A Nonlinear Diffusion Problem Relevant to Desalination by Reverse Osmosis - Williams, F. A. (SIAM journal on applied mathematics, 1969)
Various techniques are used to obtain approximate solutions to the one-dimensional, time-dependent diffusion equation in a semi-infinite, incompressible medium, subject to nonlinear boundary conditions that describe flow through a semipermeable membrane across which is applied a constant difference in hydrostatic pressure. A term describing the convective motion is included in the diffusion equation. A convective-diffusive region, in which time derivatives are negligible, develops and grows in the vicinity of the membrane, while a time-dependent diffusive region in which convection is negligible is established at large distances from the membrane. Theoretically predicted membrane fluxes are shown to be in reasonable agreement with experiment.