STABILITY OF FLUID-FLOW PAST A MEMBRANE

The stability of the flow of a fluid past a solid membrane of infinite simal thickness is investigated using a linear stability analysis. The system consists of two fluids of thicknesses R and HR and bounded by rigid walls moving with velocities V-a and V-b and separated by a memb rane of infinitesimal thickness which is flat in the unperturbed state . The fluids are described by the Navier-Stokes equations, while the c onstitutive equation for the membrane incorporates the surface tension , and the effect of curvature elasticity is also examined for a membra ne with no surface tension. The stability of the system depends on the dimensionless strain rates Lambda(a) and Lambda(b) in the two fluids, which are defined as (V-a eta/Gamma) and (-V-b eta/Gamma H) for a mem brane with surface tension Gamma, and (VaR2 eta/K) and (VbR2 eta/KH) f or a membrane with zero surface tension and curvature elasticity K. In the absence of fluid inertia, the perturbations are always stable. In the limit k --> 0, the decay rate of the perturbations is O(k(3)) sma ller than the frequency of the fluctuations. The effect of fluid inert ia in this limit is incorporated using a small wave number k much less than 1 asymptotic analysis, and it is found that there is a correctio n of O(kRe) smaller than the leading order frequency due to inertial e ffects. This correction causes long wave fluctuations to be unstable f or certain values of the ratio of strain rates Lambda(r) = (Lambda(b)/ Lambda(a)) and ratio of thicknesses H. The stability of the system at finite Reynolds number was calculated using numerical techniques for t he case where the strain rate in one of the fluids is zero. The stabil ity depends on the Reynolds number for the fluid with the non-zero str ain rate, and the parameter Sigma = (rho Gamma R/eta(2)), where Gamma is the surface tension of the membrane. It is found that the Reynolds number for the transition from stable to unstable modes, Re-t, first i ncreases with Sigma, undergoes a turning point and a further increase in the Re-t results in a decrease in Sigma. This indicates that there are unstable perturbations only in a finite domain in the Sigma - Re-t plane, and perturbations are always stable outside this domain.