STABILITY OF GRAVIEN FLOW PAST A NEO HOOKEAN DEFORMABLE SOLID LAYER

Abstract

In this report, we have analyzed the flow of a Newtonian fluid down an incline plane, which is coated with a neo-Hookean solid layer. The linear stability at zero and non-zero Reynolds number has been studied. It is shown that the two interfaces are present, liquid-solid (LS) and gas-liquid (GL), which can become unstable under certain circumstances. The objective of this work is to suppress these instabilities and to analyze the effect of solid layer deformability on the instability behavior. These GL and LS interfacial instabilities are studied in detail in order to assess the prior predictions in zero and finite Re limit. We are using an Eulerian-Lagrangian approach for the coupled fluid-solid system. We have introduced some modifications in the normal and tangential stress condition at the LS interface. One extra term will appear in each of the stress continuity equation. These additional terms arises due to the different but correct approach adopted by us. It is found that, in the creeping flow limit, the GL mode is stable at all wavenumber, but it is true only below some critical value of angle of inclination θ. The LS mode is stable at low and finite wavenumber, but instabilities are present at high wavenumber limits. This short wave instability has arisen due to the non-zero first normal stress difference at the base state. These results are at variance to those predicted earlier, where it was shown that both GL and LS modes become unstable at finite wavenumber limit for the case of creeping flows. We attribute this difference to the modified LS interfacial boundary conditions. For the case of finite Re limit, the GL mode is unstable in the finite k region. This free surface instability could be completely stabilized by the deformability of the solid layer. This is better represented by the vs. k plots, which shows stability window, in the limits of which the instabilities could be completely suppressed. On further increasing the deformability parameter , the flow becomes again unstable for both the GL and LS modes in the finite k wavenumber