Three-dimensional MHD duct flows with strong transverse magnetic fields. Part 4. Fully insulated, variable-area rectangular ducts with small divergences

Abstract

Part 3 of this study treats a prototype with insulating side walls at z = ± 1 for all x and insulating top and bottom walls a ty = ± a for x < 0 and at y = ± (a + bx) for x > 0, where the applied magnetic field is in the y direction and the flow is in the x direction. In the diverging portion (x > 0) of this duct, the entire mass flux is carried by high-velocity jets adjacent to the side walls, while the fluid elsewhere is stagnant. In the constant-area portion (x < 0), the fully developed flow is severely disturbed as it approaches the join at x = 0, and high-velocity jets occur even before the top and bottom walls begin to diverge. The analysis presented in Part 3 is not valid in the limit b → 0, and the object of this paper is to reconcile the stagnant core flow for b ≠ 0 with the fully developed flow for b = 0. Conditions are such that inertia forces are negligible.

The fist transitional stage occurs when 1 [Gt ] b [Gt ] M^−½, where M is the (large) Hartmann number. The upstream disturbance disappears, and downstream each of the O(M^−½) side-wall boundary layers splits into an O(b^−lM−^½) outer layer and an O(M^−½) inner layer. The fluid outside these layers is still stagnant and an O(bM^−½) velocity in the outer sublayers accounts for the mass flux. The viscous inner sublayers reduce the velocity in the outer sublayers to zero at the side walls.

The second transitional stage occurs when b = O(M^−½). The outer sublayers spread across the entire duct so that none of the fluid is stagnant, and an O(1) core velocity accounts for the mass flux. This analysis is valid no matter how small b becomes, and as b → 0 the fully developed solution is recovered every-where.