Entropy Production

It is instructive to rewrite the species-$s$ energy evolution equation, Equation (4.49), as an entropy evolution equation (Hazeltine and Waelbroeck 2004). The fluid definition of entropy density, which coincides with the thermodynamic entropy density in the limit that the distribution function approaches a Maxwellian, is (Reif 1965)

$$s_s = n_s\ln\left(\frac{T_s^{\,3/2}}{n_s}\right)+c,$$ (4.51)

where $c$ is a constant. The corresponding entropy flux density is written

$${\bf s}_s = s_s\,{\bf V}_s + \frac{{\bf q}_s}{T_s}.$$ (4.52)

Clearly, entropy is convected by the fluid flow, but is also carried by the flow of heat, in accordance with the second law of thermodynamics (Reif 1965). After some algebra, Equation (4.49) can be rearranged to give

$$\frac{\partial s_s}{\partial t} + \nabla\cdot{\bf s}_s = {\mit\Theta}_s,$$ (4.53)

where the right-hand side is given by

$${\mit\Theta}_s = \frac{W_s}{T_s} - \frac{\mbox{\boldmath$\pi$}_s : \nabla {\bf V}_s}{T_s} - \frac{{\bf q}_s}{T_s} \cdot \frac{\nabla T_s}{T_s}.$$ (4.54)

It follows, from our previous discussion of conservation laws, that the quantity ${\mit\Theta}_s$ can be regarded as the entropy production rate per unit volume for species $s$ . Evidently, entropy is produced by collisional heating, viscous heating, and heat flow down temperature gradients.