Momentum balance

We modify equation number (2.9) of LangbeinEtAl90 in the following way:

$$\frac{\partial (\rho_{\rm g}u_{\rm g})}{\partial t}=u_{\rm g... ...artial t}+ \rho_{\rm g} \frac{\partial u_{\rm g}}{\partial t};$$ (43)

we substitute this equality in their equation, divide by $\rho_{\rm g}$ ($u_{\rm g}$ is the variable in our code) and make use of the equation of continuity for the gas component. Thus, we get the following expression:

$$\frac{\partial u_{\rm g}}{\partial t}+u_{\rm g} \frac{\parti... ...H= \bigg( \frac{\delta u_{\rm g}}{\delta t}\bigg)_{\rm coll}$$ (44)

To get the interaction term we use the mass and momentum conservation:

$$\bigg( \frac{\delta \rho_{\rm g}}{\delta t}\bigg)_{\rm coll}... ...ac{\delta \rho_{\star}}{\delta t}\bigg)_{\rm coll}=0 \nonumber$$

$$\bigg( \frac{\delta (\rho_{\rm g} u_{\rm g})}{\delta t}\bigg... ...\delta (\rho_{\star} u_{\star})}{\delta t}\bigg)_{\rm coll}=0.$$ (45)

We know that

$$\bigg( \frac{\delta u_{\star}}{\delta t}\bigg)_{\rm coll}=0,$$ (46)

thus,

$$\bigg( \frac{\delta (\rho_{\rm g} u_{\rm g})}{\delta t}\bigg... ...lta t}\bigg)_{\rm coll} + u_{\rm g} X_{\rm coll} \rho_{\star}.$$ (47)

Therefore, the resulting interaction term is

$$\bigg( \frac{\delta u_{\rm g}}{\delta t}\bigg)_{\rm coll}= \... ...\rho_{\star}}{\rho_{\rm g}} X_{\rm coll}(u_{\star}-u_{\rm g})$$ (48)

In the case of the stellar system

$$F= \frac{1}{2}(F_{\rm r}+F_{\rm t})=\frac{5}{2}\rho_{\star}v_{\star}$$ (49)

By analogy, we now introduce $F_{\rm rad}$ in this way

$$\frac{F_{\rm rad}}{4\pi}=H=\frac{5}{2}p_{\rm g}v_{\rm g},$$ (50)

where $v_{\rm g}$ is per gas particle.

$$v_{\rm g}=\frac{2}{5} \frac{H}{p_{\rm g}}$$ (51)

As means to write the equation in its ``logarithmic variable version'', we multiply the equation by $\rho_{\rm g} r /p_{\rm g}$, as we did for the corresponding momentum balance star equation and replace $H$ by $5/2p_{\rm g}v_{\rm g}$,

$$\frac{\rho_{\rm g}r}{p_{\rm g}} \bigg( \frac{\partial u_{\rm g}}{... ...\partial \ln r}-\frac{5}{2} \frac{\kappa_{\rm ext}}{c} \rho_{\rm g}r v_{\rm g}=$$

$$\frac{r}{p_{\rm g}} \rho_{\star}X_{\rm coll} (u_{\star}-u_{\rm g})$$ (52)