On stationary solutions for free quasi-parallel mixing layers with a longitudinal magnetic field

Abstract

The paper is devoted to the theoretical investigation of the possible existence of stationary mixing layers and of their structure in nearly perfectly conducting, nearly inviscid fluids with a longitudinal magnetic field. A system of two equations is used, which generalizes the well-known Blasius equation (for flow around a semi-infinite plate) to the case under consideration. The system depends on the magnetic Prandtl number, Pm=ν/νm, where ν and νm are the usual and the magnetic viscosities, respectively.For the existence of stationary flows the ratio between the flow velocity vx and the Alfvén velocity cA=Hx/(4πρ)1/2 (ρ being the fluid density) plays a critical role. Super-Alfvén (vx>cA) flows are possible at any value of Pm and for any values of vx and Hx on the layer boundaries. Sub-Alfvén (vx<cA) stationary flows are impossible at any value of Pm and for any values of the differences in vx and Hx across the layer, except for two cases: Pm=0 and Pm=1. When Pm=0, i.e. when the fluid is strictly inviscid, ν=0, flow is possible in both the super- and sub-Alfvén regimes; however, the magnetic field must be uniform, Hx=const, Hy=0 in this case. For Pm=1 both flow regimes are also possible; however, the sub-Alfvén flow is possible only for a definite relationship between the magnetic field and velocity differences: ΔHx=−δvx (in corresponding units). For the case where the relative differences in vx and Hx across the layer are small, Δvx[Lt ]vx, ΔHx[Lt ]Hx, solutions are obtained in explicit form for arbitrary Pm (here vx and Hx are averaged over the layer). For the specific case Pm=1, exact analytical solutions of basic system are found and studied in detail.