The question I'm working on is about the Laplace equation in polar coordinates, regarding the velocity field of a perfect incompressible fluid in the plane.
The problem is to solve the Laplace equation $$\frac{1}{r}\partial_r(r\partial_r\phi)+\frac{1}{r^2}\partial_{\theta \theta}\phi=0,$$
with boundary conditions $$\partial_r\phi=0 \text{ for } r=R \qquad \text{ and } \qquad \phi \rightarrow Ur\cos\theta \text{ as } r \rightarrow \infty,$$ where $U$ is the velocity far away from the disc ($\phi$ is the flow outside of a disc of radius $R$).
What I am unclear about is the boundary condition $\phi \rightarrow Ur\cos\theta \text{ as } r \rightarrow \infty.$
This seems completely unintuitive. My interpretation reading this is that as $r$ approaching infinity, we have that a function $\phi(r,\theta)$ approaches $Ur\cos(\theta)$, and since $-1\le \cos(\theta) \le 1$ we will have that $Ur\cos(\theta)$ just oscillates between negative and positive infinity. How does this boundary condition make sense?