Question

The velocity potential function in a two-dimensional flow field is given by ϕ = x² – y²

The magnitude of velocity at point P (1, 1) is

1. Zero
2. 2
3. 2√2
4. 8

Answer 1

Correct Answer - Option 3 : 2√2

Concept:

The velocity potential function, stream function, and velocity components are related as shown below

\(U = \; - \frac{{\partial ϕ }}{{\partial x}} = \; - \frac{{\partial ψ }}{{\partial y}}\;\; ...\left( 1 \right)\)

\(V = \; - \frac{{\partial ϕ }}{{\partial y}} = \;\frac{{\partial ψ }}{{\partial x}}\;\;\; ...\left( 2 \right)\)

where ϕ - velocity potential function, ψ - stream function.

Calculation:

Given:

Velocity potential function is ϕ = x2 – y²;

Now the velocity components will be 

\(U = \; - \frac{{\partial ϕ }}{{\partial x}} = -2x\)

\(V = \; - \frac{{\partial ϕ }}{{\partial y}} = 2y\)

At P(1,1), U = -2 m/s, V = 2 m/s;

Now the magnitude of velocity at point P (1, 1) will be 

\(= \sqrt {(-2)^2 + (2)^2} = 2\sqrt2\)