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Couette and Poiseuille flows of an Oldroyd 6-constant fluid with magnetic field. (English) Zbl 1067.35074
The inadequacy of the classical Navier-Stokes model when describing rheologically complex fluids has led to the formulation of several non-Newtonian fluid models. This paper considers the Oldroyd 6-constant fluid model governed by a nonlinear differential equation. The three nonlinear boundary value problems are solved using a homotopy analysis and a numerical method. The solutions for a Navier-Stokes fluid, as well as those corresponding to an Oldroyd 3-constant fluid, and for a Maxwell fluid, are shown to appear as limiting cases.

MSC:
35Q35 PDEs in connection with fluid mechanics
76W05 Magnetohydrodynamics and electrohydrodynamics
76A05 Non-Newtonian fluids
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[1] Oldroyd, J.G., On the formulation of rheological equations of state, Proc. roy. soc. London ser. A, 200, 523-541, (1950) · Zbl 1157.76305
[2] Rajagopal, K.R.; Bhatnagar, R.K., Exact solutions for some simple flows of an Oldroyd-B fluid, Acta mech., 113, 233-239, (1995) · Zbl 0858.76010
[3] Rajagopal, K.R., On an exact solution for the flow of an Oldroyd-B fluid, Bull. tech. univ. Istanbul, 49, 617-623, (1996) · Zbl 0918.76003
[4] Pontrelli, G.; Bhatnagar, R.K., Flow of a viscoelastic fluid between two rotating circular cylinders subject to suction or injection, Internat. J. numer. methods fluids, 24, 337-349, (1997) · Zbl 0895.76007
[5] Hayat, T.; Siddiqui, A.M.; Asghar, S., Some simple flows of an Oldroyd-B fluid, Internat. J. engrg. sci., 39, 135-147, (2001)
[6] Asghar, S.; Parveen, S.; Hanif, S.; Siddiqui, A.M.; Hayat, T., Hall effects on the unsteady hydromagnetic flows of an Oldroyd-B fluid, Internat. J. engrg. sci., 41, 609-619, (2003) · Zbl 1211.76138
[7] Baris, S., Flow of an Oldroyd 6-constant fluid between intersecting planes, one of which is moving, Acta mech., 147, 125-135, (2001) · Zbl 0984.76005
[8] S.J. Liao, The proposed homotopy analysis technique for the solution of nonlinear problems, Ph.D. thesis, Shanghai Jiao Tong University, 1992
[9] Liao, S.J.; Chwang, A.T., Application of homotopy analysis method in nonlinear oscillations, ASME J. appl. mech., 65, 914-922, (1998)
[10] Liao, S.J., An explicit, totally analytic approximation of Blasius viscous flow problems, Internat. J. non-linear mech., 34, 759-778, (1999) · Zbl 1342.74180
[11] Liao, S.J., A uniformly valid analytic solution of 2D viscous flow past a semi-infinite flat plate, J. fluid mech., 385, 101-128, (1999) · Zbl 0931.76017
[12] Liao, S.J.; Campo, A., Analytic solutions of the temperature distribution in Blasius viscous flow problems, J. fluid mech., 453, 411-425, (2002) · Zbl 1007.76014
[13] Liao, S.J.; Cheung, K.F., Homotopy analysis of nonlinear progressive waves in deep water, J. engrg. math., 45, 105-116, (2003) · Zbl 1112.76316
[14] Liao, S.J., An analytic approximate technique for free oscillations of positively damped systems with algebraically decaying amplitude, Internat. J. non-linear mech., 38, 1173-1183, (2003) · Zbl 1348.74225
[15] Wang, C.; Zhu, J.M.; Liao, S.J.; Pop, I., On the explicit analytic solution of cheng – chang equation, Internat. J. heat mass trans., 46, 1855-1860, (2003) · Zbl 1029.76050
[16] Liao, S.J.; Pop, I., Explicit analytic solution for similarity boundary layer equations, Internat. J. heat mass. trans., 47, 75-85, (2004) · Zbl 1045.76008
[17] Hayat, T.; Khan, M.; Ayub, M., On the explicit analytic solutions of an Oldroyd 6-constant fluid, Internat. J. engrg. sci., 42, 123-135, (2004) · Zbl 1211.76009
[18] T. Hayat, M. Khan, S. Asghar, Magnetohydrodynamic flow of an Oldroyd 6-constant fluid, Appl. Math. Comput., in press · Zbl 1126.76388
[19] Liao, S.J., On the analytic solution of magnetohydrodynamic flows of non-Newtonian fluids over a stretching sheet, J. fluid mech., 488, 189-212, (2003) · Zbl 1063.76671
[20] Chang, C.C.; Yen, J.T., Rayleigh’s problem in magnetohydrodynamics, Phys. fluid, 2, 393-403, (1959) · Zbl 0090.42702
[21] V.J. Rossow, On flow of electrically conducting fluids over a flat plate in the presence of a transverse magnetic field, NASA, Report no. 1358, 489, 1958
[22] Shercliff, J.A., A text book of magnetohydrodynamics, (1965), Pergamon Elmsford, NY · Zbl 0134.22101
[23] Nanousis, N.D., Theoretical magnetohydrodynamic analysis of mixed convection boundary layer flow over a wedge with uniform suction or injection, Acta mech., 138, 21-30, (1999) · Zbl 0954.76096
[24] Vajravelu, K.; Rivera, J., Hydromagnetic flow at an oscillating plate, Internat. J. non-linear mech., 38, 305-312, (2003) · Zbl 1346.76213
[25] Hayat, T.; Zamurad, M.; Asghar, S.; Siddiqui, A.M., Magnetohydrodynamic flow due to non-coaxial rotations of a porous oscillating disk and a fluid at infinity, Internat. J. engrg. sci., 41, 1177-1196, (2003)
[26] Tokis, T.N., Hydromagnetic unsteady flow due to an unsteady plate, Astrophys. space sci., 59, 167-174, (1978) · Zbl 0388.76043
[27] Pop, I.; Kumari, M.; Nath, G., Conjugate MHD flow past a flat plate, Acta mech., 106, 215-220, (1994) · Zbl 0847.76096
[28] Abel, S.; Veena, P.H.; Rajagopal, K.R.; Pravin, V.K., Non-Newtonian magnetohydrodynamic flow over a stretching surface with heat and mass transfer, Internat. J. non-linear mech., 39, 1067-1078, (2004) · Zbl 1348.76010
[29] Bird, R.B.; Armstrong, R.C.; Hassager, O., Dynamics of polymeric liquids, vol. 1, fluid mechanics, (1987), Wiley New York, p. 354
[30] Huilgol, R.R., Continuum mechanics of viscoelastic liquids, (1975), Wiley New York, p. 195 · Zbl 0353.76003
[31] Rajagopal, K.R., A note on unsteady unidirectional flows of a non-Newtonian fluid, Internat. J. non-linear mech., 17, 369-373, (1982) · Zbl 0527.76003
[32] Erdogan, M.E., On the unsteady unidirectional flows generated by impulsive motion of a boundary or sudden application of a pressure gradient, Internat. J. non-linear mech., 37, 1091-1106, (2002) · Zbl 1346.76031
[33] Khaled, A.R.A.; Vafai, K., The effect of the slip condition on Stokes and Couette flows due to an oscillating wall: exact solutions, Internat. J. non-linear mech., 39, 795-809, (2004) · Zbl 1348.76060
[34] Daily, J.W.; Harleman, D.R.F., Fluid dynamics, (1966), Addison-Wesley Don Mills, ON, p. 117 · Zbl 0193.54902
[35] Jordan, P.M.; Puri, P., Stokes’ first problem for a rivlin – ericksen fluid of second grade in a porous half-space, Internat. J. non-linear mech., 38, 1019-1025, (2003) · Zbl 1348.76150
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