Verify that the energy conservation equation, for the random kinetic energy 1/2m c 2 can

Verify that the energy conservation equation, for the random kinetic energy 1/2m_αc²_α can be obtained from the viscous stress equation (see problem 8.8) by letting j = k, and summing over k.

Data from Problem 8.8

In the general transport equation, consider that the property χ(r, v, t) is the random momentum flux, that is, m_αC_αjC_αk· Show that, in this case,

where the summation convention on repeated indices is being used. Plug these results in the general transport equation to derive the following equation, known as the viscous stress equation:

Transcribed Image Text:

Ә дхi Ә at (na a) = -(na < Xvi >a) = na a=0 Әx Ət Za Pajk Ət ә дxi -(Qaijk + Uai Pajk) диак Paij ахi tk Juaj - Paik axi - >a=Pma