The two-dimensional velocity field is V=(5+a 1x+b 1y) i+(4+a 2x+b 2y) j, with constant coefficients. Which condition is required for incompressible flow?

GATE 2017 ME (ME1) – Question 17

Fluid Mechanics · Control-volume analysis of mass, momentum and energy · 1 mark · Multiple choice

The two-dimensional velocity field is $\mathbf V=(5+a_1x+b_1y)\hat{\mathbf i}+(4+a_2x+b_2y)\hat{\mathbf j}$, with constant coefficients. Which condition is required for incompressible flow?

$a_1+b_1=0$
$a_1+b_2=0$
$a_2+b_2=0$
$a_2+b_1=0$

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Show answer and explanation

Correct answer: (B) $a_1+b_2=0$

Explanation

Incompressibility requires $\nabla\cdot\mathbf V=\partial u/\partial x+\partial v/\partial y=0$. The two derivatives are $a_1$ and $b_2$. Hence the condition is $a_1+b_2=0$, B. The other coefficients appear in cross derivatives, not the divergence.

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