Question 2 of 6: Ideal Radial Flow from a Velocity Potential

Question 2: Ideal Radial Flow from a Velocity Potential (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 planar potential flow with $\phi = -\dfrac{A}{2\pi}\ln r$, $A \gt 0$; polar coordinates $(r,\theta)$, velocity $\mathbf V=\nabla\phi$ with components $V_r=\partial\phi/\partial r$ and $V_\theta=\tfrac1r\partial\phi/\partial\theta$.

Find. The stream function, the flow-net sketch, the radial velocity and flow type, and the physical meaning of the strength constant.

Approach. Get $V_r,V_\theta$ from $\phi$; integrate the Cauchy–Riemann-type relations $V_r=\tfrac1r\psi_\theta$, $V_\theta=-\psi_r$ for $\psi$; read the geometry from $\phi=$const and $\psi=$const; interpret the strength constant through the volume flux.

1. Velocity field. With $\phi=-\dfrac{A}{2\pi}\ln r$, $$V_r=\frac{\partial\phi}{\partial r}=-\frac{A}{2\pi r},\qquad V_\theta=\frac1r\frac{\partial\phi}{\partial\theta}=0 .$$ The flow is purely radial.
2. Stream function (part a). For plane incompressible flow $V_r=\tfrac1r\,\partial\psi/\partial\theta$ and $V_\theta=-\partial\psi/\partial r$. From $V_\theta=0$, $\psi=\psi(\theta)$ only; from $V_r=-A/(2\pi r)=\tfrac1r\,\psi'(\theta)$, $\psi'(\theta)=-A/(2\pi)$, hence $$\boxed{\psi=-\frac{A}{2\pi}\,\theta.}$$
3. Flow net (part b). $\phi=$const $\Rightarrow \ln r=$const $\Rightarrow r=$const: concentric circles. $\psi=$const $\Rightarrow \theta=$const: radial lines through the origin. The two families are mutually orthogonal, as required for a valid flow net (Figure 2).
4. Radial velocity and flow type (part c). $$\boxed{V_r=-\frac{A}{2\pi r}} \; (\lt 0\ \text{since } A\gt0).$$ The velocity is directed toward the origin at every radius, growing without bound as $r\to0$: this is a two-dimensional line sink of strength $A$ located at the origin.
5. Physical meaning of the strength constant (part d). The volume flow per unit depth crossing any circle of radius $r$ is $$Q=\oint \mathbf V\cdot\hat n\,d\ell = V_r\,(2\pi r)=-A ,$$ independent of $r$. Thus $A$ is the sink strength — the volumetric flow rate (per unit span, $\text{m}^2/\text{s}$) drawn into the sink. Note the circulation of this flow is $$\Gamma=\oint \mathbf V\cdot d\boldsymbol\ell = V_\theta(2\pi r)=0 ,$$ so the quantity written $(2\pi\Gamma)$ is zero here: a radial source/sink is irrotational and carries no circulation. Circulation $\Gamma$ is instead the strength of the companion potential vortex $\phi=\tfrac{\Gamma}{2\pi}\theta$; for the present flow the governing constant is the sink strength $A=|Q|$.