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

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 threedimensional cylindrical coordinates a general solution for a problem.Themethod 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.


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][6][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][11][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. Threedimensional 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][10][11][12][13][14][15][16][17][18][19][20][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 Section 2, 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.

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 in which v( , , ) = {V , V , V } ( , , ) is the fluid velocity vector, ( , , ) is the fluid pressure, is the density of the fluid, and ] and = ] 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 potentials , , and , 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: where e z is the unit vector along the direction. Substituting the above resolutions into (1) and (2), after some manipulations, the equation for the conservation of mass (2) becomes the Laplace equation: and the equation for the conservation of momentum (1) becomes the Helmholtz equations where is the Laplacian operator in polar coordinates, and time dependence has the form exp( ). The pressure can be presented by 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 ; as a result all potentials components and pressure will depend on time only through the factor exp( ).
Applying the method of separation of variables, the solution of the equations for potentials, associated with an axial wave number , radial wave number , and circumferential mode parameter , after considerable algebraic manipulations, can be shown to be and are Bessel functions of the first and second kind of order . and are modified Bessel functions of the first and second kinds of order . and are the azimuthal and axial wavenumbers. = 0, 1, 2, . . ., whereas is found by satisfying the symmetry boundary condition on the = 0 and = . Note that the velocity field v has components that are symmetric or antisymmetric in and . Following standard practice, the solutions with symmetric (antisymmetric) axial velocities are called the antisymmetric (symmetric) axial modes, respectively, with and denoting the corresponding eigenvalues. Thus, we have = ( − 1) / and = / ( = 0, 1, 2, . . .). However, the azimuthal modes corresponding to cos( ) and sin( ) are really the same, due to periodicity in the azimuthal direction; that is, there is no distinction in the values of for the two families. , , , , , and are unknown coefficients which will be determined later by imposing the appropriate boundary conditions. The radial wave number is related to the axial wave number by where = √ −1. In the end, the pressure can be obtained by insertion of (7) into (6), Using (3) and taking into account (7), (8), and (9), the fluid particle velocities in terms of the Bessel functions can be written as In the following section we consider a fluid with different boundary conditions.

Boundary Conditions and Frequency Equation
First we define Γ 1 ( = 1 ), as the inner boundary of the fluid region, and Γ 2 ( = 2 ), as the outer boundary. We begin the analysis of frequency equation for imposing the no-slip boundary conditions.

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 at Γ 1 : (2) the velocity is equal to zero at Γ 2 : Combining these boundary conditions with (12) yields for each mode number ( , ) the following linear system: where [M 1 ] is a 6 × 6 matrix whose components are given in Appendix A. For a non-trivial solution, the determinant of the matrix [M 1 ] must be equal to zero The roots of (16) give the natural frequencies of the cylindrical oscillations. Figure 16 in Appendix D shows analytical calculation of eigenvalues and using (10) we obtain the natural frequencies [rd/s].

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 in which I is a unit tensor. By applying this condition on Γ 1 we obtain Combining these boundary conditions and (14) yields for each mode number ( , ) the following linear system: where [M 2 ] is a 6 × 6 matrix whose components are given in Appendix B. For a nontrivial solution, the determinant of the matrix [M 2 ] must be equal to zero The roots of (20) give the natural frequencies of the cylindrical oscillations.

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.

Torsional Stokes Eigenmode.
The torsion mode vibration is such a mode in which the scalar components of the velocity {V , V } are zeros and only the circumferential velocity V is independent of . This condition is achieved if = 0 and = 0. Through (3) this gives for the nonvanishing components of displacement and stresses: Thus, the general solution for must be constructed from the set In this case the boundary conditions equations (13) and (14) become Then, (15) becomes [T] is a 2 × 2 matrix whose components are calculated using Appendices A and B. Solving det[T] = 0 gives the torsional modes.

Longitudinal Stokes Eigenmode.
Another simpler mode vibration is called longitudinal mode vibration in which V = 0 and {V , V } are independent of . This means that the motion is confined to planes perpendicular to the -axis, which can move, expand, and contract in their planes. The solution for the displacement field and stress vector follows from (3) and (17): Thus, the general solution for and must be constructed from the set: In this case the boundary conditions equations (14) and (18) become Then, (19) becomes [L] is a 4 × 4 matrix whose components are calculated using the Appendices A and B. Solving det[L] = 0 gives the longitudinal modes.

Flexural and Breathing Stokes
Eigenmodes. The mode shape = 1 is called flexural mode vibration in which all components of the displacement are nonvanishing and depend on , , and . The mode shape ⩾ 2 is called breathing mode vibration in which all components of the displacement are non-vanishing and depend on , , and .
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 region Γ 1 is represented by an elastic wall.

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.

Governing Equations of Elastic
Media. The wave motion in an isotropic elastic medium is governed by the classical Navier's equation: where is the density, , are the Lamé constants, and u( , , ) = { , , } ( , , ) 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 potentials Φ, Ψ, and Θ, known as the Helmholtz decomposition such that Substituting (30) into (29) leads to three sets of differential equations where = √( + 2 )/ and = √ / 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 , radial wave number ( Φ , Ψ ), and circumferential mode parameter , after considerable algebraic manipulations, can be shown to be The radial wave number ( Φ , Ψ ) is related to the axial wave number by and , , , , , and 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 and the radial and tangential stresses are given by Hooke's law as

Boundary Condition and Frequency Equation.
We define Γ 1 ( = 1 ) as the boundary contact between the fluid region and the solid region and Γ 2 ( = 2 ) 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 interface Γ 0 : (2) Kinematic boundary condition (velocity must be continuous) on the interface Γ 1 : (3) Dynamic boundary condition (normal stresses must be continuous) on the interface Γ 1 : (4) On the interface Γ 2 at the outer cylinder the normal stress is equal to zero: Combining these boundary conditions with (34)-(35) and taking into account (7)-(9) and (32) yields for each mode number ( , ) the following linear system: where {y} = { } 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: This equation indicates a relationship between the dynamic fluid viscosity , density of the fluid , and the elastic constants. The roots of (41) give the infinite natural frequencies .

Analytical, Numerical Results and Validation
Numerical calculations were performed on the example of the hollow cylinder (Figures 1, 2 With the derived eigenfrequency equations, natural frequencies = Im(] 2 ) for each pair of ( , ) are calculated in the software Mathematica. and 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         Tables 1 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 ( = 1), nine to breathing vibration ( ⩾ 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. Tables 1 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. Figures 7, 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 ( , ), where is the modal number in the circumferential direction and is the modal number in the axial direction. The modal shapes are not in order with the parameters and . 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.
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 mode , axial mode , and fluid thickness   = 2 − 1 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 mode . Figure 10 shows that the natural frequencies increase as the axial mode increases except when = 0. This value of corresponds to 2D problem of viscous oscillations.
Secondly, one investigates how the frequencies vary with the fluid thickness. Figures 11, 12, and 13 show that the fluid thickness has a strong influence on the natural frequencies.
Thirdly, one investigates how the dense fluid (added mass) affects the natural frequencies. Tables 3 and 4 show the coupled and uncoupled natural frequencies varying with circumferential and axial mode ( , ). As 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.

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 for . 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.

A.
The matrix M 1 in (16) is defined as follows: