Suppression of viscous fluid fingering: A piecewise-constant injection process

Abstract

The injection of a fluid into another of larger viscosity in a Hele-Shaw cell usually results in the formation of highly branched patterns. Despite the richness of these structures, in many practical situations such convoluted shapes are quite undesirable. In this Brief Report, we propose an efficient and easily reproducible way to restrain these instabilities based on a simple piecewise-constant pumping protocol. It results in a reduction in the size of the viscous fingers by one order of magnitude.

Two-Phase Fluid Displacement and Interfacial Instabilities Under Elastic Membranes

“…The interface remains stable since the tip of any developing finger experiences higher capillary resistance than the lagging parts of the finger owing to the tapered geometry [12,14]. A number of studies have previously focused on a variety of injection schemes to control viscous fingering [23][24][25][26]. In contrast, elastic membranes present a simple geometrical approach.…”

Minimization of Viscous Fluid Fingering: A Variational Scheme for Optimal Flow Rates

“…A second line of research [18,19] searches for mechanisms that are capable not only of restraining densebranching formation in radial Hele-Shaw cells, but that also unfavors the establishment of interfacial deformations. In this context the ultimate goal would be to obtain front propagation in the form of nearly stable, axisymmetric interfaces.…”

“…In this context the ultimate goal would be to obtain front propagation in the form of nearly stable, axisymmetric interfaces. This has been achieved by utilizing a twostage piecewise constant injection process which applies a relatively low injection rate followed by a proper, stronger injection stage [18]. An alternative suppression method [19] employs a standard constant pumping rate, but substitutes the originally rigid HS upper plate with an elastic membrane.…”

“…Despite the ability of existing controlling methods [18,19] to damp interfacial irregularities, a fundamental question still remains unanswered: if one wishes to inject a certain amount of fluid in a given time, what would be the optimal time-dependent injection rate QðtÞ for which the perturbation amplitudes could be truly minimized? Considering the multiplicity of possibilities for time-varying injection scenarios the response to this pertinent query is decidedly nontrivial.…”

“…[12][13][14][15][16][17][18][19], we present a variational scheme which allows one to systematically search for the…”