Numerical Study on Viscous Fingering Using Electric Fields in a Hele-Shaw Cell

We investigate the nonlinear dynamics of a moving interface in a Hele-Shaw cell subject to an in-plane applied electric field. We develop a spectrally accurate nu-merical method for solving a coupled integral equation system. Although the stiffness due to the high order spatial derivatives can be removed using a small scale decom-position technique, the long-time simulation is still expensive since the evolving ve-locity of the interface drops dramatically as the interface expands. We remove this physically imposed stiffness by employing a rescaling scheme, which accelerates the slow dynamics and reduces the computational cost. Our nonlinear results reveal that positive currents restrain finger ramification and promote the overall stabilization of patterns. On the other hand, negative currents make the interface more unstable and lead to the formation of thin tail structures connecting the fingers and a small inner region. When no fluid is injected, and a negative current is utilized, the interface tends to approach the origin and break up into several drops. We investigate the temporal evolution of the smallest distance between the interface and the origin and find that it obeys an algebraic law (t* -t)b, where t* is the estimated pinch-off time.