Dual Solution of MHD Stagnation-Point Flow towards a Stretching Surface

Abstract

The effect of a uniform transverse magnetic field on two-dimensional stagnation-point flow of an incompressible viscous electrically conducting fluid over a stretching surface is investigated when the surface is stretched in its own plane with a velocity proportional to the distance from the stagnation-point. This magnetohydrodynamic (MHD) flow problem is governed by the parameter b representing the ratio of the strain rate of the stagnation-point flow to that of the stretching sheet and the magnetic field parameter M. It is known from a previous paper [9] that if b > 1, the steady solution to the problem is monotonic increasing and the solution is also unique. But when 0 < b < bc (where bc (< 1) depends on M), there exists a dual solution which is non-monotonic in addition to a monotonic decreasing solution. It is found in this paper that bc decreases as M increases. Numerically it is shown that if M > 0.23919, the non-monotonic solution cannot exist and so in this case, the only solution is monotonic decreasing. A stability analysis reveals that when 0 < b < bc, the solutions along the upper branch corresponding to the monotonic solution are linearly stable while those along the lower branch for the non-monotonic solution are linearly unstable. It is also shown that the decay rate of a disturbance increases with increasing M for the stable solution but the growth rate of instability for the non-monotonic solution decreases with increasing M.

The effect of a uniform transverse magnetic field on two-dimensional stagnation-point flow of an incompressible viscous electrically conducting fluid over a stretching surface is investigated when the surface is stretched in its own plane with a velocity proportional to the distance from the stagnation-point. This magnetohydrodynamic (MHD) flow problem is governed by the parameter b representing the ratio of the strain rate of the stagnation-point flow to that of the stretching sheet and the magnetic field parameter M. It is known from a previous paper [9] that if b > 1, the steady solution to the problem is monotonic increasing and the solution is also unique. But when 0 < b < bc (where bc (< 1) depends on M), there exists a dual solution which is non-monotonic in addition to a monotonic decreasing solution. It is found in this paper that bc decreases as M increases. Numerically it is shown that if M > 0.23919, the non-monotonic solution cannot exist and so in this case, the only solution is monotonic decreasing. A stability analysis reveals that when 0 < b < bc, the solutions along the upper branch corresponding to the monotonic solution are linearly stable while those along the lower branch for the non-monotonic solution are linearly unstable. It is also shown that the decay rate of a disturbance increases with increasing M for the stable solution but the growth rate of instability for the non-monotonic solution decreases with increasing M.

Keywords

Dual Solution; Magnetohydrodynamic Stagnation-Point Flow; Stretching Surface; Stability Analysis

Dual Solution; Magnetohydrodynamic Stagnation-Point Flow; Stretching Surface; Stability Analysis

Cite this paper

nullT. Mahapatra, S. Nandy and A. Gupta, "Dual Solution of MHD Stagnation-Point Flow towards a Stretching Surface,"*Engineering*, Vol. 2 No. 4, 2010, pp. 299-305. doi: 10.4236/eng.2010.24039.

nullT. Mahapatra, S. Nandy and A. Gupta, "Dual Solution of MHD Stagnation-Point Flow towards a Stretching Surface,"

References

[1] L. J. Crane, “Flow Past a Stretching Plate,” Zeitschrift für Angewandte Mathematik und Physik, Vol. 21, 1970, pp. 645-647.

[2]
K. B. Pavlov, “Magnetohydrodynamic Flow of an Incompressible Viscous Fluid Caused by the Deformation of a Plane Surface,” Magnitnaya Gidrodinamika, Vol. 4, 1974, pp. 146-147.

[3]
A. Chakrabarti and A. S. Gupta, “Hydromagnetic Flow and Heat Transfer over a Stretching Sheet,” Quarterly of Applied Mathematics, Vol. 37, April 1979, pp. 73-78.

[4]
H. I. Andersson, “MHD Flow of a Viscoelastic Fluid Past a Stretching Surface,” Acta Mechanica, Vol. 95, 1992, pp. 227-230.

[5]
T. C. Chiam, “Stagnation-point Flow towards a Stretching Plate,” Journal of the Physical Society of Japan, Vol. 63, 1994, pp. 2443- 2444.

[6]
T. R. Mahapatra and A. S. Gupta, “Heat Transfer in Stagnation-point Flow towards a Stretching Sheet,” Heat Mass Transfer, Vol. 38, 2002, pp. 517-521.

[7]
T. R. Mahapatra and A. S. Gupta, “Magnetohydrodynamic Stagnation-point Flow towards a Stretching Sheet,” Acta Mechanica, Vol. 152, 2001, pp. 191-196.

[8]
J. Paullet and P. Weidman, “Analysis of Stagnation Point Flow towards a Stretching Sheet,” International Journal of Non-Linear Mechanics, Vol. 42, No. 9, 2007, pp. 1084-1091.

[9]
T. R. Mahapatra, S. K. Nandy, and A. S. Gupta, “Magnetohydrodynamic Stagnation-point Flow of a Power-law Fluid towards a Stretching Surface,” International Journal of Non-Linear Mechanics, Vol. 44, 2009, pp. 123-128.

[10]
J. A. Shercliff, “A Textbook of Magnetohydrodynamics,” Pergamon Press, Oxford, 1965.

[11]
J. H. Merkin, “Mixed Convection Boundary Layer Flow on a Vertical Surface in a Saturated Porous Medium,” Journal of Engineering Mathematics, Vol. 14, 1980, pp. 301-313.

[12]
S. Chandrasekhar, “Hydrodynamic and Hydromagnetic Stability,” Clarendon Press, Oxford, 1961.