The velocity field of an incompressible flow in a Cartesian system is V=2(x2-y2) i+v j+3 k. Which expression for v is valid?

GATE 2020 ME (ME1) – Question 26

Fluid Mechanics · Differential equations of continuity and momentum, velocity potential · 1 mark · Multiple choice

The velocity field of an incompressible flow in a Cartesian system is $\vec V=2(x^2-y^2)\hat i+v\hat j+3\hat k$. Which expression for v is valid?

$-4xz+6xy$
$-4xy-4xz$
$4xz-6xy$
$4xy+4xz$

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Correct answer: (B) $-4xy-4xz$

Explanation

Continuity requires $\partial u/\partial x+\partial v/\partial y+\partial w/\partial z=0$. Here 4x+∂v/∂y+0=0, so v=−4xy+F(x,z). Only **B**, with F=−4xz, satisfies this identity.