Question

A stream function is given by ψ = 3x² - y³. What is the magnitude of velocity components at the point (2, 1)? 
1. 8.52
2. 9.17
3. 10.81
4. 12.37

Answer 1

Correct Answer - Option 4 : 12.37

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:

Stream function is ψ = 3x2 – y3

Now the velocity components will be 

\(U = \; - \frac{{\partial ψ }}{{\partial y}} = -(-3y^2) =3y^2\)

\(V = \; \frac{{\partial ψ }}{{\partial x}} = 6x\)

At P(2,1),

→ U = 3 × (1)² m/s ⇒ U = 3 m/s

→ V = 6 × (2) m/s = 12 m/s

Now the magnitude  of velocity at point P (2, 1) will be 

\(= \sqrt {(3)^2 + (12)^2} = 12.37 \)

∴ Magnitude of velocity components at the point (2, 1) is 12.37 m/s