I may not be able to answer the question, but, can you please indicate in which reference this tensor has been found? (I have not seen it before either, but am now curious)

Hi,

This is reference to Korteweg Stress Tensor which arises due interface energy between two different phases.

A) First part of the answer

When the Navier-Stokes/Cahn-Hilliard model is used, the Korteweg tensor is often expressed in its equivalent potential form -C grad(mu) in Navier-Stokes equation where C is the composition and mu the chemical potential. You can see the following reference p 153 (relationship 2.9)

https://doi.org/10.1017/S0022112001004682

In equation (2.10) both terms of the Korteweg tensor appear.

To prove the equivalence, it is easier to consider the notations with the Einstein summation convention and the kronecker symbol. To guide you to prove it, you can see few details in the book http://eu.wiley.com/WileyCDA/WileyTitle/productCd-1118971337.html

calculus (9.10) p 294.

B) Second part of the answer

A quick proof of the Korteweg tensor is derived p 143-144 in Article Diffuse-Interface Methods in Fluid Mechanics

C) Last part of the answer

Another method is possible in order to derive the Korteweg tensor.

The demonstration starts from the differential of internal energy du with an important additional term : dot product of d(grad rho) and its conjugate vector (chosen K.(grad rho)) where K=Kappa is the capillary coefficient.

Next, use the conservation equations du/dt, ds/dt and du/dt (where s is the entropy) and manipulate the term K.grad(rho).d(grad rho)/dt. Finally express the entropy source and you will derive the relationships of the stress tensor T and the heat flux q with the term K.grad(rho). For T, the result is the Korteweg tensor.