Fluid Mixing from Viscous Fingering

Abstract

Viscous fingering is a well-known hydrodynamic instability that sets in when a less viscous fluid displaces a more viscous fluid [1]. When the two fluids are miscible, viscous fingering introduces disorder in the velocity field and exerts a fundamental control on the rate at which the fluids mix. We present a fluid dynamics video of the mixing process in a viscously unstable flow, generated from a high-resolution numerical simulation using a computational strategy that is stable for arbitrary viscosity ratios. We develop a two-equation dynamic model of concentration variance and mean dissipation rate to quantify the degree of mixing in such a displacement process [2]. The model reproduces accurately the evolution of these two quantities as observed in highresolution numerical simulations and captures the nontrivial interplay between channeling and creation of interfacial area as a result of viscous fingering.

Dynamics of a Highly Viscous Circular Blob in Homogeneous Porous Media

“…Nevertheless, the asymmetry never vanishes, since the tail formation persists with viscosity contrast of all orders feasible with the current numerical computations. We also calculated the degree of mixing [29,30],…”

“…At later times, the degree of mixing of the blob with the ambient fluid has a non-trivial dependence on R. For R l c < R < R u c , the rear interface of the blob undergoes VF, in addition to a tail formation at the frontal interface. Consequently, the area of contact between the two fluids increases [29], which eventually increases mixing. However, as R is increased beyond R u c , the front interface deforms into a tail and no VF is observed.…”

Interface evolution during radial miscible viscous fingering

“…Other studies have also focused on the effects of anisotropic dispersion [31,33,36], medium heterogeneity [32,[37][38][39][40], gravity [41][42][43][44][45][46], chemical reactions [3,[47][48][49], absorption [50], and flow configuration [51][52][53][54][55] on the viscous fingering instability. Despite the extensive work done, the effect of viscous fingering on mixing has only recently been investigated numerically for a rectilinear geometry [14,56]. While the dynamics of the interface between two miscible fluids is crucial to explain and predict the rate of mixing [1,16], a solid understanding of the temporal evolution of the viscously unstable fluid-fluid interface is still missing.…”

“…We define the fluid-fluid interface γ as the set of points where ∇χ is locally maximum, which corresponds to where the mixing rate is also locally maximum [14,19]. Here, we focus on understanding the mechanisms controlling the temporal evolution of the length of the interface γ , and we propose an effective model able to describe and predict its dynamics.…”

“…In such mixing-driven systems, the flow responsible for fluid displacement reflects the heterogeneity of the host medium structure [8][9][10][11] or the physical properties of the fluids, such as density [3,12,13] or viscosity [14,15]. Mixing takes place at the fluid-fluid interface and is determined by the combined action of molecular diffusion, which acts to reduce the local concentration gradients, and advection, which controls the interface dynamics [16][17][18][19][20][21].…”