Hydrodynamic instability and wave formation of a viscous film flowing down a slippery inclined substrate: Effect of odd-viscosity

Abstract

Hydrodynamic instability and formation of waves for thin viscous film flowing down a slippery inclined substrate with broken time-reversal-symmetry have been discussed in this present study. The effect of slip on the substrate is modeled using the Navier slip boundary condition. We have derived a nonlinear evolution model in the framework of the long-wave expansion technique. The one equation model can track the free surface evolution and involve the viscosity, gravity, surface tension, and the effect of the slip of the wall. The odd part of the Cauchy stress tensor with an odd-viscosity coefficient significantly modifies the characteristic of the film flow. We performed the linear stability analysis with the Orr Sommerfeld technique and classical temporal analysis from the model. Comparing the results shows a satisfactory agreement between Orr Sommerfeld and the model when the velocity scale is chosen as twice the free surface velocity. Analyzing the traveling wave solutions and studying different bifurcation analyzes and phase diagrams, we focused on and discussed two significant wave families, namely γ1 (low-wave-speed) and γ2 (high-wave-speed). The spatio-temporal study of the model has been performed numerically for different odd-viscosity and slip parameters. Results show that the odd-viscosity plays a stabilizing role while the slip on the substrate gives a destabilizing effect within the parameter range of our consideration.