Boundary Layer Flow and Heat Transfer due to an Exponentially Shrinking Sheet with Variable Magnetic Field

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1 International Journal of Scientific Research Engineering & Technology (IJSRET), ISSN Volume 4, Issue 6, June Boundary ayer Flow and Heat Transfer due to an Exponentially Shrinking Sheet with Variable Magnetic Field Rakesh Kumar Department of Mathematics, Central University of Himachal Pradesh, India ABSTRACT An analysis is carried out to study the effects of variable magnetic field on the flow field and heat transfer over an exponentially shrinking sheet. Using the exponential similarity transformations, the governing equations are transformed into self similar nonlinear ordinary differential equations. The transformed differential equations are solved using a power series expansion to obtain a closed form solution. The numerical results are depicted through graphs to illustrate the influence of pertinent parameters of the problem. It is found that suction parameter and magnetic field have substantial effect on velocity and temperature profiles. Keywords: Exponentially shrinking sheet, variable magnetic field, Heat transfer. I. INTRODUCTION The study of boundary layer flow and heat transfer over a stretching/shrinking surface has attracted the attention of several researchers due to its various applications in the fields of technology and industry. The extraction and manufacturing of polymer sheets, hot rolling and glass-fiber production, paper production, cooling of metallic sheets or electronic chips, processing of magnetic materials, MHD electrical power generation and purification of crude oil are some examples for the application of these flows (Fang et al.[]). In shrinking sheet problems, the surface is stretched towards a slot, causing a velocity away from the sheet. The physical grounds reveals that vorticity flow over the shrinking sheet does not remain confined and the flow is unlikely to exist (ok et al.[]). Mahapatra and Nandy [3] in their stability analysis found that an adequate suction or stagnation flow is required to confine the vorticity with in the boundary layer. Wang [4] during his investigation of liquid film behaviour on an unsteady stretching sheet observed this unusual flow due to shrinking surface. Miklavcic and Wang [5] obtained the existence and uniqueness conditions for the similarity solution of viscous fluid over shrinking surfaces and showed that the behaviour of fluid depends on the externally imposed mass suction. A new of Blasius solution over shrinking sheet was reported by Fang et al. [6]. Hayat et al. [7] analyzed the three dimensional rotating flow due to a shrinking sheet. Fang [8] also presented a solution for the boundary layer flow over a shrinking sheet with power law velocity. Fang and Zhang [9] obtained an analytical solution for the heat transfer over a shrinking sheet. VanGorder and Vajravelu [0] discussed about the multiple solutions for the MHD flow over a stretching or shrinking sheet. The dual and triple solutions for MHD slip flow of non-newtonian fluid over a shrinking surface were obtained by Turkyilmazoglu []. Further, the dual solutions for unsteady stagnation flow over a shrinking sheet were investigated by Bhattacharyya []. In the above research paper, the authors consider either linear or non-linear sheets, and there is lesser number of research papers available in literature for flows over exponentially shrinking surfaces. Moreover, the dynamics of fluid flows over shrinking surfaces is still unknown. Magyari and Keller [3] are assumed to be the first one to study the boundary flow over an exponentially stretching sheet. In this sequence, the other paper on the boundary layer flow and heat transfer over an exponentially shrinking sheet was reported by Bhattacharyya [4]. Bachok et al. [5] considers an exponentially stretching/shrinking sheet in nanofluids to study the stagnation point flow and corresponding heat transfer. The stagnation point flow and heat transfer over an exponentially shrinking sheet was investigated by Bhattacharyya and Vajravelu [6] and was extended by Rohni et al. [7] by considering the effects of free convection and mass suction. Bhattacharyya et al. [8] in his paper found that a strong magnetic field has predominant effect on the flow field and due to this; the similarity solution of the flow over a shrinking sheet is always unique. In some of the above said papers the uniform strength of magnetic field was assumed. But in general, its non-linear characteristics can be utilized to predict and control the behaviour of fluid flows over shrinking surfaces. Motivated by this, the objective of the present study is to analyze the effect of variable magnetic field on the flow and heat transfer of a viscous fluid over an exponentially shrinking sheet.

2 International Journal of Scientific Research Engineering & Technology (IJSRET), ISSN Volume 4, Issue 6, June II. MATHEMATICA FORMUATION A steady two-dimensional boundary layer flow of a viscous, incompressible, electrically conducting fluid is considered over an exponentially shrinking sheet. A magnetic field of the following exponential type B 0, By,0 where By B0 expx / and B 0 is a constant is applied normal to the plane of the sheet. The magnetic Reynolds number is assumed to be small to neglect the induced magnetic field. The governing boundary layer equations of the present problem are Equation of continuity: u u 0 () x y Equation of momentum: u u u By u v u () x y y Equation of energy: T T k T u v (3) x y C p y The boundary conditions are given by u U w x, v v w, T T w x T T0 exp x / at y 0 (4) u 0, T 0 as y The shrinking sheet velocity U w is given by U wx c exp x /, where c 0 is shrinking constant. Here,,, k, C p, T 0, T w and T are the electrical conductivity, kinematic viscosity, density, thermal conductivity, specific heat at constant pressure, characteristic length of the sheet, mean temperature, temperature of the sheet and ambient temperature of the fluid respectively. We introduce the following similarity variables c f expx /, T T Tw T, (5) where is the similarity variable defined by c y expx / (6) and is the stream function which is defined in the classical form as following expressions as u cexp x / f ' and v expx / f f ' u and y c (7) where prime denotes differentiation with respect to. This suggests that, we can assume c vwx expx / S, (8) where S 0 is the dimensionless suction parameter. Using equation (5) to (7) in equations () and (3), we obtain the following ordinary differential equations f ff f Mf 0 (9) Pr f f 0 (0) The boundary conditions transform to v.thus we have the x

3 International Journal of Scientific Research Engineering & Technology (IJSRET), ISSN Volume 4, Issue 6, June f f S, f, at 0 0, 0 as 0 The physical parameters of interest in the present problem, the skin friction coefficient C f and the Nusselt number Nu, are defined by u C f () U y w y0 T Nu (3) Tw T y y0 Substituting (5) to (7) into above two equations, we get the following expressions of skin friction and Nusselt number: C f Re expx / f '' 0 (4) / Nu ' 0 / Re exp x (5) B Here M 0 C p c (Hartmann number), Pr (Prandtl number) and Re (Reynolds number) are the c k dimensionless parameters introduced in the above equations. The differential equations (9) and (0) under the boundary conditions () are solved using the series expansion method as suggested by Singh and Dikshit [9]. et us define S, f S F and (6) The equations (9) to () becomes F FF F M F 0 (7) G Pr FG FG (8) F F 0 S, F, G at 0 0, G 0 as 0 where prime denotes the differentiation with respect to. For large suction, S assumes large positive values so that is small. Therefore, F and G can be expanded in terms of small perturbation quantity as F F 3 0 F F F3... (0) G G 3 0 G G G3... () Substituting (0) and () into (7),(8) and (9), we obtain the following sets of ordinary differential equations along with the corresponding boundary conditions : Zeroth Order O(): F 0 F0 F0 F 0 0 () G 0 PrF0G0 F0 G0 0 (3) F0 0, F0 0 0, F0 0 G00, G0 0 (4) First-OrderO : F F0 F F F0 M F0 0 (5) () (9)

4 International Journal of Scientific Research Engineering & Technology (IJSRET), ISSN Volume 4, Issue 6, June PrF G0 F0G F0 G F G , F 0, F 0 0 0, G 0 G (6) F G (7) Second-OrderO : F F F0 F0 F F F 4F0 F F M F 0 (8) G PrF0G F G FG0 F0 G F G F G0 0 (9) F 0 0, F 0 0, F 0 G0 0, G 0 (30) Third-OrderO 3 : F 3 F0 F3 F F F F F3 F0 4F F M F 0 (3) G 3 PrF0G3 F G FG F3G0 F0 G3 F G F G F3 G0 0 (3) F3 0 0, F3 0, F3 0 G30 0, G3 0 (33) The obtained solutions of the above equations under the corresponding boundary conditions are: F 0 (34) F exp (35) 5 8M 3 4M F exp exp 4 4 (36) M exp F exp 4 exp exp exp (37) 6 4 exp 7 exp G 0 exp Pr (38) Pr Pr G exp exp Pr Pr exp Pr (39) Pr G exp Pr 9 exp Pr 0 exp Pr exp (40) Pr The velocity and temperature profiles can be calculated from the following expressions f F F F3, (4) G 0 G G. (4) In order to obtain more accurate results for velocity and temperature profiles, we have evaluated the expression up to the third order. III. RESUTS AND DISCUSSION The numerical values of the similarity solutions obtained in the previous section are computed to demonstrate the effects of suction parameter and magnetic field on the velocity, temperature profiles, skin-friction coefficient and Nusselt number. Since the momentum equation is independent from the Prandtl number, therefore it has no effect on velocity profiles. The

5 International Journal of Scientific Research Engineering & Technology (IJSRET), ISSN Volume 4, Issue 6, June 05 6 influence of Prandtl number on the temperature profiles and heat flux are also targeted. The pertinent parameters of the problem are given arbitrary values. The Figures and present the influence of suction parameter on velocity profiles. It is found that the velocity is increased with the increasing suction parameter. Further, it is clear from these Figures that boundary layer thickness decreases with the increasing suction when for S. 0(app.), whereas it is increased when S. 0. Thus, the critical suction parameter determines the transition in boundary layer thickness. The effect of magnetic field on velocity profiles with certain combination of suction parameter has been depicted by the Figures 3, 4 and 5. The Figures 3 and 4 shows that the velocity profiles are enhanced with the increasing strength of the magnetic field. These Figures also reveal that the boundary layer thickness becomes thinner with the increasing magnetic field for S 4 and M 4. 8(app.), and becomes thicker for S 4 and M 4. 8(app.). However, the Figure 5 is illustrating the opposite behaviour as the velocity profiles are reduced with the increase of magnetic field strength when S. Hence, the stronger magnetic field along with certain range of suction parameter can be utilized in removing the uncertainty in the flow dynamics due to its resisting nature, that is, it makes the similarity solution unique. Figure : Velocity profiles for suction parameter with M Figure : Velocity profiles for suction parameter with M The Figures 6 and 7 are plotted to ensure the smallness of skin-friction coefficients with respect to the suction parameter. The Figure 6 demonstrates the decreasing nature of skin-friction when the strength of magnetic field is enhanced up to M.6(app.). Whereas the Figure 7 is depicting the reverse process, that is, the skin-friction increases for M.76 (app.). Also at certain distance from the sheet, the behaviour is again noted to be reversed. Thus, the critical magnetic parameter is also helpful in diminishing the skin-friction coefficient.

6 International Journal of Scientific Research Engineering & Technology (IJSRET), ISSN Volume 4, Issue 6, June 05 6 Figure 3: Velocity profiles for Hartmann number with S 4 Figure 4: Velocity profiles for Hartmann number with S 4 Figure 5: Velocity profiles for Hartmann number with S Figure 6: Skin friction coefficient

7 International Journal of Scientific Research Engineering & Technology (IJSRET), ISSN Volume 4, Issue 6, June Figure 7: Skin friction coefficient Figure 8: Temperature profiles with M and Pr 0. 7 The Figures 8 and 9 cover the effects of suction parameter and Prandtl number S on temperature profiles respectively. In both the Figures, the temperature profiles are found to be reduced with their increasing strength. Also the thermal boundary layer thickness is decreased with the increasing suction and Prandtl number. In general, the thermal boundary layer thickness becomes thinner with the increase in Prandtl number. This is due to the physical fact that the increasing Prandtl number decreases the thermal conductivity of the fluid, hence causes a reduction in the thermal boundary layer thickness. Figure 9: Temperature profiles with M and S 3

8 International Journal of Scientific Research Engineering & Technology (IJSRET), ISSN Volume 4, Issue 6, June Figure 0: Rate of heat transfer with Pr 0. 7 The rate of heat transfer with the effect of magnetic field and Prandtl number with respect to suction parameter is shown in Figures 0 and. The Figure 0 clearly shows that Nusselt number increases for magnetic of strength up to M , thereafter it decreases. The Figure shows that Nusselt number increases with the increase in Prandtl number. Figure : Rate of heat transfer with M IV. CONCUSION The effects of suction parameter, magnetic field, and Prandtl number on the boundary layer flow and heat transfer over shrinking sheet has been investigated. The closed form similarity solutions have obtained by perturbation technique. The conclusions of the study are: The velocity profiles are increased for all range of suction parameters, but the boundary layer thickness decreases when the suction is greater than the critical suction parameter, and it is reversed for suction less the critical value. The magnetic field is found to have both the increasing and decreasing effects on the velocity profiles with certain combination of suction parameter. The temperature profiles are reduced with the suction and Prandtl number. The rate of heat transfer is enhanced for magnetic fields of weaker strength, and reduced for strong magnetic field. However, the opposite phenomenon is observed for skin-friction coefficient. V. APPENDIX Pr PrPr Pr, 5 8M 3 4M, 3, 4, Pr Pr 4 5 M, 6 Pr Pr 4, 7 Pr Pr 4 Pr 9 Pr 3 Pr, 7 0, 4 3Pr, Pr Pr, 8 5

9 International Journal of Scientific Research Engineering & Technology (IJSRET), ISSN Volume 4, Issue 6, June Pr 6 Pr 3 Pr8,, Pr Pr Pr, M 5, 4 M 3 M, 5 M 4 M, 6 5, 6 7, , , , 5 7, 4 6. REFERENCES [] Fang, T.; Yao, S.; Pop, I., (0). Flow and Heat transfer over a generalized stretching/shrinking wall problemexact solution of the Navier-Stokes equations. Int. J. of Nonlinear Mech., 46, 6-7. [] ok, Y. Y.; Ishak, A.; Pop, I., (0). Stagnation point flow with suction towards a shrinking sheet. Sains Malays., 40, [3] Mahapatra, T. R.; Nandy, S. K., (03). Stability of dual solutions in stagnation point flow and heat transfer over a porous shrinking sheet with thermal radiation. Meccanica, 48, 3-3. [4] Wang, C. Y., (990). iquid film on an unsteady stretching sheet. Q. Appl. Math., 48, [5] Miklavcic, M.; Wang, C. Y.; (006). Viscous flow due to a shrinking sheet. Q. Appl. Math., 64, [6] Fang, T.; iang, W.; ee, C. F., (008). A new solution branch for the Blasius equation-a shrinking sheet problem. Computat, Math. Appl., 56, [7] Hayat, T.; Abbas, Z.; Javed, T.; Sajad, M., (009). Three dimensional rotating flow induced by a shrinking sheet for suction. Chaos Solitons fractals, 39, [8] Fang, T., (008). Boundary layer flow over a shrinking sheet with power law velocity. Int. J. Heat Mass Transfer, 5, [9] Fang, T.; Zhang, J., (00). Heat transfer over shrinking sheet-an analytical solution. Acta Mechanica, 09, [0] VanGorder, R. A.; Vajravelu, K., (0). Multiple solutions for hydromagnetic flow of a second grade fluid over a stretching or shrinking sheet. Quart. Appl. Math., 69, [] Turkyilmazoglu, M., (0). Dual and triple solution for MHD slip flow on non-newtonian fluid over a shrinking surface. Computers and Fluids, 70, [] Bhattacharyya, K., (0). Dual solutions in unsteady stagnation point flow over a shrinking sheet. Chin. Phy. ett., 8(8), [3] Magyari, E.; keller, B., (999). Heat and mass transfer in the boundary layers on an exponentially stretching continuous surface. J. Phys. D. Appl. Phys., 3, [4] Bhattacharyya, K., (0). Boundary layer flow and heat transfer over an exponentially shrinking sheet. Chin. Phy. ett., 8(7), [5] Bachok, N.; Ishak, A.; Pop, I., (0). Boundary layer stagnation point flow and heat transfer over an exponentially stretching/shrinking sheet in a nanofluid. Int. J. Heat Mass Trans., 55, 8-8. [6] Bhattacharyya, K.; Vajravelu, K., (0). Stagnation point flow and heat transfer over an exponentially sheet. Comm. Nonlin. Sci. Numer. Simul., 7, [7] Rohni, A. M.; Ahmed, S.; Pop, I., (04). Flow and heat transfer at a stagnation-point over an exponentially shrinking sheet with suction. Int. J. of Thermal Sci., 75, [8] Bhattacharyya, K.; Hayat, T.; Alsaedi, A., (03). Analytic solution for Magnetohydrodynamic boundary layer flow of casson fluid over a stretching/shrinking sheet with mass transfer. Chin. Phys. B., (), [9] Singh, A. K.; Dikshit, C. K., (988). Hydromagnetic flow past a continuously moving semi-infinite plate for large suction. Astrophysics and Space Science, 48,