The interaction between stirring and osmosis. Part 2

A semipermeable membrane forms part of a plane boundary which separates pure solvent from a solution of concentration Cb. Stirring motions in the solution cause it to flow parallel to the plane with uniform wall shear rate α. The velocity of osmotic solvent flow across the membrane into the solution at any point is PΔC, where P is the osmotic permeability of the membrane and ΔC is the local concentration difference across it. ΔC is reduced below Cb by a position-dependent factor γ because of the concentration boundary layer over the membrane, which is thicker than in the absence of osmosis as a result of advection by the osmotic flow itself. The concentration boundary layer is analysed, on the assumption that it is two-dimensional, for both small and large values of the dimensionless longitudinal co-ordinate \[ \xi = PC_b (9x/\alpha D^2)^{\frac{1}{3}}, \] where x is the dimensional co-ordinate and D is the solute diffusivity. These expansions are used to compute γι, the average value of the flux-reduction factor γ over the whole membrane, as a function of β′, which is the value of ξ at the downstream end of the layer, x = l. It is shown that the standard physiological model, in which the layer has a given thickness δ and the stirring motions are not explicitly considered, gives accurate results for γl as a function of β = PCb δ/D as long as δ is given by \[ \delta = 1.22(Dl/\alpha)^{\frac{1}{3}}, \] so that β is proportional to β′.