Consider a uniform mixture of different fluids (all spatial derivatives vanish), with no external forces, such that the equation of motion for the a species becomes

(a) Show that the time rate of change of the total fluid kinetic energy density, w_k, is given by

where

(b) Consider now the total fluid thermal energy density,

If the energy equation for a homogeneous fluid mixture, with no external forces, is

then show that the time rate of change of W_T is given by

Thus, the total thermal energy density W_T increases at exactly the same rate as the total kinetic energy density W_k decreases. Notice that for any function that is summed over two indices, the result is unchanged if we interchange the indices, so that

Transcribed Image Text:

dua dt - Σναβ(Μα - ug) · β