Numerical simulations of viscous fingering patterns in a Hele-Shaw cell whose geometry has been manipulated

A commonly used experimental tool for studying viscous fingering instabilities is the classical Hele-Shaw cell, which is an apparatus that is made up of two parallel plates held close together. Typically, when a less viscous fluid penetrates a more viscous fluid, the interface is unstable and fingering patterns and structures form. More recently, there is interest in manipulating the physical geometry of the Hele-Shaw cell with a view to promote or inhibit viscous fingering, for example by tapering the plates, making the gap thickness time-dependent, or by replacing one of the plates with an elastic membrane. Here, we discuss fully-nonlinear numerical simulations of some of these non-standard scenarios, computed via a level set method. We shall summarise various control strategies that are derived from applying linear stability analysis and demonstrate how these strategies can be implemented numerically.