A velocity field is v=x i+y j+p(z) k, with p(0)=0. If its divergence is zero, find p(z).

GATE 2022 MN – Question 12

Engineering Mathematics · Vector Calculus · 1 mark · Multiple choice

A velocity field is $\mathbf v=x\mathbf i+y\mathbf j+p(z)\mathbf k$, with p(0)=0. If its divergence is zero, find p(z).

0
$-2z$
2
$2z$

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Correct answer: (B) $-2z$

Explanation

Divergence is ∂x/∂x+∂y/∂y+dp/dz=2+p′(z). Setting it to zero gives p′=−2. Integrate to obtain p=−2z+C, and use p(0)=0 to get C=0. Thus B. The boundary condition rules out an arbitrary additive constant.

GATE 2022 MN – Question 12

Explanation

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