Linear temporal and spatiotemporal stability analysis of two-layer falling films with density stratification

Instabilities of a free bilayer flowing on an inclined porous medium

The instabilities of a free bilayer flowing on an inclined Darcy-Brinkman porous layer have been explored. The bilayer is composed of a pair of immiscible liquid films with a deformable liquid-liquid interface and a liquid-air free surface. An Orr-Sommerfeld analysis of the governing equations and boundary conditions uncovers that this configuration can be unstable by a pair of long-wave interfacial modes at the free surface and at the interface together with a couple of finite wave-number shear modes originating from the inertial influences at the liquid layers. In particular, one of the shear modes originates beyond a threshold flow rate owing to the slippage at the porous-liquid interface and is found to be the dominant one even when the porous medium is moderately thin, porous, and permeable. The strength of the porous media mediated mode (a) grows with increase in porosity, (b) grows and then remains invariant with increase in thickness, and (c) initially grows and then decays with increase in the permeability of the porous layer. Further, the presence of a lower layer with smaller viscosity and a thicker upper layer is found to facilitate the growth of this newly identified porous media mode. Importantly, beyond a threshold upper to lower thickness and viscosity ratios and the angle of inclination the porous media mode dominates over all the other interfacial or shear modes, highlighting its importance in the bilayer flows down an inclined porous medium. The study showcases the importance of a porous layer in destabilizing a free bilayer flow down an inclined plane, which can be of importance to improve mixing, emulsification, and heat and mass transfer characteristics in the microscale devices.

Stability of two-layer shear-thinning film flows

The stability of a two-layer film flow of non-Newtonian fluids is studied with a linear temporal approach. Shear-thinning fluids are considered, which follow the four-parameter inelastic Carreau model. A modified Orr-Sommerfeld equation system is obtained, which is solved by using a spectral Tau collocation method based on Chebyshev polynomials. The effects of density and viscosity stratification are considered, as well as the influence of the shear-thinning properties of the fluid. It is found that, when the viscosity is stronger in the upper layer, the base flow and the stability properties are almost not influenced by the change of the shear-thinning properties in this upper layer. In the other situations, the shear-thinning properties have an influence on the different instabilities, the long-wave surface instability and the short- and long-wave interface instabilities.