b are spaced 90° apart from the third elongated member 320c and the fourth elongated member 320d around the circumference of the boss 314.

[0193] Fig.31 A vibration transducer is shown in which eight elongated members 370a, 370b, 370c, 370d, 370e, 370f, 370g, 370h are connected to the boss 314. The first elongated member 370a is connected to the third axial section of the boss 314 and has the form of a cone extending in a direction inclined relative to the axis 312 of the shaft 310. The second elongated member 370b is connected to the second axial section of the boss 314 and extends radially outward from the boss 314 from its proximal end. The second elongated member 370b is not straight, and includes two straight sections that are connected at right angles to form an "L" shaped member. The distal section of the second elongated member 370b extends in a direction parallel to the axis 312. The third elongated member 370c is connected to the second axial section of the boss 314 and has the form of a "T" shape. The proximal section of the third elongated member 370c is connected to the boss 314 and extends radially outward from the boss 314. The distal section of the third elongated member 370c is connected to the distal end of the proximal section of the third elongated member 370c and extends in a direction parallel to the axis 312, that is, perpendicular to the direction of the proximal section. The fourth elongated member 370d is connected to the second axial section of the boss 314 and includes a proximal section and a distal section, the proximal section is connected to the boss 314 and extends in an inclined direction relative to the axis 312 of the shaft 310, and the end of the distal section is connected to the distal end of the proximal section, and the distal section extends in a direction parallel to the axis 312. The fifth elongated member 370e is connected to the second axial section of the boss 314. The fifth elongated member 370e is not straight, but extends outward from the boss 314 in a wide radial direction, but it has a wide circular cross-section along its length. The sixth elongated member 370f is connected to the second axial section of the boss 314 and extends radially outward from the boss 314. At the distal end of the sixth elongated member 370f, another elongated member is connected, which has the form of a closed loop, and specifically the form of a circular ring. The seventh elongated member 370g is connected to the second axial section of the boss 314 and extends radially outward from the boss 314. At the distal end of the seventh elongated member 370g, another elongated member is connected, which is curved around an arc concentric with the circumference of the second axial section of the boss 314 in the circumferential direction. The eighth slender member 370h is connected to the third axial section of the boss 314 and includes a proximal section and a distal section, wherein the proximal section is connected to the boss 314 and extends outward from the surface of the third axial section in an inclined direction relative to the axis 312 of the shaft 310, and the end of the distal section is connected to the distal end of the proximal section, and the distal section extends in a direction parallel to the axis 312.

[0194] In the embodiments discussed above, the slender member used for geometric damping has a circular cross-section from which the radius can be easily determined. The analysis presented above uses the radius of the slender member to determine whether the conditions for geometric damping are met. However, those skilled in the art will recognize that the technology of the present disclosure does not rely entirely on slender members having a cross-section that is exactly circular. A circular cross-section may be a preferred embodiment, particularly for torsional vibrations around the axis of the slender member, since such vibrations will only produce shear waves. However, as described above, under some conditions, i.e., when the Reynolds number is low enough so that the flow is laminar, the offset of the slender member from the vibration axis can still produce monopole-like behavior in the wave field. Therefore, the requirement for the circularity of the slender member is only loose. According to the technology of the present disclosure, a slender member having a cross-section that is not exactly circular can still provide geometric damping.

[0195] One measure of the circularity of a shape is given by the following expression: Where p is the perimeter and A is the area of ​​the shape. For a perfect circle, the circularity by this metric would be equal to 1. For a square, the circularity by this metric would be equal to π / 4, which is approximately equal to 0.785. For an equilateral triangle, the circularity by this metric would be equal to It is approximately equal to 0.605. For a regular hexagon, the circularity will be equal to It is approximately equal to 0.907. In the present disclosure, reference to a "substantially" circular shape is intended to mean having a circularity by this measure in the range of 0.75 to 1 (inclusive), more preferably in the range of 0.8 to 1 (inclusive), even more preferably in the range of 0.85 to 1 (inclusive), even more preferably in the range of 0.9 to 1 (inclusive), even more preferably in the range of 0.92 to 1 (inclusive), even more preferably in the range of 0.95 to 1 (inclusive), even more preferably in the range of 0.96 to 1 (inclusive), even more preferably in the range of 0.97 to 1 (inclusive), even more preferably in the range of 0.98 to 1 (inclusive), and even more preferably in the range of 0.99 to 1 (inclusive). For non-convex shapes, it may be more appropriate to characterize circularity by convex perimeter and convex area, where the convex perimeter is the perimeter of the convex hull enclosing the shape, and the convex area is the area of ​​the convex hull enclosing the shape.

[0196] Therefore, when characterizing the technology of the present disclosure and considering the requirements for obtaining geometric damping, it may be more appropriate to consider the half-width of the slender member rather than the radius, the slender member having an axis extending along the length of the slender member and a constant or variable cross-section along the axis, the cross-section having a half-width representing the radius in the above analysis. The characteristic half-width used to characterize the geometric damping behavior of the slender member at this position along the axis can be the minimum distance from the center of mass of the cross-sectional area, the maximum distance from the center of mass, or the value between the minimum distance and the maximum distance from the center of mass (which can be the average of the minimum distance and the maximum distance), or the average of the distance from the center of mass averaged around the periphery of the cross-section. Alternatively, the half-width can be determined from the circumference of the equivalent circle itself, or if the shape is concave, it is determined from the convex circumference. For example, if the convex circumference is p, the half-width used to characterize the shape can be calculated as p / 2π. In the case of a circle, this expression will return the radius of the circle. But the shape is not a perfect circle, and the formula will return the equivalent "radius" (i.e., half-width), which will characterize the geometric damping behavior of the slender member. Alternatively, the half-width that characterizes the shape may be based on the convex area of ​​the shape of an equivalent circle, ie, √(A / π).

[0197] In some embodiments, the elongated member has a constant cross-section along its length. Therefore, the calculation of the half-width that characterizes the geometric damping behavior of the elongated member can be performed at any point along the length. In some other embodiments, the elongated member has a non-constant cross-section along its length. The radius or half-width can vary along the length of the elongated member. Therefore, it may be more appropriate to calculate the characteristic radius of the elongated member. Conservatively, if the maximum half-width along the length of the elongated member is less than the viscoelastic propagation depth, then geometric damping can be assumed to exist. However, if Figure 4 As shown, even at radii slightly exceeding the viscoelastic propagation depth, the deviation from linearity between the damping factor and viscosity may remain slightly gradual. Therefore, taking a less conservative approach, geometric damping may still be achieved if the average measure of the non-constant half-width is less than the viscoelastic propagation depth. Such an average value may be calculated from the average of the half-widths along the length of the elongated member, such as the arithmetic mean of the half-widths along the length of the elongated member, or from the volume and surface area of ​​the elongated member by the following expression: where V is the volume of the elongated member and A is the surface area; in the case of a non-convex form, these quantities may be the volume and area of ​​the convex hull of the elongated member.

[0198] Fig.32An example device for analyzing a fluid using one or more techniques of the present disclosure is schematically shown. The device includes a vibration transducer 415 in a fluid sample 400. In this case, the fluid sample 400 is a fixed volume of substantially stationary fluid in a chamber 406, the fluid having a free surface 407, and portions of the viscosity transducer pierce the free surface 407 from above.

[0199] Although chamber 406 is Fig.32 405. The chamber 407 is drawn as having a closed top wall, as may be necessary if its contents are pressurized, but the chamber may likewise be open from above. Thus, the fluid sample may be at atmospheric pressure. In such a configuration, the vibration transducer may be located above the chamber 407 so that at least a portion of the vibration transducer extends into the upper opening of the chamber 407 and contacts the fluid sample 405 instead.

[0200] The vibration transducer 415 includes a vibrating element configured to oscillate in a torsional mode. The vibrating element is immersed in a fluid, and the viscosity is determined by a correlation with the damping (i.e., Q factor) experienced by the element. In particular, the vibration transducer includes a sensor mount 413, a semi-rigid connecting member 411, a shaft 410, and a boss or "sensing element" 414. The shaft 410 and the boss 414 are driven to vibrate torsionally at an angular frequency ω around the longitudinal axis of the shaft 410. The boss 414 has a relatively large mass, and the shaft 410 and the boss 414 are at least substantially and possibly entirely formed of a metallic material such as stainless steel. In addition to the presence of a slender member 420 extending radially outward from the circumference of the boss 414 according to the technology disclosed herein, the boss 414 and the shaft 410 both have a circular cross-section. The entire slender member 420 deviates from the longitudinal axis of the shaft 410.

[0201] Boss 414 and elongated member 420 are exposed to the viscous effects of the fluid in sample 400. Increasing the viscosity of the fluid causes increased vibration damping in the sensor, resulting in a measurable decrease in the vibration efficiency of the system. Measurements of quantities indicative of the degree of damping in the system, such as a Q factor or loss factor, are provided from the vibration transducer to processor 418, which is configured to process the measurements and determine a physical property of the fluid sample 400, such as viscosity. Processor 418 may be integrated with vibration transducer 415 or connected to the vibration transducer via a data interface.

[0202] Fig.33 Another example apparatus for analyzing a fluid using one or more techniques of the present disclosure is schematically shown. The apparatus includes Fig.32The same vibration transducer 415 as in FIG. 4 is used, but in this case, the fluid sample 400 flows in the conduit 418 at an upstream average velocity of, for example, 1 m / s. The vibration transducer 415 extends from above through the wall of the conduit 408, and the portion extending through the wall of the conduit 408 contacts the fluid sample 400 as it flows past the vibration transducer 415. Fig.32 4 , the vibration transducer 415 includes an elongated member 420 according to the disclosed technology, which extends radially outward from the circumference of the boss 414. The entire elongated member 420 is offset from the longitudinal axis of the shaft 410.

[0203] It should be emphasized that although Fig.32 and Fig.33 The vibration transducer including the boss 414 is shown in each case, but the technology of the present disclosure is not limited to the vibration transducer including the boss.

[0204] Fig.34 A flow chart of a method according to the technology of the present disclosure is shown. The method includes a first step 510 of vibrating a vibration transducer element in a fluid at a vibration frequency, wherein the vibration transducer element includes a slender member contacting the fluid, the slender member having a half-width less than the propagation depth of the shear wave in the fluid at the vibration frequency. The method includes a second step 520 of measuring the vibration of the vibration transducer element in the fluid at the vibration frequency. The method includes a third step 530 of determining the physical properties of the fluid based on the measurement of the vibration. According to the technology of the present disclosure, determining the physical properties of the fluid based on the measurement of the vibration may include determining one or more of the following: viscosity, viscoelasticity, density, fluid stiffness, loss tangent, storage modulus, loss modulus, and yield stress.

[0205] Fig.35 A flow chart of another method according to the technology of the present disclosure is shown. The method includes a first step 610 of vibrating a vibration transducer element in a fluid at a vibration frequency, first at a first vibration amplitude and then at a second vibration amplitude, wherein the vibration transducer element includes a slender member contacting the fluid. The method includes a second step 620 of determining a respective first quantity and a second quantity indicating a degree of damping based on the vibration of the vibration transducer element in the fluid at the first amplitude and the second amplitude, respectively, wherein the first quantity and the second quantity may be a Q factor in some embodiments. The method includes a third step 630 of determining a property of the fluid based on a difference between the first quantity and the second quantity, which may be, for example: i) determining a dipole of a wave field around the vibration transducer at the vibration frequency based on the difference between the first quantity and the second quantity, ii) determining a Reynolds number of the fluid based on the difference between the first quantity and the second quantity, or iii) determining a vibration velocity relative to the fluid, a viscosity of the fluid, or a density of the fluid based on the difference between the first quantity and the second quantity.

[0206] In some embodiments, Fig.35 The method described in Fig.34 A combination of the methods described in Fig.35 The method is used to determine the dipole property of the wave field around a vibration transducer to confirm or to determine the Reynolds number of the fluid for the purpose of determining or confirming the presence of geometric damping.

[0207] A vibration transducer according to the techniques described herein can be used to determine a physical property of a fluid by vibrating the vibration transducer in the fluid at a vibration frequency and determining an amount indicative of the degree of damping based on the vibration. For example, to make a measurement of viscosity, the Q factor of the vibration can be determined. The Q factor is a dimensionless parameter that indicates the damping level of a resonator, where the damping level is a function of viscosity. In particular, it indicates the degree to which the resonator is underdamped. On a graph of the frequency response, a high Q factor provides a high and narrow peak at the resonant frequency, while a low Q factor provides a low and wide peak. Due to the variation in the width of the peak with damping, the Q factor can be defined as the ratio of the resonant frequency to the resonant bandwidth: where ω R is the resonant frequency in radians per second, Δω is the Full Width at Half Maximum (FWHM), the bandwidth over which the vibration power is greater than half of the maximum value (or equivalently, the vibration amplitude is greater than the maximum amplitude at resonance divided by √2), that is, the bandwidth between the 3dB points. Fluid viscosity is a function of the Q factor. In the absence of geometric damping, the fluid viscosity is inversely proportional to the square of the Q factor, and any proportionality constant required to calculate the viscosity measurement can be obtained by calibration with a reference fluid of known viscosity. In the case of applicable geometric damping, the fluid viscosity is inversely proportional to the Q factor. Fluid viscosity can be determined under geometric damping conditions based on an amount that is inversely proportional to the measured or estimated value of the Q factor (assuming that the mechanical damping of the vibrating system is small or negligible relative to the damping provided by the fluid load).

[0208] It should be noted that the viscosity measurement at or corresponding to the vibration frequency may include taking amplitude measurements at more than one frequency to estimate the Q factor, but obtaining a single viscosity measurement at a frequency corresponding to the resonant frequency or frequencies. For example, the bandwidth may be determined based on the frequency required to drop the amplitude to 1 / √2 of the maximum amplitude at resonance. As a non-limiting example, the frequency required to drop the amplitude to 1 / √2 of the maximum amplitude at resonance may be determined by performing a frequency sweep around the resonant frequency, but the skilled person will recognize that the 3dB point frequency may be identified by various other techniques.

[0209] Another way to determine the Q factor is to measure the vibration amplitude at a range of frequencies around the resonant frequency and fit a parabola to the frequency and amplitude values ​​(or their logarithms) by the least squares method. The 3dB point can then be obtained as a solution to a quadratic equation based on the best fit parabola to the measured values.

[0210] Another way to determine the Q factor is by logarithmic decrement. By stopping driving the transducer and measuring the decay of the vibration, the Q factor can be determined by monitoring the time series of the vibration and by the following expression: 1 and A 2 The Q factor is determined by taking the natural logarithm of the ratio:

[0211] As an alternative to determining the Q factor, the techniques of the present disclosure contemplate determining a "loss factor," where the loss factor is the inverse of the Q factor and can be determined by techniques corresponding to those set forth above with respect to the Q factor.

[0212] In the techniques described herein, where it is desired to determine the location of an "end" of an elongated member, the location of the end of the elongated member should be considered to be the location of the centroid of the cross-sectional area of ​​the elongated member at that end.

[0213] Although some embodiments described in this disclosure relate to determining properties of a fluid, those skilled in the art will recognize that the geometric damping techniques described herein may have broader applications.

[0214] For example, geometric damping can be used to intentionally modify or control the damping characteristics of any object that vibrates or is about to vibrate in a fluid. An example may be a shaft that vibrates in a fluid, as is often encountered in industry. According to the techniques of the present disclosure, the damping characteristics may be controlled by providing the shaft with one or more elongated members. Geometric damping may occur if the one or more elongated members are correctly dimensioned relative to the propagation depth of the shear wave in the fluid at the vibration frequency of the shaft. In addition, if the one or more elongated members are spaced apart from the longitudinal axis of the shaft, an amplification of the damping effect may be obtained, while geometric damping may be maintained if the fluid flow range allows at least partial recovery of the monopole wave field. An appropriate number of elongated members having an appropriate geometry may be provided to the shaft according to the techniques of the present disclosure to provide the desired damping characteristics independent of the fluid properties, provided that the geometric damping criteria are met. In some cases, vibration may be undesirable, so the goal of controlling the damping behavior is to provide a relatively high degree of damping so that the vibration is attenuated.

[0215] A method for controlling the damping behavior of a shaft configured to vibrate in a fluid, the method comprising: providing a shaft configured to vibrate at a vibration frequency, the shaft having a longitudinal axis; and providing a slender member connected to the shaft but not colinear with the longitudinal axis of the shaft, the slender member being characterized by a width, a half-width equal to half of the width, and a length greater than the width, the slender member and at least a portion of the shaft being configured to vibrate in the fluid at the vibration frequency, wherein the half-width of the slender member is less than the propagation depth of a shear wave in the fluid at the vibration frequency.

[0216] As previously described, the propagation depth may be the distance at which the amplitude of a shear wave propagating in a fluid at the vibration frequency decreases by 1 / e and / or may be the viscoelastic propagation depth according to (Equation 1).

[0217] The elongated member may be any of the elongated members described herein, including Figures 7 to 31 In some embodiments, the elongated member has a first end and a second end, wherein one or both of the first end and the second end are spaced apart from the longitudinal axis of the shaft by an offset distance greater than half the width of the elongated member.

[0218] The elongate member may be arranged such that during vibration of the shaft at the vibration frequency, fluid flow around the elongate member is laminar. This may influence the selection of dimensions of the elongate member in order to obtain laminar flow around the elongate member during vibration of the shaft.

[0219] In some embodiments, the elongated member is configured such that during vibration of the shaft in the fluid at the vibration frequency, the Reynolds number Re of the fluid flow around the elongated member is less than 1, wherein the Reynolds number is given by: Where μ is the viscosity of the fluid, ρ is the density of the fluid, R is the half-width of the elongated member, and v is the maximum velocity of the elongated member relative to the fluid during vibration of the shaft. The choice of the length and half-width of the elongated member will affect the Reynolds number, and the Reynolds number will therefore represent a constraint on the size of the elongated member.

[0220] For example, if the dimensions of the elongated member meeting the Reynolds number criteria do not provide the desired level of damping, the shaft may be provided with a plurality of elongated members connected to the shaft, each elongated member being non-colinear with the longitudinal axis of the shaft, each elongated member having a half-width less than the propagation depth of the shear wave in the fluid at the vibration frequency. In this manner, the damping characteristics of the shaft are affected by the combined effect of the plurality of elongated members.

[0221] Those skilled in the art will further recognize that these techniques are not limited to providing an elongated member on a vibration axis, but can include providing one or more elongated members on any vibrating object, such as a plate configured to vibrate in an in-plane direction at a vibration frequency to generate shear waves. Providing one or more elongated members connected to the plate (e.g., extending vertically or obliquely outward from the plate) can result in the occurrence of geometric damping as described herein, the elongated member having a half-width that is less than the propagation depth of the shear wave in the fluid at the vibration frequency.

[0222] Those skilled in the art will further appreciate that the various illustrative logic blocks, configurations, modules, circuits, and algorithmic steps described in conjunction with the embodiments disclosed herein can be implemented as electronic hardware, computer software, or a combination of the two. To clearly illustrate this interchangeability of hardware and software, various illustrative components, blocks, configurations, modules, circuits, and steps have been generally described above with respect to their functionality. Whether such functionality is implemented as hardware or software depends on the specific application and the design constraints imposed on the entire system. Those skilled in the art may implement the described functionality in different ways for each specific application, but such implementation decisions should not be interpreted as resulting in a departure from the scope of the present disclosure.

[0223] The steps of the method or algorithm described in conjunction with the embodiments disclosed herein may be directly embodied in hardware, in a software module executed by a processor, or in a combination of the two. The software module may reside in a random access memory (RAM), a flash memory, a read-only memory (ROM), a programmable read-only memory (PROM), an erasable programmable read-only memory (EPROM), an electrically erasable programmable read-only memory (EEPROM), a register, a hard disk, a removable disk, a compact disc read-only memory (CD-ROM), or any other form of storage medium known in the art. An exemplary storage medium is coupled to a processor so that the processor can read information from the storage medium and write information to the storage medium. In an alternative, the storage medium may be a component of the processor. The processor and the storage medium may reside in an application specific integrated circuit (ASIC). The ASIC may reside in a computing device or a user terminal. In an alternative, the processor and the storage medium may reside in a computing device or a user terminal as discrete components.

[0224] The previous description of the disclosed embodiments is provided to enable those skilled in the art to make or use the disclosed embodiments. Various modifications to these embodiments will be apparent to those skilled in the art, and the principles defined herein may be applied to other embodiments without departing from the scope of the present disclosure. Therefore, the present disclosure is not intended to be limited to the embodiments shown herein, but to be as wide as possible consistent with the principles and novel features defined by the appended claims.

Claims

1. A method for determining a physical property of a fluid, the method comprising: causing a vibrating transducer element to vibrate in a fluid at a vibration frequency, wherein the vibrating transducer element comprises an elongated member contacting the fluid, the elongated member being characterized by a width, a half-width equal to half of the width, and a length greater than the width, wherein the half-width is less than a propagation depth of a shear wave in the fluid at the vibration frequency; measuring vibration of the vibration transducer element in the fluid at the vibration frequency; and A physical property of the fluid is determined based on the measurement of the vibrations.

2. The method according to claim 1, wherein: Determining the physical properties of the fluid based on the measurement of vibrations includes determining one or more of: viscosity, viscoelasticity, density, fluid stiffness, loss tangent, storage modulus, loss modulus, and yield stress.

3. The method according to claim 2, wherein: Measuring the vibration includes determining a quantity indicative of a degree of damping of the vibration transducer element in the fluid at the vibration frequency, and wherein determining a physical property of the fluid based on the measurement of the vibration includes determining a viscosity of the fluid based on the quantity indicative of the degree of damping.

4. The method according to claim 2, wherein: Measuring the vibration includes determining a first quantity indicative of a degree of damping of the vibration transducer element in the fluid at the vibration frequency, wherein the method further includes vibrating the vibration transducer element in the fluid at another vibration frequency and determining a second quantity indicative of a degree of damping of the vibration transducer element in the fluid at the another vibration frequency, wherein determining the physical property of the fluid based on the measurement of the vibration includes determining the viscoelasticity of the fluid based on the quantities indicative of the degree of damping at the vibration frequency and at the another vibration frequency.

5. The method according to claim 2, wherein: Measuring the vibrations includes determining a resonant frequency of the vibrating transducer element in the fluid, wherein determining a physical property of the fluid based on the measurement of the vibrations includes determining a density of the fluid based on the resonant frequency.

6. A method according to any one of the preceding claims, wherein: The propagation depth is the distance over which the amplitude of a shear wave propagating in the fluid at the vibration frequency decreases by 1 / e, where e is the base of the natural logarithm.

7. A method according to any one of the preceding claims, wherein: The propagation depth of a shear wave propagating in the fluid at the vibration frequency is given by the following expression: Where μ is the viscosity of the fluid, ρ is the density of the fluid, ω is the angular frequency of vibration, and Δ varies between 0 and π / 2 and is defined by the loss tangent tanΔ, and where tanΔ is equal to the following expression where G' is the storage modulus of the fluid:

8. A method according to any one of the preceding claims, wherein: The half width of the elongated member is less than 50% of the propagation depth.

9. A method according to any one of the preceding claims, wherein: The vibration transducer element includes a shaft having a longitudinal axis, wherein the elongated member is connected to the shaft, and wherein the elongated member is not co-linear with the longitudinal axis of the shaft.

10. The method according to claim 9, wherein: During vibration of the vibration transducer element at the vibration frequency, the flow of the fluid around the elongate member is laminar.

11. The method according to claim 9 or claim 10, wherein: The elongated member has a first end and a second end, wherein one or both of the first end and the second end are spaced from the longitudinal axis of the shaft by an offset distance greater than the half-width of the elongated member.

12. The method according to any one of claims 9 to 11, wherein: During vibration of the vibration transducer element in the fluid at the vibration frequency, the Reynolds number Re of the fluid flow around the elongated member is less than 1000, preferably less than 100, more preferably less than 10, even more preferably less than 1, wherein the Reynolds number is given by: wherein μ is the viscosity of the fluid, ρ is the density of the fluid, R is the half width of the elongated member, and v is the maximum velocity of the elongated member relative to the fluid during vibration of the vibration transducer.

13. The method according to any one of claims 9 to 12, wherein: The shaft includes a boss, and the elongated member is connected to the shaft at the boss.

14. The method according to any one of claims 9 to 13, wherein: The vibration transducer element includes a plurality of elongated members connected to the shaft, each elongated member being non-colinear with a longitudinal axis of the shaft, each elongated member having a half-width less than a propagation depth of a shear wave in the fluid at the vibration frequency.

15. The method according to claim 14, wherein: A half-width of a first elongated member of the plurality of elongated members is different than a half-width of a second elongated member of the plurality of elongated members.

16. A method according to any one of the preceding claims, wherein: The circularity of the cross-section of the elongated member is in the range of 0.75 to 1 along at least 50% of the length of the elongated member, wherein the circularity of the cross-section is calculated by the following formula: Where A is the convex area of ​​the cross section, and p is the convex perimeter of the cross section.

17. A method according to any one of the preceding claims, wherein: The elongate member may have a constant cross-section along at least 90% of its length, or a constant cross-section along no more than 10% of its length.

18. A method according to any one of the preceding claims, wherein: The elongated member is straight or non-straight, or comprises one of the following: a cylinder, a cone, a frustum of a cone, a torus, and an arcuate portion of a torus.

19. A method according to any one of the preceding claims, wherein: Vibrating the vibration transducer element comprises vibrating the vibration transducer element with an oscillating rotational motion and / or an oscillating linear motion and / or an oscillating curvilinear motion.

20. The method according to claim 19, wherein: The elongated member is straight, and wherein vibrating the transducer element comprises vibrating the elongated member in an oscillatory rotational motion about an axis along the length of the elongated member.

21. A method according to any one of the preceding claims, wherein: The length of the elongated member is greater than twice the width of the elongated member.

22. A method according to any one of the preceding claims, wherein: The half width of the elongated member is greater than 0.5 mm, wherein the viscosity of the fluid is greater than 100 Pa·s, wherein the density of the fluid is between 500 kg / m 3 With 1500kg / m 3 and wherein the vibration frequency is less than 10 kHz.

23. An apparatus for determining a physical property of a fluid, the apparatus comprising: A vibration transducer comprising a shaft configured to vibrate at a vibration frequency, the shaft having a longitudinal axis, wherein the vibration transducer further comprises an elongated member connected to the shaft but not co-linear with the longitudinal axis of the shaft, the elongated member being characterized by a width, a half-width equal to half of the width, and a length greater than the width, wherein the apparatus is configured to determine a physical property of the fluid by vibrating the shaft in the fluid at the vibration frequency when the elongated member is in contact with the fluid, wherein at the vibration frequency, the half width of the elongated member is less than a propagation depth of a shear wave in the fluid, The device is configured to measure the vibration of the vibration transducer in the fluid at the vibration frequency, and determine the physical property of the fluid based on the measurement of the vibration at the vibration frequency.

24. The device according to claim 23, wherein: At least a portion of the elongated member is offset from the longitudinal axis by an offset distance greater than the half-width of the elongated member.

25. The device according to claim 24, wherein: The elongated member has a first end and a second end, wherein one or both of the first end and the second end are spaced from the longitudinal axis of the shaft by an offset distance greater than the half-width.

26. The device according to any one of claims 23 to 25, wherein: The circularity of the cross-section of the elongated member is in the range of 0.75 to 1 along at least 50% of the length of the elongated member, wherein the circularity of the cross-section is calculated by the following formula: Where A is the convex area of ​​the cross section, and p is the convex perimeter of the cross section.

27. The device according to any one of claims 23 to 26, wherein: The elongate member may have a constant cross-section along at least 90% of its length, or a constant cross-section along no more than 10% of its length.

28. The device according to any one of claims 23 to 27, wherein The elongated member is straight or non-straight, or comprises one of the following: a cylinder, a cone, a frustum of a cone, a torus, and an arcuate portion of a torus.

29. The device according to any one of claims 23 to 28, wherein The length of the elongated member is greater than twice the width of the elongated member.

30. The device according to any one of claims 23 to 29, wherein: The half width of the slender member is greater than 0.5 mm, and / or greater than 1 mm, and / or greater than 2 mm, and / or greater than 5 mm, and / or greater than 10 mm, and / or greater than 20 mm, and / or greater than 50 mm.

31. Apparatus according to any one of claims 23 to 30, comprising a plurality of elongate members connected to the shaft, each elongate member having a respective width, half-width and length.

32. The device according to claim 31, wherein A half-width of a first elongated member of the plurality of elongated members is different than a half-width of a second elongated member of the plurality of elongated members.

33. The device according to any one of claims 23 to 32, wherein: The shaft is configured to vibrate torsionally about its longitudinal axis, and / or to vibrate longitudinally along its longitudinal axis and / or transversely to its longitudinal axis.

34. The device according to any one of claims 23 to 33, wherein: The shaft includes a boss, and the elongated member is connected to the shaft at the boss.

35. A method of determining a property of a fluid, the method comprising: causing a vibration transducer element to vibrate in a fluid at a vibration frequency first at a first vibration amplitude and then at a second vibration amplitude, wherein the vibration transducer element comprises an elongated member contacting the fluid, the elongated member being characterized by a width, a half-width equal to half of the width, and a length greater than the width; determining a first quantity indicative of a degree of damping based on vibration of the vibrating transducer element in the fluid at the first amplitude; determining a second quantity indicative of a degree of damping based on vibration of the vibrating transducer element in the fluid at the second amplitude; A property of the fluid is determined based on a difference between the first amount and the second amount.

36. The method of claim 35, wherein: Determining the properties of the fluid includes determining whether the half-width of the slender member is smaller than the propagation depth of the shear wave in the fluid at the vibration frequency and / or the extent to which the half-width of the slender member is smaller than the propagation depth of the shear wave in the fluid at the vibration frequency based on the difference between the first quantity and the second quantity.

37. A method according to claim 35 or claim 36, wherein: Determining the property of the fluid includes determining a Reynolds number of the fluid based on a difference between the first quantity and the second quantity.

38. A method according to any one of claims 35 to 37, wherein: Determining the property of the fluid includes determining a vibration velocity relative to the fluid, a viscosity of the fluid, or a density of the fluid based on a difference between the first quantity and the second quantity.

39. The method according to any one of claims 35 to 38, wherein: Determining the first quantity and the second quantity includes determining a first Q factor and a second Q factor.

40. The method according to any one of claims 35 to 39, wherein: The determined property of the fluid is a property of the flow of the fluid due to the vibration of the vibrating transducer element in the fluid at a vibration frequency of one or both of the first amplitude and the second amplitude.

41. The method of any one of claims 35 to 40, further comprising performing the method of any one of claims 1 to 22.

42. The method according to any one of claims 35 to 41, wherein: The method is performed using an apparatus according to any one of claims 23 to 34.

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