Singular effects of surface tension in evolving Hele-Shaw flows

Siegel, Michael; Tanveer, S.; Dai, Wei

doi:10.1017/s0022112096000894

Cited by 50 publications

(77 citation statements)

References 19 publications

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“…The daughter singularity impact time can be estimated by solving an ordinary differential equation. 22,23 Since the neck formation and the bulging of the finger are observed around t c , we rule out the effects of the daughter singularity for the particular flow we consider here.…”

Section: Further Discussion and Conclusionmentioning

confidence: 99%

Numerical study of Hele-Shaw flow with suction

Ceniceros

Hou

1999

Physics of Fluids

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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.

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Section: Further Discussion and Conclusionmentioning

confidence: 99%

Numerical study of Hele-Shaw flow with suction

Ceniceros

Hou

1999

Physics of Fluids

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“…10,[13][14][15][16][17] Moreover, the ZST problem is ill-posed in the sense that small perturbations may cause large effects in the resulting interface evolution. 18,19 The present work uses numerical smoothing to regularise the problem enabling convergent numerical results to be computed. The choice to use smoothing rather than surface tension offers an alternative way to understand the selection problem for bubbles in an unbounded Hele-Shaw cell.…”

Section: Introductionmentioning

confidence: 99%

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

2015

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Unsteady propagating bubbles in an unbounded Hele-Shaw cell are considered numerically in the case of zero surface tension. The instability of elliptical bubbles and their evolution toward a stable circular boundary, with speed twice that of the fluid speed at infinity, is studied numerically and by stability analysis. Numerical simulations of bubbles demonstrate that the important role played by singularities of the Schwarz function of the bubble boundary in determining the evolution of the bubble. When the singularity lies close to the initial bubble, two types of topological change are observed: (i) bubble splitting into multiple bubbles and (ii) a finite fluid blob pinching off inside the bubble region. C 2015 AIP Publishing LLC.

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“…Indeed, it would be of great interest to explore the evolution of daughter singularities in the present problem where there exist two competing physical mechanisms which might affect their evolution and determine whether they first impact the inner or outer interface. Moreover, an impacting daughter singularity cluster is likely to more readily explain the typical finger-like formations (as opposed to cusps) that are seen to develop on the interfaces in the experiments (see, for example, [18]). This interesting problem is left for the future, and the present results are expected to form a basis for any developments in this direction.…”

Section: « J\c\=pmentioning

confidence: 94%

Theory of exact solutions for the evolution of a fluid annulus in a rotating Hele-Shaw cell

Crowdy¹

2002

Quart. Appl. Math.

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Abstract.Motivated by a series of recent experiments on the evolution of fluid annuli in rotating Hele-Shaw cells, this paper presents a new class of exact time-dependent solutions to a mathematical model of this nonlinear free boundary problem. For a certain class of initial conditions, the free boundary problem is reduced to the solution of a finite set of coupled nonlinear ordinary differential equations. These solutions can be explicitly studied and, despite the fact that the model problem is mathematically ill-posed, display the same qualitative features as the recent experiments.It is discussed how the present exact solutions might form an important basis for further study of the appropriately regularized model problem.

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Singular effects of surface tension in evolving Hele-Shaw flows

Cited by 50 publications

References 19 publications

Numerical study of Hele-Shaw flow with suction

Numerical study of Hele-Shaw flow with suction

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

Theory of exact solutions for the evolution of a fluid annulus in a rotating Hele-Shaw cell

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