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 boundary integral method where a coupled integral equation system is formulated. Although the stiffness due to the high order spatial derivatives can be removed, the long-time simulation is still expensive since the evolving velocity 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 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 flux 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 $\displaystyle (t_*-t)^b$, where $t_*$ is the estimated pinch-off time.
翻译:我们调查了Hele-Shaw 电池中移动界面的非线性动态,该界面在机内应用电场。我们开发了一个光谱精确的边界整体方法,其中设计了一个组合整体方程系统。虽然可以去除高顺序空间衍生物的僵硬性,但随着界面扩展,界面不断演变的速度急剧下降,长期的模拟费用仍然很高。我们采用一个调整方案,加速了缓慢的动态并降低了计算成本,从而消除了这种物理硬性。我们的非线性结果显示,正流抑制了手指拉动,并促进了模式的总体稳定。另一方面,负流使界面更加不稳定,导致将手指和小内部区域连接起来的细尾结构形成。当没有注入通量,使用负流时,界面倾向于接近源头并分解成几滴。我们调查了界面和源之间最小距离之间的时间变化,并发现它符合平方法律 $\ diplaysteg (t\-t) ⁇ b$, 其中, $t\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\\