On the geometry of streamlines in hydromagnetic fluid flows when the magnetic field is along a fixed direction

Abstract

1. Introduction. In this paper we discuss three-dimensional steady flows of inviscid magnetic fluids. We assume that the magnetic field is along a fixed direction. The main purpose is to investigate various dynamical and kinematical relations connecting the flow and the field quantities with the geometrical parameters of the streamlines. The expressions for the tangent, principal normal and binormal vectors and the curvature and torsion of the streamlines are given in terms of the velocity components, the pressure, the density and the magnitude of the magnetic field. The variations of the hydromagnetic pressure along the streamlines, along their principal normals and along their binormals are, obtained. It is observed that the hydromagnetic pressure along the binormals remains constant. The Bernoulli function is defined and it is found that if this function exists then the Bernoulli surfaces contain both the streamlines and the vortex lines. We shall determine an intrinsic relation satisfied by the Bernoulli function. From this relation we obtain a necessary and sufficient condition for the Bernoulli surfaces to be a family of parallel surfaces. We find that the Bernoulli surfaces exist in the case of incompressible fluid and they form the surfaces on which the sum of the fluid pressure, the kinetic energy and the magnetic energy is constant. In the case of isentropic flows, the variation of the fluid pressure along the streamline is expressed in terms of the Mach number, the magnetic field, the divergence of the unit vector along the streamlines, the density and the magnitude of velocity. In the three-dimensional nonmagnetic gas flows this result reduces to the result of Kanwal [1]. Finally, we shall obtain a class of circular helical flows. 2. The equations. Let xi (j= 1, 2, 3) denote the variables of a system of Cartesian orthogonal coordinates. In order to obtain the summation convention, we shall write indices in covariant and contravariant positions. We shall write i = 9/9xi. The equations of continuity and motion and the Maxwell relations for an inviscid, conductive fluid are [2 ]