UPSC 2024 Maths Optional Paper 2 Q7c — Step-by-Step Solution

20 marks · Section B

Euler's equation of motion for inviscid flow · Mechanics & Fluid Dynamics · asked 2× in 14 yrs · Read the full method →

Question

The velocity field $u=B(x^2-y^2)/(x^2+y^2)^2$ , $v=2Bxy/(x^2+y^2)^2$ , $w=0$ satisfies the equations of inviscid incompressible flow. Determine the associated pressure.

Technique

Verify incompressibility ( $\nabla\cdot\vec v=0$ ) and irrotationality ( $\nabla\times\vec v=0$ ); then apply Bernoulli. The key simplification: $(x^2-y^2)^2+4x^2y^2=(x^2+y^2)^2$ .

Solution

Step 1 — Incompressibility.

\frac{\partial u}{\partial x}=\frac{2Bx(3y^2-x^2)}{(x^2+y^2)^3},\quad\frac{\partial v}{\partial y}=\frac{2Bx(x^2-3y^2)}{(x^2+y^2)^3}.

Sum: $\nabla\cdot\vec v=0$ . ✓

Step 2 — Irrotationality.

\frac{\partial v}{\partial x}=\frac{2By(y^2-3x^2)}{(x^2+y^2)^3},\quad\frac{\partial u}{\partial y}=\frac{2By(y^2-3x^2)}{(x^2+y^2)^3}.

$\omega_z=\partial v/\partial x-\partial u/\partial y=0$ . ✓

Step 3 — Speed squared.

|\vec v|^2=u^2+v^2=\frac{B^2[(x^2-y^2)^2+4x^2y^2]}{(x^2+y^2)^4}=\frac{B^2(x^2+y^2)^2}{(x^2+y^2)^4}=\frac{B^2}{(x^2+y^2)^2}.

Step 4 — Bernoulli’s equation.

For steady, inviscid, incompressible, irrotational flow:

p+\tfrac{1}{2}\rho|\vec v|^2=p_\infty.

Answer

\boxed{\;p=p_\infty-\frac{\rho B^2}{2(x^2+y^2)^2}.\;}