Question 2 of 6: Ideal Flow from the Potential $\phi=-\Gamma\ln r$

Question 2: Ideal Flow from the Potential $\phi=-\Gamma\ln r$ (20 marks)

Question text not reproduced: the examination questions are © Engineers and Geoscientists BC. Open the official past paper (linked at the top of this page) to read the question, then follow the worked solution below.

Given. A plane potential flow with velocity potential $\phi=-\Gamma\ln r$ in polar coordinates $(r,\theta)$, with $\Gamma\gt 0$ a constant. In polar form $V_r=\partial\phi/\partial r$, $V_\theta=\tfrac{1}{r}\partial\phi/\partial\theta$, and the stream function satisfies $V_r=\tfrac{1}{r}\partial\psi/\partial\theta$, $V_\theta=-\partial\psi/\partial r$.

Find. $\psi$; the sketch of equipotentials and streamlines; $V_r$ and the flow type; and the meaning of $2\pi\Gamma$.

(a) Stream function. The velocity components follow directly from the potential: $$V_r=\frac{\partial\phi}{\partial r}=-\frac{\Gamma}{r},\qquad V_\theta=\frac{1}{r}\frac{\partial\phi}{\partial\theta}=0.$$ Matching to the stream-function definitions, $\tfrac{1}{r}\partial\psi/\partial\theta=V_r=-\Gamma/r$ gives $\partial\psi/\partial\theta=-\Gamma$, and $-\partial\psi/\partial r=V_\theta=0$ confirms $\psi$ is independent of $r$. Integrating, $$\boxed{\psi=-\Gamma\,\theta.}$$

(b) Sketch. Equipotential lines are $\phi=\text{const}\Rightarrow \ln r=\text{const}\Rightarrow r=\text{const}$: a family of concentric circles about the origin. Streamlines are $\psi=\text{const}\Rightarrow\theta=\text{const}$: radial rays through the origin. Equipotentials and streamlines cross at right angles, as required for a potential flow (see figure), with the flow directed radially inward.

(c) Radial velocity and flow pattern. From part (a), $$\boxed{V_r=-\frac{\Gamma}{r},\qquad V_\theta=0.}$$ Because $\Gamma\gt 0$, $V_r\lt 0$ everywhere: the fluid moves toward the origin along the radial streamlines, and the speed grows as $1/r$ approaching the centre. This is a plane (two-dimensional) line sink located at the origin — the mirror image of a source. The origin is a singular point where the potential-flow model breaks down.

(d) Meaning of $2\pi\Gamma$. The volumetric flow rate per unit depth crossing any circle of radius $r$ is $$q=\int_0^{2\pi}V_r\,r\,d\theta=\int_0^{2\pi}\left(-\frac{\Gamma}{r}\right)r\,d\theta=-2\pi\Gamma,$$ independent of $r$ (mass conservation). The magnitude $\boxed{2\pi\Gamma=|q|}$ is therefore the strength of the sink — the volume of fluid (per unit span, units $\text{m}^2/\text{s}$) swallowed by the sink each second. The negative sign marks it as inflow (a source would give $+2\pi\Gamma$).