Instability of Oldroyd-B Liquid Films with Odd Viscosity on Porous Inclined Substrates

In this paper, we investigate the effect of singular viscosity on the stability of a thin film of Oldroyd-B viscoelastic fluid flowing along a porous inclined surface under the influence of a normal electric field. First, we derive the governing equations and boundary conditions for the flow of the film and assume that the film satisfies the Beavers-Joseph sliding boundary condition when it flows on a porous inclined surface. Second, through the long-wave approximation, we derive the nonlinear interfacial evolution equation. Then, linear and nonlinear stability analyses are performed for the interfacial evolution equation. The stability analyses show that the singular viscosity has a stabilizing effect on the flow of the film, while the strain delay time of the Oldroyd-B fluid, the electric field, and the parameters of the porous medium all have an unsteady effect on the flow of the film. Interestingly, in the linear stability analysis, the parameters of the porous medium have an unsteady effect on the flow of the film after a certain value is reached and a stabilizing effect before that value is reached. In order to verify these results, we performed numerical simulations of the nonlinear evolution equations using the Fourier spectral method, and the conclusions obtained are in agreement with the results of the linear stability analysis, i.e., the amplitude of the free surface decreases progressively with time in the stable region, whereas it increases progressively with time in the unstable region.