Question

A velocity field in Cartesian coordinate system is expressed as
\[ \mathbf{v} = x\,\hat{i} + y\,\hat{j} + p(z)\,\hat{k}, \quad \text{where } p(0)=0. \] If div $\mathbf{v} = 0$, $p(z)$ is

• A. 0
• B. $-2z$
• C. 2
• D. $2z$

Hint:

For divergence-free flows, the sum of the partial derivatives of the velocity components must be zero.

Answer 1

The Correct Option is B

The divergence of the velocity field is:
\[ \nabla \cdot \mathbf{v} = \frac{\partial x}{\partial x} + \frac{\partial y}{\partial y} + \frac{d p(z)}{d z} = 1 + 1 + p'(z) \] Given $\nabla \cdot \mathbf{v} = 0$:
\[ 1 + 1 + p'(z) = 0 \] \[ p'(z) = -2 \] Integrating:
\[ p(z) = -2z + C \] Given $p(0) = 0$:
\[ C = 0 \] So,
\[ p(z) = -2z. \] Final Answer: $p(z) = -2z$