Scaling in concentration-driven convection boundary layers with transpiration

We study concentration-driven natural convection boundary layers on horizontal surfaces, subjected to a weak, surface normal, uniform blowing velocity Vi for three orders of range of the dimensionless blowing parameter 10-8 ≥ J = Re3x/Grx ≥ 10-5, where Rex and Grx are the local Reynolds and Grashof numbers at the horizontal location x, based respectively on Vi and ΔC, the concentration difference across the boundary layer. We formulate the integral boundary layer equations, with the assumption of no concentration drop within the species boundary layer, which is valid for weak blowing into the thin species boundary layers that occur at the high Schmidt number (Sc - 600) of concentration-driven convection. The equations are then numerically solved to show that the species boundary layer thickness δd = 1.6 x(Rex/Grx )1/4, the velocity boundary layer thickness δv = δdSc1/5, the horizontal velocity u = Vi(Grx/Rex )1/4f (n), where n = y/δv, and the drag coefficient based on Vi, CD = 2.32/vJ.We find that the vertical profile of the horizontally averaged dimensionless concentration across the boundary layer becomes, surprisingly, independent of the blowing and the species diffusion effects to follow aGr2/3y scaling, where Gry is the Grashof number based on the vertical location y within the boundary layer.We then show that the above profile matches the experimentally observed mean concentration profile within the boundary layers that form on the top surface of a membrane, when a weak flow is forced gravitationally from below the horizontal membrane that has brine above it and water below it. A similar match between the theoretical scaling of the species boundary layer thickness and its experimentally observed variation is also shown to occur.