Rotationally-axisymmetric Motion of a Binary Mixture with a Flat Free Boundary at Small Marangoni Numbers

Rotationally-axisymmetric motion of a binary mixture with a flat free boundary at small Marangoni numbers is investigated. The problem is reduced to the inverse linear initial-boundary value problem for parabolic equations. Using Laplace transformation properties the exact analytical solution is obtained. It is shown that a stationary solution is the limiting one with the growth of time if there is a certain relationship between the temperature of the solid wall and the external temperature of the gas. If there is no connection, the convergence to the stationary solution is broken. Some examples of numerical reconstruction of the temperature, concentration and velocity fields are given, which confirm the theoretical conclusions

Shear-layers in magnetohydrodynamic spherical Couette flow with conducting walls

We consider the steady axisymmetric motion of an electrically conducting fluid contained within a spherical shell and permeated by a centred axial dipole magnetic field, which is strong as measured by the Hartmann number M. Slow axisymmetric motion is driven by rotating the inner boundary relative to the stationary outer boundary. For M ≫ 1, viscous effects are only important in Hartmann boundary layers adjacent to the inner and outer boundaries and a free shear-layer on the magnetic field line that is tangent to the outer boundary on the equatorial plane of symmetry. We measure the ability to leak electric current into the solid boundaries by the size of their relative conductance ɛ. Since the Hartmann layers are sustained by the electric current flow along them, the current inflow from the fluid mainstream needed to feed them increases in concert with the relative conductance, because of the increasing fraction ℒ of the current inflow leaked directly into the solids. Therefore the nature of the flow is sensitive to the relative sizes of ɛ−1 and M.The current work extends an earlier study of the case of a conducting inner boundary and an insulating outer boundary with conductance ɛo = 0 (Dormy, Jault & Soward, J. Fluid Mech., vol. 452, 2002, pp. 263–291) to other values of the outer boundary conductance. Firstly, analytic results are presented for the case of perfectly conducting inner and outer boundaries, which predict super-rotation rates Ωmax of order M1/2 in the free shear-layer. Successful comparisons are made with numerical results for both perfectly and finitely conducting boundaries. Secondly, in the case of a finitely conducting outer boundary our analytic results show that Ωmax is O(M1/2) for ɛo−1 ≪ 1 ≪ M3/4, O(ɛo2/3M1/2) for 1 ≪ ɛo−1 ≪ M3/4 and O(1) for 1 ≪ M3/4 ≪ ɛo−1. On increasing ɛo−1 from zero, substantial electric current leakage into the outer boundary, ℒo ≈ 1, occurs for ɛo−1 ≪ M3/4 with the shear-layer possessing the character appropriate to a perfectly conducting outer boundary. When ɛo−1 = O(M3/4) the current leakage is blocked near the equator, and the nature of the shear-layer changes. So, when M3/4 ≪ ɛo−1, the shear-layer has the character appropriate to an insulating outer boundary. More precisely, over the range M3/4 ≪ ɛo−1 ≪ M the blockage spreads outwards, reaching the pole when ɛo−1 = O(M). For M ≪ ɛo−1 current flow into the outer boundary is completely blocked, ℒo ≪ 1.

To the theory of the barothermic effect in gases

The formulation of the problem of temperature processes at filtration of gas in a porous medium under axisymmetric motion. The equation for the pressure field is obtained for the barothermal effect. The results of calculations of the barothermic effect at various pressure compressibilities. The results can be used to solve various problems of oil and gas geology.