Past exam of the fluids and waves course of the University of Oxford / 2017 / 1 / b

Work away from the line source, so $r>0$. In polar coordinates, $u_r=Q/(2\pi r)$ and $u_{\theta }=0$. The supplied divergence and curl formulas giveHence the field defines an incompressible flow and an irrotational flow for $r>0$. The restriction matters: the outward volume flux through any enclosing circle isThe origin is the excluded injection point. Applying the divergence theorem to a disk through that singularity would violate its smoothness hypothesis.

A velocity potential satisfies $\mathbf{u}=\nabla \varphi$. Integration of ${\varphi }_r=Q/(2\pi r)$ and $r^{-1}{\varphi }_{\theta }=0$ givesChoose the streamfunction convention $u={\psi }_y$, $v=-{\psi }_x$. Its streamfunction in polar coordinates satisfies $u_r=r^{-1}{\psi }_{\theta }$ and $u_{\theta }=-{\psi }_r$, soAdditive constants are immaterial. This complex potential of a two-dimensional potential flow is defined on a chosen logarithm branch. Its complex velocity $w^{\prime }=Q/(2\pi z)=u-iv$ is single-valued away from the origin. Continuing once around the line source increases $\psi$ by $Q$, consistently with its nonzero volume flux.

The streamlines are the level curves of the streamfunction, so $\theta$ is constant: radial rays directed outwards from the origin. Their arrows point outwards because $Q>0$. A complete straight line through the origin comprises two distinct rays separated by the excluded line source. Equivalently, $r\dot{\theta }=u_{\theta }=0$ and $\dot{r}=Q/(2\pi r)>0$ along each pathline; the steady flow makes these the same curves as the streamlines.