A Review of One-Phase Hele-Shaw Flows and a Level-Set Method for Nonstandard Configurations

Abstract

The classical model for studying one-phase Hele-Shaw flows is based on a highly nonlinear moving boundary problem with the fluid velocity related to pressure gradients via a Darcy-type law. In a standard configuration with the Hele-Shaw cell made up of two flat stationary plates, the pressure is harmonic. Therefore, conformal mapping techniques and boundary integral methods can be readily applied to study the key interfacial dynamics, including the Saffman–Taylor instability and viscous fingering patterns. As well as providing a brief review of these key issues, we present a flexible numerical scheme for studying both the standard and nonstandard Hele-Shaw flows. Our method consists of using a modified finite-difference stencil in conjunction with the level-set method to solve the governing equation for pressure on complicated domains and track the location of the moving boundary. Simulations show that our method is capable of reproducing the distinctive morphological features of the Saffman–Taylor instability on a uniform computational grid. By making straightforward adjustments, we show how our scheme can easily be adapted to solve for a wide variety of nonstandard configurations, including cases where the gap between the plates is linearly tapered, the plates are separated in time, and the entire Hele-Shaw cell is rotated at a given angular velocity.

Interfacial instabilities in confined displacements involving non-Newtonian fluids

“…These studies clearly demonstrate the important role of fluid rheology in pattern formation. There are many other experimental setups used to study fluid-fluid displacements [87] that we have not discussed due to space constraints. We would like to refer the reader to the literature involving lifting Hele-Shaw cells, wherein the cell gap is time dependent [88,89] and inertial effects influence the growth of the emergent interfacial patterns [90].…”