Formação de padrão na célula de Hele - Shaw girante: um estudo computacional através do método de phase- field

Abstract

In the past few decades many scientific papers were published about hydrodynamic instabilities in the Hele-Shaw cell and pattern formation through viscous fingering in the interface between two immiscible fluids. This problem was analyzed both by experimental and theoretical physicists, due its prototypical role in describing interfacial growth, such as oil recovery, solidification, electrical discharges, etc. When a less viscous fluid displaces a more viscous one, these instabilities arise due to the difference in viscosity between the two confined liquids (Saffman-Taylor instability), and the driving force of this destabilization is the fluid injection rate. However, other physical mechanisms may destabilize an initially circular interface in this effectively two-dimensional geometry and produce qualitatively different viscous fingering phenomena. In this work, we approach the problem of a rotating Hele-Shaw cell, where the instabilities are now induced by the centrifugal force and depend on the density difference between the fluids. This variant version of the original Saffman-Taylor problem was previously studied in the literature trough analytical, numerical and experimental investigations, and several morphological and dynamical properties of viscous fingering were explored. Here, we revisit this system with the goal of reproducing several of these observations through computational simulations utilizing the Phase-Field method for two-dimensional flow with finite surface tension. This method consists in rewriting the equations in terms of an auxiliary scalar field that regularizes the interface between the fluids by introducing an artificial finite thickness to the phase separation region. By proceeding this way, we are able to numerically integrate partial differential equations that automatically implement the boundary conditions of the problem, without the necessity of tracking the moving interface. Via this technique, we have performed numerical simulations of the time evolution of initially circular droplets in situations where we introduce symmetric perturbations at its interface and other cases where we use asymmetric perturbations. In this study, we analyze the role of both dimensionless parameters that govern the system’s dynamics: the dimensionless surface tension B, which measures the ratio between capillary and centrifugal forces; and the dimensionless viscous contrast A between the fluids. Our observations compare well with the theoretical predictions of the linear perturbation analysis, both qualitatively and quantitatively, and are in agreement with previous results contained in the scientific literature on the subject.