← 2015 Paper 2

UPSC 2015 Maths Optional Paper 2 Q5d — Step-by-Step Solution

10 marks · Section B

Sources, sinks, doublets · Mechanics & Fluid Dynamics · asked 8× in 13 yrs · Read the full method →

Question

Consider a uniform flow $U_0$ in the positive $x$-direction. A cylinder of radius $a$ is located at the origin. Find the stream function and the velocity potential. Find also the stagnation points.

Technique

Standard “uniform flow + image doublet” complex potential; separate real and imaginary parts; impose $\psi=0$ on cylinder surface ($r=a$); stagnation points where $\nabla\phi=0$.

Solution

Setup. 2D incompressible irrotational flow past a circular cylinder. Use polar coordinates $(r,\theta)$ with origin at cylinder centre.

For uniform flow $U_0$ in the $+x$ direction superposed with a doublet at the origin (so the cylinder surface $r=a$ is a streamline), the standard result is:

Step 1 — Complex potential

(Uniform flow $U_0 z$ + doublet $U_0 a^2/z$.)

Step 2 — Velocity potential $\phi$ and stream function $\psi$

$w=\phi+i\psi$ with $z=re^{i\theta}$:

So

Step 3 — Verify cylinder is a streamline

At $r=a$: $\psi=U_0(a-a)\sin\theta=0$. So the cylinder surface $r=a$ is the streamline $\psi=0$ ✓.

Step 4 — Stagnation points

Velocity components:

Stagnation: $v_r=v_\theta=0$.

$v_r=0\Rightarrow$ either $r=a$ or $\cos\theta=0$ (i.e. $\theta=\pm\pi/2$). $v_\theta=0\Rightarrow \sin\theta=0$ (i.e. $\theta=0$ or $\pi$); $1+a^2/r^2>0$ always.

Combining: $v_\theta=0$ requires $\sin\theta=0$, then $v_r=U_0(1-a^2/r^2)\cos\theta=0$ requires $r=a$ (since $\cos\theta=\pm 1\ne 0$).

Stagnation points: $r=a$, $\theta=0$ or $\pi$ — i.e. $(a,0)$ (front of cylinder) and $(-a,0)$ (rear).

In Cartesian: $(a,0)$ and $(-a,0)$.

Answer