On Singularity Formation in a Hele-Shaw Model

Peter Constantin, Tarek Elgindi, Huy Nguyen, Vlad Vicol

Research output: Contribution to journalArticle

Abstract

We discuss a lubrication approximation model of the interface between two immiscible fluids in a Hele-Shaw cell, derived in Constantin et al. (Phys Rev E 47(6):4169–4181, 1993) and widely studied since. The model consists of a single one dimensional evolution equation for the thickness 2h = 2h(x, t) of a thin neck of fluid,∂th+∂x(h∂x3h)=0, for x∈(-1,1)andt≥0. The boundary conditions fix the neck height and the pressure jump: h(±1,t)=1,∂x2h(±1,t)=P>0. We prove that starting from smooth and positive h, as long as h(x, t) > 0, for x ∈ [−1, 1], t ∈ [0, T ], no singularity can arise in the solution up to time T. As a consequence, we prove for any P > 2 and any smooth and positive initial datum that the solution pinches off in either finite or infinite time, i.e., inf [ - 1 , 1 ] × [ 0 , T )h= 0 , for some T∈ (0 , ∞]. These facts have been long anticipated on the basis of numerical and theoretical studies.

Original languageEnglish (US)
Pages (from-to)139-171
Number of pages33
JournalCommunications in Mathematical Physics
Volume363
Issue number1
DOIs
StatePublished - Oct 1 2018

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Hele-Shaw
Lubrication Approximation
Singularity
Immiscible Fluids
fluids
lubrication
fixing
Evolution Equation
Jump
boundary conditions
Boundary conditions
Fluid
Cell
cells
approximation
Model

ASJC Scopus subject areas

  • Statistical and Nonlinear Physics
  • Mathematical Physics

Cite this

On Singularity Formation in a Hele-Shaw Model. / Constantin, Peter; Elgindi, Tarek; Nguyen, Huy; Vicol, Vlad.

In: Communications in Mathematical Physics, Vol. 363, No. 1, 01.10.2018, p. 139-171.

Research output: Contribution to journalArticle

Constantin, P, Elgindi, T, Nguyen, H & Vicol, V 2018, 'On Singularity Formation in a Hele-Shaw Model', Communications in Mathematical Physics, vol. 363, no. 1, pp. 139-171. https://doi.org/10.1007/s00220-018-3241-6
Constantin, Peter ; Elgindi, Tarek ; Nguyen, Huy ; Vicol, Vlad. / On Singularity Formation in a Hele-Shaw Model. In: Communications in Mathematical Physics. 2018 ; Vol. 363, No. 1. pp. 139-171.
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