Question

The condition for irrotational flow is given by:

• A. $\frac{\partial u}{\partial y} = \frac{\partial v}{\partial x}$
• B. $\frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} = 0$
• C. $\frac{\partial v}{\partial y} = \frac{\partial w}{\partial x}$
• D. $\frac{\partial w}{\partial x} + \frac{\partial v}{\partial y} = 0$

Hint:

In irrotational flow, the vorticity components must vanish. For example: \[ \frac{\partial v}{\partial y} = \frac{\partial w}{\partial x}. \] Keep this in mind when analyzing three-dimensional flows.

Answer 1

The Correct Option is C

Irrotational flow necessitates zero vorticity ($\vec{\omega}$): \[ \vec{\omega} = abla \times \vec{v} = 0. \] In three-dimensional flow, this implies that all vorticity components must vanish. Specifically for the z-component: \[ \omega_z = \frac{\partial v}{\partial y} - \frac{\partial w}{\partial x} = 0. \] Therefore, for irrotational flow: \[ \frac{\partial v}{\partial y} = \frac{\partial w}{\partial x}. \]