Irrotational flow
= Irrotational flow
{title2=$\nabla\times\mathbf u=\mathbf0$}
{wiki}
Use <irrotational flow> to identify where a <velocity potential> can describe the <velocity field> locally. An <irrotational flow> has $\boldsymbol\omega=\nabla\times\mathbf u=\mathbf0$ throughout the fluid region. For a continuously differentiable <velocity field>, the <Poincaré lemma> supplies a <velocity potential> on sufficiently small neighborhoods. A global single-valued <velocity potential> requires zero <circulation> around every <closed curve>; a <simply connected domain> is a sufficient domain assumption.
The potential <line vortex> $\mathbf u=K\mathbf e_\theta/r$ is an <irrotational flow> on the punctured plane, or on an <annulus> $a<r<b$ with $a>0$, even though its <pathlines> are circular. Every sufficiently small loop bounding a disk in that fluid has zero <circulation>, by <Stokes theorem>. A loop winding once counterclockwise around the excluded origin has <circulation> $2\pi K$. Thus local <irrotational flow> and nonzero <circulation> around a hole are compatible: the enclosing disk passes outside the domain of the <velocity field>.
An <irrotational flow> is a kinematic condition on a <velocity field>, independent of whether a particular model neglects <viscosity> or assumes <incompressible flow>. Adding <incompressible flow> makes the local <velocity potential> satisfy the <Laplace equation>. This reduces the number of unknowns and makes <potential flow> a powerful starting point for flow around bodies and for <gravity waves>. https://ocw.mit.edu/courses/2-016-hydrodynamics-13-012-fall-2005/c472432debcf6ee250209b68cf18cc12_2005reading4.pdf[A. H. Techet, MIT Hydrodynamics reading 4, pp. 1–2]{ref}