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The velocity field in a flow system is given by $\mathbf{v}=2\mathbf{i}+(x+y)\mathbf{j}+(xyz)\mathbf{k}$. The acceleration of the fluid at $(1,1,2)$ is

1. $2\mathbf{i}+10\mathbf{k}$
2. $4\mathbf{i}+12\mathbf{k}$
3. $\mathbf{j}+\mathbf{k}$
4. $4\mathbf{j}+10\mathbf{k}$

Correct answer: $4\mathbf{j}+10\mathbf{k}$

Solution

Acceleration is given by the material derivative $\frac{D\mathbf{v}}{Dt}=\frac{\partial \mathbf{v}}{\partial t}+(\mathbf{v}\cdot\nabla)\mathbf{v}$. Here the field is steady, so only convective terms contribute. Evaluating at $(1,1,2)$ gives $4\mathbf{j}+10\mathbf{k}$.

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