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1 Z. Naturforsch. 5; 7()a: 8 9 Kambiz Vafai Ambreen Afsar Khan Saba Sajjad and Rahmat Ellahi* The Stud of Peristaltic Motion of Third Grade Fluid under the Effects of Hall Current and Heat Transfer Abstract: This article is concerned with the peristaltic pumping of an incompressible electricall conducting third grade fluid in a uniform channel. The Hall effect under the influence of wall properties and heat transfer is taken into account. Mathematical modelling is based upon continuit momentum and energ equations. Closed form solutions for velocit temperature concentration and heat transfer coefficient are obtained. Effects of pertinent parameters such as third grade parameter Γ Hall parameter M amplitude ratio ε Brickman number Br Soret number Sc wall tension E and elasticit parameters E and E on the velocit u temperature θ concentration φ and heat transfer coefficient Z are discussed through graphs. Kewords: Analtical Solutions; Hall Current; Heat Transfer; Peristaltic Flow; Perturbation Method; Third Grade Fluid. DOI.55/zna-- Received November ; accepted Februar 5; previousl published online March 8 5 Introduction Peristalsis is the process of transportation of fluid in a channel/tube due to successive contractions of walls. This process appears in the stomach intestine and oesophagus. Due to wave propagation of peristaltic motion the movement of organ walls propels food and liquid to mi the contents within each organ. This wave propagation along the fleible walls of a channel can *Corresponding author: Rahmat Ellahi Department of Mechanical Engineering Universit of California Riverside CA 95 USA; and Department of Mathematics and Statistics IIUI H- Sector Islamabad Pakistan rellahi@engr.ucr.edu rahmatellahi@ahoo.com Kambiz Vafai: Department of Mechanical Engineering Universit of California Riverside CA 95 USA Ambreen Afsar Khan and Saba Sajjad: Department of Mathematics and Statistics IIUI H- Sector Islamabad Pakistan induce flow without an eternal pressure gradient. Such flows are encountered in several parts of human bodies including the transportation of urine from kidne to the bladder food through the digestive tract and bile from the gall bladder into the duodenum movement of ovum in the fallopian tube. For reasons such as these this flow has been the subject of man studies in the past. During the last four decades researchers have etensivel focused on the peristaltic flow of Newtonian and nonnewtonian fluids; for instance Mitra and Prasad [] studied the peristaltic flow of a Newtonian fluid with elastic walls. Davies and Carpenter [] analsed the instabilities of plane channel flow between the compliant walls. Muthu et al. [] etended the analsis of Mitra and Prasad to micropolar fluid in a circular clindrical tube. Contributions regarding this have been mentioned b a few authors [ 9]. Moreover the Hall effect plas an important role especiall when the Hall parameter is ver high. This occurs when either the densit of an electricall conducting fluid is low and/or the applied magnetic field is strong. Customaril the Hall term is ignored in Ohm s law as it has no manifest character for small and adequate values of the magnetic field. Nevertheless the current tendenc with a strong magnetic field is alwas noticeable and has a great influence on the electromagnetic force. Under these circumstances the Hall current has an abundant impact and noticeable effects on the magnitude direction of the current densit and eventuall on the magnetic-force term. Therefore it is of great interest to stud the influence of the Hall current on the flow as the Hall current determines the flow features of the problem and induces secondar flow in the flow field. The interaction of Hall current and heat transfer has also received some attention as it might be able to determine the efficienc of some devices such as power generators and heat echangers []. The flow of rheological fluids with Hall current in the presence of heat and mass transfer is also important because nutrients diffuse out from blood especiall when simultaneous effects of heat and mass transfer are considered; then a complicated relationship occurs between the flues and the driving potential. It also occurs in man Brought to ou b Universit of California - Riverside Download Date /9/5 : PM

2 8 K. Vafai et al.: Peristaltic Motion of Third Grade Fluid industrial processes such as membrane separation processes reverse osmosis distillation process and diffusion of chemical impurities [ ]. Furthermore it is known that all homogeneous fluids follow the Newtonian s law of viscosit or Hookian s relation in which shear stress is directl and linearl proportional to the rate of shear strain. Nevertheless there are man other fluids such as bubble columns bubble absorption blood at low shear rate and fermentation that reveal a direct but nonlinear relationship between shear stress and the rate of strain. These fluids are designated as nonnewtonian fluids. The flows of nonnewtonian fluids occur widel in practical applications such as paints oghurt mud ketchup cla coatings polmer melts shampoo greases etc. In view of the abundant applications of nonnewtonian fluids man scientists have been involved in carring out the flow analses of nonnewtonian fluids under various conditions. Several constitutive epressions for these fluids have been suggested. In nonnewtonian fluids the associated model and corresponding differential equations are known as second and third grade fluids. No doubt sufficient attempts have been made for the stud of the simplest subclass of nonnewtonian fluids called second grade fluids. A second grade fluid is able to describe the normal stress but it is inadequate to eplain shear thinning and shear thickening effects which are speculated b man liquids. The third grade fluid can onl predict shear thinning/shear thickening effects even in cases of stead flows over a rigid boundar but the resulting equations of third grade fluids in general are more complicated and of a higher order in comparison to Newtonian and second grade fluids. Some relevant studies on the topic can be seen in the list of references [ ]. The aforementioned studies and literature surve bear witness that the peristaltic pumping of an incompressible electricall conducting third grade fluid in a channel under the influence of wall properties and heat transfer has not been presented et. The stud of third grade fluid is much more complicated because of the nonlinear relationship between the shear stress and the rate of strain and this is due to the fact that most real world phenomena are essentiall nonlinear and usuall described in nonlinear equations. It is ver eas to solve a linear problem but finding the solutions of nonlinear problems is still a ver challenging task. In order to fill the gap of the eisting literature the phsical problem is first modelled and then solved via regular perturbation methods to get analtical solutions. It is worth mentioning that in applied sciences analtical solutions of an phsical model are of great importance if available because analtical solutions not onl lead to drawing correct phsical interpretations but are also ver helpful in the validating of numerical investigations. The organisation of this article is as follows. Section comprises the mathematical formulation of the problem. Analtical solutions are discussed in Section. Section deals with results and discussion. Concluding remarks are given in Section 5. Finall the role of sundr parameters such as Hartmann s number Brickman s number elasticit parameters and Soret s number are demonstrated graphicall at the end. Mathematical Formulation of the Problem Consider the peristaltic flow in a channel of uniform thickness d filled with a third grade fluid. The temperatures and concentrations of the lower and upper walls of the channel are maintained at T C and T C respectivel. The wall s deformation due to the propagation of sinusoidal wave is π ( ) ( ) =± t =± d+ asin ct λ where c is the wave speed a is the wave amplitude and λ is the wave length. The governing equations for an incompressible fluid are V= dv ρ = div T+ J B dt where V is the velocit vector T is the Cauch stress tensor ρ is the fluid densit d/dt is the material time derivative J is the current densit and B = ( B ) is the constant magnetic field. The generalised Ohm s law in the presence of a Hall current is σb ( ) ( ) σb ˆ J B = Mv ui Mu + v ˆ j () + M + M where the Hall parameter M is obtained as e () () σb M =. () e Due to the compleit of nonnewtonian fluids there is no single model which can describe all of its properties. Therefore several constitutive equations have been proposed which can describe all the behaviours of non- Newtonian fluids. Amongst the man models there is a third grade model which is the most popular. This Brought to ou b Universit of California - Riverside Download Date /9/5 : PM

3 K. Vafai et al.: Peristaltic Motion of Third Grade Fluid 8 is particularl due to the fact that one can reasonabl eplain shear thinning/shear thickening properties even for stead and unidirectional flows. The Cauch stress in a third grade fluid is given b T= pi+ µ A + α A + α A + β A + β ( AA + AA) + β( tra ) A where μ is the coefficient of viscosit p is hdrostatic pressure T is Cauch stress tensor pi is the spherical stress due to the constraint of incompressibilit α i (i = ) are material constants β β and β are third grade parameters. The first three Rivlin Ericksen kinematical tensors A A and A are defined b ( ) ( ) t A = gradv + grad V () = dan t A ( grad ) ( grad ) A for n. n n n dt A V + V > (7) If all the motions of the fluid are to be compatible with thermodnamics in the sense that these motions satisf the Clausius Duhem inequalit and if it is assumed that the specific Helmholtz free energ is a minimum when the fluid is locall at rest then thermodnamic properties impose the following constraints [ ]: µ α α + α µβ β = β = β. (8) It is proved that the constitutive relation given in (5) not onl predicts the normal stress differences but also predicts the shear-thickening phenomenon for the third grade parameter β >. In the present stud we assume that the fluid is thermodnamicall compatible and therefore () reduces to T= pi+ µ A + α A + α A + β( tra ) A. (9) In a laborator framework the equations governing two-dimensional motion of an incompressible third grade fluid are (5) u v + = () du p u u σb ρ µ = + ( Mv u) dt M u v u v u v + t + + α + u v u v u u v v t + u ( α+ α u v ) + α + u u v v β u v u v v () dv p v v σb ρ µ = + ( Mu v) dt M v u v + ( α+ α ) + α + u v u v u u v v + t α u v v u+ v t u v u v v β v u v v dt T T u v ρ Cp = k + + µ + dt µ u v α u u v u t v u v α u v u v t u v + + u u v v + α v u v v u u v t u v u v v β u v u v v dc C C DK T T T = D dt T m () () () Equations () and () ield the following compatibilit relationship: Brought to ou b Universit of California - Riverside Download Date /9/5 : PM

4 8 K. Vafai et al.: Peristaltic Motion of Third Grade Fluid d ρ u v µ u u v v dt = + + u v u v v u + t + + α u v u v t u u v v +. u v u v v β u v u v v σb + ( ) ( Mv u Mu v + + ) M + (5) The equation corresponding to the compliant wall is obtained as τ + + = t t m C p p () where τ is the tension in the membrane m is the mass per unit area C is the coefficient of the viscous damping force S and p is the pressure on the outside surface of the wall. Assuming p = and the walls of the channel are not etendable their lateral motion can onl take place provided that the horizontal displacement of the wall is zero. Further continuit of stresses requires that at the interface of the walls and fluid p must be the same as that which acts on the fluid at = ±. Using () and () along with no slip condition we obtained Introducing stream function ψ ψ u= v= (8) and utilising the following set of dimensionless quantities ψ ct dp ψ = = = t = p = = cd λ d λ cλµ d c T T CC α = α θ= φ= Br = E Pr µ d T T C C τd mcd Cd µ C p E = E = E = Pr = λµ c λµ λµ k cd µ ρ( T T ) DK c = = = = ν ρ D µ T ( C C ) C ( T T ) m p = T Re Sc Sr E α α µ c c d βc σ α = α δ= Γ= γ = Bd d µ d λ µ d µ (9) u u u σb τ m C µ ρ + + = ( Mv u) t t + + t + M u v u v u v + + t α u v u v u u v v t u u v ( α α. + + ) + α + u u v 8 v + + β u v u v 8 v (7) Brought to ou b Universit of California - Riverside Download Date /9/5 : PM

5 K. Vafai et al.: Peristaltic Motion of Third Grade Fluid 85 Equations () through (5) under the assumptions of long wavelength and low Renolds number approimations are finall reduced to the following forms: Substituting (8) into nonlinear coupled problems given in () () we have the following sstem of equations. ψ ψ ψ + Γ N = (). Zeroth-Order Sstem θ ψ ψ + Br + Γ = () ψ N = ψ (9) φ θ + Sr = Sc () θ ψ + Br = () where N = γ + M The corresponding boundar conditions are. ( ( )) ψ = at = ± = ± + ε sin π t ( ( )) θ = at = ± = ± + εsin π t φ = at =± =± ( + εsin π( t) ) ψ E + E + E t = t at ( =± ). ψ ψ + Γ N () () (5) () (7) In all these equations Br is the Brickman number Pr is the Prandtl number E is the Eckert number Sc is the Schmidt number Sr is the Soret number Γ is the third grade parameter and E E and E are the elasticit parameters. Solution of the Problem For analtical solutions we epand ψ θ φ and Z in terms of third grade parameter Γ as ψ= ψ + Γψ + Γψ +. θ= θ + Γθ + Γθ +.. φ= φ + Γφ + Γφ +. Z = Z + ΓZ + Γ Z +. (8) φ θ + Sr =. Sc () The boundar conditions can also be obtained b substituting (8) into () (7). Because Γ appears eplicitl and implicitl therefore b epanding the Talor series about the mean position one can obtain the following boundar conditions associated with the zeroth-order deformation problems given in (9) () ψ = θ = φ at = =± t ψ ψ = N at =±. E + E + E t () () Solving (9) through () with the use of boundar conditions of () and () we obtain the zeroth-order solutions of velocit u temperature θ concentration φ and heat transfer coefficient Z as sinhn ψ = c Ncosh () BrL [coshmcoshm M ( )] θ = + + 8cosh M ScSrBrc coshn cosh φ = + + 8cosh + N ( ) (5) () The coefficient of the heat transfer at the wall is obtained: BrNc Z = θ ( ) = [ sinh ]. + cosh (7) Brought to ou b Universit of California - Riverside Download Date /9/5 : PM

6 8 K. Vafai et al.: Peristaltic Motion of Third Grade Fluid. First-Order Sstem The governing equations and the appropriate boundar conditions for the first-order are ψ ψ ψ N + = θ ψ ψ ψ = Br + φ θ =Sr Sc ψ = θ = φ = at =± t E + E + E t ψ ψ ψ = + N at. =± (8) (9) () () () Solving (8) () with the use of boundar conditions () and () we obtain 9Brc N Brc N φ( ) = ScSr cosh cosh Brc N Brc N (coshn cosh ) + cosh cosh 9Brc N Brc N + (coshn cosh ). cosh cosh BrL N + ( sinhnsinh ) cosh Brc cn Brc N + ( ) cosh 8cosh (5) The heat transfer coefficient at the wall is calculated b the relation 5 8Nc sinhn Z = θ ( ) = cosh 57 Brc N Brc N Brc N Brc N + sinh + cosh cosh cosh cosh Brc N Brc N BrccN cosh cosh cosh (). cn cn sinh ψ( ) = ( cosh 8 cosh ) + c cosh cosh cn cnsinh + (cosh+ 8sinh+ 9cosh ) cosh cosh c sinhn c N + (NcoshN + sinh N) cosh cosh () 9Brc N Brc N θ( ) = (coshn cosh ) cosh cosh Brc N Brc N 9Brc N Brc N + + cosh cosh cosh cosh Brc N (coshn cosh ) + ( sinhn sinh ) cosh Brc N Brc cn + ( ) 8 cosh cosh (). Second-Order Sstem The governing equations of second-order are ψ ψ ψ ψ N + = (7) θ ψ ψ ψ ψ ψ =Br Br Br 8 (8) φ θ =Sr. (9) Sc Brought to ou b Universit of California - Riverside Download Date /9/5 : PM

7 K. Vafai et al.: Peristaltic Motion of Third Grade Fluid 87 The appropriate boundar conditions are ψ = θ = φ = at =± (5) E E E N + + = at ( ). =± t t ψ ψ ψ + (5) Solving (7) and (9) with the help of boundar conditions given in (5) and (5) we get 9Brc N θ cosh + c ( sinhn sinh ) = c (coshn cosh N ) + ( coshn cosh N ) Brc N + ( ) + c ( ) + c ( ) cosh 7Brc N sinh + (coshn cosh ) + c sinh 5 8cosh N + c ( coshn cosh ) 7 5 7Brc N + ( coshncosh ) cosh 7Brc N sinh + sinh N c (coshn cosh ) 8 cosh + 5 8cN + ( sinhn si nh ) 8cosh 9Brc N + ( sinhn sinh ) 8cosh (5) φ ( ) = ScSr[ θ ( )] (5) 9 Brc N( cosh sinh ) Z = θ ( ) = Nc sinh + c sinhn 7 cosh Brc N 7Brc N sinh + Nc cosh+ + c + c + Nc cosh 5 cosh cosh 5 c sinh 7Brc sinh 7Brc N cosh 5 + c sinh + 7 8cosh cosh 5 7Brc N sinh 8c N sinh 8cN cosh + Nc sinh + 8 cosh 8cosh cosh 5 9Brc sinh 9Brc N cosh 9Brc N sinh + + cosh cosh 8cosh (5) Brought to ou b Universit of California - Riverside Download Date /9/5 : PM

8 88 K. Vafai et al.: Peristaltic Motion of Third Grade Fluid 5 c cn ψ ( ) = cosh ( Ncosh+ cosh ) 5 N cosh 5 5 7cN c cosh cosh 7 sinh cosh 5sinh ( ) cosh + 5 cosh N N N N 5 9cN [ Nsinh (5 cosh+ 9cosh ) + 8N sinh ] 5 cosh + N cosh (sinh ) + sinhcosh Nsinh [ N sinh N (9N 7 N )cosh N N sinh N ] cosh sinh + + Nc sinh cosh( c ) N N c c 5 sinhn + coshn Nsinh Ncosh 5 c cn c cosh ( Ncosh+ cosh ) 5 cosh N cosh N cn 7cN c ( Ncosh+ cosh ) cosh 7 cosh cosh [N sinh + (9N + 7 N )cosh N sinh ] 7 sinhn cosh 5sinh ( ) cosh + 5 cosh N N N N 5 9cN ( N(sinh )( 5 cosh+ 9cosh ) + 8N sinh 5 cosh + N cosh sinh+ sinh cosh cosh sinh c + c N N 5 Nsinh N + + 5RsinhN 7RcoshN +. N 8N (55) The coefficients c i (i = 8) consisting of emerging parameters can be easil obtained through the routine calculations. Results and Discussion Γ = Γ = Γ = To see the effects of emerging parameters such as third grade parameter Γ Hall parameter M amplitude ratio ε Brickman number Br Soret number Sc wall tension E and elasticit parameters E and E on the velocit u temperature θ concentration φ and heat transfer coefficient Z Figures have been displaed. The effects of pertinent parameters on velocit are shown in Figures. Figure is prepared to discuss the influence of third grade parameter Γ on the velocit u. One can see that the velocit increases b increasing the value of Γ. It is noted that the velocit in third grade fluid is larger when compared with the U Figure : Variation of Γ on velocit u. Brought to ou b Universit of California - Riverside Download Date /9/5 : PM

9 K. Vafai et al.: Peristaltic Motion of Third Grade Fluid U 5 U.5 E E E =..5 E E E =.9.. E E E = V M =.8 M = M = Figure : Variation of E E E on velocit u. Figure : Variation of M on velocit u. Newtonian fluid. Figure studies the effect of wall tension E and elasticit parameters E and E on the velocit u. It is observed that the velocit increases with an increase in wall tension E whereas velocit decreases with an increase of E and E. Figures and depict the effects of amplitude ratio ε and the Hall parameter M on the velocit u. It is found that the velocit increases with an increase in amplitude ratio whereas the Hall parameter M leads to decrease in the velocit. This is in accordance with the phsical epectation because the Hall parameter is the ratio between the electron cclotron frequenc and the electron atom-collision frequenc. Figures 5 9 have been prepared to eplain the effects of sundr parameters Γ M Br ε E E and E on the temperature profile. Figure U θ.5 ε =.9 ε =. ε = Γ =. Γ =. Γ = Figure : Variation of ε on velocit u. Figure 5: Variation of Γ on temperature θ. Brought to ou b Universit of California - Riverside Download Date /9/5 : PM

10 9 K. Vafai et al.: Peristaltic Motion of Third Grade Fluid.5.5 θ.5 θ.5 M = M =. M = E E E =.7.. E E E =.. E E E = Figure 9: Variation of E E E on θ. Figure : Variation of M on temperature θ..5.5 θ φ E E E =.9.. E E E =.8..5 E E E = Br = Br =. Br = Figure : Variation of E E E on φ. Figure 7: Variation of Br on temperature θ..5 Γ =. Γ =. Γ =..5.5 θ θ ε =. ε =. ε = Figure 8: Variation of ε on temperature θ. Figure : Variation of Γ on concentration. Brought to ou b Universit of California - Riverside Download Date /9/5 : PM

11 K. Vafai et al.: Peristaltic Motion of Third Grade Fluid 9.5 Br = Br =. Br =..5 ε =. ε =. ε =..5 φ φ Figure : Variation of Br on concentration. gives the variation of the third grade parameter Γ on the temperature profile θ. The temperature is increased as Γ increases. The effects of M on the temperature can be seen in Figure. The temperature decreases with an increase in the Hall parameter M. Figures 7 and 8 elucidate the effects of the Brickman number Br and the amplitude ratio ε on the temperature θ respectivel. It is noted that Br and ε have the opposite effect on θ as observed in case of Hall parameter M. Figure 9 illustrates the effects of wall tension E and elasticit parameters E and E on the temperature profile. It is seen that θ decreases b increasing the values of E E and E. Figures 5 have been sketched to see Figure : Variation of ε on concentration. the behaviour of wall tension E elasticit parameters E and E third grade parameter Γ Brickman number Br Soret number Sc Hall parameter M and amplitude ratio ε on concentration φ. Figure represents the behaviour of E E and E on the concentration field. It is noticed that the concentration field increases b increasing the values of E E and E. In Figures it is observed that the concentration field decreases with an increase in Γ Br Sc and ε. Figure 5 shows the influence of M on the concentration field. It is noted that the concentration field.5 Sc = Sc =. Sc =..5 M =.9 M =. M = φ φ Figure : Variation of Sc on concentration. Figure 5: Variation of M on concentration. Brought to ou b Universit of California - Riverside Download Date /9/5 : PM

12 9 K. Vafai et al.: Peristaltic Motion of Third Grade Fluid 5.5 Br =. Br =. Br = ε =. ε =.5 ε =..5 Z Z X Figure : Variation of Br on heat transfer. increases b increasing the values of M. The effects of emerging parameters Brickman number Br third grade parameter Γ amplitude ratio ε wall tension E elasticit parameters E and Hall parameter M on heat transfer coefficient Z have been shown in Figures. It is found that the oscillator behaviour of heat transfer coefficient occurs due to the sinusoidal waves along the walls of the channels. From Figures 7 to 9 one can clearl see that the magnitude of heat transfer coefficient increases X Figure 8: Variation of ε on heat transfer coefficient. b increasing the values of Γ ε E E and E whereas the magnitude of Z decreases b increasing Br and M as shown in Figures and. 5 Conclusions In this article we derived and analsed a mathematical model subject to low Renolds number and long 5 Γ =. Γ =. Γ =. 5 8 E E E =.7.. E E E =... E E E =... Z Z X Figure 7: Variation of Γ on heat transfer X Figure 9: Variation of E E E on heat transfer. Brought to ou b Universit of California - Riverside Download Date /9/5 : PM

13 K. Vafai et al.: Peristaltic Motion of Third Grade Fluid 9 Z M = M =.5 M = X Figure : Variation of M on heat transfer. wavelength approimations in order to stud the peristaltic pumping of an incompressible electricall conducting third grade fluid in a channel under the influence of wall properties. Analsis was carried out in the presence of heat transfer and Hall current. Analtical solutions were obtained b regular perturbation method. The present theoretical model ma be considered as mathematical representation to the case of bile duct with stones gall bladder and dnamics of blood flow in living creatures. References [] T. K. Mitra and S. N. Prasad J. Bio. Mech. 8 (97). [] C. Davies and P. W. Carpenter J. Fluid Mech. 5 5 (997). [] P. Muthu B. V. R. Kumar and P. Chandra Appl. Math. Model. 9 (8). [] N. S. Akbar and S. Nadeem Z. Naturforsch. A 8a (). [5] M. Y. Malik A. Hussain and S. Nadeem Z. Naturforsch. A 7a 55 (). [] C. Fetecau M. Rana and C. Fetecau Z. Naturforsch. A 8a (). [7] Kh. S. Mekheimer and Y. Abd Elmaboud Phs. Lett. A (8). [8] O. U. Mehmood N. Mustapha S. Shafia and C. Fetecau Int. J. Appl. Mechanics 58 (). [9] F. M. Abbasi T. Haat and B. Ahmad G. Q. Chen Z. Naturforsch. 9a 5 (). [] H. A. Attia Canadian Phs. 8 7 (). [] S. Srinivas and M. Kothandapani Appl. Math Comp. 97 (9). [] S. Nadeem N. S. Akbar and J. Taiwan Int. Chem. Eng. 58 (). [] S. Nadeem N. S. Akbar N. Bibi and S. Ashiq Commun. Non-Linear Sci. Numer. Simul. 5 9 (). [] K. R. Rajagopal Thermodnamics and Stabilit of Non- Newtonian Fluids Ph.D. thesis Universit of Minnesota 978. [5] K. R. Rajagopal Int. J. Non-Linear Mech. 7 9 (98). [] C. Fetecau and C. Fetecau Int. J. Eng. Sci. 788 (). [7] K. R. Rajagopal and A. S. Gupta Meccanica 9 58 (98). [8] W. C. Tan and T. Masuoka Phs. Fluids 7 (5). [9] W. C. Tan and T. Masuoka Int. J. Non-Linear Mech. 5 (5). [] W. C. Tan and T. Masuoka Phs Lett. A 5 (7). [] R. L. Fosdick and K. R. Rajagopal Proc. R. Soc. Lond. A 9 5 (98). [] J. E. Dunn and K. R. Rajagopal Int. J. of Eng. Sci. 89 (995). Brought to ou b Universit of California - Riverside Download Date /9/5 : PM