Effect of tangential interface motion on the viscous instability in fluid flow past flexible surfaces

The stability of linear shear flow of a Newtonian fluid past a flexible mem brane is analysed in the limit of low Reynolds number as well as in the int ermediate Reynolds number regime for two different membrane models. The obj ective of this paper is to demonstrate he importance of tangential motion i n the membrane on the stability characteristics of the shear flow. The firs t model assumes the wall to be a "spring-backed" plate membrane, and the di splacement of the wall is phenomenologically related in a linear manner to the change in the fluid stresses at the wall. In the second model, the memb rane is assumed to be a two-dimensional compressible viscoelastic sheet of infinitesimal thickness, in which the constitutive relation for the shear s tress contains an elastic part that depends on the local displacement field and a viscous component that depends on the local velocity in the membrane . The stability characteristics of the laminar flow in the limit of low Rr. are crucially dependent on the tangential motion in the membrane wall. In both cases, the flow is stable in the low Reynolds number limit in the abse nce of tangential motion in the membrane. However, the presence of tangenti al motion in the membrane destabilises the shear flow even in the absence o f fluid inertia. In this case, the non-dimensional velocity (Lambda (t)) re quired for unstable fluctuations is proportional to the wavenumber k (Lambd a (t) similar to k) in the plate membrane type of wall while it scales as k (2) in the viscoelastic membrane type of wall (Lambda (t) similar to k(2)) in the limit k --> 0. The results of the low Reynolds number analysis are e xtended numerically to the intermediate Reynolds number regime for the case of a viscoelastic membrane. The numerical results show that for a given se t of wall parameters, the flow is unstable only in a finite range of Reynol ds number, and it is stable in the limit of large Reynolds number.