Tangential energy equation

To conclude the set of equations of the star component with the interaction terms, we have the following equation:

$$\frac{\partial{p_{\rm t}}}{\partial {t}} + \frac{1}{r^2} \fr... ...ial}{\partial r}(r^2F_{\rm t})+\frac{2F_{\rm t}}{r}= \nonumber$$

$$\bigg( \frac{\delta p_{\rm t}}{\delta t}\bigg)_{\rm drag}+\bigg( \frac{\delta p_{\rm t}} {\delta t}\bigg)_{\rm coll},$$ (36)

where

$$\bigg( \frac{\delta p_{\rm t}}{\delta t}\bigg)_{\rm drag}=-2... ...X_{\rm coll} \rho_{\star} \tilde{\sigma_{\rm t}}^2 \epsilon.$$ (37)

We follow the same path like in the last case and so we get the following logarithmic variable equation:

$$\frac{\partial \ln p_{\rm t}}{\partial t}+(u_{\star}+2v_{\rm t}) ... ...\partial}{\partial r}(u_{\star}+2v_{\rm t})+ \frac{4}{r}(u_{\star}+2v_{\rm t})-$$ (38)

$$\frac{4}{5} \frac{2\frac{p_{\rm r}}{p_{\rm t}}-1}{t_{\rm relax}}=... ... \frac{1}{p_{\rm t}} \bigg( \frac{\delta p_{\rm t}} {\delta t}\bigg)_{\rm coll}$$