Some pictures of a solution

Figure 1: Solution for $\varepsilon =1.0$.

Figure 2: Solution for $\varepsilon =0.2$.

Figure 3: Solution for $\varepsilon =0.04$.

Figure 4: Structure of the boundary and internal layer.

Below in Figure 4, in black, the structure of the layers is shown by the first exponential factors in the asymptotic expansions for small $\varepsilon$.
In the wake: $\sqrt{r^2-1} + r\cos\theta + \vert\theta\vert - \pi/2 - \arccos(1/r)= -C$.
Outside the wake: $r\cos\beta-2\cos\alpha -r\cos\theta = -C$, where $\alpha$ and $\beta$ solve the equations $\beta+\pi-\theta=2\alpha$ and $r \sin\beta = \sin\alpha$.

The lines are given for constant exponential factors, for the values $C=0.2$, $0.1$, $0.05$, $0.02$, $0.001$ and $0.0001$.

In red we show the structure of the internal layers, away from the circle.