Modelling of miscible liquids with the Korteweg stress
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 5, p. 741-753

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 exist during some time. They can lead to the appearance of an “effective interfacial tension”. To study these phenomena we use the mathematical model consisting of the diffusion equation with convective terms and of the 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 converges to the uniform composition distribution with zero velocity field. We also present numerical simulations of miscible drops and show how transient interfacial phenomena can change their shape.

Classification:  35K50,  76D05
Keywords: miscible liquids, Korteweg stress, drops
     author = {Kostin, Ilya and Marion, Martine and Texier-Picard, Rozenn and Volpert, Vitaly A.},
     title = {Modelling of miscible liquids with the Korteweg stress},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {37},
     number = {5},
     year = {2003},
     pages = {741-753},
     doi = {10.1051/m2an:2003042},
     zbl = {pre02029413},
     mrnumber = {2020862},
     language = {en},
     url = {}
Kostin, Ilya; Marion, Martine; Texier-Picard, Rozenn; Volpert, Vitaly A. Modelling of miscible liquids with the Korteweg stress. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 37 (2003) no. 5, pp. 741-753. doi : 10.1051/m2an:2003042.

[1] D.M. Anderson, G.B. Mcfadden and A.A. Wheeler, Diffuse interface methods in fluid mechanics. Annu. Rev. Fluid Mech. 30 (1998) 139-165.

[2] L.K. Antanovskii, Microscale theory of surface tension. Phys. Rev. E 54 (1996) 6285-6290.

[3] J.W. Cahn and J.E. Hilliard, Free energy of a nonuniform system. I. Interfacial Free Energy. J. Chem. Phys. 28 (1958) 258-267.

[4] D. Joseph and M. Renardy, Fundamentals of two-fluid dynamics, Vol. II. Springer, New York (1992). | Zbl 0784.76003

[5] D.J. Korteweg, 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. | JFM 32.0756.02

[6] O.A. Ladyzhenskaya, Mathematical theory of viscous incompressible flow. Gordon and Breach (1963). | MR 155093 | Zbl 0121.42701

[7] J.L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires. Gauthier-Villars, Paris (1969). | MR 259693 | Zbl 0189.40603

[8] 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).

[9] 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).

[10] P. Petitjeans, Une tension de surface pour les fluides miscibles. C. R. Acad. Sci. Paris Sér. I Math. 322 (1996) 673-679.

[11] R. Temam, Navier-Stokes equations. Theory and numerical analysis. North-Holland Publishing Co., Amsterdam-New York, Stud. Math. Appl. 2 (1979). | Zbl 0426.35003

[12] R. Temam, Navier-Stokes equations and nonlinear functional analysis. SIAM (1983). | Zbl 0833.35110

[13] J.S. Rowlinson, 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.

[14] V. Volpert, J. Pojman and R. Texier-Picard, Convection induced by composition gradients in miscible liquids. C. R. Acad. Sci. Paris Sér. I Math. 330 (2002) 353-358. | Zbl 1076.76597