Hydrodynamic instability of odd-viscosity-induced shear-imposed falling film

Abstract

In this paper, an analysis of linear and weakly nonlinear stability for an odd-viscosity-induced shear-imposed falling film over an inclined plane is performed. Using the Chebyshev spectral collocation approach, the linear effect for disturbance of arbitrary wavenumbers is numerically examined by solving the Orr–Sommerfeld eigenvalue problem within the framework of normal mode analysis. The study reveals that instability rises with increasing external shear in the streamwise direction. However, as external shear rises in the reverse flow direction, wave energy is dissipated, and the surface wave stabilizes. Furthermore, the longwave expansion method is applied to calculate the nonlinear surface deformation expression, and it is found that the odd viscosity has the ability to stabilize the fluid flow instability caused by a positive shear force. The investigation of weakly nonlinear stability is also performed using the multiple scale method, which led to the Ginzburg–Landau equation of the nonlinear surface deformation equation. The corresponding results confirm the significant effect of both imposed shear and odd viscosity coefficient on the existent subcritical unstable and supercritical stable zones along with unconditional and explosive zones near the threshold of the film flow instability. The bandwidth of the subcritical stable zone mitigates for the higher viscosity ratio while it enhances the flow-directed potent imposed shear. Additionally, the amplitude and phase speed of nonlinear waves in the supercritical stable regime rise with increasing induced shear in the fluid flow direction and gradually decrease with increasing the value of the odd viscosity coefficient.