Compressible Jeffery-Hamel Flow of a Plasma

Abstract

A similarity transformation is given, which reduces the partial, nonlinear differential equations describing a compressible, polytropic plasma flow across an azimuthal magnetic field in a duct with plane inclined walls to an ordinary nonlinear differential equation of second order. The latter is solved rigorously in terms of a hyperelliptic integral. The form of the plasma flow fields in pure outflows (diffuser) is discussed analytically in dependence of the Reynolds (R) and Hartmann (H) numbers and the polytropic coefficient (γ) for given duct angles θ0 . The realizable Mach numbers are shown to be eigenvalues of the nonlinear boundary-value problem, M=MX{R, H, γ, θ0). The flow solutions are different in type for Hartmann numbers H 1) below and 2) above a critical Hartmann number Hc defined by Hc 2= [2(γ - 1)/(γ +1)]R+ [2 γ/(γ +1)]2. Some of the eigenvalues Mx are calculated and the associated velocity profiles are represented graphically for prescribed flow parameters.