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

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