Asymptotic integration of the equation governing magnetohydrodynamic flow with aligned velocity and magnetic fields
The planar flow of an infinitely conducting fluid moving in such a way that the velocity and magnetic fields are always aligned can be analyzed by means of the hodograph technique. The partial differential equation for the stream function in hodograph variables is linear and may be solved by separation of variables. One of the two resulting ordinary differential equations is immediately solvable in terms of trigonometric functions; the other is, for general values of the adiabatic exponent, a second-order linear differential equation with very complicated variable coefficients. In this Memorandum the asymptotic behavior of the solutions of this latter equation for large values of the separation constant is obtained. The asymptotic expansion of the derivative of the solution is also given explicitly. The results reduce to the well-known asymptotic forms of the Chaplygin function and its derivative when the magnetic field is set equal to zero. 27 pp. Bibliog.