Fluid flow beneath a semipermeable membrane during drying processes.

The dynamic interactions between a semipermeable membrane and a long, thin layer of liquid beneath it are investigated in the context of drying processes. The membrane separates two aqueous solutions of sugar, and the transport of water across the membrane is driven by concentration and pressure gradients across it. A model is formulated using a long-wave approximation that includes the effects of volume loss due to water transport across the membrane, the incompressibility and bending stiffness of the membrane, and the dynamical effects that arise owing to the viscous stresses generated by the fluid flow. This model is first applied to study the desiccation of a sessile vesicle that is clamped to a rigid substrate and then also to study the behavior of blisters on laminated substrates. For each problem, equilibrium membrane shapes are obtained and their bifurcation structures are described as the sugar concentration above the membrane is varied. It is demonstrated that a wrinkled membrane coarsens to lessen the frequency of wrinkles and that if the membrane is clamped symmetrically so that it meets the substrate at a nonzero angle, then the membrane favors an asymmetric shape as water is drawn out through it.