← 2023 Paper 2

UPSC 2023 Maths Optional Paper 2 Q5e — Step-by-Step Solution

10 marks · Section B

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

Question

In a fluid motion, there is a source of strength $2m$ placed at $z=2$ and two sinks of strength $m$ are placed at $z=2+i$ and $z=2-i$. Find the streamlines.

Technique

Superposition of source/sink complex potentials; combine logs; extract imaginary part for the stream function.

Solution

Strategy. In two-dimensional ideal-fluid flow, the complex potential of a source of strength $Q$ at $z=z_0$ is $w(z)=Q\log(z-z_0)$ (sink: replace $Q$ by $-Q$). Add contributions by superposition; the streamlines are the level curves of $\psi=\operatorname{Im}w$.

Step 1 — Build the complex potential.

Source $+2m$ at $z=2$; sink $-m$ at $z=2+i$; sink $-m$ at $z=2-i$:

Combine logarithms:

The denominator $(z-2-i)(z-2+i)=(z-2)^{2}+1$. Hence

Step 2 — Stream function $\psi=\operatorname{Im}w$.

Translate: let $\zeta=z-2$, $\zeta=X+iY$ with $X=x-2,\,Y=y$.

Step 3 — Algebraic form of $\psi=\text{const}$.

Compute the real and imaginary parts of $\dfrac{\zeta^{2}}{\zeta^{2}+1}$ (multiplying by the conjugate of the denominator):

$\zeta^{2}=(X^{2}-Y^{2})+2iXY$, $\zeta^{2}+1=(X^{2}-Y^{2}+1)+2iXY$.

Numerator (after multiplying by $\overline{\zeta^{2}+1}$):

• Real: $(X^{2}-Y^{2})(X^{2}-Y^{2}+1)+4X^{2}Y^{2}=(X^{2}+Y^{2})^{2}+X^{2}-Y^{2}$.
• Imaginary: $2XY(X^{2}-Y^{2}+1)-2XY(X^{2}-Y^{2})=2XY$.

(Real-part simplification: $(X^{2}-Y^{2})^{2}+4X^{2}Y^{2}=(X^{2}+Y^{2})^{2}$, then add the $(X^{2}-Y^{2})$ remainder.)

Therefore

Streamlines $\psi=$ const correspond to the family

Answer