For the outflow boundary, perhaps something like

\rho(r,\theta,\phi,t) = \rho_1 f_1(\theta) + \rho_2 f_2(\theta)

\vec{v}(r,\theta, \phi,t) = v_1 g_1(\theta) \hat{r} + v_2 g_2 (\theta) \hat{r}

T(r, \theta, \phi,t) = T_1h(\theta)

which is held fixed within some region - usually spherical or conical determined by R_0 and possibly \theta_0

and for the initial conditions, perhaps something like

\rho(r, \theta, \phi) = \rho_0

\vec{v}(r, \theta, \phi) = 0

T(r, \theta, \phi) = T_0