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}
}
  • Xuming Xie
  • Published 1 October 2016
  • Mathematics
  • European Journal of Applied Mathematics
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
A REVIEW OF ONE-PHASE HELE-SHAW FLOWS AND A LEVEL-SET METHOD FOR NONSTANDARD CONFIGURATIONS
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
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 StefanExpand
Saffman-Taylor fingers with kinetic undercooling.
TLDR
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
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 asymptoticExpand
Analytic theory for the selection of a symmetric Saffman-Taylor finger in a Hele-Shaw cell
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 arbitraryExpand
Analytic theory of the Saffman-Taylor fingers.
TLDR
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
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 evidenceExpand
Regularization by Kinetic Undercooling of Blow-up in the Ill-posed Stefan Problem
TLDR
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
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 anyExpand
Kinetic undercooling in Hele-Shaw flows.
TLDR
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
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 theExpand
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