0.5 Equations of motion and energy in cartesian coordinates  (Page 4/12)

This may be simplified by subtracting from it the expression we already have for the rate of change of kinetic energy, using Reynolds transport theorem, and Green's theorem.

Since this is valid for any arbitrary material volume, we have assumed continuity of the integrand

We assume Fourier's law for the conduction of heat.

We assume a Newtonian fluid for the dissipation of energy.

Substituting this back into the energy balance we have

Physically we see that the internal energy increases with the influx of heat, the compression and the viscous dissipation.

If we write the equation in the form

the left-hand side can be transformed by one of the fundamental thermodynamic identities. For if S is the specific entropy,

Substituting this into the last equation for internal energy gives

Giving an equation for the rate of change of entropy. Dividing by $T$ and integrating over a material volume gives

The second law of thermodynamics requires that the rate of increase of entropy should be no less than the flux of heat divided by temperature. The above equation is consistent with this requirement because the volume integral on the right-hand side cannot be negative. It is zero only if $k$ or $\nabla T$ and are zero. This equation also shows that entropy is conserved during flow if the thermal conductivity and viscosity are zero.

Assignment 6.2

The effect of compressibility (batcehlor, 1967)

Isentropic flow. The condition of zero viscosity and thermal conductivity results in conservation of entropy during flow or isentropic flow. This ideal condition is useful for illustration the effect of compressibility on fluid dynamics. The conservation of entropy during flow implies that density, pressure, and temperature are changing in a reversible manner during flow. The relation between entropy, density, temperature, and pressure is given by thermodynamics.

These relations may be combined with the condition that the material derivative of entropy is zero to obtain a relation between temperature and pressure during flow.

The equation of state expresses the density as a function of temperature and pressure. During isentropic flow the pressure and temperature are not independent but are constrained by constant entropy or adiabatic compression and expansion. The density in this case is given as

We now have as many equations as unknowns and the system can be determined. The simplifying feature of isentropic flow is that exchanges between the internal energy and other forms of energy are reversible, and internal energy and temperature play passive roles, merely changing in response to the compression of a material element. The continuity equation and equation of motion governing isotropic flow may now be expressed as follows.