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Modelling of Miscible Liquids with the Korteweg Stress

Published online by Cambridge University Press:  15 November 2003

Ilya Kostin ,
Martine Marion ,
Rozenn Texier-Picard  and
Vitaly A. Volpert
Ilya Kostin
Affiliation:
Université de Franche-Comté, UMR 6623 CNRS, 25030 Besançon, France. kostin@pdmi.ras.ru.
Martine Marion
Affiliation:
École Centrale de Lyon, UMR 5585 CNRS, 69134 Ecully Cedex, France. Martine.Marion@ec-lyon.fr.
Rozenn Texier-Picard
Affiliation:
Université Lyon 1, UMR 5585 CNRS, 69622 Villeurbanne Cedex, France. texier@maply.univ-lyon1.fr., volpert@maply.univ-lyon1.fr.
Vitaly A. Volpert
Affiliation:
Université Lyon 1, UMR 5585 CNRS, 69622 Villeurbanne Cedex, France. texier@maply.univ-lyon1.fr., volpert@maply.univ-lyon1.fr.
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Abstract

When two miscible fluids, such as glycerol (glycerin) and water,are brought in contact, they immediately diffuse in each other. However if the diffusion is sufficiently slow, large concentration gradients existduring some time. They can lead to the appearance of an “effective interfacial tension”. To study these phenomena we use the mathematical modelconsisting of the diffusion equation with convective terms and ofthe Navier-Stokes equations with the Korteweg stress.We prove the global existence and uniqueness of the solution for the associated initial-boundary value problem in a two-dimensional bounded domain.We study the longtime behavior of the solution and show that it convergesto the uniform composition distribution with zero velocity field.We also present numerical simulations of miscible drops and show howtransient interfacial phenomena can change their shape.

Keywords

Miscible liquidsKorteweg stressdrops.
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 37 , Issue 5 , September 2003 , pp. 741 - 753
Copyright
© EDP Sciences, SMAI, 2003

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