Time Dependent Pressure Gradient Effect on Unsteady MHD Couette Flow and Heat Transfer of a Casson Fluid

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1 Engineering, 011, 3, doi:10.436/eng Published Online January 011 ( Tie ependent Pressure Gradient Effect on Unsteady MH Couette Flow and Heat Transfer of a Casson Fluid Abstract M. E. Sayed-Ahed, Haze A. Attia, Kare M. Ewis epartent of Engineering Matheatics and Physics, Faculty of Engineering, Fayou University, Fayou, Egypt E-ail: es00@fayou.edu.eg Received Noveber 1, 009; revised July 8, 010; accepted August 3, 010 The unsteady agnetohydrodynaic flow of an electrically conducting viscous incopressible non-newtonian Casson fluid bounded by two parallel non-conducting porous plates has been studied with heat transfer considering the Hall effect. The fluid is acted upon by a unifor and exponential decaying pressure gradient. An external unifor agnetic field is applied perpendicular to the plates and the fluid otion is subjected to a unifor suction and injection. The lower plate is stationary and the upper plate is suddenly set into otion and siultaneously suddenly isotherally heated to a teperature other than the lower plate teperature. Nuerical solutions are obtained for the governing oentu and energy equations taking the Joule and viscous dissipations into consideration. The effect of unsteady pressure gradient, the Hall ter, the paraeter describing the non-newtonian behavior on both the velocities and teperature distributions have been studied. Keywords: MH Flow, Heat Transfer, Non-Newtonian Fluids, Unsteady Pressure, Nuerical Solution 1. Introduction The study of Couette flow in a rectangular channel of an electrically conducting viscous fluid under the action of a transversely applied agnetic field has iediate applications in any devices such as agnetohydrodynaic (MH) power generators, MH pups, accelerators, aerodynaics heating, electrostatic precipitation, polyer technology, petroleu industry, purification of crude oil and fluid droplets sprays. Channel flows of a Newtonian fluid with heat transfer were studied with or without Hall currents by any authors [1-10]. These results are iportant for the design of the duct wall and the cooling arrangeents. The ost iportant non- Newtonian fluid possessing a yield value is the Casson fluid, which has significant applications in polyer processing industries and bioechanics. Casson fluid is a shear thinning liquid which has an infinite viscosity at a zero rate of shear. Casson s constitutive equation represents a nonlinear relationship between stress and rate of strain and has been found to be accurately applicable to silicon suspensions, suspensions of bentonite in water and lithographic varnishes used for printing inks [11-13]. Many authors [14-0] studied the flow or/and heat transfer of a non-newtonian fluids in different geoetries. The effect of tie dependent pressure gradient on unsteady dusty fluid was studied by Rukangadachari [1] in a rectangular duct and Gireesha et al. [] in a nholonoic coordinates syste. Attia [10] studied the influence of the Hall current on the velocity and teperature fields of an unsteady Hartann flow of a conducting Newtonian fluid between two infi- nite non-conducting horizontal parallel and porous plates. Attia and Sayed-Ahed [17] studied the influence of the Hall current on the velocity and teperature fields of an unsteady Couettee flow of a conducting Casson fluid between two infinite non-conducting horizontal parallel plates with constant pressure gradient. The extension of such proble to the case of Couettee flow of non-newtonian Casson fluid has been done in the present study. The upper plate is oving with a unifor velocity while the lower plate is stationary. The fluid is acted upon by an exponentially decaying pressure gradient, unifor suction and injection fro above and be- Copyright 011 SciRes.

2 M. E. SAYE-AHME ET AL. 39 low, respectively. The fluid is also, subjected to a unifor agnetic field perpendicular to the plates. The Hall current is taken into consideration while the induced agnetic field is neglected by assuing a very sall agnetic Reynolds nuber [5]. The two plates are kept at two different but constant teperatures. This configuration is a good approxiation of soe practical situations such as heat exchangers, flow eters, and pipes that connect syste coponents. The Joule and viscous dissipations are taken into consideration in the energy equation. The governing oentu and energy equations are solved nuerically using the finite difference approxiations. The inclusion of unsteady pressure gradient, the Hall current, the suction and injection, and the non-newtonian fluid characteristics leads to soe interesting effects on both the velocity and teperature fields.. Forulation of the Proble The geoetry of the proble is shown in Figure 1. The fluid is assued to be lainar, incopressible and obeying a Casson odel and flows between two infinite horizontal plates located at the y = h planes and extend fro x = - to and fro z = - to. The upper plate is suddenly set into otion and oves with a unifor velocity U o while the lower plate is stationary. The upper plate is siultaneously subjected to a step change in teperature fro T 1 to T. Then, the upper and lower plates are kept at two constant teperatures T and T 1 respectively, with T > T 1. The fluid is acted upon by an exponentially decaying pressure gradient p/ x in the x-direction, and a unifor suction fro above and injection fro below which are applied at t = 0. A unifor agnetic field B o is applied in the positive y-direction and is assued undisturbed as the induced agnetic field is neglected by assuing a very sall agnetic Reynolds nuber. The Hall effect is taken into consideration and consequently a z-coponent for the velocity is expected to arise. The unifor suction iplies that the y-coponent of the velocity v 0 is constant. Thus, the fluid velocity vector is given by, v uiv j wk The fluid otion starts fro rest at t = 0, and the no-slip condition at the plates in z-direction iplies that the fluid velocity has no z-coponent at y = h. The initial teperature of the fluid is assued to be equal to T 1. Since the plates are infinite in the x and z- directions, the physical quantities do not change in these directions. The flow of the fluid is governed by the oentu equation v vpj B o (1) t o Figure 1. Geoetry of the proble. where is the density of the fluid and is the apparent viscosity of the odel and is given by u w K c o y y where K c is the Casson s coefficient of viscosity and o is the yield stress. If the Hall ter is retained, the current density J is given by J vbo JBo (3) where is the electric conductivity of the fluid and is the Hall factor [5]. Equation (3) ay be solved in J to yield Bo JBo uwiwuk 1 (4) where is the Hall paraeter and = B o. Thus, the two coponents of the oentu Equation (1) read u u p u Bo vo uw t y x y y 1 (5) w w w Bo vo wu t y y y 1 (6) p dp at where e is the unsteady pressure gradient x dx The energy equation with viscous and Joule dissipations is given by T T T u w cp cpvo k t y y y y Bo u w 1 (7) 1 () Copyright 011 SciRes.

3 40 M. E. SAYE-AHME ET AL. where c p and k are, respectively, the specific heat capac- ity and the theral conductivity of the fluid. The second and third ters on the right-hand side represent the vis- cous and Joule dissipations respectively. We notice that each of these ters has two coponents. This is because the Hall effect brings about a velocity w in the z-direc- tion. The initial and boundary conditions of the proble are given by u w 0 at t 0, and w 0 at y =-h and y =h for t > 0, u = 0 at y =-h for t > 0, u = U o at y =h for t > 0, (8) T = T 1 at t 0, T = T at y =h and T = T 1 at y = -h for t > 0 (9) It is expedient to write the above equations in the non-diensional for. To do this, we introduce the following non-diensional quantities x y z tuo u w x, y, z, t, u, w, h h h h Uo Uo p T T1 p,,, U o T T1 Kc dp is the constant pressure gradient. dx h a a is the decaying paraeter in the unsteady u0 pressure gradient oh is the Casson nuber (diensionless yield KcUo stress) Uh o Re is the Reynolds nuber, Kc vh o S is the suction paraeter, Kc cuh p o Pr is the Prandtl nuber, k UK o c Ec is the Eckert nuber, cpht T1 Bh o Ha is the Hartann nuber squared, Kc In ters of the above non-diensional variables and paraeters Eqs.(5-9) and () are, respectively, written as (where the hats are dropped for convenience); u S u at 1 u Ha e uw t Re y Re y y 1 (10) w S w 1 w Ha wu t Re y Re y y 1 (11) S 1 u w Ec t Re y Pr y y y Ha Ec u w 1 u w0 for t 0 and u w0 at y =-1, w 0, u 1 at y =1 for t 0 (1) (13) =0 for t 0 and =0 at y 1, =1 at y = 1 for t 0 (14) u w 1 / y y where is the constant pressure gradient a is the decaying paraeter. 3. Nuerical Solution 1/ (15) dp dx and Equations (10,11,15) represent coupled syste of nonlinear partial differential equations which are solved nuerically under the initial and boundary conditions (13) using the finite difference approxiations. A linearization technique is first applied to replace the nonlinear ters at a linear stage, with the corrections incorporated in subsequent iterative steps until convergence is reached. Then the Crank-Nicolson iplicit ethod is used at two successive tie levels [3]. An iterative schee is used to solve the linearized syste of difference equations. The solution at a certain tie step is chosen as an initial guess for next tie step and the iterations are continued till convergence, within a prescribed accuracy. Finally, the resulting block tridiagonal syste is solved using the generalized Thoas-algorith [3]. The energy Equation (1) is a linear non-hoogeneous second-order partial differential equation whose right- hand side is known fro the solutions of the flow Equations (10,11,15) subject to the conditions (13). The values of the velocity coponents are substituted in the right-hand side of Equation (1) which is solved nuerically with the initial and boundary conditions (14) using central differences and Thoas-algorith to obtain the teperature distribution. Finite difference equations relating the variables are obtained by writing the equations at the id-point of the coputational cell and then replacing the different ters by their second order central difference approxiations in the y-direction. The diffusion ters are replaced by the average of the central differences at two successive tie-levels. The coputational doain is Copyright 011 SciRes.

4 M. E. SAYE-AHME ET AL. 41 divided into eshes of diension t and y in tie and space respectively as shown in Figure. We define the variables v=u y,b=w y, H y and = y to reduce the second order differential Equations (10,11,1) to first order differential equations. The finite difference representations for the resulting first order differential Equations (10,11) take the for (see Equations (16,17)). where i 1, j1 i 1, ji, j1 i, j i 1/, j1/ and 4 i1, j1 i1, j i, j1 i, j i1/, j1/ 4 The variables with bars are given initial guesses fro the previous tie step and an iterative schee is used at every tie to solve the linearized syste of difference equations. Then the finite difference for for the energy Equation (1) can be written as i1, j1 i, j1 i 1, j i, j S Hi1, j1 Hi, j1 Hi 1, j Hi, j t Re 4 1 ( Hi 1, j1 Hi, j1) ( Hi 1, jhi, j) ISP Pr y (18) where ISP represents the Joule and viscous dissipation ters which are known fro the solution of the oentu equations and can be evaluated at the id point (i+1/,j+1/) of the coputational cell. Coputations have been ade for α = 5, Pr = 1, Re = 1, Ha = 3 and Ec = 0.. Grid-independence studies show that the coputational doain 0<t< and 1< y <1 can be divided into intervals with step sizes t = and y = for tie and space respectively. The truncation error of the central difference schees of the governing equations is O t, y. Stability and rate of convergence are func- Figure. Mesh layout. tions of the flow and heat paraeters. Saller step sizes do not show any significant change in the results. Convergence of the schee is assued when all of the unknowns u, v, w, B, θ and H for the last two approxiations differ fro unity by less than 10-6 for all values of y in 1 < y < 1 at every tie step. Less than 7 approxiations are required to satisfy this convergence criteria for all ranges of the paraeters studied here. 4. Results and iscussions Figures 3-5 show the variation of the velocity coponents u and w and the teperature θ at the central ui1, j1 ui, j1 ui 1, jui, j S vi1, j1 vi, j1 vi 1, jvi, j t Re 4 i1, j1 i, j1 i1, j i, j v v v v i 1/, j 1/ i 1/, j 1/ vi 1, j 1 vi, j 1 v at i 1, j v i, j e Re y Re 4 Ha ui1, j1 ui, j1 ui 1, j ui, j wi 1, j1 wi, j1 wi 1, j wi, j, 1 4Re 4Re wi1, j1 wi, j1 wi 1, jwi, j S Bi1, j1 Bi, j1 Bi 1, jbi, j t Re 4 i1, j1 i, j1 i1, j i, j B B B B i1/, j1/ i1/, j1/ Bi 1, j1 Bi, j1 Bi 1, j Bi, j Re y Re 4 Ha ui 1, j1 ui, j1 ui 1, jui, j wi 1, j1 wi, j1 wi 1, jwi, j, 1 4Re 4Re (16) (17) Copyright 011 SciRes.

5 4 M. E. SAYE-AHME ET AL. (a) 0 (a) 0 (b) 0.05 (b) 0.05 (c) 0.1 Figure 3. Effect of decaying paraeter a on u at y = 0 for various values of τ ( = 3, S = 1). plane of the channel (y = 0) with tie. These figures show the results for various values of the decaying paraeter a = 0, 1 and and for yield stress = 0.0, 0.05 and 0.1. In these figures S = 1 and = 3. Figure 3 shows that u decreases with increasing a for all values of. It is observed also that the tie at which u reaches its steady state value decreases with increasing a for a > 0 but that occurs earlier for constant pressure gradient (a = 0). Increasing increases u for all values of a but with (c) 0.1 Figure 4. Effect of decaying paraeter a on w at y = 0 for various values of τ ( = 3, S = 1). sall differences. In Figure 4, the velocity coponent w decreases with increasing a. This figure indicates that the influence of on w depends on t and becoes ore clear when the decaying paraeter a = 0 but this influence is sall for large a. It is observed that increasing ore decreases w for a = 0. Figure 5 shows that the influence of a on θ depends on t. It is observed that increasing a decreases θ Copyright 011 SciRes.

6 M. E. SAYE-AHME ET AL. 43 (a) 0 (a) t = 0. (b) (b) t = 1 (c) 0. 1 Figure 5. Effect of decaying paraeter a on the tie developent of q at y = 0 for various values of τ ( = 3, S = 1). while it is not greatly affected by changing. The figure shows also that the tie at which θ reaches its steady state value decreases with increasing a while it is not greatly affected by changing. Figures 6-8 show the profiles of the velocity coponents u and w and the teperature θ, respectively, for various values of tie a and for t = 0., 1, and. The figures are evaluated for =3, = 0.05 and S = 1. It is clear fro Figures 6 and 7 that the effect of decaying (c) t = Figure 6. Effect of decaying paraeter a on the tie developent of the velocity u for various values of t ( = 3, S = 1, τ = 0.05 ). paraeter a on u and w depends on t and y. Figure 6 shows that, for sall t, increasing a decreases u (with sall differences) for all values of y and a. It is also observed that increasing a (a > 0) decreases u for all values of y with significant differences at ediu and large t. It is also observed that the constant pressure gradient (a = 0) is greatly different fro unsteady pressure gradients (a > 0). This can be attributed to the fact that increasing a will decrease the pressure gradient which ainly generates Copyright 011 SciRes.

7 44 M. E. SAYE-AHME ET AL. (a) t = 0. (a) t = 0. (b) t = 1 (b) t = 1 (c) t = (c) t = Figure 7. Effect of decaying paraeter a on the distribution of w with y for various values of t ( = 3, S = 1, τ = 0.05 ). the velocity u. Figure 7 shows that, for sall t, increasing a increases w (with no significant differences ) for all y. For large t, increasing a decreases w (with no signifycant difference between a=1 and a=) for all y. The figures show also that the velocity coponents u and w do not reach their steady state onotonically. The velocities u and w increase with tie up till a axiu value and then decrease up to the steady state. Figure 8 shows that the teperature profile does not reach its steady state Figure 8. Effect of decaying paraeter a on the distribution of w q with y for various values of t ( = 3, S = 1, τ = 0.05 ). onotonically. Increasing a decreases θ for all y and t. It is observed also that the velocity coponent u reaches the steady state faster than w which, in turn, reaches the steady state faster than θ. This is expected as u is the source of w, while both u and w act as sources for the teperature. Figures 9-11 show the variation of the velocity coponents u and w and the teperature θ at the central Copyright 011 SciRes.

8 M. E. SAYE-AHME ET AL. 45 (a) 0 (a) 0 (b) (b) (c) 0. 1 (c) 0. 1 Figure 9. Effect of the Hall current on u at y = 0 for various values τ (S = 0, a=1). plane of the channel (y = 0) with tie for various values of the Hall paraeter and for = 0.0, 0.05, and 0.1. In these figures S = 0. Figure 9 shows that u increases with increasing for all values of as the effective conductivity ( /(1+ )) decreases with increasing which reduces the agnetic daping force on u. It is observed also fro the figure that the tie at which u reaches its steady state value increases with increasing. Increasing Figure 10. Effect of the Hall current on w at y = 0 for various values τ (S = 0, a = 1). increases u for all and its effect on u becoes ore pronounced for higher values of. In Figure 10 the velocity coponent w increases with increasing as w is a result of the Hall effect. On the other hand, at sall ties, w decreases when increases. This happens due to the fact that, at sall ties w is very sall and then the source ter of w is proportional to ( u/(1+ ) ) which decreases with increasing ( > 1 ). This accounts for Copyright 011 SciRes.

9 46 M. E. SAYE-AHME ET AL. (a) 0 (a) 0 (b) (b) (c) 0. 1 (c) 0. 1 Figure 11. Effect of on t at y = 0 for various values τ (S = 0, a = 1). the crossing of the curves of w with t for all values of. Figure 10 indicates that the influence of on w depends on t and becoes ore clear when is large. It is observed that, increasing decreases w and increasing increases w. Figure 11 shows that the influence of on θ depends on t. Increasing decreases θ at sall ties, Figure 1. The variation of u profiles with t for various values of τ ( = 3, S = 1, a = 1). but this is reversed at large ties. This is due to the fact that, for sall ties, u and w are sall and an increase in increases u but decreases w. Then, the Joule dissipation which is also proportional to (1/1+ ) decreases. For large ties, increasing increases both u and w and, in turn, increases the Joule and viscous dissipations. This accounts for the crossing of the curves of θ with tie for Copyright 011 SciRes.

10 M. E. SAYE-AHME ET AL. 47 (a) 0 (a) 0 (b) (b) (c) 0. 1 Figure 13. The variation of w profiles with t for various values of τ ( = 3, S = 1, a = 1). all values of. It is also observed that increasing decreases the teperature θ for all values of. This is because increasing decreases both u and w and their gradients which decreases the Joule and viscous dissipations. The figure shows also that the tie at which θ reaches its steady state value increases with increasing while it is not greatly affected by changing. Figures (1-14) present the profiles of the velocity coponents u and w and the teperature θ for variousvalues of tie t and for = 0.0, 0.05 and 0.1. The fig- (c) 0. 1 Figure 14. The variation of q profiles with t for various values of τ ( = 3, S = 1, a = 1). ures are evaluated for = 3 and S = 1. It is clear fro Figures 1 and 13 that the effect of yield stress on u and w depends on t and y. Figure 1 shows that, for sall t, increasing the yield stress decreases u for sall y, but this is reversed for large y. As tie develops, increasing increases u for all y. Figure 13 shows that increasing increases w for all values of y, but increasing ore decreases w for large y. Copyright 011 SciRes.

11 48 M. E. SAYE-AHME ET AL. For large t, increasing decreases w for sall t and all values of y. This can be attributed to the fact that increasing will delay the attainent of axia of u and w. It is also observed, fro Figures 1 and 13, that the velocity coponents u and w do not reach their steady state onotonically. Figure 14 shows that the teperature profile does not reach its steady state onotonically. Increasing increases θ for all y and t as a result of increasing the dissipations. It is observed also that the velocity coponent u reaches the steady state faster than w which, in turn, reaches the steady state faster than θ. This is expected as u is the source of w, while both u and w act as sources for the teperature. 5. Conclusions A finite difference ethod is used to solve the transient Couette flow and heat transfer of a Casson non-newtonian fluid under the influence of unsteady pressure gradient and unifor agnetic field. In the present work, we study Hall effect. The effects of the decaying paraeter a, Casson yield stress, and the Hall paraeter on the velocity and teperature distributions are studied. The decaying paraeter a affects the ain velocity coponents u and w and the teperature θ. The Hall ter affects the ain velocity coponent u in the x-direction and gives rise to another velocity coponent w in the z-direction. The results show that the influence of the paraeters a and on u and w depend on tie and the Hall paraeter. It is also found that the effect of on w and θ depends on tie for all values of which accounts for a crossover in the w-t and θ-t graphs for various values of. The effect of on the agnitude of θ depends on and becoes ore pronounced in case of sall. It is also found that the effect of a on the agnitude of θ depends on and becoes ore pronounced in case of sall. 6. References [1] I. N. Tao, Magnetohydrodynaic Effects on the Foration of Couette Flow, Journal of Aerospace Science, Vol. 7, 1960, pp [] S.. Niga and S. N. Singh, Heat Transfer by Lainar Flow Between Parallel Plates under the Action of Transverse Magnetic Field, The Quarterly Journal of Mechanics and Applied Matheatics, Vol. 13, 1960, pp doi: /qja/ [3] R. A. Alpher, Heat Transfer in Magnetohydrodynaic Flow between Parallel Plates, International Journal of Heat and Mass Transfer, Vol. 3, No., 1961, pp doi: / (61) [4] I. Tani, Steady Motion of Conducting Fluids in Channels under Transverse Magnetic Fields with Consideration of Hall Effect, Journal of Aerospace Science, Vol. 9, 196, pp [5] G. W. Sutton and A.Sheran, Engineering Magnetohydrodynaics, McGraw-Hill, New York, [6] V. M. Soundalgekar, N. V. Vighnesa and H. S. Takhar, Hall and Ion-Slip Effects in MH Couette Flow with Heat Transfer, IEEE Transactions on Plasa Science, Vol. PS-7, No. 3, 1979, pp doi: /tps [7] V. M. Soundalgekar and A. G. Uplekar, Hall Effects in MH Couette Flow with Heat Transfer, IEEE Transactions on Plasa Science, Vol. PS-14, No. 5, 1986, pp doi: /tps [8] E. M. H. Abo-El-ahab, Effect of Hall Currents on Soe Magnetohydrodynaic Flow Probles, Master Thesis, Helwan University, Ain Helwan, [9] H. A. Attia and N. A. Kotb, MH Flow between Two Parallel Plates with Heat Transfer, Acta Mechanica, Vol. 117, No. 1-4, 1996, pp doi: /bf [10] H. A. Attia, Hall Current Effects on the Velocity and Teperature Fields of an Unsteady Hartann Flow, Canadian Journal of Physics, Vol. 76, No. 9, 1998, pp doi: /cjp [11] N. Casson, A Flow Equation for Pigent Oil-Suspensions of the Printing Ink Type, In: C. C. Mill, Ed., Rheolgy of isperse Systes, Pergaon Press, London, 1959, p. 84. [1] B. Taaashi, Consideration of Certain Heorheological Phenoena fro the Stand-Point of Surface Cheistry, In: A. L. Copley, Ed., Heorheology, Pergaon Press, London, 1968, p. 89. [13] W. P. Walawander, T. Y. Chen and. F. Cala, An Approxiate Casson Fluid Model for Tube Flow of Blood, Biorheology, Vol. 1, No., 1975, pp [14] R. L. Batra and B. Jena, Flow of a Casson Fluid in a Slightly Curved Tube, International Journal of Engineering Science, Vol. 9, No. 10, 1991, pp doi: /000-75(91)9008- [15] B. as and R. L. Batra, Secondary Flow of a Casson Fluid in a Slightly Curved Tube, International Journal of Non-Linear Mechanics, Vol. 8, No. 5, 1993, pp doi: / (93)90048-p [16] M. E. Sayed-Ahed and H. A. Attia, Magnetohydrodynaic Flow and Heat Transfer of a Non-Newtonian Fluid in an Eccentric Annulus, Canadian Journal of Physics, Vol. 76, No.5, 1998, pp doi: /cjp [17] H. A. Attia and M. E. Sayed-Ahed, Hydrodynaic Ipulsively Lid-riven Flow and Heat Transfer of a Casson Fluid, Takang Journal of Science and Engineering, Vol. 9, No. 3, 006, pp [18] I. A. Abdalla, H. A. Attia and M. E. Sayed-Ahed, Adoian s Polynoial Solution of Unsteady Non- Newtonian MH Flow, Applied Matheatics and Inforation Science, Vol. 1, No. 3, 007, pp [19] H. A. Attia, Unsteady MH Couette Flow and Heat Copyright 011 SciRes.

12 M. E. SAYE-AHME ET AL. 49 Transfer of usty Fluid with Variable Physical Properties, Applied Matheatics and Coputation, Vol. 177, No. 1, 006, doi: /j.ac [0] A. Mehood and A. Ali, The Effect of Slip Condition on Unsteady MH Flow Oscillatory Flow of a Viscous Fluid in a Planner Channel, Roanian Journal of Physics, Vol. 5, No. 1-, 007, pp [1] E. Rukangadachari, usty Viscous Flow through a Cylinder of Rectangular Cross-Section under Tie ependent Pressure Gradient, efense Science Journal, Vol. 31, No., 1981, pp [] B. J. Gireesha, C. S. Bagewadi and B. C. Prasanna Kaara, Flow of Unsteady usty Fluid under Varying Pulastile Pressure Gradient in Anholonoic Coordinate Syste, Electronic Journal of Theoretical Physics, Vol. 4, No. 14, 007, pp [3] M. Antia, Nuerical Methods for Scientists and Engineers, Tata McGraw-Hill, New elhi, Copyright 011 SciRes.