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.

Comparison of Surface Tension Models for the Volume of Fluid Method

“…Raeini et al [76] Introduced a sharp surface force formulation to calculate the capillary force, which is then filtered to reduce the spurious currents (known as FSF model). Neglecting the filtering terms in the FSF model provides a sharp surface formulation of surface tension known as SSF, which is described in [42]. The SSF has been reported to be reduce the spurious currents by two to three orders in comparison to CSF [42].…”

“…Neglecting the filtering terms in the FSF model provides a sharp surface formulation of surface tension known as SSF, which is described in [42]. The SSF has been reported to be reduce the spurious currents by two to three orders in comparison to CSF [42]. The FSF model has been reported to provide periodic bursts in the velocity fields but lower spurious current than SSF [42].…”

“…The SSF has been reported to be reduce the spurious currents by two to three orders in comparison to CSF [42]. The FSF model has been reported to provide periodic bursts in the velocity fields but lower spurious current than SSF [42]. The approach has been used to model bubbles in microfluidic devices [51] and flow through porous media [52].…”

“…Some examples of flow phenomena used to compare surface tension models are rising bubbles whose diameters are in the order of few millimetres [33][34][35], translating and rotating bubbles [64], oscillating droplets or bubbles [34], stagnant bubbles or droplets [34,35,39,64], Rayleigh-Taylor instability [37,38], Taylor bubbles [64], falling films [41], droplet splashing [38,39], capillary rise [42] and bubble evolution [37,40]. These typically compare the CSF model with height functions [33,34,64], PROST [37], PLIC [42], CLSVOF and its variants [37][38][39][40]64], FSF and SSF [42], and CSS [35,41] models. Although the flow scenarios that are used to compare surface tension models are diverse, they can be broadly categorised based on the dominance of surface tension in the flow using the Capillary number (Ca), which is defined as the ratio of viscous to surface tension forces in the system.…”

A model of spontaneous flow driven by capillary pressure in nanoporous media

“…Thus, it is essential to understand the dynamic characters of spontaneous imbibition in these media with nano-pores. A great interest has been focused on this topic for more than a century, with many models developed (Handy, 1960;Morrow and Mason, 2001;Alava et al, 2004;Schmid et al, 2016;Pavuluri et al, 2018;Abd et al, 2019).…”