On the Stability of a Compressible Axial Flow with an Axial Magnetic Field

We consider the stability problem of inviscid compressible axial flows with axial magnetic fields following the work of Dandapat and Gupta (Quarterly of Applied Mathematics, 1975). A numerical study of the stability of some basic flows has been carried out and it is found that an increase in the magnetic field strength has a stabilizing effect on subsonic flows and a destabilizing effect on supersonic flows. An analytical study of the stability problem has also been done in the present paper, but this analytical study is restricted by the approximation and , where is the Mach number and is the imaginary part of the complex phase velocity . A semicircular region depending on the magnetic field parameter and the Mach number is found for subsonic disturbances and as a consequence it is found that sufficiently strong magnetic field stabilizes all subsonic disturbances. Under a weak magnetic field, it is shown that short subsonic disturbances are stable. 1. Introduction The stability of inviscid shear flows of a compressible fluid was studied in Blumen [1]. In order to simplify the stability problem, Blumen [1] focused attention on the stability of basic parallel shear flow of a perfect gas whose thermodynamic state is constant. From the equations of motion, the pressure of the shear flow is a constant only and if the basic flow temperature is also a constant, it follows that the basic flow density is also a constant. In the context of magnetogasdynamics, Dandapat and Gupta [2] studied the stability of a parallel flow of an inviscid perfectly conducting gas in the presence of a uniform magnetic field. Following Blumen [1], Dandapat and Gupta [2] also considered the case of a constant basic thermodynamic state. In Dandapat and Gupta [3] the stability of a nondissipative axial flow of a compressible conducting fluid between two concentric cylinders in the presence of a uniform axial field was studied. In cylindrical polar coordinates they considered the basic flow with velocity , magnetic field , where is a constant, is the pressure, and is the density. In the absence of the swirl component of the velocity in the basic flow it follows from the equations of magnetohydrodynamics that is a constant. In their stability analysis, Dandapat and Gupta [3] noticed that the equation involving the density perturbation is not needed in deriving the stability equation. Dandapat and Gupta [3] explained this by observing that there is no basic swirl velocity and therefore the mechanism of the centrifugal acceleration playing the role of a radial effective gravity is absent