Vibration Analysis of an Infinite Poroelastic Circular Cylindrical Shell Immersed in Fluid

The purpose of this paper is to study the effect of presence of fluid within and around a poroelastic circular cylindrical shell of infinite extent on axially symmetric vibrations. The frequency equation each for a pervious and an impervious surface is obtained employing Biot’s theory. Radial vibrations and axially symmetric shear vibrations are uncoupled when the wavenumber is vanished. The propagation of axially symmetric shear vibrations is independent of presence of fluid within and around the poroelastic cylindrical shell while the radial vibrations are affected by the presence of fluid. The frequencies of radial vibrations and axially symmetric shear vibrations are the cut-off frequencies for the coupled motion of axially symmetric vibrations. The non-dimensional phase velocity as a function of ratio of thickness to wavelength is computed and presented graphically for two different types of poroelastic materials for thin poroelastic shell, thick poroelastic shell and poroelastic solid cylinder.


Introduction
Gazis [1] discussed the propagation of free harmonic waves along a hollow elastic circular cylinder of infinite extent and presented numerical results.Bjorno and Ram Kumar [2] presented theoretical and experimental results of propagation of axially symmetric waves in submerged elastic rods.Chandra et al. [3] studied the axially symmetric vibrations of cylindrical shells immersed in an acoustic medium.Employing Biot's [4] theory, Tajuddin and Sarma [5] studied the torsional vibrations of poroelastic cylinders.Wisse et al. [6,7] presented the experimental results of guided wave modes in porous cylinders and extended the classical theory of wave propagation in elastic cylinders to poroelastic mandrel modes.Chao et al. [8] studied the shock-induced borehole waves in porous formations.Vashishth and Poonam Khurana [9] presented the solutions of elastic wave propagation along a cylindrical borehole in an anisotropic poroelastic solid and derived frequency equations for empty and fluidfilled boreholes.Farhang et al. [10] investigated the wave propagation in transversely isotropic cylinders.Tajuddin and Ahmed Shah [11,12] studied the circumferential waves and torsional vibrations of infinite hollow poroelastic cylinders in presence of dissipation.Ahmed Shah [13,14] studies the axially symmetric vibrations of fluidfilled poroelastic circular cylindrical shells and spherical shells of various wall-thicknesses.
In the present analysis, the axially symmetric vibrations of poroelastic circular cylindrical shells of infinite extent immersed in an acoustic medium are investigated employing Biot's [4] theory.Biot's model consists of an elastic matrix permeated by a network of interconnected spaces saturated with liquid.The frequency equation of such vibrations is derived each for a pervious surface and an impervious surface.Cut-off frequencies when the wavenumber is zero are obtained both for pervious and impervious surfaces.For zero wavenumber, the frequency equations of axially symmetric shear vibrations and radial vibrations are uncoupled.Axially symmetric shear vibrations are independent of nature of surface as well as presence of fluid within and around the poroelastic cylindrical shell.The radial vibrations are dependent on nature of surface and these are affected by the presence of fluid within and around poroelastic cylindrical shell.Nondimensional phase velocity for propagating modes is computed in absence of dissipation for cylindrical shells immersed in an acoustic medium each for a pervious and an impervious surface.The cut-off frequency as a function of h/r 1 is determined.The results are presented graphically for two types of poroelastic materials and then discussed.By ignoring the liquid effects, and after rearrangement of terms, results of purely elastic solid are shown as a particular case considered by Chandra et al.
[3], Bjorno and Ram Kumar [2].The considered problem is applicable to deep sea sound sources and transducers, petrochemical industries, acoustic waveguides, ultrasonic delay-lines and frequency control devices.

Governing Equations
The equations of motion of a homogeneous, isotropic poroelastic solid (Biot,[4]) in presence of dissipation b are where  2 is the Laplacian,   u, v, w u and are displacements of solid and liquid respectively, e and  are the dilatations of solid and liquid; A, N, Q, R are all poroelastic constants and  ij (i, j = 1, 2) are the mass coefficients following Biot [4].The poroelastic constants A, N corresponds to familiar Lame' constants in purely elastic solid.The coefficient N represents the shear modulus of the solid.The coefficient R is a measure of the pressure required on the liquid to force a certain amount of the liquid into the aggregate while total volume remains constant.The coefficient Q represents the coupling between the volume change of the solid to that of liquid.


The equation of motion for a homogeneous, isotropic, inviscid elastic fluid is where  is displacement potential function and V f is the velocity of sound in the fluid.The displacement of fluid is .

 
The stresses  ij and the liquid pressure s of the poroelastic solid given by Biot [4] are where  ij is the well-known Kronecker delta function.
The fluid pressure P f is given by In Equation ( 4),  f is the density of the fluid.
The subscript "if" or "of" associated with a quantity represents that the quantity is related to inner or outer fluid.For example, V if is the velocity of sound in the inner fluid and P of is the outer fluid pressure.

Solution of the Problem
Let (r, , z) be the cylindrical polar coordinates.Consider a homogeneous, isotropic, infinite poroelastic cylindrical shell immersed in an inviscid elastic fluid.Let the inner and outer radii of the poroelastic cylindrical shell be r 1 and r 2 respectively so that the thickness of shell is h [= (r 2 -r 1 ) > 0].The axis of the poroelastic shell is in the direction of z-axis.The fluid column within the poroelastic cylindrical shell extends from zero to infinity in axial direction and zero to r 1 in the radial direction.The outer fluid extends from r 2 to infinity in radial direction and zero to infinity in axial direction.Then for axially symmetric vibrations, the displacement of solid that can readily be evaluated from field Equation ( 1) is (as shown in the bottom of this page).
In Equation ( 5),  is the frequency of wave, k is wavenumber, C 1 , C 2 , C 3 , C 4 , A and B are constants, J 0 (x), Y 0 (x) are Bessel functions of first and second kind each of order zero, J 1 (x), Y 1 (x) are Bessel functions of first and second kind each or order one.Here i is complex unity or i 2 = -1 and (6) where V i (i = 1, 2) are dilatational wave velocities of first and second kind respectively, V 3 is shear wave velocity.
The displacement of inner fluid column u if = (u if , 0, w if ) for axially symmetric vibrations is where A if is constant and with the help of displacement potential function, the pressure of the inner fluid column is given by Similarly, the displacement and the outer fluid pressure are given by equations where A of is constant, is Hankel function of first kind and order n and velocity {/k} is less than V of the Hankel function of first kind is replaced by the modified Bessel function of second kind K 0 ( of r).
Substituting the displacement function u and w from Equation ( 5), fluid pressures from Equations ( 9) and ( 10), into Equation (3) together with Equation ( 7), the relevant displacement, liquid pressure and stresses are For imaginary values of  of , that is, when the phase where

Equation
For perfect contact between the poroelastic cylindrical shell and the fluids, we assume that the normal and stresses and radial displacements are continuous at r = r 1 ditions in case of a perconditions in case of an impervious surface are s P 0, 0, s 0, u u 0, at r r , s P 0, 0, s 0, u u 0, at r r .

Frequency
The boundary Substitution of Equations ( 12)-( 14) and ( 16)-( 18) into the Equation ( 22) result in a system of eight homogene ous algebraic equations in eight constants C 1 , C 2 , C 3 , C 4 , A, B, A if and A of .For a non-trivial solution, the determ nant of the coefficients must vanish.By eliminating t co ihese nstants, the frequency equation of axially symmetric vibrations of poroelastic circular cylindrical shell immersed in fluid in case of a pervious surface is In Equation (24), the elements A ij are w .( 12), ( 13), ( 15

Frequency Equation for Poroelastic Solid Cylinder
When the ratio of thickness to inner radius of the poroelastic cylindrical shell i.e., h/r 1  as r 1 0 with finite thickness, it reduce to a poroelastic solid cylinder of ra- where the elements P ij are In Equation (29), J     Equations ( 28) and 0) are the frequency equations of (3 axially symmetric ibrations a poroelas der immersed in fluid, for a pervious and an i pervious surface, respectively.
By eliminating liquid effects and after s e rearrangement of term in Equat elastic solid consi red by B ar v of tic solid cylin m om s ion (28), the results of purely de jorno and Ram Kumar (1972) e recovered as a special case.Frequency equation of an impervious surface (30) has no counterpart in purely elastic solid.

Cut-Off Frequencies
The frequencies obtained by equating wavenumber to zero are referred to as the cut-off frequencies.Thus for k = 0, the frequency equation of pervious surface (24) reduce to the product of two determinants as where D 1 and D 2 are  (32) it is ve the frequencies of radial vibrations of poroelastic cylindrical shells immersed in an acoustic medium, for a pervious surface while the frequency equation does not depend on fluid param etric shear vibrations which are independent of presence of fluid within and around the poroelastic cylindrical shell.The radial vibrations are affected by the presence of fluid within and around the poroelastic cylindrical shell while the axially symmetric t affected as can be seen from Equations (34) and (35).
Similarly, the frequency equation of an impervious surface (26), when k = 0 is reduced to the product of two determinants where D and D are The elements appearing in D 3 and D 4 are defined in Equation ( 27) are now evaluated for k = 0. From Equation (36) it is clear that either D 3 = 0 or D 4 = 0. T tion corresponds to frequencies of radial vibrations of a poroelastic cylindrical shell immersed in an acoustic medi e cut-off frequencies independent of presence of fluid.Also it is seen th und the poroelastic surface, that is, pervious g non-dimensional vari- he equaum in case of an impervious surface, while the equation yield th at Equations ( 35) and (39) are same by virtue of Equation ( 27).Hence Equation ( 39) is independent of nature of surface, that is, pervious or impervious.Therefore, the cur-off frequencies given by Equation (39) are independent of presence of fluid within and aro cylindrical shell and nature of or impervious.Equation (35) is the frequency equation of axially symmetric shear vibrations.From Equation (32), it is clear that the radial vibrations and axially symmetric shear vibrations are uncoupled for poroelastic cylindrical shell immersed in an acoustic medium in case of a pervious surface.Similarly, these are uncoupled for an impervious surface as can be seen from Equation (36).The cut-off frequencies of poroelastic solid cylinder for pervious and impervious surfaces are obtained in a similar way as obtained in case of poroelastic cylindrical shells.

Non-Dimensionalization of Frequency Equation
For the purpose of numerical computation we set b = 0, and the wavenumber k is real.The phase velocity C is the ratio of frequency to wavenumber, that is, C=/k.To analyze the frequency Equations ( 24) and (26) it is convenient to introduce the followin ables: where  is n sional phase velocity mersed in an

Results and Discussions
Two types o stic materials are considered to carry out the computational work, one is sandstone saturated with kerose f poroela ne, say M one satu ), whos in Table 1.
aterial-I (Fatt, water, ensional p l, fr iona thick poroelasti [15]), the ot uency Equations ed using Eq c cylindrical her M eq liz ua- numerically to compute either the phase velocity or the frequency, following the analysis of Gazis [1].The counterpart of frequency Equation (34) was not solved numerically for elastic medium by Chandra et al. [3] while the author solved these equations for poroelastic medium in a different paper.
The phase velocity of axially symmetric vibrations of poroelastic cylindrical shells immersed in an acoustic medium is presented in Figures 1-3 for material-I and II each for a pervious and an impervious surface.Figure 1 shows the phase velocity for materials-I and II in case of pervious and impervious surfaces.From Figure 1 it is clear that the phase velocity for a pervious surface is higher than that of an impervious surface in 0    0.5 one is sandst and Jogi, [16] eters are given    for material-I while beyond  = 0.5 it is less than or equal to the phase velocity of an impervious surface.T he phase velocity of pervious and impervious surfaces is almost is same in case of material-II.The phase velocity for material-I, in general, is higher than that of material-II both for pervious and impervious surfaces.Thus it can be inferred that presence of mass-coupling parameter increases the phase velocity for thin poroelastic cylindrical shells immersed in an acoustic medium.
Figure 2 shows the phase velocity of thick poroelastic cylindrical shells immersed in an acoustic medium in case of materials-I and II each for a pervious and an impervious surface.It is seen from Figure 2 that the phase velocity of a pervious surface in case of material-I isnd 0.7    1.In 0.2    0.5 the phase velocity of a pervious surface is higher than that of an surface while in 0.5    0.7 it is less than that of an im .Ag as in case of a t poroelastic cylindrical shell, the phase velocity is sa pe nd impervious surfa hi oel ylin ell.In general, the phase velocity for thick poroelastic cylindrical shell is higher in case of material-I than that of material-II.The phase velocity decreased with the increase of thickness for a pervious surface in case of material-I.In case of an impervious surface, in general, the phase velocity increases with the increase of thickness.Increase of thickness has no significant effect on phase velocity in case of material-II for pervious and impervious surfaces.
Figure 3 shows the phase velocity of poroelastic solid cylinders immersed in an acoustic medium each for a pervious and an impervious surface in case of materials-I locity of a pervious surface, in general, is higher than that ic solid cylinder.Also the presence of mass-coupling parameter increases the phase velocity of an impervious surface of the poroelastic solid cylinder.In general, the phase velocity is less in poroelastic solid cylinder than that of either a thin shell or a thick shell both for pervious and impervious surfaces and for both the considered materials.

Concluding Remarks
The study of axially symmetric vibrations of poroelastic cylindrical shells immersed in an acoustic medium has lead to following conclusions: ar vibrations is independent of nature of surface and presence of fluid within and around the poroelastic cylindrical shell.
3) The phase velocity is same for pervious and impervious surfaces in case of material-II each for thin and thick poroelastic cylindrical shell.
4) In general, the phase velocity is higher for material-I than that of material-II each for a pervious and an impervious surface.
5) The frequency of radial vibrations of poroelastic cylindrical shell immersed in an acoustic medium for a pervious surface is higher than that of an impervious surface in case of material-I.
uency of an impervious surface is higher pervious surface in case of material-II.

dgements
author thankful to the Editor-in-Chief, the reviewers, and particularly the Editorial Assistant Mr. Judy Liu, Cylinders," al most same to that of an impervious surface in 0    0.2 that the absence of mass-coupling parameter increases the phase velocity of a pervious surface of a poroelast a impervious 6) The freq than that of a pervious surface ain hin me for rvious a ces for t ck por astic c drical sh and II.From Figure 3 it is clear that the phase velocity of pervious and impervious surfaces vary in a staggered way for material-I.In case of material-II, the phase ve of an impervious surface unlike in case of poroelastic thin and thick cylindrical shells.Therefore it is inferred for their suggestions and cooperation in improving the quality of this paper.

1
The elements A ij of D 1 and D 2 are defined in Equation (25) are now evaluated for k = 0. From Equation clear that either D 1 = 0 or D 2 = 0 and these two equations give the cut-off frequencies of axially symmetric vibrations.The frequency equation D

Figure 1 .
Figure 1.Phase velocity as a function of wavelength (Mat-I, Mat-II, Thin-Shell) axially symmetric vibrations of poroelastic cylindrical shells immersed in an acoustic medium.

Figure 2 .
Figure 2. Phase velocity as a function of wavelength (Mat-I, Mat-II, Thick-Shell) axially symmetric vibrations of poroelastic cylindrical shells immersed in an acoustic medium.rial Parameters.m 11 m 12 m 22

Figure 3 .
Figure 3. Phase velocity as a function of wavelength (Mat-I, Mat-II, Solid cylinder) axially symmetric vibrations of poroelastic solid cylinders immersed in an acoustic medium.