A review of one-phase Hele-Shaw flows and a level-set method for non-standard 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 standard and non-standard 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 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 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.