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

Van Gorder, Robert; Vajravelu, K.

doi:10.1090/S0033-569X-2011-01211-1

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
Full-text PDF Free Access

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.

References

\bibliographystyle{amsplain}

C. Truesdell and W. Noll, The non-linear field theories of mechanics, Handbuch der Physik, Band III/3, Springer-Verlag, Berlin, 1965, pp. 1–602. MR 0193816
K. R. Rajagopal, On boundary conditions for fluids of the differential type, Navier-Stokes equations and related nonlinear problems (Funchal, 1994) Plenum, New York, 1995, pp. 273–278. MR 1373221
K. R. Rajagopal and P. N. Kaloni, Some remarks on boundary conditions for flows of fluids of the differential type, Continuum mechanics and its applications (Burnaby, BC, 1988) Hemisphere, New York, 1989, pp. 935–942. MR 1051699
K. R. Rajagopal and A. S. Gupta, An exact solution for the flow of a non-Newtonian fluid past an infinite porous plate, Meccanica–J. Italian Assoc. Theoret. Appl. Mech. 19 (1984), no. 2, 158–160 (English, with Italian summary). MR 767046, DOI https://doi.org/10.1007/BF01560464
D. W. Beard and K. Walters, Elastico-viscous boundary-layer flows. I. Two-dimensional flow near a stagnation point, Proc. Cambridge Philos. Soc. 60 (1964), 667–674. MR 171475
V. K. Garg and K. R. Rajagopal, Flow of a non-Newtonian fluid past a wedge, Acta Mech. 88 (1991), no. 1-2, 113–123. MR 1111096, DOI https://doi.org/10.1007/BF01170596
Ganjam K. Rajeswari and Seekote L. Rathna, Flow of a particular class of non-Newtonian visco-elastic and visco-inelastic fluids near a stagnation point, Z. Angew. Math. Phys. 13 (1962), 43–57 (English, with German summary). MR 141330, DOI https://doi.org/10.1007/BF01600756
R. S. Rivlin and J. L. Ericksen, Stress-deformation relations for isotropic materials, J. Rational Mech. Anal. 4 (1955), 323–425. MR 68413, DOI https://doi.org/10.1512/iumj.1955.4.54011
J. E. Dunn and K. R. Rajagopal, Fluids of differential type: critical review and thermodynamic analysis, Internat. J. Engrg. Sci. 33 (1995), no. 5, 689–729. MR 1321925, DOI https://doi.org/10.1016/0020-7225%2894%2900078-X
K. R. Rajagopal, A. S. Gupta, and T. Y. Na, A note on the Falkner-Skan flows of a non-Newtonian fluid, Internat. J. Non-Linear Mech. 18 (1983), no. 4, 313–320 (English, with French and German summaries). MR 718753, DOI https://doi.org/10.1016/0020-7462%2883%2990028-8
K. Vajravelu and D. Rollins, Heat transfer in a viscoelastic fluid over a stretching sheet, J. Math. Anal. Appl. 158 (1991), no. 1, 241–255. MR 1113413, DOI https://doi.org/10.1016/0022-247X%2891%2990280-D
M. S. Sarma and B. Nageswara Rao, Heat transfer in a viscoelastic fluid over a stretching sheet, J. Math. Anal. Appl. 222 (1998), no. 1, 268–275. MR 1623919, DOI https://doi.org/10.1006/jmaa.1998.5940
William C. Troy, Edward A. Overman II, G. B. Ermentrout, and James P. Keener, Uniqueness of flow of a second-order fluid past a stretching sheet, Quart. Appl. Math. 44 (1987), no. 4, 753–755. MR 872826, DOI https://doi.org/10.1090/S0033-569X-1987-0872826-3
Wen-Dong Chang, The nonuniqueness of the flow of a viscoelastic fluid over a stretching sheet, Quart. Appl. Math. 47 (1989), no. 2, 365–366. MR 998108, DOI https://doi.org/10.1090/S0033-569X-1989-0998108-7
P. Sam Lawrence and B. Nageswara Rao, Reinvestigation of the nonuniqueness of the flow of a viscoelastic fluid over a stretching sheet, Quart. Appl. Math. 51 (1993), no. 3, 401–404. MR 1233521, DOI https://doi.org/10.1090/qam/1233521
Wen-Dong Chang, Nicholas D. Kazarinoff, and Chunqing Lu, A new family of explicit solutions for the similarity equations modelling flow of a non-Newtonian fluid over a stretching sheet, Arch. Rational Mech. Anal. 113 (1990), no. 2, 191–195. MR 1079187, DOI https://doi.org/10.1007/BF00380417

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.

K. Vajravelu and D. Rollins, Hydromagnetic flow of a second grade fluid over a stretching sheet, Appl. Math. Comput. 148 (2004), no. 3, 783–791. MR 2024543, DOI https://doi.org/10.1016/S0096-3003%2802%2900942-6

A.D. Barinberg, A.B. Kapusta, B.V. Chekin, Magnitnaya Gidrodinamika (English translation) 11 (1975) 111–121.

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.

T. Fang, J. Zhang, S. Yao, Viscous Flow over an Unsteady Shrinking Sheet with Mass Transfer, Chinese Phys. Lett. 26 (2009) 014703.

T. Fang, Boundary layer flow over a shrinking sheet with power-law velocity, International Journal of Heat and Mass Transfer 51 (2008) 5838–5843.

Tiegang Fang, Wei Liang, and Chia-fon F. Lee, A new solution branch for the Blasius equation—a shrinking sheet problem, Comput. Math. Appl. 56 (2008), no. 12, 3088–3095. MR 2474564, DOI https://doi.org/10.1016/j.camwa.2008.07.027

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.

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.

T. Hayat, T. Javed, and M. Sajid, Analytic solution for MHD rotating flow of a second grade fluid over a shrinking surface, Phys. Lett. A 372 (2008), no. 18, 3264–3273. MR 2414279, DOI https://doi.org/10.1016/j.physleta.2008.01.069

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.

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.

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.

C.Y.Wang, Stagnation flow towards a shrinking sheet, International Journal of Non-Linear Mechanics 43 (2008) 377–382.

M. Miklavčič and C. Y. Wang, Viscous flow due to a shrinking sheet, Quart. Appl. Math. 64 (2006), no. 2, 283–290. MR 2243864, DOI https://doi.org/10.1090/S0033-569X-06-01002-5

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.

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.

J. A. Shercliff, A textbook of magnetohydrodyamics, Pergamon Press, Oxford-New York-Paris, 1965. MR 0185961

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.

Shi-Jun Liao, A new branch of solutions of boundary-layer flows over a permeable stretching plate, Internat. J. Non-Linear Mech. 42 (2007), no. 6, 819–830. MR 2328735, DOI https://doi.org/10.1016/j.ijnonlinmec.2007.03.007

L.J. Crane, Flow past a stretching plate, Z. Angew. Math. Phys. 21 (1970) 645.

M. Subhas Abel and Mahantesh M. Nandeppanavar, Heat transfer in MHD viscoelastic boundary layer flow over a stretching sheet with non-uniform heat source/sink, Commun. Nonlinear Sci. Numer. Simul. 14 (2009), no. 5, 2120–2131. MR 2474469, DOI https://doi.org/10.1016/j.cnsns.2008.06.004

C. Y. Wang, Exact solutions of the steady-state Navier-Stokes equations, Annual review of fluid mechanics, Vol. 23, Annual Reviews, Palo Alto, CA, 1991, pp. 159–177. MR 1090331

A. Chakrabarti, A.S. Gupta, Hydromagnetic flow and heat transfer over a stretching sheet, Quart. Appl. Math. 37 (1979) 73–78.

K. Vajravelu, D. Rollins, Heat transfer in an electrically conducting fluid over a stretching sheet, Int. J. Non-Linear Mech. 27 (1992) 265–277.

H. I. Andersson, An exact solution of the Navier-Stokes equations for magnetohydrodynamic flow, Acta Mech. 113 (1995), no. 1-4, 241–244. MR 1361702, DOI https://doi.org/10.1007/BF01212646

Ioan Pop and Tsung-Yen Na, A note on MHD flow over a stretching permeable surface, Mech. Res. Comm. 25 (1998), no. 3, 263–269. MR 1618160, DOI https://doi.org/10.1016/S0093-6413%2898%2900037-8

Shi-Jun 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, DOI https://doi.org/10.1017/S0022112003004865