Effects of radiation and heat source/sink on unsteady MHD boundary layer flow and heat transfer over a shrinking sheet with suction/injection

Abstract

In this paper, an investigation is made to study the effects of radiation and heat source/sink on the unsteady boundary layer flow and heat transfer past a shrinking sheet with suction/injection. The flow is permeated by an externally applied magnetic field normal to the plane of flow. The self-similar equations corresponding to the velocity and temperature fields are obtained, and then solved numerically by finite difference method using quasilinearization technique. The study reveals that the momentum boundary layer thickness increases with increasing unsteadiness and decreases with magnetic field. The thermal boundary layer thickness decreases with Prandtl number, radiation parameter and heat sink parameter, but it increases with heat source parameter. Moreover, increasing unsteadiness, magnetic field strength, radiation and heat sink strength boost the heat transfer.

Keywords

MHD boundary layer / unsteady flow / heat transfer / thermal radiation / heat source/sink / shrinking sheet / suction/injection

References

[1] Crane L J. Flow past a stretching plate. Zeitschrift für Angewandte Mathematik und Physik, 1970, 21(4): 645–647

[2] Gupta P S, Gupta A S. Heat and mass transfer on a stretching sheet with suction and blowing. Canadian Journal of Chemical Engineering, 1977, 55(6): 744–746

[3] Wang C Y. The three dimensional flow due to a stretching flat surface. Physics of Fluids, 1984, 27(8): 1915–1917

[4] McLeod J B, Rajagopal K R. On the uniqueness of flow of a Navier-Stokes fluid due to a stretching boundary. Archive for Rational Mechanics and Analysis, 1987, 98(4): 385–393

[5] Rajagopal K R, Na T Y, Gupta A S. Flow of a viscoclastic fluid over a stretching sheet. Rheologica Acta, 1984, 23(2): 213–215

[6] Andersson H I,Bech K H,Dandapat B S. Flow of a power-law fluid over a stretching sheet. International Journal of Non-Linear Mechanics, 1992, 27(6): 929–936

[7] Pavlov K B. Magnetohydrodynamic flow of an incompressible viscous fluid caused by deformation of a surface. Magnetohydrodynamics, 1974, 10(4): 146–148

[8] Chakrabarti A, Gupta A S. Hydromagnetic flow and heat transfer over a stretching sheet. Quarterly of Applied Mathematics, 1979, 37: 73–78

[9] Andersson H I. MHD flow of a viscoelastic fluid past a stretching surface. Acta Mechanica, 1992, 95(1-4): 227–230

[10] Pop I, Na T Y. A note on MHD flow over a stretching permeable surface. Mechanics Research Communications, 1998, 25(3): 263–269

[11] Hayat T, Javed T. On analytic solution for generalized three-dimensional MHD flow over a porous stretching sheet. Physics Letters A, 2007, 370(3-4): 243–250

[12] Wang C Y. Liquid film on an unsteady stretching sheet. Quarterly of Applied Mathematics, 1990, 48: 601–610

[13] Miklavčič M, Wang C Y. Viscous flow due a shrinking sheet. Quarterly of Applied Mathematics, 2006, 64: 283–290

[14] Hayat T, Abbas Z, Sajid M. On the analytic solution of magnetohydrodynamic flow of a second grade fluid over a shrinking sheet. J Appl Mech, 2007, 74(6): 1165–1171

[15] Hayat T, Javed T, Sajid M. Analytic solution for MHD rotating flow of a second grade fluid over a shrinking surface. Physics Letters A, 2008, 372(18): 3264–3273

[16] Muhaimin R K, Khamis A B. Effects of heat and mass transfer on nonlinear MHD boundary layer flow over a shrinking sheet in the presence of suction. Applied Mathematics and Mechanics, 2008, 29(10): 1309–1317

[17] Fang T, Zhang J. Closed-form exact solution of MHD viscous flow over a shrinking sheet. Communications in Nonlinear Science and Numerical Simulation, 2009, 14(7): 2853–2857

[18] Sajid M, Hayat T. The application of homotopy analysis method for MHD viscous flow due to a shrinking sheet. Chaos, Solitons and Fractals, 2009, 39(3): 1317–1323

[19] Hayat T, Abbas Z, Javed T, Sajid M. Three-dimensional rotating flow induced by a shrinking sheet for suction. Chaos, Solitons and Fractals, 2009, 39(4): 1615–1626

[20] Hayat T, Abbas Z, Ali N. MHD flow and mass transfer of a upper-convected Maxwell fluid past a porous shrinking sheet with chemical reaction species. Physics Letters A, 2008, 372(26): 4698–4704

[21] Noor N F M, Kechil S A, Hashim I. Simple non-perturbative solution for MHD viscous flow due to a shrinking sheet. Communications in Nonlinear Science and Numerical Simulation, 2010, 15(2): 144–148

[22] Hayat T, Iram S, Javed T, Asghar S. Shrinking flow of second grade fluid in a rotating frame: An analytic solution. Communications in Nonlinear Science and Numerical Simulation, 2010, 15(10): 2932–2941

[23] Fang T, Zhang J. Thermal boundary layers over a shrinking sheet: an analytic solution. Acta Mechanica, 2010, 209(3-4): 325–343

[24] Devi C D S, Takhar H S, Nath G. Unsteady three-dimensional boundary layer flow due to a stretching surface. International Journal of Heat and Mass Transfer, 1986, 29(12): 1996–1999

[25] Smith S H. An exact solution of the unsteady Navier-Stokes equations resulting from a stretching surface. J Appl Mech, 1994, 61(3): 629–633

[26] Pop I, Na T Y. Unsteady flow past a stretching sheet. Mechanics Research Communications, 1996, 23(4): 413–422

[27] Elbashbeshy E M A, Bazid M A A. Heat transfer over an unsteady stretching surface. Heat and Mass Transfer, 2004, 41(1): 1–4

[28] Tsai R, Huang K H, Huang J S. Flow and heat transfer over an unsteady stretching surface with a non-uniform heat source. International Communications in Heat and Mass Transfer, 2008, 35(10): 1340–1343

[29] Nazar R, Amin N, Filip D, Pop I. Unsteady boundary layer flow in the region of the stagnation point on a stretching sheet. International Journal of Engineering Science, 2004, 42(11-12): 1241–1253

[30] Mukhopadhyay S, Andersson H I. Effects of slip and heat transfer analysis of flow over an unsteady stretching surface. Heat and Mass Transfer, 2009, 45(11): 1447–1452

[31] Hayat T, Mustafa M, Asghar S. Unsteady flow with heat and mass transfer of a third grade fluid over a stretching surface in the presence of chemical reaction. Nonlinear Analysis: Real World Applications, 2010, 11(4): 3186–3197

[32] Hayat T, Qasim M, Abbas Z. Homotopy solution for the unsteady three-dimensional MHD flow and mass transfer in a porous space. Communications in Nonlinear Science and Numerical Simulation, 2010, 15(9): 2375–2387

[33] Tie-Gang F, Ji Z, Shan-Shan Y. Viscous flow over an unsteady shrinking sheet with mass transfer. Chinese Physics Letters, 2009, 26(1): 014703

[34] Merkin J H, Kumaran V. The unsteady MHD boundary-layer flow on a shrinking sheet. European Journal of Mechanics - B/Fluids, 2010, 29(5): 357–363

[35] Viskanta R, Grosh R J. Boundary layer in thermal radiation absorbing and emitting media. International Journal of Heat and Mass Transfer, 1962, 5(9): 795–806

[36] Chen T S, Ali M M, Armaly B F. Natural convection-radiation interaction in boundary layer flow over horizontal surfaces. AIAA Journal, 1984, 22(12): 1797–1803

[37] Elbashbeshy E M A. Radiation effect on heat transfer over a stretching surface. Canadian Journal of Physics, 2000, 78(12): 1107–1112

[38] Ouaf M E M. Exact solution of thermal radiation on MHD flow over a stretching porous sheet. Applied Mathematics and Computation, 2005, 170(2): 1117–1125

[39] Hayat T, Abbas Z, Sajid M, Asghar S. The influence of thermal radiation on MHD flow of a second grade fluid. International Journal of Heat and Mass Transfer, 2007, 50(5-6): 931–941

[40] Hayat T, Nawaz M, Sajid M, Asghar S. The effect of thermal radiation on the flow of a second grade fluid. Computers & Mathematics with Applications (Oxford, England), 2009, 58(2): 369–379

[41] Mohamed A E A. Radiation effect on the flow and heat transfer over an unsteady stretching sheet. International Communications in Heat and Mass Transfer, 2009, 36(5): 521–524

[42] 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, 2009, 52(13-14): 3261–3265

[43] Hayat T, Qasim M, Abbas Z. Radiation and mass transfer effects on the magnetohydrodynamic unsteady flow induced by a stretching sheet. Zeitschrift fur Naturforschung. Section A. Physical Science, 2010, 65a: 231–239

[44] Hayat T, Qasim M. Radiation and magnetic field effects on the unsteady mixed convection flow of a second grade fluid over a vertical stretching sheet. International Journal for Numerical Methods in Fluids, 2010, 66(7), 820–832

[45] Brewster M Q. Thermal Radiative Transfer Properties. John Wiley and Sons, 1972

[46] Andersson H I, Aarseth J B, Dandapat B S. Heat transfer in a liquid film on an unsteady stretching surface. International Journal of Heat and Mass Transfer, 2000, 43(1): 69–74

[47] Bellman R E, Kalaba R E. Quasilinearization and nonlinear boundary value problem. New York: American Elsevier Publishing Co Inc, 1965

Acknowledgments

The author would like to thank the referee for his useful suggestion to improve the paper. The author also gratefully acknowledges the financial support of National Board for Higher Mathematics (NBHM), DAE, Mumbai, India for pursuing this work.

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2014 Higher Education Press and Springer-Verlag Berlin Heidelberg