Instabilities and singularities in Hele-Shaw flow

Raymond E. Goldstein, Adriana I. Pesci, Michael Shelley

Research output: Contribution to journalArticle

Abstract

A mechanism by which smooth initial conditions evolve towards a topological reconfiguration of fluid interfaces is studied in the context of Darcy's law. In the case of thin fluid layers, nonlinear PDEs for the local thickness are derived from an asymptotic limit of the vortex sheet representation. A particular example considered is the Rayleigh-Taylor instability of stratified fluid layers, where the instability of the system is controlled by a Bond number B. It is proved that, for a range of B and initial data "subharmonic" to it, interface pinching must occur in at least infinite time. Numerical simulations suggest that "pinching" singularities occur generically when the system is unstable, and in particular immediately above a bifurcation point to instability. Near this bifurcation point an approximate analytical method describing the approach to a finite-time singularity is developed. The method exploits the separation of time scales that exists close to the first instability in a system of finite extent, with a discrete spectrum of modes. In this limit, slowly growing long-wavelength modes entrain faster short-wavelength modes, and thereby, allow the derivation of a nonlinear evolution equation for the amplitudes of the slow modes. The initial-value problem is solved in this slaved dynamics, yielding the time and analytical structure of a singularity that is associated with the motion of zeros in the complex plane, suggesting a general mechanism of singularity formation in this system. The discussion emphasizes the significance of several variational principles, and comparisons are made between the numerical simulations and the approximate theory.

Original languageEnglish (US)
Pages (from-to)2701-2723
Number of pages23
JournalPhysics of Fluids
Volume10
Issue number11
StatePublished - 1998

Fingerprint

Fluids
fluids
vortex sheets
Bond number
Wavelength
nonlinear evolution equations
pulse detonation engines
Taylor instability
Initial value problems
Computer simulation
variational principles
boundary value problems
wavelengths
Vortex flow
derivation
simulation

ASJC Scopus subject areas

  • Fluid Flow and Transfer Processes
  • Computational Mechanics
  • Mechanics of Materials
  • Physics and Astronomy(all)
  • Condensed Matter Physics

Cite this

Goldstein, R. E., Pesci, A. I., & Shelley, M. (1998). Instabilities and singularities in Hele-Shaw flow. Physics of Fluids, 10(11), 2701-2723.

Instabilities and singularities in Hele-Shaw flow. / Goldstein, Raymond E.; Pesci, Adriana I.; Shelley, Michael.

In: Physics of Fluids, Vol. 10, No. 11, 1998, p. 2701-2723.

Research output: Contribution to journalArticle

Goldstein, RE, Pesci, AI & Shelley, M 1998, 'Instabilities and singularities in Hele-Shaw flow', Physics of Fluids, vol. 10, no. 11, pp. 2701-2723.
Goldstein RE, Pesci AI, Shelley M. Instabilities and singularities in Hele-Shaw flow. Physics of Fluids. 1998;10(11):2701-2723.
Goldstein, Raymond E. ; Pesci, Adriana I. ; Shelley, Michael. / Instabilities and singularities in Hele-Shaw flow. In: Physics of Fluids. 1998 ; Vol. 10, No. 11. pp. 2701-2723.
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