The Pe ~ 1 regime of convection across a horizontal permeable membrane

Abstract

In natural convection, driven by an unstable density difference due to a heavier fluid (brine) above a lighter fluid (water) across a horizontal permeable membrane, we discover a new regime of convection, where the Sherwood number (Sh) scales approximately as the Rayleigh number (Ra). Inferring from the planforms of plume structure on the membrane and the estimates of velocity through the membrane, we show that such a regime occurs when advection balances diffusion in the membrane, i.e. the Péclet number based on the membrane thickness (Pe) is of order one. The advection is inferred to be caused by the impingement of the large-scale flow on the membrane. Utilizing mass balance and symmetry assumptions in the top and the bottom fluids, we derive an expression for the concentration profile in the membrane pore in the new regime by solving the convection–diffusion equation in the membrane pore; this helps us to obtain the concentration drops above and below the membrane that drive the convection. We find that the net flux, normalised by the diffusive flux corresponding to the concentration drop on the side opposite to the impingement of the large-scale flow remains constant throughout the new regime. On the basis of this finding, we then obtain an expression for the flux scaling in the new regime which matches with the experiments; the expression has the correct asymptotes of flux scaling in the advection and the diffusion regimes. The plume spacings in the new regime are distributed lognormally, and their mean follows the trend in the advection regime.