RETRACTED: Influence of the odd viscosity on a falling film down a slippery inclined plane (Retracted Article)

The stability of a thin viscous Newtonian fluid with broken time-reversal-symmetry draining down a slippery inclined plane is examined. The presence of the odd part of the Cauchy stress tensor with an odd viscosity coefficient brings new characteristics in fluid flow as it gives rise to new terms in the pressure gradient of the flow. By odd viscosity, it is meant that apart from the well-known coefficient of shear viscosity, a classical liquid with broken time-reversal symmetry is endowed with a second viscosity coefficient. The model implements a Navier slip condition at the solid-liquid interface with the slip length being the parameter that measures the deviation from the no-slip condition. The classical long-wave expansion technique is performed and a nonlinear evolution equation of Benney-type is derived in terms of film thickness h(x, t), which is significantly modified due to the presence of odd viscosity in the liquid. The parameters governing the film flow system and the slippery substrate strongly influence the waveforms and their amplitudes and hence the stability of the fluid. The linear stability analysis is performed using the normal mode approach and a critical Reynolds number is obtained. The results of the linear stability analysis reveal that larger odd viscosity leads to the higher critical Reynolds number while the higher slip length makes the critical Reynolds number lower. In other words, odd viscosity has a stabilizing effect while the slip length promotes instability. Based on the method of multiple scales, a weakly nonlinear stability analysis is carried out, which shows that there is a range of wave numbers with a supercritical bifurcation and a range of larger wave numbers with a subcritical bifurcation. Different instability zones are also demarcated. The weakly nonlinear study shows that with an increase in the odd viscosity, the supercritical stable region and the explosion area shrink, whereas the unconditional stable and the subcritical unstable regions increase. It has also been shown that the spatial uniform solution corresponding to the sideband disturbance may be stable in the unstable region. The spatiotemporal evolution of the model has been analyzed numerically by employing the Crank-Nicolson method in a periodic domain for different values of the odd viscosity and slip length. The nonlinear simulations are found to be in good agreement with the linear and weakly nonlinear stability analysis.