The general method of a self-similar solutions construction for 2D MHD flows

A. Bartulis - V. N. Kremenetskii - E. V. Shcherbinin

Institute of Physics, University of Latvia, Salaspils-1, LV_2169, Latvia

Abstract
The general properties of 2D MHD flows have been defined. A supposition that flows is 2D allow us to introduce three stream functions: hydrodynamic, electrical and magnetic. It is shown that these stream functions define two component of velocity, three component of electrical current and three component of magnetic field. The differential equations for these stream functions have been obtained.The pressure was excluded from the equations by using rot operation. It is shown that in 2D MHD flows the z component of electrical field must be constant. Further investigations deal with similar solutions, when the system of the partial derivative differential equations is restricted to the system of the ordinary differential equations. It is proved that self-similarity of the problem is possible only in the cases when functions depending on one of coordinates has power or exponential form. It is shown that all 2D MHD boundary layer problems are described by two universal equation coefficients of that are dependent from one constant. To describe a specific flow it is necessary to know two quantitative dimension characteristic values. The types of applied magnetic field, which do not violate the possibilities of self-similar solution, have been found and the equations for this approach are obtained. The exact solutions in the similarity approach have been found. The examples of well known in hydrodynamics specific flows, which are suitable for the approach to be offered, are given and also are formulated for magnetohydrodynamic cases. Figs 6, Refs 10.