Stability analysis of the interface between two weak viscoelastic liquids under periodic oscillations

Abstract

We consider the motion and the linear hydrodynamic instabilities of two immiscible viscoelastic liquids above a horizontal solid surface induced by the periodic oscillations of the horizontal plate along its plane. A planar interface, parallel to the oscillating plate, separates the lower layer from the other viscoelastic fluid that extends vertically to infinity. The two-dimensional motion of these fluids is studied together with the conditions under which the flow becomes unstable, deforming the planar interface and promoting the mixing of both liquids. The study extends the previous work by Isakova et al. [“A model for the linear stability of the interface between aqueous humor and vitreous substitutes after vitreoretinal surgery,” Phys. Fluids 26, 124101 (2014)] by considering non-Newtonian fluids, particularly liquids with weak viscoelasticity (neglecting normal stress differences), which may model more accurately the physical behavior of the aqueous humor and, especially, the vitreous humour substitute in the vitreous chamber of the eye after vitrectomy. A novel approach to the quasi-steady stability analysis of unsteady flows of Maxwell liquids is developed in the present paper. We focus on the effect of the small Deborah numbers on the motion and on the hydrodynamic instability of the two fluids as the other non-dimensional parameters are varied within the range of interest for the biofluiddynamics of the eye. The special case in which the lower layer modelling the aqueous humor is a Newtonian liquid and the upper vitreous substitute is a Maxwell liquid is considered with detail. We find that, even for a very small Deborah number of the vitreous substitute, the dynamics and the hydrodynamic stability of the two fluids can be qualitatively very different to the Newtonian case, especially as the viscosity ratio is varied, showing that weak viscoelasticity may change dramatically the dynamics of the eye. An exhaustive characterization of the influence of the different parameters on the hydrodynamic stability is given, which may be useful also for the study of the dynamics of other systems where two viscoelastic liquids in contact are subjected to periodic oscillations.