Relativistic blast waves in medium of non uniform density The relativistic outflow of a gamma ray burst drives a shock into the circumburst medium. This medium could have a uniform density $n_0$ or a density profile of the form, $n(r) \propto r^{-2}$.
For the uniform case, the Lorentz factor of the relativistic outflow $\gamma(t)$ can be found, as in Eq (125) here which draws on the work of Blandford & Mckee (1976). 
Can anyone point me in the direction of a similar expression for the $n(r) \propto r^{-2}$ case?
Thanks
 A: Thorough Approach
So I read through Blandford & McKee, [1976].
In their Section IV. D. they discuss density gradients of $n(r) \propto r^{-k}$.  The special case of $k$ = 2 corresponds to

...a constant power source feeding a blast wave in a constant velocity wind...

In that case, the continuity equation (their equation 16):
$$
\partial_{t} n' + \frac{1}{r^{2}} \partial_{r} \left( r^{2} \beta n' \right) = 0 \tag{1}
$$
where $\partial_{\alpha}$ is the partial derivative with respect to variable $\alpha$, $r$ is the radial distance, $\beta = v/c$, and $n' = \gamma^{-1} n$ or the density in the co-moving frame with Lorentz factor $\gamma$.  If we can assume that $v \sim constant$, then Equation 1 goes to:
$$
\gamma^{-1} \partial_{t} n + \frac{\beta}{\gamma \ r^{2}} \partial_{r} \left( r^{2} n \right) = 0 \tag{2}
$$
If we substitute in the assumed density gradient, we get:
$$
\partial_{t} r^{-k} + \frac{\beta}{r^{2}} \partial_{r} \left( r^{2 - k} \right) = 0 \tag{3}
$$
where the left-hand side goes to zero in the limit that $k$ = 2.
Easy Approach
One could just as easily apply a much simpler approach and again assume mass density continuity.  Since the problem is one-dimensional (i.e., only depends upon radial distance), we can get away with a similar zeroth order approximation to that is done in fluid flow problems.  Since we ignore sources/losses and assume steady state (i.e., $\partial_{t} \rightarrow 0$), then the continuity equation in integral form is:
$$
\int \ dV \ \nabla \cdot \left( \rho \ \mathbf{u} \right) = \oint \ dA \ \hat{\mathbf{n}} \cdot \left( \rho \ \mathbf{u} \right) = 0 \tag{4}
$$
where $dV$ is the volume element, $dA$ the area element, $\rho$ the mass density, $\hat{\mathbf{n}}$ unit normal vector to surface area element $dA$, and $\mathbf{u}$ bulk fluid flow velocity.  We can see that Equation 4 simplifies to the following in a one-dimensional flow (e.g., flow only along radial vector):
$$
\rho \ u \ A = constant 
$$
where in the case of spherical symmetry and constant flow speed results in $\rho \propto r^{-2}$.  Note that $\rho(r)$ can be easily exchanged for $n(r)$ in most cases.
