Thermal stress tensor in fluids
 

I am looking at a compressible viscous fluid that compresses under heat applied to it around the edges. I know what the thermal stress tensor for an elastic material, but things are different for a viscous fluid. Does anyone know the correct mathematical equation for a viscous fluid?

As long as we are talking about thermodynamic equilibrium and not going into the statistical thermodynamics or QED then the usual compressible navier stokes is sufficient (it's already compressible)

Unfortunately, it isn't. I have my material in an oven and its compressibility is determined by the heat is coming from an oven. So unfortunately that doesn't apply here. I need to add it in.

Compressibility is already taken care of by the EOS.

For gases anyway, the only thing you can do better is to use the real gas law instead of ideal gas law which has the isothermal/volumetric compressibility in it. Note that the ideal gas model constrains the isothermal compressibility

Where it typically breaks down is for non-newtonian behavior because you are heating a slurry and not a gas but I doubt that is your question.

So you don't know. I know what the continuity equation is. That's not what I'm asking.

I get what you're asking, what is the stress tensor look like for a jello

So can you say what is your substance?


My point is you are just looking for a constitutive relation for your specific substance, which is usually trade information, and it's not really a theoretical problem.

I'll use use the isotropic thermal stress tensor: and hopefully that will give me what I need.

Not sure what you want, in fluid dynamics we have a foundation about the fact that the stress tensor and the velocity gradient are linked by means of the Newtonian model.

Thus, temperature acts in the stress tensor by means of the two viscosity coefficients but, in general, the Stokes hypothesis reduces to only the viscosity function mu(T) and a functional relation like Sutherland is used.


However, the disregarded contribution to the tensor appears like mu2*(Div v)I