determined by comparing leading-order terms in the boundary layer thickness to the inviscid terms in the gas energy equation. Setting the left-hand side of equation to $\sim e'\Gamma_jv_j/z$, we find, as in Paper I,

$$\delta^2 \sim \frac{1}{\rho'\,\kappa\,{z}\,\Gamma_j\,v_j} \label{deltaeq}.$$

Here $v_j$ and $\Gamma_j = (1-v_j^2)^{-1/2}$ are the jet velocity and Lorentz factor, respectively, measured at some characteristic length along the jet axis $z_0$, and $\rho'$ is a characteristic density of scatterers throughout the outflow. Due to the fact that the boundary layer thickness is the same, the boundary layer equations governing the outflow are identical to those found in paper I, which can be compactly written as

$$\nabla_{\mu}\bigg{[}\bigg{(}\rho'+\frac{4}{3}e'\bigg{)}U^{\mu}U^{\nu}\bigg{]}+\frac{1}{3}g^{\mu\nu}\partial_{\mu}e' = \frac{8}{27}\frac{\partial}{\partial{y}}\bigg{[}\frac{e'}{\rho'\kappa}\frac{\partial{U^{\nu}}}{\partial{y}}\bigg{]} \label{bleqs},$$

$$\nabla_{\mu}\bigg{[}\rho'U^{\mu}\bigg{]} = 0.$$

In paper I we dealt with the $\nu = y$ and $\nu = z$ components of equation and its contraction with $U_{\nu}$ – the gas energy equation . For the free-streaming jet problem, however, it will be more convenient to deal with the $\nu = y$ component of equation , the gas energy equation, and the contraction of equation with the projection tensor $\Pi^{z}_{\,\,\nu}$, which, as we will see, is the relativistic, viscous counterpart of the Bernoulli equation. Performing a few manipulations, we find that these equations 