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On exact solution of unsteady MHD flow of a viscous fluid in an orthogonal rheometer. (English) Zbl 1304.35568
Summary: This paper studies the unsteady MHD flow of a viscous fluid in which each point of the parallel planes are subject to the non-torsional oscillations in their own planes. The streamlines at any given time are concentric circles. Exact solutions are obtained and the loci $$\Gamma$$ of the centres of these concentric circles are discussed. It is shown that the motion so obtained gives three infinite sets of exact solutions in the geometry of an orthogonal rheometer in which the above non-torsional oscillations are superposed on the disks. These solutions reduce to a single unique solution when symmetric solutions are looked for. Some interesting special cases are also obtained from these solutions.
##### MSC:
 35Q35 PDEs in connection with fluid mechanics 76W05 Magnetohydrodynamics and electrohydrodynamics 76U05 General theory of rotating fluids
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##### References:
 [1] Berker, RA: Sur Quelques cas d’ intéegration des équations du mouvement d’un fluide visqueux incompressible. Théses françaises de l’entre-deux-guerres (1936) · Zbl 1210.76215 [2] Berker, R, Intégration des équations du mouvement d’un fluide visqueux incompressible, No. VIII/2, 1-384, (1963), Berlin [3] Berker, R, A new solution of the Navier-Stokes equation for the motion of a fluid contained between two parallel plates rotating about the same axis, Arch. Mech. Stosow., 31, 265-280, (1979) · Zbl 0415.76026 [4] Berker, R, An exact solution of the Navier-Stokes equation. the vortex curvilinear axis, Int. J. Eng. Sci., 20, 217-230, (1982) · Zbl 0487.76039 [5] Rajagopal, KR; Gupta, AS, Flow and stability of second grade fluid between rotating planes, Arch. Mech., 33, 663-674, (1981) · Zbl 0501.76006 [6] Rajagopal, KR; Gupta, AS, Flow and stability of a second grade fluid between two parallel plates rotating about noncoincident axes, Int. J. Eng. Sci., 19, 1401-1409, (1981) · Zbl 0469.76003 [7] Rajagopal, KR, On the flow of a simple fluid in an orthogonal rheometer, Arch. Ration. Mech. Anal., 79, 39-47, (1982) · Zbl 0513.76002 [8] Siddiqui, AM; Rana, MA; Ahmed, N, Effects of Hall current and heat transfer on MHD flow of a burgers’ fluid due to a pull of eccentric rotating disks, Commun. Nonlinear Sci. Numer. Simul., 13, 1554-1570, (2008) · Zbl 1221.76244 [9] Siddiqui, AM; Rana, MA; Ahmed, N, Magnetohydrodynamics flow of a burgers’ fluid in an orthogonal rheometer, Appl. Math. Model., 34, 2881-2892, (2010) · Zbl 1201.76316 [10] Rao, AR; Kasiviswanathan, SR, On exact solutions of the unsteady Navier-Stokes equations-the vortex with instantaneous curvilinear axis, Int. J. Eng. Sci., 25, 337-349, (1987) · Zbl 0609.76021 [11] Abbott, TNG; Walters, K, Rheometrical flow systems, part 2. theory for the orthogonal rheometer, including an exact solution of the Navier-Stokes equations, J. Fluid Mech., 40, 205-213, (1970) · Zbl 0184.52102 [12] Thornley, C, On Stokes and Rayleigh layers in a rotating system, Q. J. Mech. Appl. Math., 21, 451-461, (1968) · Zbl 0164.28301 [13] Kasiviswanathan, SR; Rao, AR, On exact solutions of unsteady MHD flow between eccentrically rotating disks, Arch. Mech., 39, 411-418, (1987) · Zbl 0637.76126 [14] Mohanty, HK, Hydromagnetic flow between two rotating disks with noncoincident parallel axes of rotation, Phys. Fluids, 15, 1456-1458, (1972) · Zbl 0246.76113 [15] Erkman, CS, Magnetohydrodynamic flow of a viscous, incompressible fluid between rotating parallel plates, Lett. Appl. Eng. Sci., 3, 51-59, (1975) [16] Murthy, SN; Ram, RKP, MHD flow and heat transfer due to eccentric rotations of a porous disk and a fluid at infinity, Int. J. Eng. Sci., 16, 943-949, (1978) [17] Rao, AR; Rao, PR, On the magnetohydrodynamic flow between eccentrically rotating disks, Int. J. Eng. Sci., 21, 359-372, (1983) · Zbl 0511.76109 [18] Ersoy, HV, MHD flow of an Oldroyd-B fluid between eccentric rotating disks, Int. J. Eng. Sci., 37, 1973-1984, (1999) · Zbl 1210.76215 [19] Rajagopal, KR, Flow of viscoelastic fluids between rotating plates, Theor. Comput. Fluid Dyn., 3, 185-206, (1992) · Zbl 0747.76018 [20] Sherchiff, J: A Textbook of Magnetohydrodynamics. Pergamon, Oxford (1965) [21] Davidson, PA: An Introduction to Magnetohydrodynamics, 1st edn. Cambridge University Press, Cambridge (2001) · Zbl 0974.76002 [22] Sutton, GW, Sherman, A: Engineering Magnetohydrodynamics. McGraw-Hill, New York (1965)
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