Tabulated solutions of the equation governing magnetohydrodynamic flow with aligned velocity and magnetic fields
When an infinitely conducting gas flows in a plane so that velocity and magnetic fields are everywhere aligned, the governing magnetohydrodynamic equations can be linearized by transforming to the hodograph plane. The problem can then be reduced to the solution of a second-order linear partial differential equation for the stream function. When separable solutions are sought, it is found that one of the resulting ordinary differential equations is immediately integrable, but the second cannot be integrated in closed form. The memorandum discusses this second equation in detail, presenting a series solution valid for small velocities and asymptotic formulas for use when the separation constant takes on larger values. A numerical integration of the equation for small values of the separation constant is derived, and results are tabulated for velocities between zero and the smallest value for which the differential equation again becomes singular. Finally, values of the quantities appearing in the asymptotic formulas are given.