Computational analysis of interfacial instabilities in Hele-Shaw cells with small gap gradient

We present a theoretical and numerical study on the stability of the interface between two fluids in a Hele-Shaw cell. Specifically, we consider the effect of a geometric taper in the direction of flow, across a range of capillary numbers $Ca$. We supplement linear stability results with fully-resolved 3D simulations (thus computing the flow field in the Hele-Shaw cell) carried out using the InterFoam solver in OpenFOAM, which employs the volume-of-fluid method to evolve the fluid-phase-field and enhances accuracy via an interface-compressing term in the continuity equation.
Three types of Hele-Shaw cells are considered: diverging, converging and parallel. Al-Housseiny et al. found a critical $Ca$ in the converging cell, below which the interface is stabilized. We extend this analysis by introducing a local $Ca$, which varies along the flow direction in tapered cells. Based on the difference between the critical $Ca$ and the inlet or outlet $Ca$, the (in)stability scenarios are divided into three regimes for each cell. Results from our 3D simulations show good agreement with the theoretically predicted (in)stability regimes and linear growth rates well with this theoretical analysis, validating our classification.