Computation of a shrinking interface in a Hele-shaw cell^\ast

In this paper, we present an adaptive rescaling method for computing a shrinking interface in a Hele-Shaw cell with a time increasing gap width b(t). We focus our study on a one-phase interior Hele-Shaw problem where a blob of fluid, surrounded by air, dynamically responds to the changing gap width. Linear analysis suggests that there exist transient fingering instabilities and noncircular self-similar evolutions depending on the dynamics of the gap b(t). Using linear theory, we identify a critical dynamic gap thickness bc_k(t) that separates stable shrinking behavior (shrinkage like a circle) from unstable shrinkage (shrinkage like a fingering pattern), where k is the wavenumber of the perturbation. The gap bc_k tends to infinity at a finite time. To explore the full nonlinear interface dynamics, we develop a spectrally accurate boundary integral method in which a new time and space rescaling is implemented. In the rescaled frame, the motion of the interface is slowed down, while the area/volume enclosed by the interface remains unchanged. This method, for the first time, enables us to adaptively remove the severe numerical stiffness imposed by the rapidly shrinking interface (especially at late times) and accurately compute the dynamics to far longer times than could previously be accomplished. Numerical tests demonstrate that the method is stable, efficient, and accurate. We perform nonlinear simulations for different dynamics of gap widths and, while the transient interface dynamics can be very complex, we find behavior generally consistent with the predictions of linear theory regarding the critical gap width. In particular, we find that when the b(t) increases exponentially in time, the nonlinear interface undergoes transient and sometimes dramatic morphological instabilities but eventually shrinks as a circle. When b = bck, or larger gap widths are used, our simulations reveal that at long times, the interface exhibits novel, strikingly thin k-fold morphologies that do not vanish as the interface shrinks, suggesting there exists mode selection in the nonlinear regime though the evolution is not self-similar.