Question

For a two dimensional potential flow, the velocity potential is given by : ϕ = 4x(3y - 4) The numerical value of stream function at the point (2, 3) is
1. 10 Units
2. 20 Units
3. 18 Units
4. 16 Units

Answer 1

Correct Answer - Option 3 : 18 Units

Concept:

Velocity potential function:

\(u = - \frac{{\partial \phi }}{{\partial x}};\;v = - \frac{{\partial \phi }}{{\partial y}}\)

Stream function:

\(u = \frac{{\partial ψ }}{{\partial y}};\;v = -\frac{{\partial ψ }}{{\partial x}}\)

Calculation:

Given:

ϕ = 4x(3y - 4)

\(u=-\frac{\delta \phi}{\delta x}\) and \(v=-\frac{\delta \phi}{\delta y}\)

Therefore, 

\(u=-\frac{\delta \phi}{\delta x}=\frac{\delta ψ}{\delta y}\) And \(v=-\frac{\delta \phi}{\delta y}=-\frac{\delta ψ}{\delta x}\)

∴ u = -4 (3y - 4) = \(\frac{\delta ψ}{\delta y}\) 

⇒ ψ_y = 16y - 6y² + C₁  

And

v = -12x = \(-\frac{\delta ψ}{\delta x}\)

ψx = 6x² + C₂

Therefore, ψ = ψ_x + ψ_y

∴ ψ = 6(x² – y²) + 16y

\({\left. ψ \right|_{\left( {2,3} \right)}} = 6\left( {4 - 9} \right) + 16 \times 3\)

= 6 × - 5 + 48

= 18 units