A velocity field in a boundary layer is given by $\vec{V} = (4x^2 y)\hat{i} - (2xy^2)\hat{j} + (3z)\hat{k}$. What is the divergence of the velocity field, $\nabla \cdot \vec{V}$, evaluated at the spatial point $(1, 2, 1)$?

Answer Key:

11

### 5-Stage Derivation & Analytical Solution: 1. **Stage 1: Gradient Operator:** $\nabla \cdot \vec{V} = \frac{\partial(4x^2 y)}{\partial x} + \frac{\partial(-2xy^2)}{\partial y} + \frac{\partial(3z)}{\partial z}$. 2. **Stage 2: Differentiation:** $= 8xy - 4xy + 3 = 4xy + 3$. 3. **Stage 3: Evaluation at $(1, 2, 1)$:** $4(1)(2) + 3 = 11$. 4. **Stage 4: Correct Answer:** 11.

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