The velocity potential function in a two dimensional flow field is given by φ = x² – y². The magnitude of velocity at point (1,1) is
- A2
- B2√2Correct
- C4
- D4√2
Solution
Concept
Velocity components from velocity potential
Formula
u = −∂φ/∂x (or +∂φ/∂x as per convention), v = −∂φ/∂y; V = √(u² + v²)
Step-by-step solution
1. φ = x² − y². 2. u = ∂φ/∂x = 2x = 2 at (1,1); v = ∂φ/∂y = −2y = −2 at (1,1). 3. V = √(2² + (−2)²) = √8 = 2√2.
Explanation
Velocity components are the partial derivatives of the velocity potential. The sign convention does not change the magnitude, so V = 2√2 at the point (1,1).
Shortcut
Differentiate, substitute, take the resultant: √(u² + v²).
Common mistake
Giving only one component, or forgetting that the derivative of −y² is −2y.
Correct answer: B) 2√2