Question

Magnitude of the component of velocity at point (1,1) for a stream function \( \Psi = x^2 - y^2 \) is equal to ............

• A. 2
• B. 4
• C. \( 2\sqrt{2} \)
• D. \( 4\sqrt{2} \)

Hint:

For 2D incompressible flows, use \( u = \frac{\partial \Psi}{\partial y} \) and \( v = -\frac{\partial \Psi}{\partial x} \) to compute velocity components from the stream function.

Answer 1

The Correct Option is C

The stream function \( \Psi = x^2 - y^2 \) represents a 2D incompressible flow.
To find the velocity components from the stream function: \[ u = \frac{\partial \Psi}{\partial y} = -2y, \quad v = -\frac{\partial \Psi}{\partial x} = -2x \] At point (1,1): \[ u = -2(1) = -2, \quad v = -2(1) = -2 \] Now, the magnitude of the velocity vector is: \[ |\vec{V}| = \sqrt{u^2 + v^2} = \sqrt{(-2)^2 + (-2)^2} = \sqrt{4 + 4} = \sqrt{8} = 2\sqrt{2} \] \[ \boxed{2\sqrt{2}} \]