the continuity equation by introducing the stream function $\psi$ via

$$\Gamma{v_z} = \frac{\partial\psi}{\partial{\tau}} \label{psi1},$$

$$\rho'\kappa\,\Gamma{v_y} = -\frac{\partial\psi}{\partial{z}} \label{psi2},$$

where the factor of $\kappa$ ensures that $\psi$ remains dimensionless. In paper I we showed that there exist self-similar solutions for $\psi$, the velocity, the comoving density of scatterers and comoving density of photons in terms of the variable $\alpha = y/\delta{y}$, where $\delta{y} \sim \sqrt{z}$. One difference between the two-stream problem and this type of outflow, however, is that we expect the jet to expand into the ambient medium, entraining material in the process. Therefore, as we look farther along the $z$-direction, the amount of inertia – predominantly in the form of radiation for the systems that we are considering – contained in the flow will increase. Owing to the conservation of momentum, the $z$-component of the velocity should thus be a decreasing function of $z$.

In light of this observation, we will assume that the $z$-component of the four-velocity scales as

$$\Gamma{v_z} = \Gamma_jv_j\bigg{(}\frac{z}{z_0}\bigg{)}^{-m}\frac{df}{d\xi} \equiv \zeta\frac{df}{d\xi} \label{vzsim},$$

where $z_0$ is a characteristic length scale in the $z$-direction, $m$ is a positive constant, for brevity we defined $\zeta \equiv \Gamma_jv_j(z/z_0)^{-m}$, and $f$ is a function of the self-similar variable $\xi \simeq 