UPSC 2024 Maths Optional Paper 2 Q5e — Step-by-Step Solution

10 marks · Section B

Potential flow · Mechanics & Fluid Dynamics · asked 10× in 13 yrs · Read the full method →

Question

The velocity potential of a flow is $\phi=\tfrac{1}{2}(x^2+y^2-2z^2)$ . Determine the streamlines.

Compute $\vec v=\nabla\phi$ ; write $dx/u=dy/v=dz/w$ ; integrate pairwise to obtain two independent first integrals.

Step 1 — Velocity field.

u=\phi_x=x,\quad v=\phi_y=y,\quad w=\phi_z=-2z.

Step 2 — Streamline equations.

\frac{dx}{x}=\frac{dy}{y}=\frac{dz}{-2z}.

Step 3 — Integrate in pairs.

From $dx/x=dy/y$ :

\ln|x|=\ln|y|+c_1\;\Rightarrow\;\frac{y}{x}=C_1.

From $dx/x=dz/(-2z)$ :

2\ln|x|+\ln|z|=\text{const}\;\Rightarrow\;x^2 z=C_2.

Streamlines in 3D are curves — the intersection of two surfaces.

\boxed{\;\frac{y}{x}=C_1\quad\text{and}\quad x^2 z=C_2.\;}