Topic: Fluid Mechanics
For a two dimensional potential flow, the velocity potential is given by: $\phi=4x(3y-4)$. The numerical value of stream function at the point (2,3) is
- A. 10 Units
- B. 20 Units
- C. 18 Units
- D. 16 Units
Answer: C. 18 Units
Explanation: From the velocity potential $\phi$, the velocity components are $u = -\frac{\partial\phi}{\partial x} = -4(3y-4)$ and $v = -\frac{\partial\phi}{\partial y} = -12x$. Using the stream function relations, $\frac{\partial\psi}{\partial y} = u = -12y + 16$ and $\frac{\partial\psi}{\partial x} = -v = 12x$. Integrating with respect to x gives $\psi = 6x^2 + g(y)$. Differentiating with respect to y gives $g'(y) = -12y + 16$, which integrates to $g(y) = 16y - 6y^2$. Thus, $\psi = 6(x² - y²) + 16y$. At (2,3), $\psi = 6(4 - 9) + 16(3) = -30 + 48 = 18$ units.