Numerical study of Hele-Shaw flow with suction

Abstract

We investigate numerically the effects of surface tension on the evolution of an initially circular blob of viscous fluid in a Hele-Shaw cell. The blob is surrounded by less viscous fluid and is drawn into an eccentric point sink. In the absence of surface tension, these flows are known to form cusp singularities in finite time. Our study focuses on identifying how these cusped flows are regularized by the presence of small surface tension, and what the limiting form of the regularization is as surface tension tends to zero. The two-phase Hele-Shaw flow, known as the Muskat problem, is considered. We find that, for nonzero surface tension, the motion continues beyond the zero-surface-tension cusp time, and generically breaks down only when the interface touches the sink. When the viscosity of the surrounding fluid is small or negligible, the interface develops a finger that bulges and later evolves into a wedge as it approaches the sink. A neck is formed at the top of the finger. Our computations reveal an asymptotic shape of the wedge in the limit as surface tension tends to zero. Moreover, we find evidence that, for a fixed time past the zero-surface-tension cusp time, the vanishing surface tension solution is singular at the finger neck. The zero-surface-tension cusp splits into two corner singularities in the limiting solution. Larger viscosity in the exterior fluid prevents the formation of the neck and leads to the development of thinner fingers. It is observed that the asymptotic wedge angle of the fingers decreases as the viscosity ratio is reduced, apparently towards the zero angle ͑cusp͒ of the zero-viscosity-ratio solution.

On the motion of unsteady translating bubbles in an unbounded Hele-Shaw cell

“…The numerical method gives rise to "spurious oscillations" similar to those reported in other BIE methods, 30,[38][39][40] and these errors may grow in time as the interface evolves. 39 Aitchison and Howison 39 note the introduction of small wavelength instabilities into the solution of the numerical problem is a result of rounding errors and approximate solution techniques and shows that the frequency of the error oscillations scale with N. The difficulty being the more mesh points that are included (i.e., increasing N) shorter wavelength errors are permitted, and it is these which grow fastest in time.…”

“…Usually, increasing values of γ increases the smoothing effect on the solution. The filtering technique employed here is analogous to that of spectral methods, 30 for example, where the Fourier series is truncated and modes for the high frequency, small amplitude oscillations are neglected.…”

“…Since the time stepping procedure is explicit in time, a stability constraint applies 30 in the form ∆t ≤ c(∆S) n for some positive number n.…”

“…26 Another popular method used to formulate an integral equation on the free boundary is the vortex sheet method. 29,30 The approach taken in the present work is also to formulate as a boundary integral equation in the complex plane using Cauchy's integral formula.…”

Topological reconfiguration in expanding Hele—Shaw flow

“…In addition, the possibility of singularity formation demands extremely high resolution. Here we employ a spectrally accurate (infinite order) boundary integral method with fourth-order time integration [9] that uses the small-scale decomposition technique of Hou et al [10] to remove the high-order stability constraint induced by surface tension. Our highly accurate numerics reveal that in the case of an air bubble, surface tension regularizes the cusped-flow and the solution appears to exist for all times.…”