Shear-imposed falling thin Newtonian film over a porous slippery surface

Abstract

The stability of a Newtonian thin film flow over a porous slippery wall approximated by Darcy's law is investigated. The modified Orr–Sommerfeld system is derived for the frequency-dependent linear stability analysis and energy-budget analysis. Moreover, in the longwave regime, both linear and weakly nonlinear stability analyses are conducted for small aspect ratios. In addition, the multiple scale approach is performed directly in the nonlinear deformation equation of the free surface to predict the extraordinary behavior of the amplitude and speed of the nonlinear disturbance in the subcritical and supercritical regimes. The study finds that the larger slip-velocity and externally imposed shear on the thin film increase the total kinetic energy of the infinitesimal perturbations. In a longwave regime, the critical conditions of the primary instability are described as a function of imposed shear stress that destabilizes the film flow for low critical Reynolds number. Furthermore, in the supercritical stable zone, both the nonlinear wave amplitude and phase speed increase with an increase in induced shear in the flow direction and velocity slip, and a reverse trend is observed in applying the imposed shear in the opposite flow direction. On the other hand, the nonlinear wave amplitude in the subcritical unstable zone increases and decreases, corresponding to the larger values of imposed shear and slip parameters, respectively.