Stability of the interface between two immiscible fluids over a periodically oscillating flat surface

We consider a flat solid surface located at y = 0, performing sinusoidal oscillations along the x–direction, with (x, y) being a Cartesian system of coordinates. We assume that two immiscible fluids occupy the region of space y>= 0. The interface between the two fluids is at y = d; fluid 1 occupies the region 0 <=y<=d, and fluid 2 extends from d to infinity. We study the linear stability of the interface using the normal mode analysis and assuming quasi-steady flow conditions, e.g. assuming that perturbations evolve on a time scale significantly smaller than the period of oscillations of the basic flow. The stability problem leads to two Orr-Sommerfeld equations for the streamfunctions in fluids 1 and 2, coupled with suitable boundary conditions. The resulting eigenvalue problem is solved numerically employing a second order finite-difference scheme and using an inverse iteration approach. The results show that instability of the interface is possible for long enough waves. We study how stability conditions depend on the dimensionless controlling parameters, showing, in particular, the relevant role played by the surface tension between the two fluids. The work represents a first attempt to understand the instability of the aqueous humour–vitreous substitute instability in vitrectomised eyes. The simple geometrical configuration considered in this work well represents the real case when the thickness of the aqueous layer in contact with the retina is much smaller than the radius of the eye, which is often the case. Our results suggest that shear instability at the aqueous humour–vitreous substitute interface is a plausible mechanism responsible for the onset of emulsification in the vitreous chamber.