Singularity formation in thin jets with surface tension

M. C. Pugh, Michael Shelley

Research output: Contribution to journalArticle

Abstract

We derive and study asymptotic models for the dynamics of a thin jet of fluid that is separated from an outer immiscible fluid by fluid interfaces with surface tension. Both fluids are assumed to be incompressible, inviscid, irrotational, and density-matched. One such thin jet model is a coupled system of PDEs with nonlocal terms - Hubert transforms - that result from expansion of a Biot-Savart integral. In order to make the asymptotic model well-posed, the Hilbert transforms act upon time derivatives of the jet thickness, making the system implicit. Within this thin jet model, we demonstrate numerically the formation of finite-time pinching singularities, where the width of the jet collapses to zero at a point. These singularities are driven by the surface tension and are very similar to those observed previously by Hou, Lowengrub, and Shelley in large-scale simulations of the Kelvin-Helmholtz instability with surface tension and in other related studies. Dropping the nonlocal terms, we also study a much simpler local model. For this local model we can preclude analytically the formation of certain types of singularities, though not those of pinching type. Surprisingly, we find that this local model forms pinching singularities of a very similar type to those of the nonlocal thin jet model.

Original languageEnglish (US)
Pages (from-to)731-795
Number of pages65
JournalCommunications on Pure and Applied Mathematics
Volume51
Issue number7
StatePublished - Jul 1998

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Surface Tension
Surface tension
Singularity
Fluids
Fluid
Model
Kelvin-Helmholtz Instability
Finite-time Singularities
Immiscible Fluids
Hilbert Transform
Term
Coupled System
Transform
Derivatives
Derivative
Zero
Demonstrate
Simulation

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Singularity formation in thin jets with surface tension. / Pugh, M. C.; Shelley, Michael.

In: Communications on Pure and Applied Mathematics, Vol. 51, No. 7, 07.1998, p. 731-795.

Research output: Contribution to journalArticle

Pugh, M. C. ; Shelley, Michael. / Singularity formation in thin jets with surface tension. In: Communications on Pure and Applied Mathematics. 1998 ; Vol. 51, No. 7. pp. 731-795.
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