Spontaneous imbibition in a microchannel: analytical solution and assessment of volume of fluid formulations

Abstract

The simulation of multi-phase flow at low capillary numbers (Ca) remains a challenge. Approximate computations of the capillary forces tend to induce parasitic currents (PC) around the interface. These PC induce additional viscous dissipation and shear forces that potentially lead to wrong calculations of the general flow dynamics. Here, we focus on the case of spontaneous imbibition in a microchannel of Hele-Shaw cell symmetry with capillarity being the only driving force. We extend the Lucas–Washburn equation to account for arbitrary viscosity ratios and assess four volume-of-fluid (VOF) formulations against the analytical solution. More specifically, we evaluate the continuum surface force (CSF) formulation, the sharp surface force (SSF) formulation, the filtered surface force (FSF) formulation and the piecewise linear interface calculation (PLIC) formulation extended by a higher order discretisation of the interface curvature through a height function with respect to accuracy, performance and heuristic parameters. We quantify PC for each formulation and investigate their impact on flow with Ca < 10 - 2. The magnitude of PC are largest for CSF and are reduced two fold by SSF. FSF reduces PC considerably more but shows periodic short bursts in the velocity field. PLIC shows no PC for the studied Ca and viscosity ratios. However, it fails when a denser fluid displaces a lighter fluid. Despite PC, all formulations are accurate within 10%. PLIC is suited to serve as a reference but relies on a structured mesh and is computationally expensive. FSF requires more heuristic parameters. Together with periodic bursts, this prevents a conclusive statement on the best choice between SSF and FSF.