Numerical investigation of controlling interfacial instabilities in non-standard Hele-Shaw configurations

Abstract

Viscous fingering experiments in Hele-Shaw cells lead to striking pattern formations which have been the subject of intense focus among the physics and applied mathematics community for many years. In recent times, much attention has been devoted to devising strategies for controlling such patterns and reducing the growth of the interfacial fingers. We continue this research by reporting on numerical simulations, based on the level set method, of a generalised Hele-Shaw model for which the geometry of the Hele-Shaw cell is altered. First, we investigate how imposing constant and time-dependent injection rates in a Hele-Shaw cell that is either standard, tapered or rotating can be used to reduce the development of viscous fingering when an inviscid fluid is injected into a viscous fluid over a finite time period. We perform a series of numerical experiments comparing the effectiveness of each strategy to determine how these non-standard Hele-Shaw configurations influence the morphological features of the inviscid-viscous fluid interface. Tapering plates in either converging or diverging directions leads to reduced metrics of viscous fingering at the final time when compared to the standard parallel configuration, especially with carefully chosen injection rates; for the rotating plate case, the effect is even more dramatic, with sufficiently large rotation rates completely stabilising the interface. Next, we illustrate how the number of non-splitting fingers can be controlled by injecting the inviscid fluid at a time-dependent rate while increasing the gap between the plates. Simulations compare well with previous experimental results for various injection rates and geometric configurations. Further, we demonstrate how the fully nonlinear dynamics of the problem affect the number of fingers that emerge and how well this number agrees with predictions from linear stability analysis.

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Cited by in corpus (14)

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• Gas compression systematically delays the onset of viscous fingering
• A review of one-phase Hele-Shaw flows and a level-set method for non-standard configurations
• Computational Analysis of Interfacial Dynamics in Angled Hele-Shaw Cells: Instability Regimes
• Viscous fingering patterns for Hele--Shaw flow in a doubly connected geometry driven by a pressure differential or rotation
• Modelling finger propagation in elasto-rigid channels
• Tuning a magnetic field to generate spinning ferrofluid droplets with controllable speed via nonlinear periodic interfacial waves
• Growth morphology and symmetry selection of interfacial instabilities in anisotropic environments
• Moving boundary problems for quasi-steady conduction limited melting
• Delayed Hopf bifurcation and control of a ferrofluid interface via a time-dependent magnetic field
• Long-wave equation for a confined ferrofluid interface: Periodic interfacial waves as dissipative solitons
• Numerical study on viscous fingering using electric fields in a Hele-Shaw cell
• Linear stability analysis on the onset of viscous fingering due to a non monotonic viscosity profile for immiscible fluids