Stability of fluid flow past a membrane

Abstract:The stability of the flow of a fluid past a solid membrane of infinitesimal thickness is investigated using a linear stability analysis. The system consists of two fluids of thicknesses R and H R and bounded by rigid walls moving with velocities Va and Vb, and separated by a membrane of infinitesimal thickness which is flat in the unperturbed state. The fluids are described by the Navier-Stokes equations, while the constitutive equation for the membrane incorporates the surface tension, and the effect of curvature elasticity is also examined for a membrane with no surface tension. The stability of the system depends on the dimensionless strain rates Λa and Λb in the two fluids, which are defined as (Vaη/Γ) and (‒Vbη/ΓH) for a membrane with surface tension Γ, and (VaR2η/K) and (VbR2η/KH) for 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(k3) smaller than the frequency of the fluctuations. The effect of fluid inertia in this limit is incorporated using a small wave number k ≪ 1 asymptotic analysis, and it is found that there is a correction of O(kRe) smaller than the leading order frequency due to inertial effects. This correction causes long wave fluctuations to be unstable for certain values of the ratio of strain rates Λr =(Λb/Λa) and ratio of thicknesses H. The stability of the system at finite Reynolds number was calculated using numerical techniques for the case where the strain rate in one of the fluids is zero. The stability depends on the Reynolds number for the fluid with the non-zero strain rate, and the parameter Σ = (ρΓR/η2), where Γ is the surface tension of the membrane. It is found that the Reynolds number for the transition from stable to unstable modes, Ret, first increases with Σ, undergoes a turning point and a further increase in the Ret results in a decrease in Σ. This indicates that there are unstable perturbations only in a finite domain in the Σ ‒ Ret plane, and perturbations are always stable outside this domain.