Hostname: page-component-cd9895bd7-jn8rn Total loading time: 0 Render date: 2024-12-24T03:09:29.091Z Has data issue: false hasContentIssue false
  • English
  • Français

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. [email protected].
Martine Marion
Affiliation:
École Centrale de Lyon, UMR 5585 CNRS, 69134 Ecully Cedex, France. [email protected].
Rozenn Texier-Picard
Affiliation:
Université Lyon 1, UMR 5585 CNRS, 69622 Villeurbanne Cedex, France. [email protected]., [email protected].
Vitaly A. Volpert
Affiliation:
Université Lyon 1, UMR 5585 CNRS, 69622 Villeurbanne Cedex, France. [email protected]., [email protected].
Get access

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

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Anderson, D.M., McFadden, G.B. and Wheeler, A.A., Diffuse interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30 (1998) 139-165. CrossRef
Antanovskii, L.K., Microscale theory of surface tension. Phys. Rev. E 54 (1996) 6285-6290. CrossRef
Cahn, J.W. and Hilliard, J.E., Free energy of a nonuniform system. I. Interfacial Free Energy. J. Chem. Phys. 28 (1958) 258-267. CrossRef
D. Joseph and M. Renardy, Fundamentals of two-fluid dynamics, Vol. II. Springer, New York (1992).
Korteweg, D.J., Sur la forme que prennent les équations du mouvement des fluides si l'on tient compte des forces capillaires causées par des variations de densité considérables mais connues et sur la théorie de la capillarité dans l'hypothèse d'une variation continue de la densité. Arch. Néerl. Sci. Exactes Nat. Ser. II 6 (1901) 1-24.
O.A. Ladyzhenskaya, Mathematical theory of viscous incompressible flow. Gordon and Breach (1963).
J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Gauthier-Villars, Paris (1969).
J. Pojman, N. Bessonov, R. Texier, V. Volpert and H. Wilke, Numerical simulations of transient interfacial phenomena in miscible fluids, in Proceedings AIAA, Reno, USA (January 2002).
J. Pojman, Y. Chekanov, J. Masere, V. Volpert, T. Dumont and H. Wilke, Effective interfacial tension induced convection in miscible fluids, in Proceedings of the 39th AIAA Aerospace Science Meeting, Reno, USA (January 2001).
Petitjeans, P., Une tension de surface pour les fluides miscibles. C. R. Acad. Sci. Paris Sér. I Math. 322 (1996) 673-679.
R. Temam, Navier-Stokes equations. Theory and numerical analysis. North-Holland Publishing Co., Amsterdam-New York, Stud. Math. Appl. 2 (1979).
R. Temam, Navier-Stokes equations and nonlinear functional analysis. SIAM (1983).
Rowlinson, J.S., Translation of J.D. van der Waals' ``The thermodynamic theory of capillarity under hypothesis of a continuous variation of density''. J. Statist. Phys. 20 (1979) 197. CrossRef
Volpert, V., Pojman, J. and Texier-Picard, R., Convection induced by composition gradients in miscible liquids. C. R. Acad. Sci. Paris Sér. I Math. 330 (2002) 353-358.