Countercurrent convection in a double-diffusive boundary layer

Countercurrent flow may be induced by opposing buoyancy forces associated with compositional gradients and thermal gradients within a fluid. The occurrence and structure of such flows is investigated by solving the double-diffusive boundary-layer equations for steady laminar convection along a vertical wall of finite height. Non-similar solutions are derived using the method of matched asymptotic expansions, under the restriction that the Lewis and Prandtl numbers are both large. Two sets of asymptotic solutions are constructed, assuming dominance of one or the other of the buoyancy forces. The two sets overlap in the central region of the parameter space; each set matches up with neighbouring unidirectional similarity solutions at the respective borderlines of incipient counterflow.Interaction between the buoyancy mechanisms is controlled by their relative strength R and their relative diffusivity Le. Flow in the outer thermal boundary layer deviates from single-diffusive thermal convection, depending upon the magnitude of the parameter RLe. Flow in the inner compositional boundary layer deviates from single-diffusive compositional convection, depending upon the magnitude of $RLe^{\frac{1}{3}}$.