182 QUARTERLY OF APPLIED MATHEMATICS JULY, 1975 ON THE STABILITY OF SWIRLING FLOW IN MAGNETOGASDYNAMICS* By B. S. DANDAPAT and A. S. GUPTA (Indian Institute of Technology, Kharagpur) 1. Introduction. The stability of the steady circular nondissipative flow of an incompressible fluid betwen two concentric cylinders was first studied by Rayleigh [1] who assumed the disturbances to be axisymmetric. He showed that this problem has a remarkable analogy with that of the stability of a density stratified fluid at rest under gravity. Michael [2] extended this problem to the case of a perfectly conducting liquid with an electric current distribution parallel to the axis of cylinders and found that Rayleigh's analogy holds in a slightly modified form. Using this analogy, Howard and Gupta [3] investigated the stability of nondissipative swirling flow of an incom- pressible fluid between two concentric cylinders with respect to axisymmetric dis- turbances. They found that stability is ensured if a Richardson number based on the swirl velocity and the shear in the axial flow exceeds ^ everywhere. Recently Howard [4] using a modification of the analysis due to Chimonas [5] on compressible stratified shear flow, derived a Richardson-number theorem for the linear stability to axisym- metric perturbations of compressible nondissipative swirling flow. The present note is an extension of Howard's [4] problem to the case of a perfectly conducting fluid permeated by an axial distribution of electric current. It is important to note that in a compressible swirling flow, the swirl velocity distribution V(r) not only plays a role similar to that in incompressible flows, but also gives rise, through the centrifugal acceleration V2/r, to a radial effective gravity which, combined with a radial density stratification, affects the perturbations significantly. We discuss the axisymmetric stability of pure axial flow of a compressible perfectly conducting fluid between two concentric cylinders permeated by a uniform axial mag- netic field. We show that the complex wave speed for any unstable wave lies in a semi- circle in the upper half plane, having the same range of axial velocity as the diameter. 2. Compressible swirling flow with an axial current. Consider the steady swirling flow of an inviscid, compressible and perfectly conducting fluid between two concentric cylinders of radii a and b (a < b), the flow being subjected to a volume distribution of current parallel to the axis of the cylinders. Using cylindrical coordinates (r, d, z), we take the basic velocity and the magnetic field as [0, V(j), W(r)] and [0, H0(r), 0]. The radial momentum equation in the undisturbed state gives PoV2 ^H2 I" 02T m ~r ~ sr + L"°+ ~s^J• (1) where p0(r) and p0(r) denote the basic pressure and density distribution and a prime denotes derivative with respect to r. Let a perturbed state of this flow be * Received February 15, 1974; revised version received April 18, 1974.