Numerical Study on Viscous Fingering Using Electric Fields in a Hele-Shaw Cell (Communications in Computational Physics, 2023)
We investigate the nonlinear dynamics of a moving interface in a Hele-Shawcell subject to an in-plane applied electric field. We develop a spectrally accurate numerical method for solving a coupled integral equation system. Although the stiffnessdue to the high order spatial derivatives can be removed using a small scale decomposition technique, the long-time simulation is still expensive since the evolving velocity of the interface drops dramatically as the interface expands. We remove thisphysically imposed stiffness by employing a rescaling scheme, which accelerates theslow dynamics and reduces the computational cost. Our nonlinear results reveal thatpositive currents restrain finger ramification and promote the overall stabilization ofpatterns. On the other hand, negative currents make the interface more unstable andlead to the formation of thin tail structures connecting the fingers and a small innerregion. When no fluid is injected, and a negative current is utilized, the interface tendsto approach the origin and break up into several drops. We investigate the temporalevolution of the smallest distance between the interface and the origin and find that itobeys an algebraic law $(t_∗−t)^b,$ where $t_∗$ is the estimated pinch-off time.