# Three-Dimensional Investigation of the Stokes Eigenmodes in Hollow Circular Cylinder

Authors: Adil El Baroudi; Fulgence Razafimahery
Journal: Advances in Acoustics and Vibration
 (2013)

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## Abstract

This paper studies the influence of boundary conditions on a fluid medium of finite depth. We determine the frequencies and the modal shapes of the fluid. The fluid is assumed to be incompressible and viscous. A potential technique
is used to obtain in three-dimensional cylindrical coordinates a general solution for a problem. The method consists
in solving analytically  partial differential equations obtained from the linearized Navier-Stokes equation. A finite
element analysis is also used to check the validity of the present method. The results from the proposed method
are in good agreement with numerical solutions. The effect of the fluid thickness on the Stokes eigenmodes is also
investigated. It is found that frequencies are strongly influenced.


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## Body

## 1. Introduction

Flow modeling in confined domains leads mostly to Stokes models [1]. The use of this model in the field of microfluidics and nanofluidics [2, 3] and in the understanding the movement of microorganisms [4] currently in full swing. This recent years, several authors have focused on the modal analysis of this model. We can cite in particular the aspects related to problems arising from geophysics. These studies highlight the existence of slow wave or Stoneley waves [5–7]. Other authors [8, 9] have focused on the dynamic aspects in the context of fluid-structure interaction. The numeric aspects for coupled (or not) modal problem were discussed in [10–12].The theory of potential flow of viscous fluid was introduced by [13]. All of his work on this topic is framed in terms of the effects of viscosity on the attenuation of small amplitude waves on a liquid-gas surface. The problem treated by Stokes was solved exactly using the linearized Navier-Stokes equations, without assuming potential flow, and was solved exactly by [14]. Reference [15] has identified the main events in the history of thought about potential flow of viscous fluids. The problem of Stokes flow in cylindrical domain has been investigated by several authors. References [16, 17] studied Stokes flow in a cylindrical container by an eigenfunction expansion procedure without the compressibility effect.Potential flows through different kinds of geometry have been studied by many investigators for several applications. For example in vascular fluid dynamics, [18] presented the role of curvature in the wave propagation and in the development of a secondary flow. Reference [19] has studied the flow and pressure dynamics of the cerebrospinal fluid flow. Three-dimensional CSF flow studies have also been reported [20].The knowledge of Stokes eigenmodes in a three-dimensional confined domain, in a square/cube [10], or in any bounded domain could provide some insight into the understanding or analysis of turbulent instantaneous flow field in a geometry as simple as, for instance, the driven cavity.This paper deals with analytical and numerical modeling of three-dimensional incompressible viscous fluid using the potential technique and the finite element method. The analytical formulation is based upon a convenient decomposition of the velocity field into two contributions [9–21]: one is related to the scalar potential and the other is the vector potential.The rest of the paper is organized as follows. In Section2, we write the linearized Navier-Stokes equation, a simple representation for the velocity field in terms of the potential. The implementation of analytical solutions is discussed in detail. Section 3 describes boundary condition treatment for modeling Stokes eigenmodes. Section 4 investigates the case of fluid-solid interaction. Section 5 is completely devoted to the analytical and numerical results. Finally, the paper is ended by some conclusions.


## 2. Governing Equations

From conservations of mass and momentum and assuming that the fluid is viscous and incompressible, the motion of the fluid flowing in the cylindrical waveguide in the absence of body force is governed by(1)ρ∂v(r,θ,z)∂t=-∇p(r,θ,z)+η∇2v(r,θ,z),(2)∇·v(r,θ,z)=0,
in which v(r,θ,z)={vr,vθ,vz}T(r,θ,z) is the fluid velocity vector, p(r,θ,z) is the fluid pressure, ρ is the density of the fluid, and ν and η=ρν are the kinematic and dynamic fluid viscosity, respectively.The obtained equations of motion are highly complex and coupled. However, a simpler set of equations can be obtained by introducing scalar potentialsϕ, ψ, and χ, known as the Helmholtz decomposition. The Helmholtz decomposition theorem [22] states that any vector v can be written as the sum of two parts: one is curl-free and the other is solenoidal. In flow fields, the velocity is thereby decomposed into a potential flow and a viscous flow. In other words, the velocity v can be decomposed into the following form:
(3)v=∇ϕ+∇×(ψez)+∇×∇×(χez),
where ez is the unit vector along the z direction. Substituting the above resolutions into (1) and (2), after some manipulations, the equation for the conservation of mass (2) becomes the Laplace equation:
(4)∇2ϕ(r,θ,z)=0,
and the equation for the conservation of momentum (1) becomes the Helmholtz equations
(5)∇2ψ(r,θ,z)-1ν∂ψ(r,θ,z)∂t=0,∇2χ(r,θ,z)-1ν∂χ(r,θ,z)∂t=0,
where ∇2=(∂2/∂r2)+(1/r)(∂/∂r)+(1/r2)(∂2/∂θ2)+(∂2/∂z2) is the Laplacian operator in polar coordinates, and time dependence has the form exp(jωt). The pressure can be presented by
(6)p=-ρ∂ϕ∂t.
Thus, the Navier-Stokes equations are reduced to formulations (4), (5), and (6). Of course, it does not mean that in any case such a decomposition of fluid’s velocity field gives considerable simplifications in solving the problem because boundary conditions for stresses are not separated. The vibrations are harmonic, with constant angular frequency ω; as a result all potentials components and pressure will depend on time only through the factor exp(jωt).Applying the method of separation of variables, the solution of the equations for potentials, associated with an axial wave numberkz, radial wave number kψ, and circumferential mode parameter n, after considerable algebraic manipulations, can be shown to be
(7)ϕ=[AIn(kzr)+BKn(kzr)]{sin(nθ)cos(nθ)}×{cos(kzz)sin(kzz)},(8)ψ=[CJn(kψr)+DYn(kψr)]{cos(nθ)sin(nθ)}×{cos(kzz)sin(kzz)},(9)χ=[EJn(kψr)+FYn(kψr)]{sin(nθ)cos(nθ)}×{sin(kzz)cos(kzz)}.Jn and Yn are Bessel functions of the first and second kind of order n. In and Kn are modified Bessel functions of the first and second kinds of order n. n and kz are the azimuthal and axial wavenumbers. n=0,1,2,…, whereas kz is found by satisfying the symmetry boundary condition on the z=0 and z=l. Note that the velocity field v has components that are symmetric or antisymmetric in θ and z. Following standard practice, the solutions with symmetric (antisymmetric) axial velocities are called the antisymmetric (symmetric) axial modes, respectively, with kza and kzs denoting the corresponding eigenvalues. Thus, we have kza=(m-1)π/l and kzs=mπ/l (m=0,1,2,…). However, the azimuthal modes corresponding to cos(nθ) and sin(nθ) are really the same, due to periodicity in the azimuthal direction; that is, there is no distinction in the values of n for the two families. A, B, C, D, E, and F are unknown coefficients which will be determined later by imposing the appropriate boundary conditions. The radial wave number kψ is related to the axial wave number kz by
(10)kψ2=α2-kz2,ω=jνα2,
where j=-1. In the end, the pressure can be obtained by insertion of (7) into (6),
(11)p=ηα2[AIn(kzr)+BKn(kzr)]cos(nθ)cos(kzz).Using (3) and taking into account (7), (8), and (9), the fluid particle velocities in terms of the Bessel functions can be written as
(12)vr={AIn′(kzr)+BKn′(kzr)-nrCJn(kψr)-nrDYn(kψr)v,r+kzEJn′(kψr)+kzFYn′(kψr)nr}b,,,×sin(nθ)cos(kzz),vθ={nkzrnrAIn(kzr)+nrBKn(kzr)-CJn′(kψr)vθ=,,-DYn′(kψr)+nkzr[EJn(kψr)+FYn(kψr)]}×cos(nθ)cos(kzz),vz={kψ2EJn(kψr)+kψ2FYn(kψr)-kzAIn(kzr)mmm,-kzBKn(kzr)kψ2}sin(nθ)sin(kzz).In the following section we consider a fluid with different boundary conditions.


## 3. Boundary Conditions and Frequency Equation

First we defineΓ1 (r=R1), as the inner boundary of the fluid region, and Γ2 (r=R2), as the outer boundary. We begin the analysis of frequency equation for imposing the no-slip boundary conditions.


### 3.1. Stokes Eigenmodes in the Case of “No-Slip-No-Slip” Boundary Conditions

The no-slip hypothesis of fluid mechanics states that liquid velocity at a solid surface is equal to the velocity of the solid surface. Hence the no-slip boundary condition can be written as(1)
the velocity is equal to zero atΓ1:
(13)vr=vθ=vz=0,(2)
the velocity is equal to zero atΓ2:
(14)vr=vθ=vz=0.Combining these boundary conditions with (12) yields for each mode number (n,m) the following linear system:
(15)[M1]{x}={0},{x}={ABCDEF}T,
where [M1] is a 6×6 matrix whose components aij are given in Appendix A. For a non-trivial solution, the determinant of the matrix [M1] must be equal to zero
(16)det[M1]=0.The roots of (16) give the natural frequencies ω of the cylindrical oscillations. Figure 16 in Appendix D shows analytical calculation of eigenvalues α and using (10) we obtain the natural frequencies ω [rd/s].


### 3.2. Stokes Eigenmodes in the Case of "No-Slip-Normal Stress-Free" Boundary Conditions

The pressure and velocity boundary conditions on the free surface are both formulated from the dynamic constraint of continuity of normal momentum flux across the free surface. The component of the stress tensor in the outward normal direction is therefore(17)σ·n={-pI+η[∇v+(∇v)T]},n=0,
in which I is a unit tensor. By applying this condition on Γ1 we obtain
(18)σrr=σrθ=σrz=0.Combining these boundary conditions and (14) yields for each mode number (n,m) the following linear system:
(19)[M2]{x}={0},{x}={ABCDEF}T,
where [M2] is a 6×6 matrix whose components bij are given in Appendix B. For a nontrivial solution, the determinant of the matrix [M2] must be equal to zero
(20)det[M2]=0.The roots of (20) give the natural frequencies ω of the cylindrical oscillations.


### 3.3. Torsional, Flexural, and Breathing Stokes Eigenmode

The results presented in (16) and (20) are a general natural frequencies equation. For some simpler modes, the abovementioned method can be simplified. For example, we have the following.


#### 3.3.1. Torsional Stokes Eigenmode

The torsion mode vibration is such a mode in which the scalar components of the velocity{vr,vz} are zeros and only the circumferential velocity vθ is independent of θ. This condition is achieved if ϕ=0 and χ=0. Through (3) this gives for the nonvanishing components of displacement and stresses:
(21)vθ=-∂ψ∂r.Thus, the general solution forψ must be constructed from the set
(22)ψ(r,z)=[CJ0(kψr)+DY0(kψr)]sin(kzz).In this case the boundary conditions equations (13) and (14) become
(23)vθ=0atr=R1,vθ=0atr=R2.Then, (15) becomes
(24)[T]{x}={0},{x}={CD}T.[T] is a 2×2 matrix whose components are calculated using Appendices A and B. Solving det[T]=0 gives the torsional modes.


### 3.4. Longitudinal Stokes Eigenmode

Another simpler mode vibration is called longitudinal mode vibration in whichvθ=0 and {vr,vz} are independent of θ. This means that the motion is confined to planes perpendicular to the z-axis, which can move, expand, and contract in their planes. The solution for the displacement field and stress vector follows from (3) and (17):
(25)vr=∂ϕ∂r+kz∂χ∂rvz=-kzϕ+kψ2χ,σrr=2η{∂2ϕ∂r2-α22ϕ+kz∂2χ∂r2},σrz=η{-2kz∂ϕ∂r+(kψ2-kz2)∂χ∂r}.Thus, the general solution forϕ and χ must be constructed from the set:
(26)ϕ=[AJ0(kzr)+BY0(kzr)]sin(kzz),χ=[EJ0(kψr)+FY0(kψr)]cos(kzz).In this case the boundary conditions equations (14) and (18) become
(27)σrr=σrz=0atr=R1,vr=vz=0atr=R2.Then, (19) becomes
(28)[L]{x}={0},{x}={ABEF}T.[L] is a 4×4 matrix whose components are calculated using the Appendices A and B. Solving det[L]=0 gives the longitudinal modes.


#### 3.4.1. Flexural and Breathing Stokes Eigenmodes

The mode shapen=1 is called flexural mode vibration in which all components of the displacement are nonvanishing and depend on r, θ, and z. The mode shape n⩾2 is called breathing mode vibration in which all components of the displacement are non-vanishing and depend on r, θ, and z.In the following we will introduce the interest of the fluid-structure interaction and we will study the influence of viscosity on the eigenmodes of an elastic solid. For this, the inner boundary of the fluid regionΓ1 is represented by an elastic wall.


## 3.1. Stokes Eigenmodes in the Case of “No-Slip-No-Slip” Boundary Conditions

The no-slip hypothesis of fluid mechanics states that liquid velocity at a solid surface is equal to the velocity of the solid surface. Hence the no-slip boundary condition can be written as(1)
the velocity is equal to zero atΓ1:
(13)vr=vθ=vz=0,(2)
the velocity is equal to zero atΓ2:
(14)vr=vθ=vz=0.Combining these boundary conditions with (12) yields for each mode number (n,m) the following linear system:
(15)[M1]{x}={0},{x}={ABCDEF}T,
where [M1] is a 6×6 matrix whose components aij are given in Appendix A. For a non-trivial solution, the determinant of the matrix [M1] must be equal to zero
(16)det[M1]=0.The roots of (16) give the natural frequencies ω of the cylindrical oscillations. Figure 16 in Appendix D shows analytical calculation of eigenvalues α and using (10) we obtain the natural frequencies ω [rd/s].


## 3.2. Stokes Eigenmodes in the Case of "No-Slip-Normal Stress-Free" Boundary Conditions

The pressure and velocity boundary conditions on the free surface are both formulated from the dynamic constraint of continuity of normal momentum flux across the free surface. The component of the stress tensor in the outward normal direction is therefore(17)σ·n={-pI+η[∇v+(∇v)T]},n=0,
in which I is a unit tensor. By applying this condition on Γ1 we obtain
(18)σrr=σrθ=σrz=0.Combining these boundary conditions and (14) yields for each mode number (n,m) the following linear system:
(19)[M2]{x}={0},{x}={ABCDEF}T,
where [M2] is a 6×6 matrix whose components bij are given in Appendix B. For a nontrivial solution, the determinant of the matrix [M2] must be equal to zero
(20)det[M2]=0.The roots of (20) give the natural frequencies ω of the cylindrical oscillations.


## 3.3. Torsional, Flexural, and Breathing Stokes Eigenmode

The results presented in (16) and (20) are a general natural frequencies equation. For some simpler modes, the abovementioned method can be simplified. For example, we have the following.


### 3.3.1. Torsional Stokes Eigenmode

The torsion mode vibration is such a mode in which the scalar components of the velocity{vr,vz} are zeros and only the circumferential velocity vθ is independent of θ. This condition is achieved if ϕ=0 and χ=0. Through (3) this gives for the nonvanishing components of displacement and stresses:
(21)vθ=-∂ψ∂r.Thus, the general solution forψ must be constructed from the set
(22)ψ(r,z)=[CJ0(kψr)+DY0(kψr)]sin(kzz).In this case the boundary conditions equations (13) and (14) become
(23)vθ=0atr=R1,vθ=0atr=R2.Then, (15) becomes
(24)[T]{x}={0},{x}={CD}T.[T] is a 2×2 matrix whose components are calculated using Appendices A and B. Solving det[T]=0 gives the torsional modes.


## 3.3.1. Torsional Stokes Eigenmode

The torsion mode vibration is such a mode in which the scalar components of the velocity{vr,vz} are zeros and only the circumferential velocity vθ is independent of θ. This condition is achieved if ϕ=0 and χ=0. Through (3) this gives for the nonvanishing components of displacement and stresses:
(21)vθ=-∂ψ∂r.Thus, the general solution forψ must be constructed from the set
(22)ψ(r,z)=[CJ0(kψr)+DY0(kψr)]sin(kzz).In this case the boundary conditions equations (13) and (14) become
(23)vθ=0atr=R1,vθ=0atr=R2.Then, (15) becomes
(24)[T]{x}={0},{x}={CD}T.[T] is a 2×2 matrix whose components are calculated using Appendices A and B. Solving det[T]=0 gives the torsional modes.


## 3.4. Longitudinal Stokes Eigenmode

Another simpler mode vibration is called longitudinal mode vibration in whichvθ=0 and {vr,vz} are independent of θ. This means that the motion is confined to planes perpendicular to the z-axis, which can move, expand, and contract in their planes. The solution for the displacement field and stress vector follows from (3) and (17):
(25)vr=∂ϕ∂r+kz∂χ∂rvz=-kzϕ+kψ2χ,σrr=2η{∂2ϕ∂r2-α22ϕ+kz∂2χ∂r2},σrz=η{-2kz∂ϕ∂r+(kψ2-kz2)∂χ∂r}.Thus, the general solution forϕ and χ must be constructed from the set:
(26)ϕ=[AJ0(kzr)+BY0(kzr)]sin(kzz),χ=[EJ0(kψr)+FY0(kψr)]cos(kzz).In this case the boundary conditions equations (14) and (18) become
(27)σrr=σrz=0atr=R1,vr=vz=0atr=R2.Then, (19) becomes
(28)[L]{x}={0},{x}={ABEF}T.[L] is a 4×4 matrix whose components are calculated using the Appendices A and B. Solving det[L]=0 gives the longitudinal modes.


### 3.4.1. Flexural and Breathing Stokes Eigenmodes

The mode shapen=1 is called flexural mode vibration in which all components of the displacement are nonvanishing and depend on r, θ, and z. The mode shape n⩾2 is called breathing mode vibration in which all components of the displacement are non-vanishing and depend on r, θ, and z.In the following we will introduce the interest of the fluid-structure interaction and we will study the influence of viscosity on the eigenmodes of an elastic solid. For this, the inner boundary of the fluid regionΓ1 is represented by an elastic wall.


## 3.4.1. Flexural and Breathing Stokes Eigenmodes

The mode shapen=1 is called flexural mode vibration in which all components of the displacement are nonvanishing and depend on r, θ, and z. The mode shape n⩾2 is called breathing mode vibration in which all components of the displacement are non-vanishing and depend on r, θ, and z.In the following we will introduce the interest of the fluid-structure interaction and we will study the influence of viscosity on the eigenmodes of an elastic solid. For this, the inner boundary of the fluid regionΓ1 is represented by an elastic wall.


## 4. Fluid-Structure Interaction

Fluid-structure interaction problems have long since attracted the attention of engineers and applied mathematics. The most important applications of this theory, are probably, structural acoustics [23], vibrations of fluid-conveying pipes [24, 25], and biomechanics. As these problems are rather complicated, some simplifications are typically adopted to facilitate their solving. In particular, it is quite typical to ignore viscosity effects (especially in structural acoustics) or to use local theories of interaction, such as the one referred to as thin layer or plane wave approximation.


### 4.1. Governing Equations of Elastic Media

The wave motion in an isotropic elastic medium is governed by the classical Navier’s equation:(29)-ρsω2u=μ∇2u+(λ+μ)∇∇·u,
where ρs is the density, λ, μ are the Lamé constants, and u(r,θ,z)={ur,uθ,uz}T(r,θ,z) is the vector displacement of particles.The obtained equations of motion are highly complex and coupled. However, a simpler set of equations can be obtained by introducing scalar potentialsΦ, Ψ, and Θ, known as the Helmholtz decomposition such that
(30)u=∇Φ+∇×(Ψez)+∇×∇×(Θez).Substituting (30) into (29) leads to three sets of differential equations
(31)∇2Φ(r,θ,z)-ν2α4cL2Φ(r,θ,z)=0,∇2Ψ(r,θ,z)-ν2α4cT2Ψ(r,θ,z)=0,∇2Θ(r,θ,z)-ν2α4cT2Θ(r,θ,z)=0,
where cL=(λ+2μ)/ρs and cT=μ/ρs are the compressional and shear wave velocities in the solids, respectively. Applying the method of separation of variables, the solution of the equations for potentials, associated with an axial wave number kz, radial wave number (kΦ,kΨ), and circumferential mode parameter n, after considerable algebraic manipulations, can be shown to be
(32)Φ=[aIn(kΦr)+bKn(kΦr)]sin(nθ)cos(kzz),Ψ=[cIn(kΨr)+dKn(kΨr)]cos(nθ)cos(kzz),Θ=[eIn(kΨr)+fKn(kΨr)]sin(nθ)sin(kzz).The radial wave number(kΦ,kΨ) is related to the axial wave number kz by
(33)kΦ2=ν2α4cL2+kz2,kΨ2=ν2α4cT2+kz2
and a, b, c, d, e, and f are unknown coefficients which will be determined later by imposing the appropriate boundary conditions.Using (30) the scalar components of the displacement vector u in cylindrical coordinates can be expressed by
(34)ur=∂Φ∂r-nrΨ+kz∂Θ∂r,uθ=nrΦ-∂Ψ∂r+nkzrΘ,uz=-kzΦ-kΨ2Θ,
and the radial and tangential stresses are given by Hooke’s law as
(35)Σrr=λν2α4cL2Φ+2μ∂ur∂r,Σrθ=μ{∂uθ∂r-uθr+1r∂ur∂θ},Σrz=μ{∂ur∂z+∂uz∂r}.


### 4.2. Boundary Condition and Frequency Equation

We defineΓ1 (r=R1) as the boundary contact between the fluid region and the solid region and Γ2 (r=R2) as the outer boundary. The relevant boundary conditions can be taken as follows. (1)
The normal components of the solid stresses must be zero on the interfaceΓ0:
(36)Σrr=Σrθ=Σrz=0.(2)
Kinematic boundary condition (velocity must be continuous) on the interfaceΓ1:
(37)vr=jωur,vθ=jωuθ,vz=jωuz.(3)
Dynamic boundary condition (normal stresses must be continuous) on the interfaceΓ1:
(38)σrr=Σrr,σrθ=Σrθ,σrz=Σrz.(4)
On the interfaceΓ2 at the outer cylinder the normal stress is equal to zero:
(39)σrr=σrθ=σrz=0.Combining these boundary conditions with (34)-(35) and taking into account (7)–(9) and (32) yields for each mode number (n,m) the following linear system:
(40)[M]{y}={0},
where {y}={ABCDEFabcdef}T and [M] is an twelfth-order operator matrix, which is given in Appendix C. For nontrivial solution, the determinant of the matrix M must be equal to zero:
(41)det[M]=0.This equation indicates a relationship between the dynamic fluid viscosityη, density of the fluid ρ, and the elastic constants. The roots of (41) give the infinite natural frequencies ω.


## 4.1. Governing Equations of Elastic Media

The wave motion in an isotropic elastic medium is governed by the classical Navier’s equation:(29)-ρsω2u=μ∇2u+(λ+μ)∇∇·u,
where ρs is the density, λ, μ are the Lamé constants, and u(r,θ,z)={ur,uθ,uz}T(r,θ,z) is the vector displacement of particles.The obtained equations of motion are highly complex and coupled. However, a simpler set of equations can be obtained by introducing scalar potentialsΦ, Ψ, and Θ, known as the Helmholtz decomposition such that
(30)u=∇Φ+∇×(Ψez)+∇×∇×(Θez).Substituting (30) into (29) leads to three sets of differential equations
(31)∇2Φ(r,θ,z)-ν2α4cL2Φ(r,θ,z)=0,∇2Ψ(r,θ,z)-ν2α4cT2Ψ(r,θ,z)=0,∇2Θ(r,θ,z)-ν2α4cT2Θ(r,θ,z)=0,
where cL=(λ+2μ)/ρs and cT=μ/ρs are the compressional and shear wave velocities in the solids, respectively. Applying the method of separation of variables, the solution of the equations for potentials, associated with an axial wave number kz, radial wave number (kΦ,kΨ), and circumferential mode parameter n, after considerable algebraic manipulations, can be shown to be
(32)Φ=[aIn(kΦr)+bKn(kΦr)]sin(nθ)cos(kzz),Ψ=[cIn(kΨr)+dKn(kΨr)]cos(nθ)cos(kzz),Θ=[eIn(kΨr)+fKn(kΨr)]sin(nθ)sin(kzz).The radial wave number(kΦ,kΨ) is related to the axial wave number kz by
(33)kΦ2=ν2α4cL2+kz2,kΨ2=ν2α4cT2+kz2
and a, b, c, d, e, and f are unknown coefficients which will be determined later by imposing the appropriate boundary conditions.Using (30) the scalar components of the displacement vector u in cylindrical coordinates can be expressed by
(34)ur=∂Φ∂r-nrΨ+kz∂Θ∂r,uθ=nrΦ-∂Ψ∂r+nkzrΘ,uz=-kzΦ-kΨ2Θ,
and the radial and tangential stresses are given by Hooke’s law as
(35)Σrr=λν2α4cL2Φ+2μ∂ur∂r,Σrθ=μ{∂uθ∂r-uθr+1r∂ur∂θ},Σrz=μ{∂ur∂z+∂uz∂r}.


## 4.2. Boundary Condition and Frequency Equation

We defineΓ1 (r=R1) as the boundary contact between the fluid region and the solid region and Γ2 (r=R2) as the outer boundary. The relevant boundary conditions can be taken as follows. (1)
The normal components of the solid stresses must be zero on the interfaceΓ0:
(36)Σrr=Σrθ=Σrz=0.(2)
Kinematic boundary condition (velocity must be continuous) on the interfaceΓ1:
(37)vr=jωur,vθ=jωuθ,vz=jωuz.(3)
Dynamic boundary condition (normal stresses must be continuous) on the interfaceΓ1:
(38)σrr=Σrr,σrθ=Σrθ,σrz=Σrz.(4)
On the interfaceΓ2 at the outer cylinder the normal stress is equal to zero:
(39)σrr=σrθ=σrz=0.Combining these boundary conditions with (34)-(35) and taking into account (7)–(9) and (32) yields for each mode number (n,m) the following linear system:
(40)[M]{y}={0},
where {y}={ABCDEFabcdef}T and [M] is an twelfth-order operator matrix, which is given in Appendix C. For nontrivial solution, the determinant of the matrix M must be equal to zero:
(41)det[M]=0.This equation indicates a relationship between the dynamic fluid viscosityη, density of the fluid ρ, and the elastic constants. The roots of (41) give the infinite natural frequencies ω.


## 5. Analytical, Numerical Results and Validation

Numerical calculations were performed on the example of the hollow cylinder (Figures1, 2, and 3) with l=0.15 [m], and different dimensions such as the inner radius R1=0.07 [m] and the outer radius R2=0.09  [m] of fluid domain are used. The fluid used in the hollow cylinder for which density of 920 [kg·m−3] and dynamic’s viscosity of 0.1 [Pa·s] is assumed. The following values of parameters of the the elastic solid are in contact with a viscous liquid were assumed: ρs=1150 [kg/m3], νs=0.48, E=3·105 [Pa], and the inner radius of elastic solid R0=0.068 [m].Figure 1
Configuration of the viscous oscillations of a cylindrical incompressible fluid of lengthl in circular cylindrical coordinate system (r,θ,z). R1 and R2 are the inner and outer radius of fluid domain, respectively.Figure 2
Geometry of fluid domain.Figure 3
Geometry of fluid-solid interation model.R0 and R1 are the inner and outer radius of solid domain, respectively.With the derived eigenfrequency equations, natural frequenciesωnm=Im(ναnm2j) for each pair of (n,m) are calculated in the software Mathematica. m and n denote the mode in axial and azimuthal (propagating clockwise around the vortex) direction, respectively. To validate the analytical results, the natural frequencies and mode shapes are also computed using Comsol Multiphysics FEM Simulation Software. The natural frequencies are computed directly from (24) for torsion vibration. For longitudinal vibration, (28) can be used to determine the corresponding natural frequencies.Tables1 and 2 show the comparison of the first 18 natural frequencies and the corresponding mode shapes of viscous fluid by FEM and the present method (see (16) and (20)). For example in the case of “no-slip-normal stress-free” boundary conditions, in the first 18 natural frequencies, four correspond to flexural vibration (n=1), nine to breathing vibration (n⩾2), four to torsional vibration, and one to longitudinal vibration. The very good agreement is observed between the results of the present method and those of FEM and the relative difference ((FEM-Present)/Present) is ⩽1%. This shows that the algorithm implemented in Comsol Multiphysics [26, 27] software for numerical computation is highly reliable and accurate. This algorithm is based on the UMFPACK method [28]. Is the attention to use the numerical formulation in future for more general geometries.Table 1
Natural frequenciesω [rd/s] for various mode shapes in the case of “no-slip-no-slip” boundary conditions.
No.
(
n
,
m
)
Present
FEM
Mode shape
1
(
0,0
)
2.694
2.694
Torsional
2
(
0,1
)
2.742
2.742
Torsional
3
(
1,1
)
2.755
2.755
Flexural
4
(
2,1
)
2.800
2.800
Breathing
5
(
3,1
)
2.882
2.882
Breathing
6
(
0,2
)
2.885
2.885
Torsional
7
(
1,2
)
2.900
2.900
Flexural
8
(
2,2
)
2.948
2.948
Breathing
9
(
4,1
)
3.000
3.000
Breathing
10
(
3,2
)
3.030
3.030
Breathing
11
(
0,3
)
3.124
3.124
Torsional
12
(
1,3
)
3.140
3.140
Flexural
13
(
4,2
)
3.147
3.147
Breathing
14
(
5,1
)
3.153
3.153
Breathing
15
(
2,3
)
3.189
3.189
Breathing
16
(
3,3
)
3.271
3.271
Breathing
17
(
5,2
)
3.299
3.299
Breathing
18
(
6,1
)
3.341
3.341
BreathingTable 2
Natural nfrequenciesω [rd/s] for various mode shapes in the case of “no-slip-normal stress-free” boundary conditions.
No.
(
n
,
m
)
Present
FEM
Mode shape
1
(
1,1
)
0.889
0.889
Flexural
2
(
0,0
)
0.898
0.898
Torsional
3
(
2,1
)
0.908
0.908
Breathing
4
(
0,1
)
0.946
0.946
Torsional
5
(
0,1
)
0.951
0.951
Longitudinal
6
(
3,1
)
0.984
0.984
Breathing
7
(
1,0
)
0.990
0.990
Flexural
8
(
1,2
)
1.087
1.087
Flexural
9
(
0,2
)
1.089
1.089
Torsional
10
(
2,2
)
1.1070
1.1070
Breathing
11
(
4,1
)
1.1073
1.1073
Breathing
12
(
1,1
)
1.113
1.113
Flexural
13
(
3,2
)
1.171
1.171
Breathing
14
(
2,0
)
1.251
1.251
Breathing
15
(
5,1
)
1.271
1.271
Breathing
16
(
4,2
)
1.281
1.281
Breathing
17
(
0,3
)
1.327
1.327
Torsional
18
(
1,3
)
1.337
1.337
FlexuralTables1 and 2 and Figures 4, 5, and 6 show that natural frequencies are very sensitive to the nature of the boundary conditions. It is seen that the effect of the free surface is very interesting and decrease the frequency of a fluid confined in a rigid cylinder.Figure 4
Variations of flexural eigenmode for different boundary conditions.Figure 5
Variations of breathing eigenmode for different boundary conditions.Figure 6
Variations of torsional eigenmode for different boundary conditions.Figures7, 8, and 9 show, respectively, the two modal shapes of the torsional, flexural, and breathing vibrations. The modal shape can be regarded as the mode (n,m), where n is the modal number in the circumferential direction and m is the modal number in the axial direction. The modal shapes are not in order with the parameters n and m. This feature of cylindrical vibration is different from that of beam vibration in which the order increases with the modal parameter. Therefore in the vibration of the cylinder, one should be careful to find the right mode of the vibration.The torsional modal shapes of(n,m): streamlines of components of velocity field.
(a)
(
n
,
m
)
=
(
0
,
1
)
(b)
(
n
,
m
)
=
(
0
,
2
)The flexural modal shapes of(1,m): streamlines of components of velocity field.
(a)
(n,m)=(1,1)
(b)
(
n
,
m
)
=
(
1
,
2
)The breathing modal shapes of(2,m): streamlines of components of velocity field.
(a)
(
n
,
m
)
=
(
2
,
1
)
(b)
(
n
,
m
)
=
(
2
,
2
)In this paper, the effects of boundary conditions and of cylindrical parameters on the natural frequencies of cylindrical viscous fluid are presented with the present method. In these studies, investigations are carried out to study the effects of circumferential moden, axial mode m, and fluid thickness e=R2-R1 on the frequencies. Influence of viscosity on the natural frequencies of an elastic solid is also investigated.First, one investigates how the natural frequencies vary with the axial modem. Figure 10 shows that the natural frequencies increase as the axial mode m increases except when m=0. This value of m corresponds to 2D problem of viscous oscillations.Figure 10
Variations of the natural frequenciesω for different values m with the mode m with the mode n.Secondly, one investigates how the frequencies vary with the fluid thickness. Figures11, 12, and 13 show that the fluid thickness has a strong influence on the natural frequencies.Figure 11
Variations of the torsional eigenmode (n=0) ω for different fluid’s thickness e with the mode m.Figure 12
Variations of the flexural eigenmode (n=1) ω for different fluid’s thickness e with the mode m.Figure 13
Variations of the breathing eigenmode (n=2) ω for different fluid’s thickness e with the mode m.Thirdly, one investigates how the dense fluid (added mass) affects the natural frequencies. Tables3 and 4 show the coupled and uncoupled natural frequencies varying with circumferential and axial mode (n,m). As n increases, the difference between the coupled and uncoupled natural frequencies increases. Figures 14 and 15 show that the presence of fluid has no influence on the modal shapes of an elastic solid.Table 3
Uncoupled natural frequenciesω [rd/s] for various mode shapes of elastic solid.
No.
(
n
,
m
)
Present
FEM
Mode shape
1
(
2,0
)
0.952
0.952
Breathing
2
(
3,0
)
2.693
2.693
Breathing
3
(
4,0
)
5.157
5.157
Breathing
4
(
4,1
)
7.145
7.145
Breathing
5
(
3,1
)
7.389
7.389
Breathing
6
(
5,0
)
8.326
8.326
Breathing
7
(
5,1
)
9.431
9.431
Breathing
8
(
2,1
)
11.456
11.456
Breathing
9
(
6,0
)
12.190
12.190
Breathing
10
(
6,1
)
13.035
13.035
Breathing
11
(
5,2
)
14.379
14.379
Breathing
12
(
4,2
)
14.700
14.700
Breathing
13
(
6,2
)
16.504
16.504
Breathing
14
(
7,0
)
16.739
16.739
Breathing
15
(
7,1
)
17.500
17.500
Breathing
16
(
3,2
)
18.005
18.005
Breathing
17
(
7,2
)
20.270
20.270
Breathing
18
(
1,1
)
20.289
20.289
FlexuralTable 4
Coupled natural nfrequenciesω [rd/s] for various mode shapes.
No.
(
n
,
m
)
Present
FEM
Mode shape
1
(
0,1
)
0.535
0.535
Torsional
2
(
1,1
)
0.566
0.566
Flexural
3
(
2,1
)
0.645
0.645
Breathing
4
(
0,2
)
0.678
7.145
Torsional
5
(
1,2
)
0.699
0.699
Flexural
6
(
2,0
)
0.702
0.702
Breathing
7
(
3,1
)
0.749
0.749
Breathing
8
(
2,2
)
0.760
0.760
Breathing
9
(
0,1
)
0.780
0.780
Longitudinal
10
(
1,1
)
0.813
0.813
Flexural
11
(
3,2
)
0.854
0.854
Breathing
12
(
4,1
)
0.872
0.872
Breathing
13
(
0,3
)
0.917
0.917
Torsional
14
(
3,0
)
0.924
0.924
Breathing
15
(
1,3
)
0.935
0.935
Flexural
16
(
4,2
)
0.978
0.978
Breathing
17
(
2,3
)
0.988
0.988
Breathing
18
(
5,1
)
1.019
1.019
BreathingThe modal shapes(n,m) of elastic solid without fluid: the colours pertain to the displacement filed.
(a)
ω
=
0
.
952
(b)
ω
=
7
.
389
(c)
ω
=
11
.
456
(d)
ω
=
20
.
289The modal shapes(n,m) of elastic solid with fluid: the colours pertain to the displacement filed.
(a)
ω
=
0.702
(b)
ω
=
0.749
(c)
ω
=
0.645
(d)
ω
=
0.813Figure 16
Analytical calculation of eigenvaluesα.


## 6. Conclusion

We have presented an analytic solution for the Stokes eigenmodes of a viscous incompressible cylindrical fluid. Using the Helmholtz decomposition for the velocity field, we obtain an eigenvalue problem forω. The analytical results are in very good agreement with FEM results. This analytic method clearly distinguishes between the potential and rotational components and the contributions of each to various flow variables can be analyzed if one wishes so. The present solution is for cylindrical viscous fluid and is not limited to a cylindrical shape. The same scalar potentials can be used to obtain stokes eigenmodes in the case of a spherical geometry. Finally, a knowledge of the stokes eigenmodes of viscous fluid is likely to be of use in performing a dynamic analysis by modal projection method.



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*Source: 857821-2013-10-08.xml*