Sound-solid coupling model of bionic robotic fish based on MFC material

Abstract

The invention discloses an acoustic-solid coupling model of a bionic robotic fish based on an MFC material. The acoustic-solid coupling model comprises a bionic robotic fish main body, a bionic robotic fish motion control model, an MFC material mechanical response model, a bionic robotic fish modal analysis model and an acoustic-solid coupling analysis model, the piezoelectric effect and the driving mechanism are defined, and a theoretical model is provided for application of the bionic robotic fish made of the MFC material. According to the method, the structure behavior of the robotic fish in the sound field drainage basin is calculated through sound-solid coupling, the modal and deformation response are promoted, the calculation difficulty is reduced, and the structure of the robotic fish is further optimized. According to the method, an acoustic-solid coupling method and modal and transient analysis are combined, the underwater motion state of the robotic fish is predicted, and the propelling motion of the bionic robotic fish under the driving of sinusoidal voltage in a fluid calculation domain is simulated. By changing the amplitude and the phase difference of the driving voltage of the MFCs on the left side and the right side, straight and steering swimming movement of the bionic robotic fish is achieved.

Description

Technical Field

[0001] The invention relates to the technical field of bionic robot fish, and more particularly to an acoustic-solid coupling model of a bionic robot fish based on MFC materials. Background Art

[0002] The ocean occupies about 71% of the earth's surface area and contains rich biological and mineral resources. The development and research of marine resources has been included in the national long-term plan. In the exploration of the ocean, bionic robot fish is of great significance. After millions of years of evolution, fish have extraordinary swimming ability, which can maintain long-distance high-efficiency swimming and have good maneuverability. The fish propulsion mode is the most efficient and most adaptable propulsion mode in the underwater environment to date. Intelligent materials with functions similar to fish muscle tissue are applied to the design of bionic robot fish as a drive to provide power for the robot fish. Underwater bionic propellers that imitate the appearance and swimming mode of fish have become a powerful tool for humans to develop and survey marine resources, providing sufficient technical guarantee for exploring complex marine environments. Due to the defects of traditional driving methods such as large size, high noise and lack of flexibility, scholars in the field of underwater robot research have begun to shift their research focus to the design of small, fast and flexible robot fish. MFC has the advantages of good flexibility, large deformation and strong driving ability, and has shown great potential advantages in the field of underwater driving. The bionic robot fish driven by MFC materials can simulate the flexible movements of fish muscles, providing a new method for the design of bionic drives.

[0003] Therefore, how to provide an acoustic-solid coupling model of a bionic robotic fish based on MFC materials is a problem that technical personnel in this field urgently need to solve. Summary of the invention

[0004] In view of this, the purpose of the present invention is to provide an acoustic-solid coupling model of a bionic robotic fish based on MFC materials.

[0005] In order to achieve the above object, the present invention adopts the following technical solution:

[0006] An acoustic-solid coupling model of a bionic robotic fish based on MFC materials, including a bionic robotic fish body, a bionic robotic fish motion control model, an MFC material mechanical response model, a bionic robotic fish modal analysis model and an acoustic-solid coupling analysis model;

[0007] The bionic robotic fish body comprises two MFC plates and one CFPR plate; wherein the one CFPR plate is sandwiched between the two MFC plates;

[0008] The bionic robotic fish motion control model is used to apply an electric field to the two MFC plates to control the bionic robotic fish to move straight, turn left, and turn right;

[0009] The MFC material mechanical response model is used to convert the electric field applied by the bionic robotic fish motion control model into the deformation power of the bionic robotic fish;

[0010] The bionic robotic fish modal analysis model is used to predict the vibration shape and modal frequency of the bending deformation of the bionic robotic fish;

[0011] The acoustic-solid coupling analysis model is used to simulate the structural behavior, propulsion mode and deformation response of the robotic fish according to the predicted vibration mode and modal frequency of the bending deformation of the bionic robotic fish.

[0012] Preferably, the space region occupied by the fluid is regarded as the acoustic field region, and the acoustic unit is used to simulate the fluid to construct an acoustic-solid coupling analysis model.

[0013] Preferably, the finite element equation of the acoustic-solid coupling analysis model is as follows:

[0014]

[0015] Where M a ,C a ,K a , A represent the overall mass matrix, damping matrix, stiffness matrix and coupling matrix of the acoustic unit respectively; M S ,C S ,K S represent the overall mass matrix, damping matrix and stiffness matrix of the structure respectively.

[0016] Preferably, in order to simulate the vibration performance of the robotic fish in water, a spherical calculation domain with a diameter of 2000 mm is used as the acoustic-solid coupling calculation domain of the robotic fish; wherein the robotic fish is located in the center.

[0017] Preferably, sinusoidal voltages with the same frequency, the same amplitude and opposite phases are applied to the two MFC plates respectively to control the bionic robotic fish to move in a straight line.

[0018] Preferably, a first left-turn drive voltage and a second left-turn drive voltage are respectively applied to the left and right MFC plates to control the bionic robotic fish to turn left; wherein the first left-turn drive voltage and the second left-turn drive voltage have the same frequency and opposite phase; and the amplitude of the first left-turn drive voltage is smaller than the amplitude of the second left-turn drive voltage.

[0019] Preferably, a first right-turn drive voltage and a second right-turn drive voltage are respectively applied to the left and right MFC plates to control the bionic robot fish to turn right; wherein the first right-turn drive voltage and the second right-turn drive voltage have the same frequency and opposite phase; and the amplitude of the first right-turn drive voltage is greater than the amplitude of the second right-turn drive voltage.

[0020] Preferably, the MFC material mechanical response model realizes its conversion function based on the following formula:

[0021]

[0022] Among them, D,σ,E v ,ε respectively represent electric displacement, mechanical stress, electric field intensity and total strain vector; r T ,r S Represent the pressure dielectric constant and pressure strain constant respectively; d represents the piezoelectric strain constant matrix, e represents the piezoelectric pressure constant matrix; s and c represent the material flexibility coefficient matrix and stiffness coefficient matrix respectively; superscript T and S represent the pressure is constant or the strain is constant respectively, and superscript E represents the external applied electric field strength is constant; A area represents the area of ​​the piezoelectric unit; F represents the deformation power of the robotic fish obtained by transformation.

[0023] Preferably, the bionic robotic fish modal analysis model realizes the function of predicting the vibration shape and modal frequency of the bending deformation of the bionic robotic fish based on the robotic fish structure modal finite element equation and the robotic fish fluid finite element equation.

[0024] Preferably, the modal finite element equation of the robotic fish structure is as follows:

[0025]

[0026] Where M S and K S is the structural matrix and stiffness matrix, U represents displacement, F S represents the external force on the structure, f F represents the pressure vector describing the acoustic-solid coupling, P represents the fluid pressure, R represents the effective area matrix associated with each node at the interface between the robotic fish structure and the fluid, and N S With N F represent the finite element shape functions of the fish body structure and the fluid structure respectively, and n represents the fluid boundary vector;

[0027] The finite element equation of the robotic fish fluid is as follows:

[0028]

[0029] Where M F and K F Denote the mass matrix and stiffness matrix respectively, F F represents the addition of mass, ρ f represents the static density of the fluid.

[0030] It can be seen from the above technical solutions that, compared with the prior art, the present invention discloses an acoustic-solid coupling model of a bionic robotic fish based on MFC materials, which can achieve the following beneficial technical effects:

[0031] (1) The present invention clarifies the piezoelectric effect and driving mechanism, and provides a theoretical model for the application of bionic robotic fish using MFC materials.

[0032] (2) The present invention uses acoustic-solid coupling to calculate the structural behavior, propulsion mode and deformation response of the robotic fish in the acoustic field, which reduces the difficulty of calculation and further optimizes the structure of the robotic fish.

[0033] (3) The present invention combines the acoustic-solid coupling method, modal and transient analysis to predict the motion state of the robotic fish underwater and simulate the propulsion motion of the bionic robotic fish driven by a sinusoidal voltage in the fluid computational domain. By changing the amplitude and phase difference of the MFC driving voltage on the left and right sides, the bionic robotic fish can achieve straight and turning swimming motions. BRIEF DESCRIPTION OF THE DRAWINGS

[0034] In order to more clearly illustrate the embodiments of the present invention or the technical solutions in the prior art, the drawings required for use in the embodiments or the description of the prior art will be briefly introduced below. Obviously, the drawings described below are only embodiments of the present invention. For ordinary technicians in this field, other drawings can be obtained based on the provided drawings without paying creative work.

[0035] Figure 1 A bionic robotic fish model provided in an embodiment of the present invention;

[0036] Figure 2 The acoustic-solid coupling calculation domain of the robotic fish provided in an embodiment of the present invention;

[0037] Figure 3 A transient analysis test plane provided in an embodiment of the present invention;

[0038] Figure 4 is the swing amplitude of the tail fin of the robot fish moving straight forward provided in an embodiment of the present invention;

[0039] Figure 5 The voltage signal for driving the robot fish to move forward in a straight line provided in an embodiment of the present invention;

[0040] Figure 6 The straight-moving posture of the robotic fish provided in an embodiment of the present invention;

[0041] Figure 7 The left-turn driving voltage signal of the robot fish provided in the embodiment of the present invention;

[0042] Figure 8 The left-turn motion posture of the robotic fish provided in an embodiment of the present invention;

[0043] Fig. 9 The right-turn driving voltage signal of the robot fish provided in the embodiment of the present invention;

[0044] Fig.10 This is the right-turn motion posture of the robotic fish provided in an embodiment of the present invention. DETAILED DESCRIPTION

[0045] The following will be combined with the drawings in the embodiments of the present invention to clearly and completely describe the technical solutions in the embodiments of the present invention. Obviously, the described embodiments are only part of the embodiments of the present invention, not all of the embodiments. Based on the embodiments of the present invention, all other embodiments obtained by ordinary technicians in this field without creative work are within the scope of protection of the present invention.

[0046] The embodiment of the present invention discloses an acoustic-solid coupling model of a bionic robotic fish based on MFC materials: (1) the size of the bionic robotic fish

[0047] The main body of the bionic robot fish is composed of two MFC plates sandwiching a CFRP substrate. The MFC material is the M-8528-P1 model, the thickness of the central CFRP plate is 0.2 mm, and the other structural parameters of the bionic robot fish are shown in Table 1. The bionic robot fish model is shown in Figure 1 shown.

[0048] Table 1 Structural parameters of the robotic fish model

[0049]

[0050]

[0051] (2) Piezoelectric constitutive equation of MFC materials

[0052] The piezoelectric fibers of the MFC material are laid in the direction of the working surface, polarized along the length of the fibers, and arranged in a cross-finger electrode pattern. This enables the MFC to make full use of the external electric field and optimize the effect of the inverse piezoelectric effect. Under the drive of the same external electric field, it produces the maximum mechanical response, thereby enabling the robotic fish to obtain greater driving force and deformation. The MFC piezoelectric driver is selected as the driving unit of the bionic robotic fish. The MFC driver is controlled by an external voltage to produce deformation, realizing the propulsion mode of the tail fin swinging.

[0053] The piezoelectric equation of MFC materials is based on the thermodynamic equation and is divided into d-type and e-type piezoelectric equations. The independent variable is the electric field intensity E. v .

[0054]

[0055] Where D, σ, E v ,ε respectively represent electric displacement, mechanical stress, electric field intensity and total strain vector. T ,r S Represent the pressure dielectric constant and pressure strain constant, respectively. d represents the piezoelectric strain constant matrix, and e represents the piezoelectric pressure constant matrix. s and c represent the material flexibility coefficient matrix and stiffness coefficient matrix, respectively. The superscripts T and S represent that the pressure is constant or the strain is constant, respectively, and the superscript E represents that the externally applied electric field strength is constant.

[0056] Convert MFC stress into mechanical fish deformation force.

[0057]

[0058] Where A represents the area of ​​the piezoelectric unit.

[0059] When the MFC is subjected to external force and produces a positive piezoelectric effect, the positive and negative charges will concentrate on different surfaces of the MFC, thereby generating electric displacement inside the MFC; when an external electric field is applied to the MFC and an inverse piezoelectric effect is generated, the movement of positive and negative charges inside it will also generate electric displacement. The MFC piezoelectric strain matrix is ​​in the following form.

[0060]

[0061] In the formula, c 11 =c 22 , c 12 =c 21 , c 13 =c 31 =c 23 =c 32 , c 44 =2(c 11 -c 12 ).

[0062] The expression of the MFC stress of the robotic fish is as follows.

[0063]

[0064] Where W pitch is the electrode spacing. Due to the inverse piezoelectric effect, MFC can generate a large axial force and driving bending moment to deform the structure of the robotic fish and generate thrust.

[0065] (3) Modal analysis of bionic robotic fish

[0066] To predict the vibration mode and modal frequency of the bending deformation of the model, it is necessary to first perform modal analysis on the acoustic-solid coupling model of the robotic fish and the flow field. The modal finite element equation of the robotic fish structure is as follows.

[0067]

[0068]

[0069] Where M S and K S is the structural matrix and stiffness matrix, U represents displacement, F S represents the external force on the structure, f F represents the pressure vector describing the acoustic-solid coupling. P represents the fluid pressure, and R represents the effective area matrix associated with each node at the interface between the robotic fish structure and the fluid. S With N F They represent the finite element shape functions of the fish body structure and the fluid structure respectively, and n represents the fluid boundary vector.

[0070] The finite element equation of the robotic fish fluid is as follows.

[0071]

[0072] Where M F and K F Denote the mass matrix and stiffness matrix respectively, F F represents the addition of mass, ρ f represents the static density of the fluid.

[0073] The coupling between the structure and the flow field is described by the boundary pressure. The deformation of the structure is affected by the fluid pressure, and the displacement change of the structure is transmitted to the fluid domain to simulate the effect of fluid-solid coupling. The sound-solid coupling modal frequencies of the robot fish are shown in Table 2.

[0074] Table 2. Modal frequencies of robot fish sound-solid coupling

[0075]

[0076] The modal frequency of the robot fish in the flow field is significantly lower than that in the air, and the higher the modal order, the greater the frequency difference. This is mainly due to the existence of fluid static pressure, which causes the structure to have additional mass additions, resulting in a decrease in the corresponding modal frequency. The first-order and second-order bending deformations occur at frequencies of 6.13Hz and 29.70Hz, respectively.

[0077] The first-order bending mode occurs at the tail of the robot fish structure with the maximum displacement of the tail fin. There is almost no displacement in the z-axis direction at the connection between the tail fin and the body. The rear 1 / 3 of the robot fish body swings to generate propulsion, while the front 2 / 3 has a smaller lateral fluctuation amplitude, and its main function is to maintain the balance of the robot fish body and prevent it from tipping over sideways.

[0078] (4) Bionic Robot Fish Sound-Solid Coupling Analysis

[0079] The space occupied by the fluid is regarded as the acoustic field area, and the acoustic unit is used to simulate the fluid. The acoustic space finite element equation is written, and then combined with the structural finite element equation to solve. The finite element equation for the acoustic-solid coupling analysis of the robot fish is as follows.

[0080]

[0081] Where M a ,C a ,K a , A represent the overall mass matrix, damping matrix, stiffness matrix and coupling matrix of the acoustic unit respectively; M S ,C S ,K S represent the overall mass matrix, damping matrix and stiffness matrix of the structure respectively.

[0082] In order to simulate the vibration performance of the robotic fish in water, a spherical computational domain with a diameter of 2000 mm is used. The robotic fish is located in the center. The computational domain of the robotic fish acoustic-solid coupling is as follows: Figure 2 shown.

[0083] The robot fish is mainly in the first-order mode during propulsion, so the applied driving frequency is about 6Hz. When the load frequency is 3Hz, the robot fish has better movement and response. The driving voltage has an amplitude between -500V and 1500V, and the frequency is 3Hz. The plane at the end of the tail fin is used as the test plane. Figure 3 As shown in the figure, the tail fin swing amplitude of the robot fish is as follows: Figure 4 shown.

[0084] The displacement of the tail fin tip in one cycle changes with time close to a sine wave, and the swing period is the same as the driving voltage signal. The MFC material has strong real-time performance in driving. The peak-to-peak value of the swing on the test plane is close to 40mm, which is about 1 / 4 of the overall length of the robot fish.

[0085] (5) Straight-line motion control of bionic robotic fish

[0086] In order to achieve the complete motion posture of the bionic robot fish, the motion control of the robot fish is analyzed using the acoustic-solid coupling method. When the bionic robot fish swims straight, the driving voltages on both sides have the same DC bias, the same amplitude, and opposite phases. The driving voltage signal of the robot fish is as follows: Figure 5 As shown. Using a sinusoidal driving voltage with a frequency of 3Hz, the amplitude of the driving voltage on the left and right sides is maintained, and the driving voltages on both sides are in opposite phases. The bionic robot fish is symmetrical on the left and right, and the peak-to-peak value of the tail fin swing at the end is about 40mm. The propulsion force is obtained by the swing of the tail fin to move forward straight. The straight movement posture of the bionic robot fish is shown in Figure 6 shown.

[0087] (6) Left-turn motion control of bionic robotic fish

[0088] By changing the driving voltage of the MFC driver, the positive peak value of the left driving voltage is reduced, while the negative peak value remains unchanged. The driving voltage signal of the robot fish turning left is as follows: Figure 7 The driving voltage makes the contraction effect of the two MFC actuators the same, and the stretching effect of the left MFC actuator is smaller than that of the right MFC actuator, so that the balance point of the CFRP substrate shifts to the left, the neutral plane shifts to the left, and the left turn is completed.

[0089] When the bionic robot fish turns, the equilibrium position of the tail fin swing is changed by the different degrees of static contraction on both sides. The greater the difference in static contraction on both sides, the greater the offset of the equilibrium position relative to the central axis of the fish body, and the smaller the turning radius. The turning radius of the robot fish is controlled by controlling the amplitude difference of the driving voltage on both sides. The greater the amplitude difference, the smaller the turning radius. The left turning posture of the robot fish is as follows: Figure 8 shown.

[0090] The fish's tail fin maintains a swing with the same amplitude and period as before near the new equilibrium position, completing the turning action. When the voltage signals of the MFC drivers on both sides have an asymmetric waveform, the z-axis direction displacement of the tail fin test plane is applied after the load. The tail fin still swings in a sinusoidal form over time, and the frequency remains at 3Hz, the same as the driving voltage frequency. The positive peak value of the sinusoidal voltage signal formed by the swing of the tail fin over time decreases and the reverse peak value increases, that is, the balance point of the tail fin swing is offset to the left by about 3mm. The tail fin of the bionic robotic fish maintains a swing with the same frequency near the new equilibrium position, realizing the left turn propulsion mode.

[0091] (7) Right turn motion control of bionic robot fish

[0092] When the robot fish turns right, the right driving voltage amplitude decreases, making the tensile deformation amplitude of the right MFC smaller than that of the left MFC. In the contraction state, the deformation amplitudes of the MFCs on both sides are the same. The right-turn driving voltage signal of the robot fish is as follows: Fig. 9 The neutral plane of the robot fish is shifted to the right, and the tail fin swings near the right side of the neutral plane to complete the right turn. The robot fish's right turn movement posture is shown in Fig.10 shown.

[0093] The positive peak value of the sinusoidal voltage signal formed by the swing of the tail fin over time decreases, while the reverse peak value increases, that is, the balance point of the tail fin swing shifts to the right by about 3mm. The tail fin of the bionic robot fish keeps swinging at the same frequency near the new balance position, realizing the right turn propulsion mode.

[0094] In this specification, each embodiment is described in a progressive manner, and each embodiment focuses on the differences from other embodiments. The same or similar parts between the embodiments can be referred to each other. For the device disclosed in the embodiment, since it corresponds to the method disclosed in the embodiment, the description is relatively simple, and the relevant parts can be referred to the method part.

[0095] The above description of the disclosed embodiments enables those skilled in the art to implement or use the present invention. Various modifications to these embodiments will be apparent to those skilled in the art, and the general principles defined herein may be implemented in other embodiments without departing from the spirit or scope of the present invention. Therefore, the present invention will not be limited to the embodiments shown herein, but rather to the widest scope consistent with the principles and novel features disclosed herein.

Claims

1. An acoustic-solid coupling model of a bionic robotic fish based on MFC material, characterized in that: Including bionic robot fish body, bionic robot fish motion control model, MFC material mechanical response model, bionic robot fish modal analysis model and acoustic-solid coupling analysis model; The bionic robotic fish body comprises two MFC plates and one CFPR plate; wherein the one CFPR plate is sandwiched between the two MFC plates; The bionic robotic fish motion control model is used to apply an electric field to the two MFC plates to control the bionic robotic fish to move straight, turn left, and turn right; The MFC material mechanical response model is used to convert the electric field applied by the bionic robotic fish motion control model into the deformation power of the bionic robotic fish; The bionic robotic fish modal analysis model is used to predict the vibration shape and modal frequency of the bending deformation of the bionic robotic fish; The acoustic-solid coupling analysis model is used to simulate the structural behavior, propulsion mode and deformation response of the robotic fish according to the predicted vibration mode and modal frequency of the bending deformation of the bionic robotic fish.

2. The acoustic-solid coupling model of a bionic robotic fish based on MFC material according to claim 1, characterized in that: The spatial region occupied by the fluid is regarded as the acoustic field region, and the acoustic unit is used to simulate the fluid to construct an acoustic-solid coupling analysis model.

3. The acoustic-solid coupling model of a bionic robotic fish based on MFC material according to claim 1, characterized in that: The finite element equation of the acoustic-solid coupling analysis model is as follows: Where M a ,C a ,K a , A represent the overall mass matrix, damping matrix, stiffness matrix and coupling matrix of the acoustic unit respectively; M S ,C S ,K S represent the overall mass matrix, damping matrix and stiffness matrix of the structure respectively.

4. The acoustic-solid coupling model of a bionic robotic fish based on MFC material according to claim 1, characterized in that: In order to simulate the vibration performance of the robotic fish in water, a spherical computational domain with a diameter of 2000 mm is used as the acoustic-solid coupling computational domain of the robotic fish, in which the robotic fish is located in the center.

5. The acoustic-solid coupling model of a bionic robotic fish based on MFC material according to claim 1, characterized in that: Sinusoidal voltages with the same frequency, amplitude and opposite phase are applied to the two MFC plates to control the bionic robotic fish to move in a straight line.

6. The acoustic-solid coupling model of a bionic robotic fish based on MFC material according to claim 1, characterized in that: A first left-turn drive voltage and a second left-turn drive voltage are applied to the left and right MFC plates respectively to control the bionic robotic fish to turn left; wherein the first left-turn drive voltage and the second left-turn drive voltage have the same frequency and opposite phase; and the amplitude of the first left-turn drive voltage is smaller than the amplitude of the second left-turn drive voltage.

7. The acoustic-solid coupling model of a bionic robotic fish based on MFC material according to claim 1, characterized in that: A first right-turn drive voltage and a second right-turn drive voltage are applied to the left and right MFC plates respectively to control the bionic robot fish to turn right; wherein the first right-turn drive voltage and the second right-turn drive voltage have the same frequency and opposite phase; and the amplitude of the first right-turn drive voltage is greater than the amplitude of the second right-turn drive voltage.

8. The acoustic-solid coupling model of a bionic robotic fish based on MFC material according to claim 1, characterized in that: The MFC material mechanical response model realizes its conversion function based on the following formula: Among them, D,σ,E v ,ε respectively represent electric displacement, mechanical stress, electric field intensity and total strain vector; r T ,r S Represent the pressure dielectric constant and pressure strain constant respectively; d represents the piezoelectric strain constant matrix, e represents the piezoelectric pressure constant matrix; s and c represent the material flexibility coefficient matrix and stiffness coefficient matrix respectively; superscript T and S represent the pressure is constant or the strain is constant respectively, and superscript E represents the external applied electric field strength is constant; A area represents the area of ​​the piezoelectric unit; F represents the deformation power of the robotic fish obtained by transformation.

9. The acoustic-solid coupling model of a bionic robotic fish based on MFC material according to claim 1, characterized in that: The bionic robot fish modal analysis model realizes the function of predicting the vibration shape and modal frequency of the bending deformation of the bionic robot fish based on the robot fish structure modal finite element equation and the robot fish fluid finite element equation.

10. The acoustic-solid coupling model of a bionic robotic fish based on MFC material according to claim 9, characterized in that: The modal finite element equation of the robotic fish structure is as follows: Where M S and K S is the structural matrix and stiffness matrix, U represents displacement, F S represents the external force on the structure, f F represents the pressure vector describing the acoustic-solid coupling, P represents the fluid pressure, R represents the effective area matrix associated with each node at the interface between the robotic fish structure and the fluid, and N S With N F represent the finite element shape functions of the fish body structure and the fluid structure respectively, and n represents the fluid boundary vector; The finite element equation of the robotic fish fluid is as follows: Where M F and K F Denote the mass matrix and stiffness matrix respectively, F F represents the addition of mass, ρ f represents the static density of the fluid.

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