Existence of weak solutions for particle-laden flow with surface tension

Roman M. Taranets; Jeffrey T. Wong

doi:10.3934/dcds.2018217

Article Contents

2018, Volume 38, Issue 10: 4979-4996. Doi: 10.3934/dcds.2018217

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Existence of weak solutions for particle-laden flow with surface tension

Roman M. Taranets^1,2,3, , and
Jeffrey T. Wong^2,4,

1.
Institute of Applied Mathematics and Mechanics of the NASU, 1, Dobrovol'skogo Str., 84100, Sloviansk, Ukraine
2.
Department of Mathematics, University of California, Los Angeles, California 90095-1555, USA
3.
Vasyl' Stus Donetsk National University, 21, 600-richya Str., 21021, Vinnytsia, Ukraine
4.
Department of Mathematics, Duke University, Durham, North Carolina, 27708-0320, USA

* Corresponding author: Roman M. Taranets
* Corresponding author: Roman M. Taranets

Received: September 30, 2017

Revised: April 30, 2018

Published: September 2018

This work was supported in part by NSF grant DMS-1312543, and by a grant from Ministry of Education and Science of Ukraine (0118U003138 to Roman Taranets).

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Abstract

We prove the existence of solutions for a coupled system modeling the flow of a suspension of fluid and negatively buoyant non-colloidal particles in the thin film limit. The equations take the form of a fourth-order non-linear degenerate parabolic equation for the film height $h$ coupled to a second-order degenerate parabolic equation for the particle density $ψ$ . We prove the existence of physically relevant solutions, which satisfy the uniform bounds $0 ≤ ψ/h ≤ 1$ and $h ≥ 0$ .

Keywords:

Fourth-order degenerate parabolic equations,
existence of weak solutions,
thin liquid films,
suspensions,
multiphase flows.

Mathematics Subject Classification: Primary: 35K65, 35K55; Secondary: 76D08.

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References

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