There is an entropy inequality in fluid mechanics, and from bounds to that, a first order wave (convective wave) is given.
With assumptions of ideal gas, internal energy proportional to temperature, I think the energy balance may give a relation for entropy.
Are there other ways, also invoking statistical mechanics?
There may be other more useful variables, that are energy conjugated to temperature or gradient of temperature.
Thank you Mr L.S. That is a good approach.
@Dr Denaro
First order wave.
I meant the same as they teach, found by googling. The entire time derive for a function following a flow with velocity v. I think it is known as Eulerian description, when the velocity is x,t where x are (small) coordinates for the moving material point. Solutions depend on x-vt where v is linearised to constant.
It is in analogy with the sum of terms for acceleration in rigid body mechanics, but there it is a rotating system velocity, and also Coriolit acceleration.
What do you think about the 'linearisation'? It would be possibly to expand into one more term, depending on the situation, assuming explicit or implicit dependency on relevant properties.
I think you are talking about a first order wave equation...
however, the entropy equation is
ds/dt + v.grad s =0
for isoebtropic flow assumption. The equation says that a material particle mantains the same entropy along its path line. This is not true for flow with energy dissipation.
you have to consider the full Euler system of equations and I suggest to see the Riemann invariant
As far as your original question is concerned, consider that the NS equations are a system that is closed by the Newton and Fourier relation and the equations of state. Therefore, apparently the entropy does not appear as an equation required in the system but its dynamic is implied by the NS equations (that take into account the dissipative effetcs and the production of entropy).
Different is the non-viscous model (Euler equations) where dissipative effects are disregarded. You can derive the entropy equation by considering s-s0 = Cv log (p/rho^gamma). It must be introduced in the Euler equation and requires some manipulation. As reference, you can see the book of Zucrow and the book of Leveque.
The entropy equation for the Navier-Stokes-Fourier (NSF) equations is derived in Linear Irreversible Thermodynamics (see the classical book by de Groot & Mazur). The equation is not needed to solve the equations, but mainly to determine entropy generation. In fact, the Newton stress and Fourier heat flux are constructed to guarantee positive entropy generation. The derivation relies on the assumption of local equilibrium, which in simple terms state that thermal and caloric equation of state, and the Gibbs equation are the same as in equilibrium (note that the fluid is not in equilibrium, since there is stress and heat flux). This assumption can be verified within the framework of kinetic theory of gases, for the case of small Knudsen numbers. Hence, after solving the NSF equations, you can use the result to determine the corresponding entropy. Even if you solve linearized NSF for waves, you must consider non-linear entropy equation, since the entropy generation is quadratic in velocity and temperature gradients.
Thanks all.
Filippo. I derived an entropy equality with that relation, and the ideal gas law, together with Navier-Stokes. I have the notes, but got confused since two state laws.
Lena I suggest starting with the 1d Euler equations in primitive variable (rho, u, p). You should find ds/dt + u ds/dx=0
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