MHD flow in the vicinity of the stagnation point in a purely azimuthal magnetic field

Autor: B. I. Reznikov, Yu. N. Smyslov
Rok vydání: 1971
Předmět:
Zdroj: Fluid Dynamics. 2:1-4
ISSN: 1573-8507
0015-4628
DOI: 10.1007/bf01019527
Popis: Axisymmetric MHD flow in the vicinity of the stagnation point in the presence of a purely azimuthal nonhomogeneous magnetic field B {0, Bθ, 0} is studied. This problem belongs to the class of MHD problems whose solutions are known as solutions of the “layer type” [1]. This class also includes, in particular, the classical exact solutions of the Navier-Stokes equations. The approximate solutions of the analogous MHD problems for the limiting cases of large and small values of the diffusion number e==υ/η have been considered in [2–5]. In this case it is possible to divide the flow into the so-called viscous and current layers, for each of which the approximate equations, simpler than the exact equations, are solved numerically or in quadratures. Using this technique it is possible to avoid the basic mathematical difficulty, which is that the sought solution of the boundary-value problem must be selected from a family of two-parameter solutions. The approximate method permits dividing the problem into two stages (corresponding to the two boundary layers) in each of which one unknown parameter is determined (in place of their simultaneous determination by direct integration of the basic equations). The drawback of the approximate methods [2–5] is their nonapplicability in the most interesting case, when the thicknesses of the current and viscous layers are of comparable magnitude, i. e., when the kinematic and magnetic viscosities (υ and η) are quantities of the same order. We should also note the poor accuracy of the methods in the framework of the considered approximations for a comparatively large volume of the calculations required, which, in turn, prevents obtaining more exact solutions. The present paper presents a numerical integration of the equations describing MHD flow in the vicinity of the stagnation point over a wide range of S and e numbers (Alfven and diffusion numbers), without the assumption of their smallness, with preliminary determination of the unknowns at the zero of the derivatives of the sought functions with the aid of the method of asymptotic integration. A critical value of the Alfven number is found, for which the retardation of the fluid by the magnetic field (for the first considered configuration of the magnetic field) at the wall is so intense that the friction vanishes everywhere on the surface of the solid body. It is also found that with further increase of the number S a region of reverse flow appears near the wall, which is separated from the remaining flow by a plane on which the z-component of the velocity is equal to zero.
Databáze: OpenAIRE
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