Comments on Magnetohydrodynamic Unsteady Flow of A Non- Newtonian Fluid Through A Porous Medium

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1 Comments on Magnetohydrodynamic Unsteady Flow o A Non- Newtonian Fluid Through A Porous Medium Mostaa A.A.Mahmoud Department o Mathematics, Faculty o Science, Benha University (358), Egypt Abstract The governing equations describing the unsteady boundary layer low o a power-law non-newtonian conducting luid through a porous medium past an ininite porous lat plate are transormed to a third order non linear ordinary dierential equation. An additional boundary condition is written as (0) = 0 by (Gamal M. Abdel- Rahman [] (IASME transa-ctions (3), July 2004)). This boundary condition has not any physical meaning and is not matching to the mathematical analysis described. In this article, the second order non-linear ordinary dierential equation with the appropriate boundary condition is solved analytically using the method o successive approximations and numerically using the shooting method.. Introduction Gamal Abdel- Rahman [] has investigated the ormation o magnetohydrodynamic, unsteady low o an incompressible, non-newtonian powerlaw electrically conducting luid past an ininite porous plate in a porous medium. By assuming that the magnetic Reynolds number is small and applying a similarity solutions, Gamal has obtained a third order non- linear ordinary dierential equation or (η) (see equation (0) in []). The transormed boundary conditions or the problem are (0) = 0 and ( ) = He assumed an additional boundary condition (0) = 0 (equation () in []), which has not any physical meaning and is not matching to the mathematical analysis o the problem. Also the special cases which were mentioned by Gamal [2-3] cannot be obtained rom his analysis. For example case (): or Newtonian luid and non-porous medium, one obtains the equations o Takara, H.S., Nath, G. [2]. Takhar and Nath [2] studied the unsteady laminar incompressible boundary layer low o an electrically conducting luid in the stagnation region o two dimensional and axisymmetric bodies with an applied magnetic ield while Gamal has studied the boundary layer over a plate. Case (2): In the absence o the Newtonian luid, we obtained the equations o Helmy [2]. Helmy discussed the unsteady 2-dimensional laminar ree convection low o an incompressible, viscous, electrically conducting (Newtonian or polar) luid through a porous medium bounded by an ininite vertical plane surace o constant temperature, while Gamal used a stationary plate. The aim o this comment is to correct the mathematical ormulation and present an analytical and numerical solution or this problem. 2. Mathematical Formulation Consider unsteady hydromagnetic low o an incompressible, non-newtonian power- law electrically conducting luid past an ininite porous plate in a porous medium. In cartesian coordinate system, let x axis be alon the plate in the direction o the low and y axis normal to it. A magnetic ield is introduced normal to the direction o the low. We assume that the magnetic Reynolds number is much less than unity so that the induced magnetic ield is neglected compared to the applied magnetic ield. Further, all the luid properties are assumed constant. Under the above assumptions with the usual Boussinesq s approximation into account, the governing equations or continuity and momentum are: v y = 0 ()

2 u t + v u y = U y + k ρ y [ u y n u y ] + σm 2 H 2 (U u) + K (U u) (2) ρ ρε where u and v are the components o the velocity in x and y direction, respectively, t is the time, ρ is the density o the luid, ε is the permeability constant, k is the viscosity, µ is the magnetic permeability, σ is the electrical conductivity, H is the magnetic ield strength and U is the outer low velocity. Using the ollowing notation v = k/ρ (3) σµ 2 H 2 = N (4) ρ Then equation (2) can be written as u t + v u y = U y + ν y [ u u n ] + (N + v/ε)(u u) (5) y y Integrating equation () we have v = v 0 (t) The initial and boundary conditions are u = 0, v = v 0 (t) at y = 0, t > 0 u U as y, t > 0 (6) where v 0 (t) is the velocity o injection at the ininite plate. Assume that U = u exp[αt] (7) v 0 = [(N +α)/(νu n )] n + ) exp[α(n t] (8) n + where α and u are constants. We urther deine the ollowing similarity variable η = y[(n + α)νu n ] n+ exp[ α( n) n+ t] u = U(η) (9) where is the non dimensional velocity. From equations (7)- (9) substituting in equation (5) and simpliication leads to the ollowing nonlinear ordinary dierential equation. n n + [ + S/( + M)]( ) /( + M)][/α + η( + n)/( n)] = 0 (0) Where M = N/α is the magnetic number, S = v/αε is the parameter o permeability and the prime denotes dierentiation with respect to η the transormed boundary conditions are η = 0 : = 0 η : 3. Analytical Solution () The successive approximations method is used to obtained the solution to (0), the dierent orders are obtained rom the equation. i+ = ( n) i [{ αn(m+) + S n(m+) }( i)], i = 0,, 2,... ( n)η n(+n)(m+) } i { n + (2) Assume that the zero-approximation solution may be written as ollows 0 = α 0 ( exp( βη)) (3) where α 0 and β are two arbitrary constants chosen such that the boundary conditions are satisied in the zero- approximation 0 ( ) = and in the irst approximation (0) = 0 i.e. α 0 β = and β is given as: β n+ n(n + )(M + )(2 n) 3 + β [2( n) (2 n)( + n)(m + ) (2 n)( + n) + α S( + n)(2 n)] = 0 (4) Integrating (2), using the act that u y 0 as y and the boundary conditions () we have, i (η) = + β (2+n) n(n + )(2 n) 3 (M + ) {β2 (2 n)( + n) + (2 n)( + n)η] + β[2( n) α (2 n)(+n)(m +) [S(+n)(2 n)]}] exp[ β(2 n)η] (5)

3 4.Numerical Solution Equation (0) with the boundary conditions () were solved numerically, using the orth order Runge- Kutta method. The missing value o (0) was determined by a shooting technique. 5. Discussion Equation (0), with the boundary conditions(), has been solved numerically using the shooting method. The eect o the magnetic parameter on the velocity distribution are shown in igure.. From this igure it is clear that the velocity increases with the increasing o the magnetic parameter M, which is contradicting the behavior achieved in the discussion o Gamal []. In igure 2 we compare our solution with the numerical solution using the shooting method. From this igure one inds that the velocity distribution obtained analytically are in good agreement with that obtained numerically. The numerical investigation to the analytical solution are shown in igures 3-5. The velocity decreases as the magnetic parameterm increase as shown in igure 3. From igure 4 one sees that the velocity distribution increases with the increasing o the parameter o permeability S. Figure 5 shows that the velocity distribution decreases as the power law index n increases. From table it is clear that both analytical and numerical values o n (0) are in good agreement. Table 2 illustrates that the skin- riction coeicient increases with the increasing o the magnetic parameterm and the parameter o permeability S. The skin- riction coeicient decreases as the power law index n increases. 6. Conclusion The problem o unsteady magnetohydrodynamic boundary layer low or a power-law non-newtonian conducting luid through a porous medium past an ininite porous lat plate is investigated. A similarity transormation is used to convert the governing partial dierential equation to ordinary dierential equations.the successive approximations method is used to solve the resulting non -linear ordinary dierential equation and the results are compared with the numerical solution. It is ound that the velocity distributions decrease as either the magnetic parameter is increased or the power law index increases. Also, the luid velocity increases with the increasing o the permeability parameter. In addition, it is concluded that the skin-riction coeicient increases as either the magnetic parameter or the permeability increases, while it is decreased as the power law index is increased. S \ n An Nu An Nu Table. Comparison o analytical (An) and numerical (Nu) skin -riction coeicient or α = 0.3 and M = 3 M S n ( (0)) n Table 2. The skin- riction coeicient or dierent values o M, S and n or α = 0.3 Reerences [] Gamal M. Abdel-Rahma, Magnetohydrodynamic unsteady low o a non- Newtonian luid through a porous medium, IASME TRANS- ACTIONS, Vol., Issue 3, Pp. 545, July [2] Takhar, H.S. and Nath, G., Similarity solution o unsteady boundary layer equations with magnetic ield, Meccanica, Inter. J. Italian Assoc. Theor. Appl. Mech., 32, No.2, 57, 997. [3] Helmy, K.A., MHD unsteady ree convection low past a vertical porous plate, ZAMM, Vol. 78, Issue 4, PP. 255, 998.

4 M=8 M=3 M= η Figure. Velocity distribution or various values o M at n =.2, S = 2 and α = 0.3 Analytical Numerical η 8 Figure 2. Velocity distribution or analytical and numerical proiles at n = 0.9, S = 2, M = 3 and α = 0.3

5 M=8 M= M= η 7 Figure 3. Velocity distribution or various values o M at n = 0.75, S = 2 and α = 0.3 S= S=3 S= η 6 Figure 4. Velocity distribution or various values o S at n = 0.75, M = 3 and α = 0.5

6 n=0.5 n=0.7 n=0.9 n= η 6 Figure 5. Velocity distribution or various values o n at α=0.3, M = 3 and S = 2