The relation that must hold for the flow to be irrotational is
- A∂v/∂x − ∂u/∂y = 0Correct
- B∂u/∂x − ∂v/∂y = 0
- C∂v/∂x + ∂u/∂y = 0
- D∂²u/∂x² + ∂²u/∂y² = 0
Solution
Concept
Condition for irrotational flow
Formula
ωz = ½(∂v/∂x − ∂u/∂y) = 0
Step-by-step solution
1. Rotation about the z-axis is ωz = ½(∂v/∂x − ∂u/∂y). 2. Irrotational flow means ωz = 0. 3. Hence ∂v/∂x − ∂u/∂y = 0.
Explanation
A flow is irrotational when the vorticity (curl of velocity) is zero. For 2D flow this means ∂v/∂x = ∂u/∂y. Option A is the condition; options with ∂u/∂x + ∂v/∂y = 0 describe continuity, and the Laplace equation (D) applies to φ or ψ.
Shortcut
Irrotational: ∂v/∂x = ∂u/∂y. Continuity: ∂u/∂x = −∂v/∂y.
Common mistake
Confusing the irrotationality condition with continuity.
Correct answer: A) ∂v/∂x − ∂u/∂y = 0