Weakly nonlinear stability of viscous flow past a flexible surface

The weakly nonlinear stability of viscous fluid flow past a flexible surface is analysed in the limit of zero Reynolds number. The system consists of a Couette flow of a Newtonian fluid past a viscoelastic medium of non-dimensional thickness H (the ratio of wall thickness to the fluid thickness), and viscosity ratio $\mu^r$ (ratio of the viscosities of wall and fluid media). The wall medium is bounded by the fluid at one surface and two different types of boundary conditions are considered at the other surface of the wall medium – for ‘grafted’ gels zero displacement conditions are applied while for ‘adsorbed’ gels the displacement normal to the surface is zero but the surface is permitted to move in the lateral direction. The linear stability analysis reveals that for grafted gels the most unstable modes have \alpha \sim O(1), while for adsorbed gels the most unstable modes have \alpha \rightarrow 0, where \alpha is the wavenumber of the perturbations. The results from the weakly nonlinear analysis indicate that the nature of the bifurcation at the linear instability is qualitatively very different for grafted and absorbed gels. The bifurcation is always subcritical for the case of flow past grafted gels. It is found, however, that relatively weak but finite-amplitude disturbances do not significantly reduce the critical velocity required to destabilize the flow from the critical velocity predicted by the linear stability theory. For the case of adsorbed gels, it is found that a supercritical equilibrium state could exist in the limit of small wavenumber for a wide range of parameters $\mu^r$ and H, while the bifurcation becomes subcritical at larger values of the wavenumber and there is a transition from supercritical to subcritical bifurcation as the wavenumber is increased