ESAIM: Mathematical Modelling and Numerical Analysis

Research Article

Modelling of Miscible Liquids with the Korteweg Stress

Kostin, Ilyaa1, Marion, Martinea2, Texier-Picard, Rozenna3 and Volpert, Vitaly A.a3

a1 Université de Franche-Comté, UMR 6623 CNRS, 25030 Besançon, France. [email protected]

a2 École Centrale de Lyon, UMR 5585 CNRS, 69134 Ecully Cedex, France. [email protected]

a3 Université Lyon 1, UMR 5585 CNRS, 69622 Villeurbanne Cedex, France. [email protected], [email protected]


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.

(Received November 4 2002)

(Online publication November 15 2003)

Key Words:

  • Miscible liquids;
  • Korteweg stress;
  • drops.

Mathematics Subject Classification:

  • 35K50;
  • 76D05