Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

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Multiple solutions for hydromagnetic flow of a second grade fluid over a stretching or shrinking sheet


Authors: Robert A. Van Gorder and K. Vajravelu
Journal: Quart. Appl. Math. 69 (2011), 405-424
MSC (2000): Primary 76D03
DOI: https://doi.org/10.1090/S0033-569X-2011-01211-1
Published electronically: April 4, 2011
MathSciNet review: 2850738
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Abstract | References | Similar Articles | Additional Information

Abstract: We study a class of fourth-order nonlinear differential equations arising in the hydromagnetic flow of a second grade fluid over a stretching or shrinking sheet. Explicit exact solutions are obtained. Furthermore we show that the differential equation may admit zero or one or two physically meaningful solutions depending on the values of the physical parameters of the model. As a special case, we recover the single or the dual solutions and compare them with the available results in the literature. Also, the obtained multiple solutions for several sets of values of the parameters are presented through tables and graphs, and the qualitative behaviors are discussed.


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  • 1. C. Truesdell, W. Noll, The non-linear field theories of mechanics, in: S. Flugge (Ed.), Encyclopedia of Physics, III/3, Springer, Berlin, 1965, pp. 1-591. MR 0193816 (33:2030)
  • 2. K.R. Rajagopal, On boundary conditions for fluids of the differential type, in: A. Sequeira (Ed.), Navier-Stokes Equations and Related Nonlinear Problems, Plenum Press, New York, 1995, pp. 273-278. MR 1373221
  • 3. K.R. Rajagopal, P.N. Kaloni, Some remarks on boundary conditions for fluids of the differential type, in: G.A.C. Graham, S.K. Malik (Eds.), Continuum Mechanics and its Applications, Hemisphere, New York, 1989, pp. 935-942. MR 1051699
  • 4. K.R. Rajagopal, A.S. Gupta, An exact solution for the flow of a non-Newtonian fluid past an infinite plate, Meccanica 19 (1984) 158-160. MR 767046 (85k:76008)
  • 5. D.W. Beard, K. Walters, Elastico-viscous boundary layer flows, Proc. Camb. Phil. Soc. 60 (1964) 667-674. MR 0171475 (30:1706)
  • 6. V.K. Garg, K.R. Rajagopal, Flow of a non-Newtonian fluid past a wedge, Acta Mechanica 88 (1991) 113-123. MR 1111096
  • 7. G.K. Rajeswari, S.L. Rathna, Flow of a particular class of non-Newtonian visco-elastic and visco-elastic fluids near a stagnation point, Z. Angew. Math. Phys. 13 (1962) 43-57. MR 0141330 (25:4736)
  • 8. R.S. Rivlin, J.L. Ericksen, Stress deformation relations for isotropic materials, J. Rat. Mech. Anal. 4 (1955) 323-425. MR 0068413 (16:881a)
  • 9. J.E. Dunn, K.R. Rajagopal, Fluids of differential type: Critical review and thermodynamic analysis, Int. J. Engrg. Sci. 33 (1995) 689-729. MR 1321925 (96a:76006)
  • 10. K.R. Rajagopal, A.S. Gupta, T.A. Na, A note on the Falkner-Skan flows of a non-Newtonian fluid, Int. J. Non-Linear Mech. 18 (1983) 313-320. MR 718753 (85a:76024)
  • 11. K. Vajravelu, D. Rollins, Heat transfer in a viscoelastic fluid over a stretching sheet, J. Math. Anal. Appl. 158 (1991) 241-255. MR 1113413 (92b:76003)
  • 12. M.S. Sarma, B.N. Rao, Heat transfer in a viscoelastic fluid over a stretching sheet, J. Math. Anal. Appl. 222 (1998) 268-275. MR 1623919 (99b:76005)
  • 13. W.C. Troy, E.A. Overman, G.B. Ermentrout, J.P. Keener, Uniqueness of flow of a second-order fluid past a stretching sheet, Quart. Appl. Math. 44 (1987) 753-755. MR 872826 (87m:76009)
  • 14. W.D. Chang, The nonuniqueness of the flow of a viscoelastic fluid over a stretching sheet, Quart. Appl. Math. 47 (1989) 365-366. MR 998108 (90g:76024)
  • 15. P.S. Lawrence, B.N. Rao, Reinvestigation of the nonuniqueness of the flow of a viscoelastic fluid over a stretching sheet, Quart. Appl. Math. 51 (1993) 401-404. MR 1233521 (94e:76002)
  • 16. W.D. Chang, N.D. Kazarinoff, C. Lu, A new family of explicit solutions for the similarity equations modelling flow of a non-Newtonian fluid over a stretching sheet, Arch. Rat. Mech. Anal. 113 (1991) 191-195. MR 1079187 (91k:76009)
  • 17. K. Vajravelu, T. Roper, Flow and heat transfer in a second grade fluid over a stretching sheet, Int. J. Non-Linear Mech. 34 (1999) 1031-1036.
  • 18. K. Vajravelu, D. Rollins, Hydromagnetic flow of a second grade fluid over a stretching sheet, Applied Mathematics and Computation 148 (2004) 783-791. MR 2024543
  • 19. A.D. Barinberg, A.B. Kapusta, B.V. Chekin, Magnitnaya Gidrodinamika (English translation) 11 (1975) 111-121.
  • 20. T. Fang, J. Zhang, Closed-form exact solutions of MHD viscous flow over a shrinking sheet, Commun. Nonlinear Sci. Numer. Simulat. 14 (2009) 2853-2857.
  • 21. T. Fang, J. Zhang, S. Yao, Viscous Flow over an Unsteady Shrinking Sheet with Mass Transfer, Chinese Phys. Lett. 26 (2009) 014703.
  • 22. T. Fang, Boundary layer flow over a shrinking sheet with power-law velocity, International Journal of Heat and Mass Transfer 51 (2008) 5838-5843.
  • 23. T. Fang, W. Liang, C. F. Lee, A new solution branch for the Blasius equation - A shrinking sheet problem, Computers and Mathematics with Applications 56 (2008) 3088-3095. MR 2474564 (2009k:76054)
  • 24. T. Hayat, Z. Abbas, T. Javed, M. Sajid, Three-dimensional rotating flow induced by a shrinking sheet for suction, Chaos, Solitons and Fractals 39 (2009) 1615-1626.
  • 25. T. Hayat, Z. Abbas, N. Alib, MHD flow and mass transfer of a upper-convected Maxwell fluid past a porous shrinking sheet with chemical reaction species, Physics Letters A 372 (2008) 4698-4704.
  • 26. T. Hayat, T. Javed, M. Sajid, Analytic solution for MHD rotating flow of a second grade fluid over a shrinking surface, Physics Letters A 372 (2008) 3264-3273. MR 2414279
  • 27. S. Nadeem, M. Awais, Thin film flow of an unsteady shrinking sheet through porous medium with variable viscosity, Physics Letters A 372 (2008) 4965-4972.
  • 28. N.F.M. Noor, S. Awang Kechil, I. Hashim, Simple non-perturbative solution for MHD viscous flow due to a shrinking sheet, Communications in Nonlinear Science and Numerical Simulation (2009), doi: 10.1016/j.cnsns.2009.03.034.
  • 29. M. Sajid, T. Hayat, The application of homotopy analysis method for MHD viscous flow due to a shrinking sheet, Chaos, Solitons and Fractals 39 (2009) 1317-1323.
  • 30. C.Y.Wang, Stagnation flow towards a shrinking sheet, International Journal of Non-Linear Mechanics 43 (2008) 377-382.
  • 31. M. Miklavcic, C. Y. Wang, Viscous flow due to a shrinking sheet. Quart. Appl. Math. 64 (2006) 283-290. MR 2243864 (2007c:76021)
  • 32. I. Muhaimin, R. Kandasamy, A. B. Khamis. Effects of heat and mass transfer on nonlinear MHD boundary layer flow over a shrinking sheet in the presence of suction. Appl. Math. Mech. -Engl. Ed. 29 (2008) 1309-1317.
  • 33. M. Rahimpour, S. R. Mohebpour, A. Kimiaeifar, G. H. Bagheri. On the analytical solution of axisymmetric stagnation flow towards a shrinking sheet, International Journal of Mechanics 2 (2008) 1-10.
  • 34. J.A. Shercliff, in: A Textbook of Magnetohydrodynamics, Pergamon Press, London, 1965, pp. 45-50. MR 0185961 (32:3421)
  • 35. S.J. Liao, A new branch of solutions of boundary-layer flows over a stretching flat plate, Int. J. Heat Mass Transfer 49 (2005) 2529-2539.
  • 36. S.J. Liao, A new branch of solution of boundary-layer flows over a permeable stretching plate, Int. J. Non-Linear Mech. 42 (2007) 819-830. MR 2328735
  • 37. L.J. Crane, Flow past a stretching plate, Z. Angew. Math. Phys. 21 (1970) 645.
  • 38. M. S. Abel, M. M. Nandeppanavar, Heat transfer in MHD viscoelastic boundary layer flow over a stretching sheet with non-uniform heat source/sink, Communications in Nonlinear Science and Numerical Simulation 4 (2009) 2120-2131. MR 2474469
  • 39. C.Y. Wang, Exact solutions of the steady state Navier-Stokes equations, Ann. Rev. Fluid Mech. 23 (1991) 159-177. MR 1090331 (92a:76030)
  • 40. A. Chakrabarti, A.S. Gupta, Hydromagnetic flow and heat transfer over a stretching sheet, Quart. Appl. Math. 37 (1979) 73-78.
  • 41. K. Vajravelu, D. Rollins, Heat transfer in an electrically conducting fluid over a stretching sheet, Int. J. Non-Linear Mech. 27 (1992) 265-277.
  • 42. H.I. Andersson, An exact solution of the Navier-Stokes equations for magnetohydrodynamic flow, Acta Mech. 113 (1995) 241-244. MR 1361702 (96h:76082)
  • 43. I. Pop and T.Y. Na, A note on MHD flow over a stretching permeable surface, Mech. Res. Commun. 25 (1998) 263-269. MR 1618160
  • 44. S.J. Liao, On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet, J. Fluid Mech. 488 (2003) 189-212. MR 2019664 (2004j:76169)

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Additional Information

Robert A. Van Gorder
Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816
Email: rav@knights.ucf.edu

K. Vajravelu
Affiliation: Department of Mathematics, University of Central Florida, Orlando, Florida 32816
Email: vajravel@pegasus.cc.ucf.edu

DOI: https://doi.org/10.1090/S0033-569X-2011-01211-1
Keywords: Similarity solution, stretching sheet, shrinking sheet, Navier-Stokes equations, exact solution, hydromagnetic flow, viscoelastic fluid, second grade fluid, multiple solutions.
Received by editor(s): August 15, 2009
Published electronically: April 4, 2011
Article copyright: © Copyright 2011 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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