For a steady, incompressible, irrotational fluid flow, the velocity potential function $\phi$ satisfies which governing partial differential equation?

(A) \nabla^2 \phi = 0 \quad \text{(Laplace equation)}

(B) \nabla^2 \phi = \omega \quad \text{(Poisson equation)}

(C) \frac{\partial \phi}{\partial t} + \nabla^2 \phi = 0

(D) \nabla \times \nabla \phi = \vec{V}

Answer Key:

A

### Solution: Irrotationality guarantees $\vec{V} = \nabla\phi$. Incompressible continuity $\nabla \cdot \vec{V} = 0$ yields $\nabla \cdot (\nabla\phi) = \nabla^2 \phi = 0$ (Laplace equation). Correct: A.

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