Pressure can be derived from free energy. P=- \frac{\partial F}{\partial V}

For nonhomogeneous fluid, the pressure is not a scalar, but a tensor. Then how to get this tensor?

Who can give me some advices or tell me some paper about it?

Thanks.

Actually the pressure is ALWAYS a scalar by definition. (In a nonhomogenous fluid it's the rest of the stress tensor that's changed; i.e the extrenous stress).

Sorry, maybe I didn't state it clearly.

I'm working on the problem about gas-liquid interface, in which there is suface tension. Some papers pointed out that there were nondiagonal componants in the pressure tensor.

In the case of a gas-liquid interface with surface tension you simply have

jump in pressure [P] across the interface = T( 1/R_1 + 1/R_2)

where T is the surface tension and R_1,2 are the principal radii of curvature.

You then have the usual equations for the flow velocity and pressure in the liquid and gas separately.

(The pressure Tensor is p times the kronnecker delta where p is a scalar in all cases - even in reacting mixtures/combustion theory)

If I use the Laplace equation, I have to compute the two curvature of the interface which means I have to trace the interface.

Now I use diffuse interface model, in which the interface is not sharp and I can only get the derivative of the density.

In fact, in nonhomogeneous fluid, the free energy has some another terms about the derivative of the density and which the free energy fuction of homogeneous fluid hasn't. I wish to derive the pressure tensor from this free energy with the derivative of the density.

In fact, some papers did this work. But there is no detail.

why the pressure is ALWAYS a scalar by definition. where is the EOS in that case, could you please expalin the reason.

If you derive the Navier-Stokes equation by taking moments of the Boltzmann equation, you find that you have to evaluate a second-rank stress tensor. In the limit of a collision-dominated fluid (mean free path << length scale of the flow), this tensor reduces to the scalar pressure times the unit tensor, plus the viscous stresses. This is the limit in which we normally do thermodynamics. The surface tension at an interface is an additional physical effect, and it is also a continuum concept.