Dear Kamesh,

 Kindly,go to  the following link for your answer...

http://perso-math.univ-mlv.fr/users/danchin.raphael/benzoni-s.pdf

Consider the following simple example: two miscible fluids A and B, with different viscosities and densities. Suppose you are trying to displace fluid A in a tube by flowing  fluid B into the tube. The efficiency of the displacement process will depend on what happened at the boundary/interface between the two miscible fluid. Since the fluids are miscible, by definition there is no interfacial tension. One method for analyzing this system is to assume that the viscosity of the system depends on the volume faction of say species A. Further, since the fluids have different densities the flow field for the system is no longer solenoidal, i.e. \nabla.v is not equal to zero. The different densities also give rise to additional stresses in the bulk fluid. Korteweg accounted for these effects by adding an additional stress term that has the form ( expressed in Latex} T_{i,j}=\delta \frac{\partial \phi}{\partial x_i} \frac{\partial \phi}{\partial x_j}+\gamma \frac{\partial^2 \phi}{\partial x_i\partial x_j}

The quantities \delta and \gamma are called Korteweg coefficients and normally functions of density and volume fraction \phi. The problem then becomes solving the  conservation of mass for the system, the momentum equations and the species balance for the volume fraction. Joseph and coworkers have solved several problems dealing with Korteweg stresses and there is also a nice simulation paper by Chen and Meiburg, 2002.  Generally speaking the effects of Korteweg stresses are small but may be important in certain cases, and as Joseph has shown can lead to instabilities. But much work remains to be done.

Thank you. I was searching for the physical implications of these stresses. This was helpful!

Hello, the physical implication of Korteweg stresses may concern the stability of a transient interface between miscible fluids. Imagine to push a fluid (A) into pipe filled with another fluid (B). The stability of the interface, like the occurrence of fingering phenomena, is ruled by these stresses that may mimic a positive interfacial tension. You may be interested to look at some recent publications on the topic (see below).

Article Off-Equilibrium Surface Tension in Colloidal Suspensions

Article Bulk and interfacial stresses in suspensions of soft and hard colloids

Article Off-equilibrium surface tension in miscible fluids