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where is the specific enthalpy. However, for conservation of energy to hold in a viscous fluid subject to thermal conduction, the energy flux due to advection must be supplemented by a heat flux given by Fourier's law and a flux due to internal friction . Then the general equation for conservation of energy is:
Equation for entropy production
Note that the thermodynamic relations for the internal energy and enthalpy are given by:
We may also obtain an equation for the kinetic energy by taking the dot product of the Navier-Stokes equation with the flow velocity to yield:
The second term on the righthand side may be expanded to read:
With the aid of the thermodynamic relation for enthalpy and the last result, we may then put the kinetic energy equation into the form:
Now expanding the time derivative of the total energy, we have:
Then by expanding each of these terms, we find that:
And collecting terms, we are left with:
Now adding the divergence of the heat flux due to thermal conduction to each side, we have that:
However, we know that by the conservation of energy on the lefthand side is equal to zero, leaving us with:
The product of the viscous stress tensor and the velocity gradient can be expanded as:
Thus leading to the final form of the equation for specific entropy production:
In the case where thermal conduction and viscous forces are absent, the equation for entropy production collapses to - showing that ideal fluid flow is isentropic.
Application
This equation is derived in Section 49, at the opening of the chapter on "Thermal Conduction in Fluids" in the sixth volume of L.D. Landau and E.M. Lifshitz's Course of Theoretical Physics.[1] It might be used to measure the heat transfer and air flow in a domestic refrigerator,[4] to do a harmonic analysis of regenerators,[5] or to understand the physics of glaciers.[6]