Applied Mathematics and Mechanics

, Volume 33, Issue 12, pp 1555–1568

# DTM-BF method and dual solutions for unsteady MHD flow over permeable shrinking sheet with velocity slip

• Xiao-hong Su (苏晓红)
• Lian-cun Zheng (郑连存)
• Xin-xin Zhang (张欣欣)
Article

## Abstract

An unsteady magnetohydrodynamic (MHD) boundary layer flow over a shrinking permeable sheet embedded in a moving viscous electrically conducting fluid is investigated both analytically and numerically. The velocity slip at the solid surface is taken into account in the boundary conditions. A novel analytical method named DTMBF is proposed and used to get the approximate analytical solutions to the nonlinear governing equation along with the boundary conditions at infinity. All analytical results are compared with those obtained by a numerical method. The comparison shows good agreement, which validates the accuracy of the DTM-BF method. Moreover, the existence ranges of the dual solutions and the unique solution for various parameters are obtained. The effects of the velocity slip parameter, the unsteadiness parameter, the magnetic parameter, the suction/injection parameter, and the velocity ratio parameter on the skin friction, the unique velocity, and the dual velocity profiles are explored, respectively.

## Key words

unsteady magnetohydrodynamic (MHD) flow shrinking sheet analytical solution slip condition dual solutions

O357 O175

## 2010 Mathematics Subject Classification

34B40 34B15 76D10 76W05

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## References

1. [1]
Crane, L. J. Flow past a stretching plate. Zeitschrift für Angewandte Mathematik und Physik, 21(4), 645–647 (1970)
2. [2]
Grubka, L. J. and Bobba, K. M. Heat transfer characteristics of a continuous stretching surface with variable temperature. ASME Journal of Heat Transfer, 107(1), 248–250 (1985)
3. [3]
Elbashbeshy, E. M. A. Heat transfer over a stretching surface with variable surface heat flux. Journal of Physics D: Applied Physics, 31(16), 1951–1955 (1998)
4. [4]
Elbashbeshy, E. M. A. and Bazid, M. A. A. Heat transfer over a continuously moving plate embedded in a non-Darcian porous medium. International Journal of Heat and Mass Transfer, 43(17), 3087–3092 (2000)
5. [5]
Hayat, T. and Sajid, M. Analytical solution for axisymmetric flow and heat transfer of a second grade fluid past a stretching sheet. International Journal of Heat and Mass Transfer, 50(1–2), 75–84 (2007)
6. [6]
Sarma, M. S. Heat transfer in a viscoelastic fluid over a stretching sheet. Journal of Mathematical Analysis and Applications, 222(1), 268–275 (1998)
7. [7]
Abel, M. S., Datti, P. S., and Mahesha, N. Flow and heat transfer in a power-law fluid over a stretching sheet with variable thermal conductivity and nonuniform heat source. International Journal of Heat and Mass Transfer, 52(11–12), 2902–2913 (2009)
8. [8]
Dandapat, B. S., Singh, S. N., and Singh, R. P. Heat transfer due to permeable stretching wall in presence of transverse magnetic field. Archives of Mechanics, 56(2), 87–101 (2004)
9. [9]
Abel, M. S. and Nandeppanavar, M. M. Heat transfer in MHD viscoelastic boundary layer flow over a stretching sheet with non-uniform heat source/heat sink. Communications in Nonlinear Science and Numerical Simulation, 14(5), 2120–2131 (2009)
10. [10]
Mahapatra, T. R., Nandy, S. K., and Gupta, A. S. Magnetohydrodynamic stagnation-point flow of a power-law fluid towards a stretching surface. International Journal of Non-Linear Mechanics, 44(2), 124–129 (2009)
11. [11]
Prasad, K. V., Vajravelu, K., and Datti, P. S. The effects of variable fluid properties on the hydro-magnetic flow and heat transfer over a non-linearly stretching sheet. International Journal of Thermal Sciences, 49(3), 609–610 (2010)
12. [12]
Goldstein, S. On backward boundary layers and flow in converging passages. Journal of Fluid Mechanics, 21(1), 33–45 (1965)
13. [13]
Miklavcic, M. and Wang, C. Y. Viscous flow due to a shrinking sheet. Quarterly of Applied Mathematics, 64(2), 283–290 (2006)
14. [14]
Sajid, M. and Hayat, T. The application of homotopy analysis method for MHD viscous flow due to a shrinking sheet. Chaos, Solitons and Fractals, 39(3), 1317–1323 (2009)
15. [15]
Hayat, T., Abbas, Z., and Sajid, M. On the analytical solution of MHD flow of a second grade fluid over a shrinking sheet. ASME Journal of Applied Mechanics, 74(6), 1165–1171 (2007)
16. [16]
Hayat, T., Javed, T., and Sajid, M. Analytical solution for MHD rotating flow of a second grade fluid over a shrinking surface. Physics Letters A, 372(18), 3264–3273 (2008)
17. [17]
Fang, T. and Zhang, J. Closed-form exact solutions of MHD viscous flow over a shrinking sheet. Communications in Nonlinear Science and Numerical Simulation, 14(7), 2853–2857 (2009)
18. [18]
Fang, T., Liang, W., and Lee, C. F. A new solution branch for the Blasius equation-a shrinking sheet problem. Computers and Mathematics with Applications, 56(12), 3088–3095 (2008)
19. [19]
Noor, N. F. M., Kechil, S. A., and Hashim, I. Simple non-perturbative solution for MHD viscous flow due to a shrinking sheet. Communications in Nonlinear Science and Numerical Simulation, 15(2), 144–148 (2010)
20. [20]
Gad-el-Hak, M. The fluid mechanics of microdevices-the Freeman scholar lecture. ASME Journal of Fluids Engineering, 121(5), 5–33 (1999)
21. [21]
Pande, G. C. and Goudas, C. L. Hydromagnetic Reyleigh problem for a porous wall in slip flow regime. Astrophysics and Space Science, 243(2), 285–289 (1996)
22. [22]
Yoshimura, A. and Prudhomme, R. K. Wall slip corrections for Couette and parallel disc viscometers. Journal of Rheology, 32(1), 53–67 (1988)
23. [23]
Andersson, H. I. Slip flow past a stretching surface. Acta Mechanica, 158(1–2), 121–125 (2002)
24. [24]
Zhu, J., Zheng, L. C., and Zhang, Z. G. Effects of slip condition on MHD stagnation-point flow over a power-law stretching sheet. ASME Journal of Fluids Engineering (English Edition), 31(4), 439–448 (2010)
25. [25]
Fang, T., Zhang, J., and Yao, S. Slip MHD viscous flow over a stretching sheet-an exact solution. Communications in Nonlinear Science and Numerical Simulation, 14(11), 3731–3737 (2009)
26. [26]
Wang, C. Y. Analysis of viscous flow due to a stretching sheet with surface slip and suction. Nonlinear Analysis: Real World Applications, 10(1), 375–380 (2009)
27. [27]
Aziz, A. Hydrodynamic and thermal slip flow boundary layers over a flat plate with constant heat flux boundary condition. Communications in Nonlinear Science and Numerical Simulation, 15(3), 573–580 (2010)
28. [28]
Bhattacharyya, K., Mukhopadhyay, S., and Layek, G. C. Slip effects on boundary layer stagnation-point flow and heat transfer towards a shrinking sheet. International Journal of Heat and Mass Transfer, 54(1–3), 308–313 (2011)
29. [29]
Wang, C. Analytical solutions for a liquid film on an unsteady stretching surface. Heat and Mass Transfer, 42(8), 759–766 (2006)
30. [30]
Ishak, A., Nazar, R., and Pop, I. Heat transfer over an unsteady stretching permeable surface with prescribed wall temperature. Nonlinear Analysis: Real World Applications, 10(5), 2909–2913 (2009)
31. [31]
Mukhopadhyay, S. Unsteady boundary layer flow and heat transfer past a porous stretching sheet in presence of variable viscosity and thermal diffusivity. International Journal of Heat and Mass Transfer, 52(21–22), 5213–5217 (2009)
32. [32]
Mukhopadhyay, S. Effect of thermal radiation on unsteady mixed convection flow and heat transfer over a porous stretching surface in porous medium. International Journal of Heat and Mass Transfer, 52(13–14), 3261–3265 (2009)
33. [33]
Tsai, R., Huang, K. H., and Huang, J. S. Flow and heat transfer over an unsteady stretching surface with non-uniform heat source. International Communications in Heat and Mass Transfer, 35(10), 1340–1343 (2008)
34. [34]
Hang, X., Liao, S. J., and Pop, I. Series solutions of unsteady three-dimensional MHD flow and heat transfer in the boundary layer over an impulsively stretching plate. European Journal of Mechanics-B/Fluids, 26(1), 15–27 (2007)
35. [35]
Ali, M. E. and Magyari, E. Unsteady fluid and heat flow induced by a submerged stretching surface while its steady motion is slowed down gradually. International Journal of Heat and Mass Transfer, 50(1–2), 188–195 (2007)
36. [36]
Zheng, L. C., Wang, L. J., and Zhang, X. X. Analytical solutions of unsteady boundary flow and heat transfer on a permeable stretching sheet with non-uniform heat source/sink. Communications in Nonlinear Science and Numerical Simulation, 16(2), 731–740 (2011)
37. [37]
Pal, D. and Hiremath, P. S. Computational modeling of heat transfer over an unsteady stretching surface embedded in a porous medium. Meccanica, 45(3), 415–424 (2010)
38. [38]
Andersson, H. I., Aarseth, J. B., and Dandapat, B. S. Heat transfer in a liquid film on an unsteady stretching surface. International Journal of Heat and Mass Transfer, 43(1), 69–74 (2000)
39. [39]
Wang, C. Y. Liquid film on an unsteady stretching sheet. Quarterly of Applied Mathematics, 48(4), 601–610 (1990)
40. [40]
Merkin, J. H. and Kumaran, V. The unsteady MHD boundary-layer flow on a shrinking sheet. European Journal of Mechanics-B/Fluids, 29(5), 357–363 (2010)
41. [41]
Fan, T., Xu, H., and Pop, I. Unsteady stagnation flow and heat transfer towards a shrinking sheet. International Communications in Heat and Mass Transfer, 37(10), 1440–1446 (2010)
42. [42]
Fang, T. G., Zhang, J., and Yao, S. S. Viscous flow over an unsteady shrinking sheet with mass transfer. Chinese Physics Letters, 26(1), 014703 (2009)
43. [43]
Zhao, J. K. Differential Transformation and Its Applications for Electrical Circuits (in Chinese), Huazhong University Press, Wuhan (1986)Google Scholar
44. [44]
Ayaz, F. Solutions of the systems of differential equations by differential transform method. Applied Mathematics and Computation, 147(2), 547–567 (2004)
45. [45]
Chang, S. H. and Chang, I. L. A new algorithm for calculating two-dimensional differential transform of nonlinear functions. Applied Mathematics and Computation, 215(7), 2486–2494 (2009)
46. [46]
Abdel-Halim Hassan, I. H. Comparison differential transformation technique with Adomian decomposition method for linear and nonlinear initial value problems. Chaos, Solitons and Fractals, 36(1), 53–65 (2008)
47. [47]
Boyd, J. Padé approximant algorithm for solving nonlinear ordinary differential equation boundary value problems on an unbounded domain. Computers in Physics, 11(3), 299–303 (1997)

© Shanghai University and Springer-Verlag Berlin Heidelberg 2012

## Authors and Affiliations

• Xiao-hong Su (苏晓红)
• 1
• 2
• 3
• Lian-cun Zheng (郑连存)
• 1
Email author
• Xin-xin Zhang (张欣欣)
• 3
1. 1.Department of Mathematics and MechanicsUniversity of Science and Technology BeijingBeijingP. R. China
2. 2.Department of Mathematics and PhysicsNorth China Electric Power UniversityBaodingHebei Province, P. R. China
3. 3.Mechanical Engineering SchoolUniversity of Science and Technology BeijingBeijingP. R. China