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

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, P-m = v/v(m) where v and v(m) are the usual and the magnetic viscosities, respectively. For the existence of stationary flows the ratio between the flow velocity L,, and the Alfven velocity c(A) = H-x/(4pirho)(1/2) (rho being the fluid density) plays a critical role. Super-Alfven (v(x) > c(A)) flows are possible at any value of P-m and for any values of v(x) and H-x on the layer boundaries. Sub-Alfven (v(x) < c(A)) stationary flows are impossible at any value of P,, and for any values of the differences in v, and 1, across the layer, except for two cases: P-m = 0 and P (m)= 1. When P-m = 0, i.e. when the fluid is strictly inviscid, v = 0, flow is possible in both the super- and sub-Alfven regimes; however, the magnetic field must be uniform, H-x = const, H-y = 0 in this case. For P,, = I both flow regimes are also possible; however, the sub-Alfven flow is possible only for a definite relationship between the magnetic field and velocity differences: DeltaH(x) = -Deltav(x) (in corresponding units). For the case where the relative differences in v(x) and H-x across the layer are small, Deltav(x) much less than (v) over bar (x), DeltaH(x) much less than (H) over bar (x), solutions are obtained in explicit form for arbitrary P-m (here (v) over bar (x) and (H) over bar (x) are averaged over the layer). For the specific case P-m = 1, exact analytical solutions of basic system are found and studied in detail.