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.

We examine the sensitivity of Saffman–Taylor fingers to controlled variations in channel depth by investigating the effects of centred, rectangular occlusions in Hele-Shaw channels. For large occlusions, the geometry is known to support symmetric, asymmetric and oscillatory propagation states when air displaces a more viscous fluid from within the channel. A previously developed depth-averaged model is found to be in quantitative agreement with laboratory experiments once the aspect ratio (width/height) of the tube’s cross-section reaches a value of 40. We find that the multiplicity of solutions at finite occlusion heights arises through interactions of the single stable and multiple unstable solutions already present in the absence of the occlusion: the classic Saffman–Taylor viscous fingering problem. The sequence of interactions that occurs with increasing occlusion height is the same for all aspect ratios investigated, but the occlusion height required for each interaction decreases with increasing aspect ratio. Thus, the system becomes more sensitive as the aspect ratio increases in the sense that multiple solutions are provoked for smaller relative depth changes. We estimate that the required depth changes become of the same order as the typical roughnesses of the experimental system ($1~{\rm\mu}\text{m}$) for aspect ratios beyond 155, which we conjecture underlies the extreme sensitivity of experiments conducted in such Hele-Shaw channels.