Stability of fluid flow past a membrane

Abstract

The stability of fluid flow past a membrane of infinitesimal thickness is analysed in the limit of zero Reynolds number using linear and weakly nonlinear analyses. The system consists of two Newtonian fluids of thickness R* and HR*, separated by an infinitesimally thick membrane, which is flat in the unperturbed state. The dynamics of the membrane is described by its normal displacement from the flat state, as well as a surface displacement field which provides the displacement of material points from their steady-state positions due to the tangential stress exerted by the fluid flow. The surface stress in the membrane (force per unit length) contains an elastic component proportional to the strain along the surface of the membrane, and a viscous component proportional to the strain rate. The linear analysis reveals that the fluctuations become unstable in the long-wave (α→0) limit when the non-dimensional strain rate in the fluid exceeds a critical value Λ_t, and this critical value increases proportional to α² in this limit. Here, α is the dimensionless wavenumber of the perturbations scaled by the inverse of the fluid thickness R*⁻¹, and the dimensionless strain rate is given by Λ_t = (γ˙*R*η*/Γ*), where η* is the fluid viscosity, Γ* is the tension of the membrane and γ˙* is the strain rate in the fluid. The weakly nonlinear stability analysis shows that perturbations are supercritically stable in the α→0 limit.