# Rigorous results in existence and selection of Saffman–Taylor fingers by kinetic undercooling

@article{Xie2018RigorousRI, title={Rigorous results in existence and selection of Saffman–Taylor fingers by kinetic undercooling}, author={Xuming Xie}, journal={European Journal of Applied Mathematics}, year={2018}, volume={30}, pages={63 - 116} }

The selection of Saffman–Taylor fingers by surface tension has been extensively investigated. In this paper, we are concerned with the existence and selection of steadily translating symmetric finger solutions in a Hele–Shaw cell by small but non-zero kinetic undercooling (ε2). We rigorously conclude that for relative finger width λ near one half, symmetric finger solutions exist in the asymptotic limit of undercooling ε2 → 0 if the Stokes multiplier for a relatively simple non-linear… Expand

#### One Citation

A REVIEW OF ONE-PHASE HELE-SHAW FLOWS AND A LEVEL-SET METHOD FOR NONSTANDARD CONFIGURATIONS

- Physics
- The ANZIAM Journal
- 2021

Abstract The classical model for studying one-phase Hele-Shaw flows is based on a highly nonlinear moving boundary problem with the fluid velocity related to pressure gradients via a Darcy-type law.… Expand

#### References

SHOWING 1-10 OF 46 REFERENCES

The selection of Saffman-Taylor fingers by kinetic undercooling

- Mathematics
- 2003

The selection of Saffman-Taylor fingers by surface tension has been widely studied. Here their selection is analysed by another regularisation widely adopted in studying otherwise ill-posed Stefan… Expand

Saffman-Taylor fingers with kinetic undercooling.

- Physics, Medicine
- Physical review. E, Statistical, nonlinear, and soft matter physics
- 2015

This work treats the numerical problem of computing these Saffman-Taylor fingers with kinetic undercooling, which turns out to be more subtle than the analog with surface tension, since kinetic under Cooling permits finger shapes which are corner-free but not analytic. Expand

Rigorous Results in Steady Finger Selection in Viscous Fingering

- Mathematics
- 2003

This paper concerns the existence of a steadily translating finger solution in a Hele-Shaw cell for small but non-zero surface tension (ɛ2). Though there are numerous numerical and formal asymptotic… Expand

Analytic theory for the selection of a symmetric Saffman-Taylor finger in a Hele-Shaw cell

- Physics
- 1987

An analytic theory is presented for the selection mechanism of a symmetric finger with a discrete set of possible width from a continuum of nonsymmetric Saffman–Taylor finger solutions of arbitrary… Expand

Analytic theory of the Saffman-Taylor fingers.

- Physics, Medicine
- Physical review. A, General physics
- 1988

A complete analytic solution is presented for the selection of symmetric Saffman-Taylor fingers, in the small\char21{}surface-tension limit, and two methods are proposed to extract the selection mechanism from this asymptotic series. Expand

Corner and finger formation in Hele-Shaw flow with kinetic undercooling regularisation

- Physics, Mathematics
- European Journal of Applied Mathematics
- 2014

We examine the effect of a kinetic undercooling condition on the evolution of a free boundary in Hele-Shaw flow, in both bubble and channel geometries. We present analytical and numerical evidence… Expand

Regularization by Kinetic Undercooling of Blow-up in the Ill-posed Stefan Problem

- Mathematics, Computer Science
- SIAM J. Appl. Math.
- 2005

The effect of kinetic undercooling is considered as a regularizing mechanism to prevent the formation of such singularities and the continuation of the solution through the "near blow-up" regime is studied. Expand

Analyticity and Nonexistence of Classical Steady Hele-Shaw Fingers

- Mathematics
- 2003

This paper concerns analyticity of a classical, steadily translating nger in a Hele-Shaw cell and nonexistence of solutions when the relative nger width is smaller than 1 . It is proven that any… Expand

Kinetic undercooling in Hele-Shaw flows.

- Materials Science, Medicine
- Physical review. E, Statistical, nonlinear, and soft matter physics
- 2015

Under radial Hele-Shaw flow, it is found that kinetic undercooling delays, but does not suppress, the development of finger tip-broadening and fingertip-splitting phenomena, and this work shows in a quantitative manner that the kinetic under Cooling contribution varies as a linear function of the normal velocity at the interface. Expand

The effect of surface tension on the shape of fingers in a Hele Shaw cell

- Physics
- 1981

The experimental results of Saffman & Taylor (1958) and Pitts (1980) on fingering in a Hele Shaw cell are modelled by two-dimensional potential flow with surface-tension effects included at the… Expand