Elastic fingering in rotating Hele-Shaw flows

Abstract

The centrifugally driven viscous fingering problem arises when two immiscible fluids of different densities flow in a rotating Hele-Shaw cell. In this conventional setting an interplay between capillary and centrifugal forces makes the fluid-fluid interface unstable, leading to the formation of fingered structures that compete dynamically and reach different lengths. In this context, it is known that finger competition is very sensitive to changes in the viscosity contrast between the fluids. We study a variant of such a rotating flow problem where the fluids react and produce a gellike phase at their separating boundary. This interface is assumed to be elastic, presenting a curvature-dependent bending rigidity. A perturbative weakly nonlinear approach is used to investigate how the elastic nature of the interface affects finger competition events. Our results unveil a very different dynamic scenario, in which finger length variability is not regulated by the viscosity contrast, but rather determined by two controlling quantities: a characteristic radius and a rigidity fraction parameter. By properly tuning these quantities one can describe a whole range of finger competition behaviors even if the viscosity contrast is kept unchanged.

Nonlinear simulations of elastic fingering in a Hele-Shaw cell

“…The model also agrees with the experimental observations in the sense that a reactive system has destabilizing effect and the fingering instability is greater than that in a nonreactive system which is also confirmed in [15]. The elastic interface idea is also considered in a rotating Hele-Shaw flow [16,17]. In [17], the authors demonstrate a beautiful set of stationary morphologies depending on the competition between elastic and centrifugal forces.…”

“…To better elucidate the gains in efficiency using the rescaling scheme, we take a time dependent flux,Ĵ(t) = −105(37A 1 + A 2 ) 4R 3 (t) following Eq. (16). According to linear theory, this flux will lead the evolution to a 6-fold self-similar shape.…”

Computation of a Shrinking Interface in a Hele-Shaw Cell

“…Besides the classical Hele-Shaw setup, there are several variants related to the viscous fingering problem [37,21,20,4,12,14,13,40]. For example, Hele-Shaw cells where the top plate is lifted uniformly at a prescribed speed and the bottom plate is fixed (lifting plate problem) [5,12,35,43,47,45,46] have been used to study adhesion related problems such as debonding [16,2,38,11] and the associated probe tack test [51,25].…”

Development of tip-splitting and side-branching patterns in elastic fingering

“…As in Refs. [20][21][22][23][24] we treat the interface as an elastic membrane, presenting a curvature-dependent bending rigidity as given by Eq. (2).…”

“…Later, Carvalho and coworkers [21] utilized the curvature-dependent bending rigidity model proposed in [20] to carry out a weakly nonlinear analysis of the system, and showed that when A = 0 nonlinear effects play a crucial role to determine the general shape assumed by the resulting patterned structures [21]. Subsequently, the same group of researchers investigated the manifestation of elastic fingering in rotating Hele-Shaw cells, where still unexploited pattern-forming dynamic behaviors [22], and innovative stationary morphologies [23] have been revealed as a result of the competition between elastic and centrifugal forces.…”