Topic: Fluid Mechanics
If the governing equation for a flow field is given by $\nabla^{2}\phi=0$ and the velocity is given by $\vec{V}=\nabla\phi$, then
- A. $\nabla\times \vec{V}=0$
- B. $\nabla\times \vec{V}=1$
- C. $\nabla^{2}\times \vec{V}=1$
- D. $(\vec{V}.\nabla)\vec{V}=\frac{\partial \vec{V}}{\partial t}$
Answer: A. $\nabla\times \vec{V}=0$
Explanation: Velocity is the gradient of the potential function $\phi$. The curl of any gradient field is always zero. Div (curl $\vec{F})=0$, i.e. $\nabla \times (\nabla\phi)=0$, thus $\nabla\times \vec{V}=0$.