Singular effect of small slip an otherwise stable two fluid shear flowSaleh Tanveer, The Ohio State University, Columbus, OH&Demetrios Papageorgiou, Imperial College, London, UK*

A Navier-slip boundary condition between two immiscible fluids had been proposed to account for polymer disentanglement at the interface between two fluids that causes apparent slip [Zhao & Macosko, 2002]. Indeed, Koplik & Banavar [2006] used Molecular Dynamics simulations with interaction potentials supporting repulsion of molecules at the liquid-liquid interface, to demonstrate slip and propose a Navier-slip condition with an apparent slip coefficient that they computed.  

We consider the consequences of introducing an arbitrarily small slip in the highly stable configuration of Yih (1971), when a thin layer of less viscous fluid resides next to a moving wall in a two-layer fluid flow with very small viscosity ratio. We find that even when the viscosity ratio $m$ and a non-dimensional slip length $L$ tend to zero in some particular manner in a nonlinear thin-film model, we have destabilization of a planar interface to short wavelength (Turing type) instability. These waves were continued to finite amplitude traveling waves.

Existence of solutions was confirmed for finite amplitude waves and global smooth solutions were also shown to exist for the initial value problem. Numerical work with the Orr-Sommerfeld equations in each fluid, confirm that this Turing type instability transcends the thin-layer approximation [Katsavria & Papageorgiou, submitted, 2023].