Linear Stability Analysis of a Model Planar Free Surface Viscoelastic Displacement Flow

The linear stability of a model planar stagnation viscoelastic free surface displacement flow, utilizing the Oldroyd-B and the FENE-P models is studied via generalized eigenvalue analysis. Specifically, we have examined the effect of the ratio of solvent to total solution viscosity (η), the Weissenberg number (Wi) and the Capillary number (Ca) on the stability of the flow by monitoring the leading discrete mode. It is demonstrated that long waves are the most dangerous modes and their growth rate is a function of Wi, η, and Ca. Specifically, the disturbance growth rate is enhanced with increasing Wi and Ca and decreases with increasing η. Eigenfunction analysis shows that the perturbation normal stresses in the flow direction and the perturbation pressure, associated with the most dangerous mode, are localized near the free surface stagnation point. Furthermore, it is shown that elastic destabilization of the flow occurs due to the alteration of the interface normal stress balance at the free surface as a result of formation of an elastic stress boundary layer near the free surface stagnation point.