On the enhanced drag force induced by permeation through a filtration membrane

Permeation drag is the predominant cause for particle deposition onto filtration membranes, and it is known that at close approach to the membrane surface this force may greatly exceed the Stokes drag in an unbounded fluid. Herein, the hydrodynamic interaction between a sphere and a permeable wall is re-visited within the framework of the lubrication approximation with the goal of deriving an analytical solution. A closed-form analytical solution is found, based on a perturbation expansion in terms of the scaled permeability, which is considered a small parameter. Numerical calculations of the drag force on the sphere agree perfectly with numerical results available in the literature, as do analytical model results within a range of validity which is affected by the particle size and membrane permeability. Specifically, the presented calculations have been framed in the context of the low permeabilities and colloidal particle sizes representative of commercial membrane separations in current use. Results illustrate that for reverse osmosis and nanofiltration membranes, the hydrodynamic interaction is practically identical to that of an impermeable wall, while for ultrafiltration membranes the drag is substantially reduced; an analogous trend is observed for increasing particle sizes. The approximate solution derived herein offers a simple and direct means of performing hydrodynamic force calculations in particle–membrane systems.