Dynamic-wetting effects in finite-mobility-ratio Hele-Shaw flow

Abstract

In this paper we study the effects of dynamic wetting on the immiscible displacement of a high viscosity fluid subject to the radial injection of a less viscous fluid in a Hele-Shaw cell. The displaced fluid can leave behind a trailing film that coats the cell walls, dynamically affecting the pressure drop at the fluid interface. By considering the nonlinear pressure drop in a boundary element formulation, we construct a Picard scheme to iteratively predict the interfacial velocity and subsequent displacement in finite-mobility-ratio flow regimes. Dynamic wetting delays the onset of finger bifurcation in the late stages of interfacial growth and at high local capillary numbers can alter the fundamental mode of bifurcation, producing vastly different finger morphologies. In low mobility ratio regimes, we see that finger interaction is reduced and characteristic finger breaking mechanisms are delayed but never fully inhibited. In high mobility ratio regimes, finger shielding is reduced when dynamic wetting is present. Finger bifurcation is delayed, which allows the primary fingers to advance further into the domain before secondary fingers are generated, reducing the level of competition.

Washing wedges: capillary instability in a gradient of confinement

“…The variety of possible dissipation mechanisms makes it difficult to derive an analytical expression for the temporal evolution of the amplitude of fingering instabilities. It is still a topic of debate, and recent works discussing the complexity of patterns obtained in the radial Saffman-Taylor instability show that the resulting dynamics is far from being fully understood (Bischofberger, Ramachandran & Nagel 2014;Jackson et al 2015). Here, we discuss some possible physical mechanisms involved at the interface between the two fluids.…”

“…Equivalently, Park & Homsy (1984) have shown that in the case of a non-wetting fluid displacing a wetting phase in a Hele-Shaw cell, dynamic wetting modifies the interfacial pressure jump by adding a term of order (γ /h)Ca 2/3 w . Following the steps of Park & Homsy (1984), a number of authors (Schwartz 1986;Reinelt 1987;Maxworthy 1989;Jackson et al 2015) have included contact line dissipation in this manner, and have concluded that accounting for dynamic wetting gives better predictions of the fingering patterns in the Saffman and Taylor instability. Implementing this additional term into (3.3) leads to a modified expression for the driving pressure difference:…”

Immiscible thermo-viscous fingering in Hele-Shaw cells

“…The solution technique is the same in this paper, with changes only in the spatial form of the mobility, which is now evaluated from the transient temperature field. The BEM and RBF-FC have also been used separately in several previous works, see [27,30] and [31,32] respectively. The homogeneous pressure (13) is solved using an indirect boundary element method which explicitly tracks the interface, providing greater accuracy compared to front capturing methods, at the expense of increased meshing computational complexity.…”

“…The contact angle of the meniscus has been assumed to be zero and we neglect dynamic wetting effects. Dynamic wetting has been shown by various authors to have a considerable effect on the interfacial displacement in a Hele-Shaw cell at high capillary numbers and requires study in its own right, putting it beyond the scope of the current work ( [25,26,27]). The displacement of the outer fluid is initiated by the injection of the inner fluid with a point source of strength Q at the origin.…”

“…To ensure that mass has been conserved in the numerical method, the volume of fluid can be calculated through numerical integration of the evolving interface (using the average interface position r) and compared to the value given by equation (27). The BEM has been validated for homogeneous mobility cases in previous work by [30], meaning the mass conservation tests presented here will validate that the RBF-FC method is contributing a correct velocity (from the perturbed pressure) to displace the interface and that the corresponding temperature field transports accurately.…”