Question

Consider a velocity vector, \( \vec{V} \) in (x, y, z) coordinates given below. Pick one or more 
CORRECT statement(s) from the choices given below:
\[ \vec{V} = u\hat{i} + v\hat{j} \]

• A. z-component of Curl of velocity; \( \nabla \times \vec{V} = \left( \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \right) \hat{z} \)
• B. z-component of Curl of velocity; \( \nabla \times \vec{V} = \left( \frac{\partial u}{\partial x} - \frac{\partial v}{\partial y} \right) \hat{z} \)
• C. Divergence of velocity; \( \nabla \cdot \vec{V} = \left( \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \right) \)
• D. Divergence of velocity; \( \nabla \cdot \vec{V} = \left( \frac{\partial u}{\partial x} - \frac{\partial v}{\partial y} \right) \)

Hint:

For 2D velocity fields, divergence is \( \nabla \cdot \vec{V} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \), and curl's z-component is \( \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \).

Answer 1

The Correct Option is A, C

Given: \[ \vec{V} = u(x, y)\,\hat{i} + v(x, y)\,\hat{j} \] There is no component in the z-direction, so the flow is 2D.

Divergence of velocity: \[ \nabla \cdot \vec{V} = \frac{\partial u}{\partial x} + \frac{\partial v}{\partial y} \] Hence, option (C) is correct.

Curl of velocity (vector form): \[ \nabla \times \vec{V} = \begin{vmatrix} \hat{i} & \hat{j} & \hat{k} \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z} \\ u & v & 0 \end{vmatrix} \] Evaluating the determinant: \[ \nabla \times \vec{V} = \left( \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \right) \hat{k} \] Thus, the z-component of curl is: \[ \left( \frac{\partial v}{\partial x} - \frac{\partial u}{\partial y} \right) \hat{z} \]
So, option (A) is also correct. Options (B) and (D) are incorrect as they show incorrect expressions for curl and divergence, respectively.