# Fluid pressure, energy and temperature

1. Aug 26, 2007

### sidman

Hi everyone. This is my first post here...
I want to create a numerical simulation in software, to solve the Navier-Stokes equations (newtonian fluids, compressible and viscous flow) and I have some questions regarding fluids and their dynamics.

(1) The Navier-Stokes equation for the conservation of momentum, contains a term which represents the spatial difference in pressure within the fluid. (the pressure gradient $$\nabla$$p) I know how to calculate gradients in general, but the problem is I can't find an expression for the pressure field, to begin with! I only found expressions for hydrostatic pressure w.r.t. depth in static fluids. I also found an equation of state, relating pressure to density, temperature and molecular mass but it's called "*perfect gas* equation of state". Would it hold for arbitrary, non-static fluids? Furthermore, shouldn't velocity affect pressure? it makes sense, I think.
(In most numerical simulations, the flow is considered incompressible and the calculation of this gradient is worked-around. In fact, it's used as a constraint to impose incompressibility.)

(2) What are the energy components in a fluid, that its "total energy" comprises? how are they defined? I suppose there should be a kinetic energy contribution, a thermal one, and perhaps a "potential" one? (e.g. one that would change into kinetic, as a stationary particle would flow towards areas of lower pressure due to the pressure gradient?) Is this correct?
Also, temperature is usually cited as "a measure of average kinetic energy". What does that mean? That it's simply proportional to kinetic energy? Does this hold "locally"? like for an infinitesimal fluid parcel of known velocity and density?

(3) My question here may be clear already from any answers to (2), but... here goes. I want to find the temperature throughout the fluid, and I understand that this is usually done with a 3rd equation (apart from the mass and momentum conservation in N-S) which comes from the conservation of energy. I found several forms of it, however two seem the most useful: the first one expresses the time-rate change of total energy E, and the other contains a product of temperature T and the time-rate change of entropy S. (they are related through the maxwell relation Τ = $$\partial$$E/$$\partial$$S)
Is it possible to find the temperature from the total energy, or any of its components mentioned in (2)? Or perhaps, is there a way of finding the entropy and substituting in the 2nd equation to solve for T? Any other useful formulas, relating energy/temperature/entropy? Perhaps, a simple equation of state would do, without even using the energy equation? (velocity and density are considered known)

I guess that's about it... I'm sorry if this is somewhat... lengthy, but I've been looking for days and can't find clear, useful information. So, I thought I'd ask here...
Naturally, I'm not expecting answers to everything from a single user, so feel free to answer/comment on anything you like. All contributions are highly appreciated.