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Open Journal of Acoust i c s , 2011, 1, 15-26 doi:10.4236/oja.2011.12003 Published Online September 2011 (http://www.SciRP.org/journal/oja) Copyright © 2011 SciRes. OJA On Axially Symmetric Vibrations of Fluid Filled Poroelastic Spherical Shells Syed Ahmed Shah1, Mohammed Tajuddin2 1Department of Mathematics, Deccan College of Engineering and Technology, Hyderabad, India 2Department of Mathematics, Osmania University, Hyderabad, India E-mail: ahmed_shah67@yahoo.com Received July 27, 2011; revised August 15, 2011; accepted August 19, 2011 Abstract Employing Biot’s theory of wave propagation in liquid saturated porous media, waves propagating in a hol- low poroelastic closed spherical shell filled with fluid are studied. The frequency equation of axially sym- metric vibrations for a pervious and an impervious surface is obtained. Free vibrations of a closed spherical shell are studied as a particular case when the fluid is vanished. Frequency as a function of ratio of thickness to inner radius is computed in absence of dissipation for two types of poroelastic materials each for a pervi- ous and an impervious surface. Results of previous works are obtained as a particular case of the present study. Keywords: Biot’s Theory, Axially Symmetric Vibrations, Radial Vibrations, Rotatory Vibrations, Spherical Shell, Elastic Fluid, Pervious Surface, Impervious Surface, Frequency 1. Introduction Using exact three dimensional equations of linear elas- ticity, Ram Kumar [1] studied the axially symmetric vi- brations of fluid-filled spherical shells. Rand and Di- Maggio [2] studied the vibrations of fluid filled elastic spherical and spheroidal shells. Employing Biot’s [3] theory, Paul [4] discussed the radial vibrations of poroe- lastic spherical shells. Chao et al. [5] presented the theo- retical and experimental results regarding wave propaga- tion in porous formations. Ahmed Shah [6] investigated the axially symmetric vibrations of fluid-filled poroelas- tic circular cylindrical shells of various wall-thicknesses in the absence of dissipation. Sharma and Sharma [7] studied the three dimensional free vibrations of transra- dially thermoelastic spheres. Ahmed Shah and Tajuddin [8] studied torsional vibrations of poroelastic spheroidal shells. In the present analysis, wave propagation in fluid filled poroelastic spherical shells is studied in absence of dissipation. Frequency equation each for a pervious and an impervious surface is obtained employing Biot’s [3] theory of wave propagation in liquid saturated poroelas- tic solid. Biot’s [3] model consists of an elastic matrix permeated by a network of interconnected spaces satu- rated with liquid. Frequency equation of axially symmet- ric vibrations each for a pervious and impervious surface is obtained for fluid filled and empty poroelastic spheri- cal shells. The frequency equations of radial and rotatory vibrations are obtained as a particular case. Non- dimensional frequency as a function of ratio of thickness to inner radius is computed for pervious and impervious surfaces in each case, i.e., fluid filled and empty poroe- lastic spherical shell in absence of dissipation. The re- sults are displayed graphically for two types of poroelas- tic materials and then discussed. Results of some previ- ous works are shown as a particular case of the present investigation. By ignoring the liquid effects, and after some rearrangement of terms, results of purely elastic solid are shown as a particular case considered by Ram Kumar [1]. 2. Governing Equations The equations of motion of a homogeneous, isotropic poroelastic solid [3] in presence of dissipation b are 2 22 12111122 12 2 ΦΦΦΦ ΦΦ, t t PQ ρρ b 2 22 12121222 12 2 ΦΦΦΦ ΦΦ, t t QR ρρ b S. A. SHAH ET AL. Copyright © 2011 SciRes. OJA 16 2 2 1111122 12 2 ΨΨΨΨΨ, t t Nρρ b 2 12 122212 2 0ΨρΨ ΨΨ, t tρb (1) where 1, 2, Ψ1, Ψ2 are potential functions of r, θ and time t; P (=A + 2N), N, Q, R are all poroelastic constants and ik (i, k = 1,2) are the mass coefficients following Biot [3]. The poroelastic constants A, N correspond to familiar Lame constants in purely elastic solid. The co- efficient 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 ag- gregate while total volume remains constant. The coeffi- cient Q represents the coupling between the volume change of the solid to that of liquid and 22 2 2222 12cot . θ θ rr θ rr r (2) The equation of motion for a homogeneous, isotropic, inviscid elastic fluid is 2 2 22 f 1Φ Φ, Vt (3) where is displacement potential function and Vf is the velocity of sound in the fluid. The stresses ik and the liquid pressure s of the poroe- lastic solid are 2, ,,, ik ikik σNeAe Qδik rθφ ,sQeR (4) where ik is the well-known Kronecker delta function. The fluid pressure σf is given by 2 2 Φ, ff σρ t (5) where f is the density of the fluid. 3. Solution of the Problem Let (r, , φ) be the spherical polar coordinates. Consider a homogeneous, isotropic, poroelastic spherical shell filled with inviscid elastic fluid. Let the inner and outer radii be r1 and r2 respectively so that the thickness of poroelastic spherical shell is h [=(r2r1) > 0]. Axially sym- metric vibrations of the spherical shell is character- ized by the displacement field of solid (,,0)uuv and liquid (,,0)UUV where u, v, U and V are functions of r, θ and time t. For axially symmetric vibrations of poroelastic spherical shell, the displacement components and solid and liquid, respectively are given potential functions 1, 2, Ψ1, Ψ2 as 2 11 1 2 ΦΨ Ψ 1cot , θ urrr θ θ 2 11 1 ΦΨ Ψ 11 v, rθrθrθ 2 22 2 2 ΦΨ Ψ 1cot , θ Urrr θ θ 2 22 2 ΦΨ Ψ 11 V. rθrθrθ (6) Similarly, the displacement vector of fluid is (,,0) fff uuv , where Φ1Φ , . ff uv rrθ (7) Solution of Equation (1) is 111112222 Φ()()()()(cos )e, i ωt nnn nn Aj ξrByξrAjξrByξrP θ 22 22 2111122 222 Φ()()) (cos )e, i ωt n1n1n2n n AδjξrBδyξrAδj(ξr)B δy(ξrP θ 133 33 Ψ))(cos )e, i ωt nnn Aj(ξrBy(ξrP θ 12 23333 22 Ψ(()(cos )e. i ωt nnn KAj ξr)B yξrP θ K (8) In Equation (8), jn, yn are Spherical Bessel functions of first and second kind respectively, Pn is Legendre poly- nomial of degree n (n is the order of spherical harmonic), A1, B1, A2, B2, A3, B3 are constants, ω is circular fre- quency. Here i is complex unity or i2 =1 and 1 k, 1,2,3 k ξωVk (9) 1 222 11121222 , 1,2 kk δRKQKVPR QRKQKk (10) 11 1 11 1112122222 , , ,KρibωKρibωKρibω (11) also 11 22 ππ ()(z), ()(). 22 nn nn jzJyzY z zz (12) S. A. SHAH ET AL. Copyright © 2011 SciRes. OJA 17 In Equation (9), V1, V2 are the dilatational wave ve- locities of first and second kind, respectively and V3 is shear wave velocity. A bounded solution of Equation (3) is Φ()(cos )e, i ωt fn fn Aj ξrP θ (13) where Af is constant and 1 f. f ξωV (14) Substituting Equation (13) into Equation (5), the fluid pressure is ()(cos )e. 2i ωt fffnfn σAρω jξrP θ (15) Dilatations of solid and liquid respectively, are 22 12 eΦ, Φ. (16) Now the required displacements, stresses and liquid pressure are 1 111 122 132 143 153 16f17 ()B()()()()()()(cos )e, i ωt f n uuAMrMrAMrBMrAMrBMrAMrP θ (17) 1 211 222 23224325326f27n ()()()()()()()P(cos )e, i ωt rr f σsσAMrBMrAMrBMrAMrBMrAMr θ (18) 1 311 322332343 353 36 d(cos ) ()()()()()()sin e, d i ωt n rθ Pθ σAMrBMrAMrBMrAMrBMr θ θ (19) 1411422 432 44f47 ()()()()()(cos )e, i ωt f n sσAMrBMrAMrBMrAMrP θ (20) 1 411 422 43244f47 ()()()()()(cos )e, fi ωt n σ sANrBNrA NrBNrA NrPθ rr (21) where the elements Mik(r) and Nik(r) are 111 1121113221422 , , , , nnn n Mr ξjξrMrξyξrMrξjξrMrξyξr 153 16317 11 , , , nnfnf nn nn Mr jξrMr yξrMr ξjξr rr 2222222 1 2111111 111 11 2 1 2 () δnn n nn ξ Mr PQQ RδξjξrAQQδRδjξrAQQδRδjξr rr 2222222 1 221 11n111n11 11 2 1 2, n nn ξ Mr PQQδRδξyξrAQQδRδyξrAQQδRδyξr rr 2222222 2 23222222 n2222 2 1 2, n n nn ξ Mr PQQδRδξjξrAQQδRδjξrAQQδRδjξr rr 222 2222 2 24222n222 2222 2 1 2 ξyξ nn nn ξ Mr PQQδRδξrAQQδRδyrAQQδRδyξr rr 3 25n 33 2 2( 1)2( 1) ()() (), n Nn nξNn n Mr jξrjξr rr 32 2633 27 2 21 21 ()()(), ()(), nnfnf Nn n+ξNn n Mr yξryξrMr ρwj ξr rr 11 3111 3211 22 22 22 , , nnnn NξNξ NN Mr jξrjξrM ryξryξr rr rr 22 3322 3422 22 22 22 (), (), nnn n NξNξ NN Mr jξrjξrMryξryξr rr rr 2 2 353n 33363n 33 2 2 12 12 N r, , n n NnnNNnnN Mr ξjξjξrMrNξyξryξr rr 22 1 411 1111 2 1 2 () nn n nn ξ Mr QRδξjξrjξrjξr rr S. A. SHAH ET AL. Copyright © 2011 SciRes. OJA 18 22 1 421 1n1n11 2 1 2 () n nn ξ Mr QRδξyξryξryξr rr 22 2 4322 n222 2 1 2 nn nn ξ Mr QRδξjξrjξrjξr rr 22 2 442 2222 2 1 2, nn n nn ξ Mr QRδξyξryξryξr rr 2 4546 47 ()0, ()0, ()(), fnf Mr Mr Mrρω jξr 2 21 23 1 411 1n1n1n11 23 2ξ21 2 ξjn nn nn ξ Nr QRξjrjξrξrjξr rrr 2 21 23 1 42111n1n 11 23 2ξ21 2 () y n n nn nn ξ Nr QRδξyξrξryξryξr rrr 2 22 23 2 432 22222 23 221 2r nnn n nn ξnn ξ Nr QRδξjξrjξrjξjξr rrr 2 22 23 2 442 22222 23 221 2 () δξ r n nn n nn ξnn ξ Nr QRξyξryξryryξr rr 2 45 46 47 ()0, ()0, ()(). ffnf NrNrNr ρωξ jξr (22) In Equation (22), a prime over jn or yn represents dif- ferentiation with respect to r. 4. Boundary Conditions-Frequency Equation The outer surface of the poroelastic spherical shell is assumed to be free from traction. Continuity of radial displacement and fluid pressure is assumed at the inter- face of solid and fluid. Thus the boundary conditions for a fluid-filled poroelastic spherical shell, in case of a per- vious surface are 0, 0, 0, 0 frrfrθf uu σsσσsσ at r = r1, 0, 0, 0, rr rθ σsσs at r = r2. (23) Similarly, the boundary conditions for a fluid-filled poroelastic spherical shell, in case of an impervious sur- face are 0, s0, 0, 0, f frr frθ σ s uu σσσ rr at r = r1,0, 0, 0, rr rθ s σsσr at r = r2. .(24) Substitution of Equations (17)-(20) into the boundary conditions (23) result in a system of seven homogeneous algebraic equations in seven constants A1, B1, A2, B2, A3, B3 and Af. For a non-trivial solution, the determinant of the coefficients must vanish. By eliminating these con- stants, the frequency equation of axially symmetric vi- brations of a fluid-filled poroelastic spherical shell in case of a pervious surface is 0, ,1,2,3,7. ik Aik (25) In Equation (25), the elements Aik are 1 (); 1,2,3,4 and 1,2,7, ik ik AMr ik 522632742 (), (), (), 1,2,6, kk kk kk A MrAMrA Mrk 37 57 67 77 0, 0, 0, 0.AAAA (26) In Equation (26), the elements Mik(r) are defined in Equation (22). Arguing on similar lines, Equations (17)-(19), (21) together with the Equation (24) yield the frequency equa- tion of axially symmetric vibrations of a fluid-filled po- roelastic spherical shell in case of an impervious surface to be 0, ,1,2,7, ik Bik (27) where the elements Bik are , 1,2,3,5,6 and 1,2,7, ik ik BAi k S. A. SHAH ET AL. Copyright © 2011 SciRes. OJA 19 441 (), 1,2,7, kk BNrk 7k4k 277 N (), 1,2,6, and 0,BrkB (28) where Mik(r) and Nik(r) are defined in Equation (22). By eliminating liquid effects from frequency equation of a pervious surface (25), that is, setting b0, 120, 220, (AQ2/R) , N , Q0, R0 and after re- arrangement of terms, the results of purely elastic solid are obtained as a particular case considered by Ram Kumar [1]. The frequency equation of an impervious surface (27) has no counterpart in purely elastic solid. The order of spherical harmonic n takes different integer values. When n = 0, radial vibrations are obtained whereas n = 1 results in rotatory vibrations. For n = 2 spheroidal vibrations are obtained in which the sphere is distorted into an ellipsoid of revolution becoming prolate or oblate according to the phase of motion. 4.1. Frequency Equation for an Empty Poroelastic Spherical Shell When the fluid density is zero, that is, f = 0 the fluid- filled poroelastic spherical shell will become an empty poroelastic spherical shell. Thus, the frequency equation of pervious surface (25) under suitable boundary condi- tions reduce to 0, 2,3,4,5,6,7 and 1,2,3,4,5,6, ik Ai k (29) where the elements Aik are defined in Equation (26) are now evaluated for f = 0. Equation (29) is the frequency equation of axially symmetric vibrations of an empty poroelastic spherical shell in case of a pervious surface. Similarly, the frequency equation of axially symmetric vibrations of an empty poroelastic spherical shell for an impervious surface under suitable boundary conditions is 0, 2,3,4,5,6,7 and 1,2,3,4,5,6, ik Bi k (30) where the elements Bik are defined in Equation (28) are now evaluated for f = 0. 4.2. Frequency Equation of Radial Vibrations When the order of spherical harmonic n = 0, the fre- quency equation of axially symmetric vibrations of fluid filled spherical shell Equation (25) reduce to the fre- quency equation of radial vibrations of fluid filled spherical shell in case of a pervious surface under suit- able boundary conditions. Radial vibrations involve u and uf as the only non-zero displacement components. Thus the frequency equation of radial vibrations of fluid filled poroelastic spherical shell for a pervious surface is 1112 13 14 17 2122 23 24 27 4142 43 44 47 51 5253 54 71 7273 74 0, 0 0 AAAAA AA AAA AA AAA AA AA AAAA (31) where the elements Aik are defined in Equation (26) are now evaluated for n = 0. Similarly, the frequency equation of radial vibrations of fluid-filled poroelastic spherical shell in case of an impervious surface is 11 1213 14 17 21 22232427 41 42434447 51 5253 54 71 727374 B0, 0 0 BBBBB BBBBB BBBB BBBB BBBB (32) where the elements Bik are defined in Equation (28) are now evaluated for n = 0. When the fluid density is zero in Equation (31), it be- comes the frequency equation of radial vibration of an empty poroelastic spherical shell under suitable bound- ary conditions as 21 2223 24 41 4243 44 51 5253 54 71 7273 74 0, AAAA AAAA AAA A AAAA (33) where the elements Aik are defined in Equation (26) are now evaluated for f = 0 and n = 0. In a similar way, the frequency equation of radial vi- brations of an empty poroelastic spherical shell for an impervious surface is 21 222324 41 424344 51 525354 71 727374 0, BBBB BBBB BBBB BBBB (34) where the elements Bik are defined in Equation (28) are now evaluated for f = 0 and n = 0. The frequency equation of radial vibrations of an empty poroelastic spherical shell Equation (33) of a per- vious surface was also studied by Paul [4] with the boundary conditions 0, 0, rr σs at r = r1 and r2. 4.3. Frequency Equation of Rotatory Vibrations When the order of spherical harmonic n = 1, the fre- quency equation of axially symmetric vibrations of fluid filled spherical shell Equation (25) reduce to the fre- S. A. SHAH ET AL. Copyright © 2011 SciRes. OJA 20 quency equation of rotatory vibrations of fluid filled spherical shell in case of a pervious surface. Thus the frequency equation of rotatory vibrations of fluid filled poroelastic spherical shell for a pervious surface is 0, ,1,2,3,7. ik Aik (35) The elements Aik appearing in Equation (35) are de- fined in Equation (26) are now evaluated for n = 1. Similarly, the frequency equation of rotatory vibra- tions of a fluid filled poroelastic spherical shell in case of an impervious surface is 0, ,1,2,3,7, ik Bik (36) where elements Bik appearing in Equation (36) are de- fined in Equation (28) are now evaluated for n = 1. Setting n = 1 in Equations (29) and (30), the frequency equations of rotatory vibrations of empty poroelastic spherical shells of a pervious and an impervious surface respectively, are obtained. 5. Non-Dimensionalisation of Frequency Equation To analyze the frequency equations of radial and rotatory vibrations of fluid-filled and empty poroelastic spherical shells in the cases of pervious and impervious surfaces, it is convenient to introduce the following non-dimensional variables: 1111 12 34 , , , ,aPH aQH aRH aNH 11 11 11 1112122222 ρ, , , , f mρmρρ mρρ tρ ρ 1212 1211 010 20330 (), (VV), (), , Ω, f xVV yzVVmVVωhC (37) where is non-dimensional frequency, H = P + 2Q + R, = 11 + 2 12 + 22, C0 and V0 are the reference veloci- ties (C0 2 = N/, V0 2 = H/), h is the thickness of the poroelastic spherical shell. Let 2 112 r1 , so that 1, . r hhg gg rrg (38) Employing the non-dimensional parameters defined in Equations (37), (38) and using the relations given in A- bramowitz and Stegun [9] 01 2 sinsin cos (), (), zzz jz jz zz z 232 31 3 ()sin cos, z jzz z zz 01 2 232 coscos sin (z), (), 31 3 ()cos sin, z zzz yyz zz z yzz z z z the frequency equation of radial vibrations of a fluid filled spherical shell in case of a pervious surface Equa- tion (31) reduce to 0; ,1,2,3,4,5, ik Cik (39) where elements Cik appearing in Equation (39) are 11111211 11 11 sin()cos(), Ccos()sin(),CTT TT TT 1322 14221544 22 4 11 1 sin()cos(), cos()sin(), sin()cos(),CTTCTTCTT TT T 22 4 21122 13 11141 1 4δδ sin( )4cos( ), a CaaaaTTaT T 22 4 22122 13 11141 1 4 δδ cos( )4 sin( ), a CaaaaT TaT T 22 4 23122 23 22242 2 4δδsin()4cos( ), a CaaaaTTaT T 2 22 44 24122 23 22242254 2 24 4Ω cos()4sin(), Csin(), (1) ata CaaaδaδTTaT T TgT 22 313 12113223 111 ()sin(), ()cos(),CaδaTT Ca aδTT 2 22 4 33322223423 222354 2 4 Ω (δ)sin(), ()cos(), sin(), (1) ta CaaTTCaaδTTC T gT S. A. SHAH ET AL. Copyright © 2011 SciRes. OJA 21 C41, C42, C51, C52 = Similar expressions as C21, C22, C31, C32 with T1 replaced by T5, C43, C44, C53, C54 = Similar expressions as C23, C24, C33, C34 with T2 replaced by T6, C45 = 0, C55 = 0. (40) Similarly, the frequency equation of radial vibrations of a fluid filled poroelastic spherical shell for an imper- vious surface in non-dimensional form is 0; ,1,2,3,4,5, ik Dik (41) Where ; 1,2,4, 1,2,3,4,5, ik ik DCi k 222 3123 1113 1211 ()sin()()cos(),DaaδTT aδaT T 222 32312113 1211 ()cos()()sin(),DaδaT T aδaT T 222 3323 2223 2222 ()sin()()cos(),DaaδTT aδaT T 222 343222232222 ()cos()()sin(),DaδaT T aδaTT 22 44 354 4 22 4 ΩΩ sin()cos(), (1) (1) ta ta DTT gT g D51, D52 = Similar expressions as D31, D32 with T1 re- placed by T5, D53, D54 = Similar expressions as D33, D34 with T2 re- placed by T6, D55 = 0. (42) The frequency equation of rotatory vibrations of a fluid filled poroelastic spherical shell for a pervious sur- face Equation (35) in non-dimensional form reduce to 0, ,1,2,3,7, ik Eik (43) where the elements Eik are 1111 1211 22 11 11 222 2 1sin()cos(), 1cos()sin(),ETTE TT TT TT 1322 1422 22 22 22 222 2 1sin()cos(), 1cos()sin(),ETTE TT TT TT 1533 1633 2 2 33 33 22 22 cos()sin(), sin()cos(),ETTET T TT TT 174 4 2 4 4 22 1sin( )cos(),ETT T T 22 22 4 4 214122 13 11122 13 111 2 1 1 12 12 4sin() cos(), aa EaaaaδaδTaaaδaδTT T T 22 22 44 22412213111 2 213111 2 1 1 12 12 4cos()sin(), aa EaaaaδaδTaaaδaδTT T T 22 22 4 4 234122 23 22122 23 222 2 2 2 12 12 4δδsin( )δδ cos( ), aa EaaaaaTaaaaTT T T 22 22 44 244122 23 22122 23 222 2 2 2 12 12 4δδ cos( )(δδ)sin(), aa EaaaaaTaaaaT T T T 44 44 25433 26433 22 33 33 121212 12 4sin()cos(), 4cos()sin(), aa aa EaTTEaTT TT TT 22 44 27 44 222 44 ΩΩ cos( )sin( ), (1)(1) ta ta ETT gT gT 44 44 31411 32411 22 11 11 66 66 E2sin(T )cos(), E2cos(T )sin(), aa aa aTa T TT TT 44 44 33422 34422 22 22 22 66 66 2sin()cos(), 2cos()sin(), T aa aa EaTTEaTT T TT S. A. SHAH ET AL. Copyright © 2011 SciRes. OJA 22 4444 3543433364343337 2 2 3 3 3 3 6666 3sin()cos(), 3cos()sin(), 0, aaaa EaTaTTEaTaTTE TT TT 22 22 4131 213121141231131 211 δsinδTcos, Eδcosδsin,Eaa TaaTaaTaaTT 22 22 433222 32222442322 32222 δVsin δcos, δcos δsin,E aTa aTTE aaTa aTT 22 44 45 46 4744 22 2 44 ΩΩ 0, 0, cossin, 11 ta ta EEET T gT gT E51, E52, E61, E62, E71, E72 = Similar expressions as E21, E22, E31, E32, E41, E42 with T1 replaced by T5, E53, E54, E63, E64, E73, E74 = Similar expressions as E23, E24, E33, E34, E43, E44 with T2 replaced by T6, E55, E56, E65, E66 = Similar expressions as E25, E26, E35, E36 with T3 replaced by T7, 57 67 75 76 77 0, 0, 0, 0, 0.EEEEE (44) Similarly, the frequency equation of rotatory vibra- tions of a fluid filled poroelastic spherical shell for an impervious surface Equation (36) in non-dimensional form reduce to 0, ,1,2,37, ik Fik (45) where ; 1,2,3,5,6, 1,2,3,4,5,6,7, ik ik FEi k 22 2 413 12113 1211 ( )(2)sin()2()cos(),FaδaTT aδaT T 22 2 42312113 1211 ( )(2)cos()2( )sin(),FaδaT TaδaT T 222 433 22223 2222 ()(2)sin( )2()cos(),FaδaTT aδaT T 22 2 44322223 2222 ()(2)cos() 2()sin( ),FaδaT TaδaT T 45 46 22 44 474 4 22 2 44 0, 0, Ω2Ω 21sin()cos( ), (1) (1) FF ta ta F TT gT gT F71, F72 = Similar expressions as E41, E42 with T1 re- placed by T5, F73, F74 = Similar expressions as E43, E44 with T2 re- placed by T6, F75 = 0, F76 = 0, F77 = 0. (46) In Equations (40), (42), (44) and (46) the quantities T1, T2, T3, T4, T5, T6, T7 are 11 11 22 22 444 4 123 4 Ω() Ω()Ω()Ω() ,, , , (1) (1)(1)(1) ax ayazaz TTT T gg gmg 111 222 44 4 567 Ω() Ω() Ω() , , , (1) (1) (1) ax gay gaz g TTT ggg (47) and 22 12 , δδ in non-dimensional form are 1 22 13 112121323 12222 ()()x ,δamamaaaamam 1 22 23 112121323 12222 ()()y .δamamaa aamam (48) 6. Numerical Results and Discussion Two types of poroelastic materials are considered to find the frequency as a function of ratio of thickness of spherical shell to inner radius. One poroelastic material is sandstone saturated with kerosene designated as Mate- rial-I [10]. The other poroelastic material is sandstone saturated with water, Material-II [11]. The non-dimen- sional physical parameters of Material-I and II are given in Table 1. Frequency Equations (39), (41), (43), (45) and the cor- responding frequency equations of empty poroelastic spherical shells each for a pervious and impervious sur- face constitute a relation between non-dimensional fre- quency Ω and ratio of thickness of the poroelastic spherical shell to inner radius h/r1. When the values of h/r1 are small it represents thin poroelastic spherical shell. Thickness of the poroelastic spherical shell increases with the increase of h/r1. As h/r1→∞ or r1→0, empty poroelastic spherical shell becomes a poroelastic solid sphere. To compute the frequency of fluid filled and empty poroelastic spherical shells values of h/r1 are taken in the interval [0.5,7]. These values of h/r1 represents transition of spherical shell from a thin spherical shell to moderately thick shell and then to thick spherical shell. For fluid-filled poroelastic spherical shells, the values of m and t are taken as m = 1.5 and t = 0.4. Table 1. Material Parameters. Material/Parameter a1 a2 a3 a4 m11 m12 m22 x y z I 0.843 0.065 0.028 0.234 0.901 –0.0010.101 0.999 4.763 3.851 II 0.960 0.006 0.028 0.412 0.877 0 0.123 0.913 4.347 2.129 S. A. SHAH ET AL. Copyright © 2011 SciRes. OJA 23 Non-dimensional frequency of radial vibrations (n = 0) as a function of ratio of thickness to inner radius of a fluid-filled and an empty poroelastic spherical shell is presented in Figure 1 for Material-I each for a pervious and an impervious surface. From Figure 1 it is clear that frequency of a fluid filled shell for pervious surface decreases in case of a thin spherical shell. As the thickness of the spherical shell increases the frequency increases gradually for moderately thick spherical shell. Still with the increase of thickness of the shell, the frequency remains constant. The frequency of fluid filled spherical shell decreases for thin shell in case of an impervious surface also and it is slightly higher than the frequency of a pervious surface. Then, as the thickness of the spherical shell increases the frequency increases rapidly and for moderately thick shell it remains constant. With the increase of thickness of the shell, the frequency decreases and then remains constant. The frequency of an impervious surface is al- ways higher than that of a pervious surface in case of a fluid filled spherical shell. The frequency of an empty spherical shell slowly increases with the increase of the thickness and for thick spherical shell it is constant. The frequency of an empty spherical thick shell is higher than that of a fluid filled thick shell. This is not true in case of a thin spherical shell. Thus it is seen that the presence of fluid is decreasing the frequency of thick spherical shell. The frequency of an empty spherical shell for an imper- vious surface is almost same as that of fluid filled mod- erately thick shell. The frequency of fluid filled and empty thick shells is same. The frequency of an imper- vious surface is higher than that of a pervious surface in case of empty spherical shell also. Thus the frequency of an impervious surface is higher than that of a pervious surface for an empty and a fluid filled spherical shell. The variation of frequency of radial vibrations of a fluid filled and an empty poroelastic spherical shell each for a pervious an impervious surface is presented in Fig- ure 2 for Material-II. The variation of frequency of fluid filled shell is similar as discussed in Figure 1. But the frequency of empty spherical shell for a pervious an im- pervious surface is nearly same for thin and moderately thick spherical shells. The frequency of an impervious surface is higher than that of a pervious surface in case of thick empty spherical shells. Presence of mass-coupling parameter has no significant effect on the frequency of fluid filled spherical shell in case of a pervious surface. The absence of mass-coupling parameter in Mate- rial-II is increasing the frequency of an impervious sur- face for fluid filled spherical shells. This phenomenon is reversed in case of empty poroelastic spherical shells where the presence of mass-coupling parameter is in- creasing the frequency of an impervious surface. Also it is seen that the presence of fluid is decreasing the fre- quency in case of a pervious surface while it is increas- ing the frequency in case of an impervious surface for both the materials. Frequency of rotatory vibrations (n=1) of fluid filled and empty spherical shells each for a pervious and an impervious surface is presented in Figures 3 and 4 for Material-I and Material-II, respectively. From Figure 3 it is clear that the frequency of a fluid filled spherical shell 0 1 2 3 4 5 6 7 0.5 1.5 2.53.5 4.5 5.56.5 Ratio of thickness to i nner ra dius Frequency _____________Perv ious Surface - - - - - - - - - - -Impervious Surface Fluid-fill ed shel l Fluid-fill ed shell Empty shell Empty shell Figure 1. Frequency as a function of ratio of thickness to inner radius Radial vibrations of Fluid-filled and empty spherical shells (Material-Ⅰ, n = 0). S. A. SHAH ET AL. Copyright © 2011 SciRes. OJA 24 0 1 2 3 4 5 6 7 8 0.5 1.5 2.5 3.5 4.5 5.5 6.5 Ratio of thickness to inner radius Frequency ____________Pervious Surface - - - - - - - - - - -Im pervious Flui d-filled shell Fluid-filled shell Empty shell Empty shell Figure 2. Frequency as a function of ratio of thickness to inner radius Radial vibrations of Fluid-filled and empty spherical shells (Material-Ⅱ, n = 0). 0 1 2 3 4 5 6 0.5 1.5 2.5 3.5 4.5 5.5 6.5 Ratio of thi ckness to inner radius Frequency ____________Perv ious Surface - - - - - - - - - - -Im pervious Fluid-filled shellFluid-filled shell Empt y shell Empt y shell Figure 3. Frequency as a function of ratio of thickness to inner radius Rotatory vibrations of Fluid-filled and empty sphe rical shells (Material-Ⅰ, n = 1). varies in a staggered way when the thickness of the spherical shell is small. With the increase of thickness it remains almost constant. Frequency of thin shell is higher than that of the frequency of moderately thick shell and thick shell in case of a fluid filled shell. The frequency of an impervious surface varies in staggered form for small thickness of the fluid filled shell. But here with the increase of thickness the frequency increases. Thus the frequency of thick shell is higher than the fre- quency of thin shell in case of fluid filled spherical shell. S. A. SHAH ET AL. Copyright © 2011 SciRes. OJA 25 0 2 4 6 8 10 12 0.5 1.5 2.5 3.5 4.5 5.5 6.5 Ratio of thickness to inner radius Frequency ___________ _P er vio us Sur face - - - - - - - - - - -I mpervious Fluid-fi lled shel l Flui d-filled shell Em pty shel l Empty shell Figure 4. Frequency as a function of ratio of thickness to inner radius Rotatory vibrations of Fluid-filled and empty sphe rical shells (Material-Ⅱ, n = 1). Obviously, the frequency of an impervious surface is higher than the frequency of pervious surface for thick fluid filled spherical shells. Therefore, the presence of fluid is increasing the frequency of an impervious sur- face for thick shell. The frequency of an empty spherical shell increases with the increase of thickness for a per- vious surface. Frequency of an empty shell is higher than the frequency of fluid filled shell. Thus the presence of fluid is decreasing the frequency of a fluid filled shell for a pervious surface. Frequency of an impervious surface of empty spherical shell is lower than the frequency of a fluid filled shell. Thus the absence of fluid in the spheri- cal shell is decreasing the frequency of an empty shell in case of an impervious surface for thick shell. Figure 4 show the frequency of fluid filled and empty spherical shell each for a pervious an impervious surface in case of Material-II. Frequency of pervious and imper- vious surface is almost same for thin fluid filled shell. With the increase of thickness, the frequency of an impervious surface is higher than the frequency of a per- vious surface. In case of empty thin spherical shells, again the frequency of pervious and impervious surface is same. With the increase of thickness, the frequency of a pervious surface is higher than the frequency of an im- pervious surface. It is seen that the presence of mass- coupling parameter has no significant effect in case of thick fluid filled shell for a pervious surface. In case of an impervious surface, the presence of mass- coupling parameter is increasing the frequency of fluid filled thick spherical shells. This phenomenon is reversed for empty spherical shells where the absence of mass-coupling pa- rameter is increasing the frequency of a pervious surface while it has no effect on an impervious surface for thick empty shells. 7. Concluding Remarks The study of radial and rotatory vibrations of fluid-filled and empty poroelastic spherical shells has lead to fol- lowing conclusions: 1) The frequency of an impervious surface in case of radial vibrations is higher than that of a pervious surface in case of a fluid filled and empty spherical shells. 2) Mass-coupling parameter has no significant effect on the frequency of fluid filled spherical shell for a per- vious surface in case of radial vibrations. 3) The absence of mass-coupling parameter in Mate- rial-II is increasing the frequency of an impervious sur- face of fluid filled spherical shells in case of radial vibra- tions. 4) Presence of fluid is increasing the frequency of an impervious surface of thick spherical shell in case of rotatory vibrations. 5) Presence of fluid is decreasing the frequency of a pervious surface for rotatory vibrations. 6) Absence of fluid in the thick spherical shell is de- creasing the frequency of an impervious surface for ro- tatory vibrations. S. A. SHAH ET AL. Copyright © 2011 SciRes. OJA 26 7) Mass-coupling parameter is increasing the fre- quency of fluid filled thick spherical shells of an imper- vious surface in case of rotatory vibrations. 8. Acknowledgements The authors are thankful to the editor, reviewers and the editorial assistant for their suggestions and kind coopera- tion in improving the quality of this paper. 9. References [1] R. Kumar, “Axially Symmetric Vibrations of a Fluid- Filled Spherical Shell,” Acustica, Vol. 21, 1969, pp. 143- 149. [2] R. Rand and F. DiMaggio, “Vibrations of Fluid Filled Spherical and Spheroidal Shells,” Journal of the Acous- tical Society of America, Vol. 42, No. 6, 1967, pp. 1278- 1286. doi:10.1121/1.1910717 [3] M. A. Biot, “Theory of Propagation of Elastic Waves in Fluid-Saturated Porous Solid,” Journal of the Acoustical Society of America, Vol. 28, 1956, pp. 168-178. doi:10.1121/1.1908239 [4] S. 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