Odd-viscosity-induced instability of a thin film with variable density

Abstract

The stability of a two-dimensional gravity-driven thin viscous Newtonian fluid with broken time-reversal-symmetry draining down a uniformly heated inclined plane is discussed. The presence of the odd part of the Cauchy stress tensor with an odd viscosity coefficient brings new characteristics in fluid flow. A theoretical model is implemented, which captures the dependence of the surface tension on temperature, and the model also allows for variation in the density of the liquid with a thermal difference. The coupled effect of odd viscosity, variable density, and surface tension has been investigated both analytically and numerically. A nonlinear evolution equation of the free surface is derived by the method of systematic asymptotic expansion. A linear stability analysis is carried out, which yields the critical conditions for the onset of instability in long-wave perturbations. New interesting results illustrating how the critical Reynolds number depends on the odd viscosity as well as other various dimensionless parameters have been obtained. In addition, a weakly nonlinear stability analysis is performed based on the method of multiple scales from which a complex Ginzburg–Landau equation is obtained. It is observed that the film not only has supercritical stable and subcritical unstable zones, but also unconditional stable and explosive zones. It has been also shown that the spatial uniform solution corresponding to the sideband disturbance may be stable in the unstable region. Employing the Crank–Nicolson method in a periodic domain, the spatiotemporal evolution of the model has been analyzed numerically for different values of the odd viscosity as well as other flow parameters. Nonlinear simulations are found to be in good agreement with the linear and weakly nonlinear stability analysis. The results are conducive to the further development of related experiments.