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Singularity formation in thin jets with surface tension
Author(s) -
Pugh M. C.,
Shelley M. J.
Publication year - 1998
Publication title -
communications on pure and applied mathematics
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 3.12
H-Index - 115
eISSN - 1097-0312
pISSN - 0010-3640
DOI - 10.1002/(sici)1097-0312(199807)51:7<733::aid-cpa2>3.0.co;2-7
Subject(s) - conservative vector field , gravitational singularity , inviscid flow , singularity , jet (fluid) , surface tension , classical mechanics , mechanics , instability , mathematics , compressibility , biot number , mathematical analysis , physics , thermodynamics
We derive and study asymptotic models for the dynamics of a thin jetof fluid that is separated from an outer immiscible fluid by fluidinterfaces with surface tension. Both fluids are assumed to beincompressible, inviscid, irrotational, and density‐matched. Onesuch thin jet model is a coupled system of PDEs with nonlocalterms—Hilbert transforms—that result from expansion of aBiot‐Savart integral. In order to make the asymptotic modelwell‐posed, the Hilbert transforms act upon time derivatives of thejet thickness, making the system implicit. Within this thin jetmodel, we demonstrate numerically the formation of finite‐timepinching singularities, where the width of the jet collapses to zeroat a point. These singularities are driven by the surface tensionand are very similar to those observed previously by Hou, Lowengrub,and Shelley in large‐scale simulations of the Kelvin‐Helmholtzinstability with surface tension and in other related studies.Dropping the nonlocal terms, we also study a muchsimpler local model. For this local model we can precludeanalytically the formation of certain types of singularities, thoughnot those of pinching type. Surprisingly, we find that this localmodel forms pinching singularities of a very similar type to thoseof the nonlocal thin jet model.© 1998 John Wiley & Sons, Inc.