For a velocity potential φ and velocity V=u i+v j, which condition necessarily holds?

$u=-\partial\phi/\partial y,\ v=\partial\phi/\partial x$

GATE 2019 AE – Question 15

Aerodynamics · Two-dimensional potential flow: sources, sinks, doublets, point vortex and Bernoulli's equation · 1 mark · Multiple choice

For a velocity potential φ and velocity $\mathbf V=u\mathbf i+v\mathbf j$, which condition necessarily holds?

$\nabla^2\phi=0$
$\nabla\times\mathbf V=0$
$\nabla\cdot\mathbf V=0$
$u=-\partial\phi/\partial y,\ v=\partial\phi/\partial x$

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Show answer and explanation

Correct answer: (B) $\nabla\times\mathbf V=0$

Explanation

A differentiable velocity potential gives V=∇φ (or its negative under the alternate convention). The curl of a gradient vanishes, so B holds. Zero divergence and Laplace’s equation additionally require incompressibility; the last relation is a stream-function relation.

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