Vorticity ($\mathbf{\omega }=\nabla \times \mathbf{u}$)

Use vorticity to distinguish the local spin of a fluid element from the motion of its center along a pathline. Circular streamlines can occur both with and without vorticity: the two velocity fields below even have the same circulation around the unit circle.

For a continuously differentiable velocity field $\mathbf{u}=(u_x,u_y,u_z)$, vorticity is the curl vectorEach partial derivative divides velocity by distance, so vorticity has units ${\mathrm{s}}^{-1}$. In a planar velocity field independent of $z$, only ${\omega }_z={\partial }_x{u}_y-{\partial }_y{u}_x$ survives. Positive ${\omega }_z$ points along $+{\mathbf{e}}_z$: the right-hand rule associates it with counterclockwise spin viewed from positive $z$ toward the plane. ^[ref]

The local rotational angular velocity is half the vorticity, ${\mathbf{\Omega }}_{\mathrm{spin}}=\mathbf{\omega }/2$. To see the factor, split the velocity gradient $L_{ij}={\partial }_j{u}_i$ into its symmetric rate-of-strain tensor $E=(L+L^T)/2$ and antisymmetric spin tensor $W=(L-L^T)/2$. A nearby displacement $\delta \mathbf{x}$ has relative velocity $L\delta \mathbf{x}$; its rotational part satisfies $W\delta \mathbf{x}=(\mathbf{\omega }/2)\times \delta \mathbf{x}$. This is a proof outline of the local spin interpretation. The rate-of-strain tensor can also change the angle of a particular material line, so that line's angular velocity need not equal $\mathbf{\omega }/2$.

For solid-body rotation, $\mathbf{u}=(-\Omega y,\Omega x,0)$ gives $\mathbf{\omega }=(0,0,2\Omega )$ and ${\mathbf{\Omega }}_{\mathrm{spin}}=(0,0,\Omega )$. Taking $\Omega =1{\mathrm{s}}^{-1}$, the vorticity is $2{\mathbf{e}}_z{\mathrm{s}}^{-1}$, although the angular velocity is only $1{\mathbf{e}}_z{\mathrm{s}}^{-1}$. For the potential line vortex, $\mathbf{u}=K{\mathbf{e}}_{\theta }/r$, the planar curl is $r^{-1}{\partial }_r(r{u}_{\theta })=r^{-1}{\partial }_{r}K=0$ wherever $r>0$. Its circular pathlines therefore coexist with zero local spin.

This distinction makes vorticity useful for recognizing shear flow as well as swirling motion; taking the curl of the Euler equations also removes a pressure gradient and leads to the vorticity equation. A historical landmark is Hermann von Helmholtz's 1858 paper on fluid vortex motion, an early study devoted to vortex dynamics. ^[ref]

Away from the axis, the cylindrical coordinates formula for curl gives $r^{-1}{\partial }_r(r{F}_{\theta })=2(1-n)r^{-2n}$ when $F_{\theta }=r^{1-2n}$. Its sign changes at $n=1$, even though the tangential velocity stays positive. Thus local angular velocity, one half of vorticity, can oppose orbital circulation. For $n=1$, the field is a line vortex with zero ordinary curl off the axis and circulation $2\pi$ around it. Stokes theorem applies on annuli excluding the axis with both boundary components included; it cannot be applied to a spanning disc containing the field singularity.

Use a linear shear flow to see why straight pathlines can still carry local spin. Faster layers pull past slower ones, and a small symmetric paddle can turn clockwise while its center travels straight ahead.

For the planar simple shear flow $\mathbf{u}=(\gamma y,0,0)$ with constant signed shear rate $\gamma$, direct differentiation givesWith $\gamma =1{\mathrm{s}}^{-1}$, these are $-1{\mathrm{s}}^{-1}$ and $-0.5{\mathrm{s}}^{-1}$: both signs indicate clockwise orientation under the right-hand rule. Yet a particle initially at $(x_0,y_0)$ follows the straight pathline $x(t)=x_0+\gamma y_{0}t$, $y(t)=y_0$. The curl measures differences of velocity between neighboring particles, rather than bending of one particle's pathline.

The velocity gradient has $L_{xy}=\gamma$ as its only nonzero entry. Its spin tensor supplies clockwise rotation, while its rate-of-strain tensor supplies simultaneous deformation. For a material line making angle $\alpha$ to the horizontal, the equation $\dot{\delta \mathbf{x}}=L\delta \mathbf{x}$ gives $\dot{\alpha }=-\gamma {\sin }^2\alpha$. Initially horizontal and vertical material lines therefore have angular velocities $0$ and $-\gamma$; their mean is $-\gamma /2$. At $t=0.5\mathrm{s}$ with $\gamma =1{\mathrm{s}}^{-1}$, an initially vertical displacement becomes $(0.5,1)$ times its original length and has angle $\arctan 2\approx 63.435^{\circ }$, a clockwise change of $26.565^{\circ }$. The plotted cross deforms; an actual paddle's response also depends on its shape and interaction with the fluid. The exact local spin statement concerns the spin tensor, rather than every tracer's orientation.

A vortex line is an integral curve of the vorticity vector field: its tangent is parallel to $\mathbf{\omega }=\nabla \times \mathbf{u}$. In a two-dimensional incompressible flow, these lines are perpendicular to the flow plane and the vorticity equation gives $D\omega /Dt=0$. Since material cross-sectional area is preserved, the flux of vorticity through a material vortex tube, equal to its circulation, is constant.

A fluid vortex is a region in which velocity field circulates about an axis or center. Vorticity measures local rotation, while circulation measures the integral of velocity around a closed path. A phase vortex is instead a topological defect defined by winding of an order parameter; superfluid examples connect the two notions.

The evolution of a vorticity distribution together with the velocity field it induces. A transport equation advances vorticity; diagnostic inversion recovers velocity from it and the specified boundary conditions.

Axisymmetric inviscid flow has no azimuthal pressure torque, so each parcel conserves specific axial angular momentum $j=r{u}_{\theta }=r^2\omega$. If late draining fluid originally lay near the rim $R$ of a uniformly rotating pool, then $j\simeq {\omega }_0{R}^2$ and $\omega \simeq {\omega }_0{R}^2/r^2$. The late-origin assumption is essential; torque conservation alone gives $j={\omega }_0{r}_0^2$ for each parcel's own initial radius $r_0$. Quasistatic radial and vertical pressure balance then gives free-surface depression $h(r)={\omega }_0^2{R}^4(r^{-2}-R^{-2})/(2g)$. The singular profile applies outside a finite drain or vortex core, not at the rotation axis.

In the limiting model of infinitely strong stable density stratification, small Froude number and fixed vertical scales suppress vertical displacement. Each horizontal level evolves by two-dimensional vortex dynamics; $z$ labels independent copies of the horizontal Poisson equation. The leading normalized Ertel potential vorticity is $Q/N^2=\zeta$. This is a constrained asymptotic model: finite buoyancy frequency restores vertical motion and coupling, and arbitrarily short vertical scales invalidate the independent-layer limit. McIntyre's discussion of potential-vorticity inversion explains the limiting independent-layer picture.

For inviscid planar incompressible flow, write velocity as $(-{\psi }_y,{\psi }_x)$ and vorticity as $\zeta ={\nabla }^2\psi$. Taking curl of the Euler equations for an inviscid fluid removes pressure and gives the displayed transport equation, with $J(\psi ,\zeta )={\psi }_x{\zeta }_y-{\psi }_y{\zeta }_x$. The Poisson equation then recovers the streamfunction at each instant. A velocity tangent to the plane cannot stretch its perpendicular vorticity vector.

In a steady inviscid planar incompressible flow, vorticity is constant along streamlines. A regular stream function supplies a local transverse coordinate, so locally $\Delta \psi$ depends only on $\psi$. A global single-valued function needs the same value on every component of each level set. This can fail: $\psi =x^3-x$ gives the steady Euler shear $(u,v)=(0,1-3x^2)$ with constant pressure, but $\psi (0)=\psi (1)=0$ while $\Delta \psi$ is respectively $0$ and $6$.

An ideal straight line vortex in a two-dimensional incompressible flow has circulation $\Gamma$ and induces velocityIts stream function, with $\mathbf{u}=({\psi }_y,-{\psi }_x)$, is $\psi =-\Gamma \log |\mathbf{r}-{\mathbf{r}}_0|/(2\pi )$. Positive $\Gamma$ means counterclockwise circulation. In an ideal point-vortex description, the vortex position is advected by the regular velocity produced by the other vortices and the boundaries; its own singular velocity is excluded.

A vortex of circulation $\Gamma$ distance $d$ from a plane wall has an opposite-sign image across the wall. The wall velocity has magnitude $|\Gamma |d/[\pi (d^2+y^2)]$, giving a pressure reduction $\rho {\Gamma }^2{d}^2/[2{\pi }^2(d^2+y^2{)}^2]$ by the Bernoulli equation. Integrating this deficit gives the displayed force per unit width, directed from the wall toward the fluid and vortex. The uniform ambient pressure is subtracted to isolate the finite vortex-induced force.

For a point vortex of circulation $\Gamma$ in the wedge $0<\theta <\pi /n$, with positive integer $n$, the conformal map $z\mapsto z^n$ sends the domain to the upper half-plane. An opposite image vortex gives the complex potentialRemove the singular self-induced velocity before setting $z=c$. The resulting complex velocity isThe finite first term follows from the expansion of $(z^n-c^n)/(z-c)$ at $c$. In polar coordinates this becomes $\dot{r}=n\Gamma \cot (n\theta )/(4\pi r)$ and $\dot{\theta }=-\Gamma /(4\pi r^2)$. Therefore $r\sin (n\theta )$ is constant along a vortex trajectory. The case $n=1$ is motion parallel to a plane wall, while $n=3$ describes a sixty-degree wedge.

Let $\zeta =g(z)$ be a conformal map, and let a point vortex of circulation $\Gamma$ lie at $z=c$. Suppose the mapped complex potential of a two-dimensional potential flow iswhere $R$ is a holomorphic function near $g(c)$ and represents the other point vortices, boundaries, and any imposed flow. The physical complex velocity is $w^{\prime }=g^{\prime }{W}^{\prime }$. The point vortex follows its regular velocity, found by removing $\Gamma /[2\pi i(z-c)]$ before taking the limit. A Taylor expansion proveswhich gives the title formula. The complex conjugate occurs because the complex velocity convention is $w^{\prime }=u-iv$, whereas $\dot{c}=u+iv$.

For a mapped upper half-plane with no imposed flow, the method of images gives $R(\zeta )=-\Gamma \log (\zeta -\overline{g(c)})/(2\pi i)$, henceThe first term is essential: excluding a mapped singularity and then transforming its regular velocity does not in general equal subtracting the physical singularity after transformation.

Take vortex strength to mean circulation and $J(x,y)=(-y,x)$. Each point vortex moves with the velocity induced by the other, so for $\mathbf{d}={\mathbf{x}}_1-{\mathbf{x}}_2$ the displayed equation holds. Since $\mathbf{d}\cdot J\mathbf{d}=0$, separation is constant; weighting the two velocities also gives $d({\kappa }_1{\mathbf{x}}_1+{\kappa }_2{\mathbf{x}}_2)/dt=0$. Equal strengths at opposite points of a circle of radius $a$ rotate with angular speed $\kappa /(4\pi a^2)$.

The method of images enforces impermeability at a plane wall by reflecting a line vortex across it and reversing its circulation. The two logarithmic stream functions cancel on the wall, which is consequently a streamline. For a vortex of circulation $\Gamma$ at $(0,b)$ above $y=0$, the image of circulation $-\Gamma$ at $(0,-b)$ induces velocity $(\Gamma /(4\pi b),0)$ at the real vortex. The signed drift parallel to the wall is therefore $\Gamma /(4\pi b)$.

In the half-plane $y>0$ with an impermeable wall, place a point vortex of circulation $\Gamma$ in the background velocity $(\kappa x,-\kappa y)$, where $\kappa >0$. The image vortex at a plane wall has circulation $-\Gamma$ at the reflected location and contributes horizontal velocity $\Gamma /(4\pi y)$. Excluding the vortex's own singular velocity gives the displayed motion equations.

Put $c=\Gamma /(4\pi )$. Their first integral is $\kappa xy+c\log y$, since its time derivative is $\kappa [(\kappa x+c/y)y-\kappa xy]+c(-\kappa )=0$. The explicit trajectory is $y=y_0{e}^{-\kappa t}$ and $x=e^{\kappa t}(x_0+ct/y_0)$. These formulas show how image-induced drift changes the hyperbolic trajectories of the background flow while preserving a modified stream-function-type invariant.

In the quadrant $x,y>0$, a line vortex of circulation $\Gamma$ at $(x,y)$ has three images: circulations $-\Gamma$ at $(-x,y)$ and $(x,-y)$, and $+\Gamma$ at $(-x,-y)$. Their induced velocity isIn polar coordinates, these equations give $\dot{r}=\Gamma \cot (2\theta )/(2\pi r)$ and $\dot{\theta }=-\Gamma /(4\pi r^2)$. Direct differentiation yields $\frac{d}{dt}(r\sin 2\theta )=0$, so the trajectory obeys $r\sin 2\theta =\text{constant}$.

In an unbounded incompressible fluid with localized vorticity and no added harmonic background, velocity is reconstructed by $\mathbf{u}(\mathbf{x})=(4\pi {)}^{-1}\int \mathbf{\omega }(\mathbf{y})\times (\mathbf{x}-\mathbf{y})/|\mathbf{x}-\mathbf{y}{|}^3{d}^{3}y$. Its multipole expansion relates hydrodynamic impulse to a dipolar far-field velocity.

Hydrodynamic impulse per unit density is $\mathbf{I}=\frac{1}{2}\int \mathbf{x}\times \mathbf{\omega }dV$ for localized vorticity. In unforced all-space incompressible flow its derivative vanishes when far-field fluxes vanish. For a spherical all-space limit, $\mathbf{I}=(3/2){\lim }_R{\int }_{V_R}\mathbf{u}dV$; a dipolar velocity tail makes the distinction from a bare momentum integral important.

A localized vorticity distribution with nonzero hydrodynamic impulse induces $\mathbf{u}\sim [3\mathbf{n}(\mathbf{I}\cdot \mathbf{n})-\mathbf{I}]/(4\pi r^3)$. This curl-free exterior velocity can remain long-ranged even when the vorticity is localized. In a random ensemble it produces dipolar longitudinal and transverse correlation tails.

A vortex tube is a bundle of vortex lines bounded by a surface to which vorticity is tangent. Circulation measures its vorticity flux. Stretching a nearly incompressible tube reduces its cross-sectional area and increases its vorticity. The Burgers vortex is a local model in which this concentration balances viscous diffusion.

A vortex filament is the ideal thin-core limit of a vortex tube, concentrating a finite circulation on a curve. In an unbounded incompressible fluid with velocity tending to zero at infinity, its induced velocity field follows the vorticity version of the Biot-Savart law. The ideal field is singular on the filament; finite-core models are needed there. A closed circular filament is a thin vortex ring.

A vortex ring is a vortex tube bent into a closed loop, often approximately a circle. Its vorticity runs around the loop, while the induced velocity field passes through the centre and returns outside. For a thin ring of radius $r$ and circulation $\Gamma$, the Biot-Savart law for incompressible vorticity gives an axial velocity $u_z(z)=\Gamma r^2/[2(r^2+z^2{)}^{3/2}]$, measured relative to the ring centre, with its sign set by the circulation orientation. A vortex ring is not a loop of fluid whose velocity everywhere merely follows the circular vortex line.

A long array of equal co-oriented vortex rings, spaced by $l\ll r$, approximates an azimuthal vortex sheet on a cylinder of radius $r$. The circulation per unit axial length is $\Gamma /l$. The conditions $\nabla \cdot \mathbf{u}=0$ and $\nabla \times \mathbf{u}=\mathbf{\omega }$ are mathematically analogous to the magnetostatic equations for a solenoid. Away from its ends, an array long compared with $r$ produces nearly uniform axial flow $\Gamma /l$ inside and weak flow outside. A finite array has return flow outside connecting its two ends. The uniform-core approximation fails close to individual vortex cores and near the ends, and the field requires a stated circulation orientation and background flow.

Enstrophy measures squared vorticity: a local density is ${\omega }^2/2$, and its spatial integral or ensemble mean gives a global measure. In homogeneous incompressible flow, viscous dissipation satisfies $ϵ=\nu ⟨{\omega }^2⟩$. Vortex stretching can produce enstrophy in three dimensions; kinematic viscosity destroys it.

For unforced homogeneous incompressible flow, the vorticity equation gives $\dot{Z}=⟨{\omega }_i{\omega }_j{S}_{ij}⟩-\nu ⟨|\nabla \times \mathbf{\omega }{|}^2⟩$. Curl and full-gradient squared averages agree because the vorticity is divergence free and boundary divergences average to zero. In established high-Reynolds-number turbulence, slow integral-time evolution makes production and destruction nearly balance.

The velocity induced by planar vorticity is a convolution with this kernel, whose sign depends on the stream function and vorticity conventions. Its magnitude is $(2\pi |z|{)}^{-1}$ and its derivative is bounded by $C|z{|}^{-2}$. Splitting the convolution at radius $R$ gives $\Vert K\ast \omega {\Vert }_{\infty }\le R\Vert \omega {\Vert }_{\infty }+\Vert \omega {\Vert }_1/(2\pi R)$. The same singularity yields a log-Lipschitz modulus, enough for a Yudovich characteristic flow.

Circular streamlines and constant vorticity imply $u_{\varphi }=\omega r/2+C/r$. Zero velocity at $2a$ and positive azimuthal velocity $V$ at $a$ giveThe radial Euler equations for an inviscid fluid give $p_r=\rho u_{\varphi }^2/r$, yielding outer-minus-inner pressure $(15-16\log 2)\rho V^2/18$. The pressure difference is independent of circulation orientation.

For differentiable vector fields, the displayed identity relates vorticity to the antisymmetric part of the velocity gradient. Together with the curl of a cross product, it converts the adjoint linearized Navier-Stokes evolution between component and rotational forms.

The local vector-calculus identityseparates the convective acceleration into a kinetic-energy gradient and a vorticity cross-product contribution. Substitution into the steady Euler equations for an inviscid fluid yields the vorticity form of the Bernoulli equation.

Kinetic helicity measures the alignment and linking of a velocity field and its vorticity. It is the volume integral of $\mathbf{u}\cdot \mathbf{\omega }$, with boundary conditions needed for its conservation.

The integral measures linkage and twisting of vortex lines through the velocity and vorticity fields.

The local density of kinetic helicity is $q=\mathbf{u}\cdot \mathbf{\omega }$. Conservation of its volume integral does not imply that $q$ is a materially advected scalar.

For a transverse complex velocity amplitude $\hat{\mathbf{u}}$ and real nonzero wavevector $\mathbf{k}$, write $\hat{\mathbf{u}}=\mathbf{a}+i\mathbf{c}$. Solenoidality makes both real vectors perpendicular to $\mathbf{k}$, so $\frac{1}{2}\mathrm{Re}(i{\hat{\mathbf{u}}}^{\ast }\times \hat{\mathbf{u}})=-\mathbf{a}\times \mathbf{c}=H\mathbf{k}$. The corresponding spatial average of kinetic helicity density is $-H{k}^2$. Cross-product order determines this sign.

For smooth inviscid barotropic fluid motion without gravity or magnetic forces, let $w$ be the specific enthalpy with $\nabla w={\rho }^{-1}\nabla p$, and set $q=\mathbf{u}\cdot \mathbf{\omega }$. ThenThis follows by dotting the barotropic vorticity transport equation with $\mathbf{u}$ and using the divergence of a cross product. A volume integral is constant when the resulting boundary flux vanishes.

The kinetic helicity conservation law impliesTherefore $q$ is materially constant exactly when the right side vanishes. Incompressible flow together with constancy of $w-u^2/2$ along vortex lines is sufficient. In compressible flow, the density-normalized quantity satisfies $D(q/\rho )/Dt=-(\mathbf{\omega }/\rho )\cdot \nabla (w-u^2/2)$.

Absolute vorticity is the vorticity measured in an inertial frame. In a frame rotating with angular velocity $\mathbf{\Omega }$, it is $\mathbf{\omega }+2\mathbf{\Omega }$.

The planetary contribution to absolute vorticity is supplied by the rotating frame. Its local vertical component is the Coriolis parameter $f=2\Omega \sin \varphi$. On a beta plane, motion with northward velocity $v$ changes this contribution at the rate $\beta v$.

For a flow satisfying incompressibility, this antisymmetric tensor writes advection and vortex stretching in conservative form. With its first index as transport direction, $(\nabla \cdot T{)}_j={\partial }_i{T}_{ij}$.

When the only nonpotential force points radially, its curl has no radial component. The radial absolute vorticity therefore obeys a local flux conservation law, although vortex stretching can change it along a fluid trajectory.

Relative vorticity is the curl of velocity measured in the chosen rotating frame. Adding the planetary vorticity gives absolute vorticity.

Use circulation to summarize how much a velocity field runs along an oriented closed curve $C$, such as a circle around a fluid vortex. Its definition is the line integralwhere $d\mathbf{l}$ is tangent to $C$ in the chosen direction. Reversing that direction reverses the sign of circulation. Its units are ${\mathrm{m}}^2{\mathrm{s}}^{-1}$, because velocity is multiplied by a length. Counterclockwise circulation in the horizontal plane is positive when the chosen unit normal vector is $+{\mathbf{e}}_z$, according to the right-hand rule.

For solid-body rotation $\mathbf{u}=(-\Omega y,\Omega x,0)$, a counterclockwise circle of radius $R$ has constant tangential velocity $\Omega R$. Multiplying by its circumference givesFor the potential line vortex $\mathbf{u}=K{\mathbf{e}}_{\theta }/r$, the same circle instead gives ${\Gamma }_C=(K/R)(2\pi R)=2\pi K$, independent of $R$. Stipulate $\Omega =1{\mathrm{s}}^{-1}$, $R=1\mathrm{m}$ and $K=1{\mathrm{m}}^2{\mathrm{s}}^{-1}$: both velocity fields have ${\Gamma }_C=2\pi {\mathrm{m}}^2{\mathrm{s}}^{-1}\approx 6.283185{\mathrm{m}}^2{\mathrm{s}}^{-1}$, although their vorticities in the fluid are $2{\mathbf{e}}_z{\mathrm{s}}^{-1}$ and zero respectively. On a circle of radius $2\mathrm{m}$, their circulations are $8\pi {\mathrm{m}}^2{\mathrm{s}}^{-1}$ and $2\pi {\mathrm{m}}^2{\mathrm{s}}^{-1}$. The comparison plot in vorticity shows this difference in radius dependence.

The bridge between circulation and vorticity is Stokes theorem:Here the velocity field must be continuously differentiable on a neighborhood of the entire spanning surface $S$ inside the fluid, and its boundary orientation must match the unit normal vector $\mathbf{n}$. For solid-body rotation, the disk is admissible and its surface integral is $2\Omega \pi R^2$. For the potential line vortex, the origin is singular and excluded from the fluid: a disk crossing it is not admissible. On an annulus, Stokes theorem instead gives ${\Gamma }_{R_{\mathrm{out}}}-{\Gamma }_{R_{\mathrm{in}}}=0$, with both circulations here measured counterclockwise; the induced inner boundary direction is clockwise. Nonzero circulation therefore does not imply nonzero local curl in the available fluid. ^[ref]

This is useful because Stokes theorem lets a local vorticity measurement determine a loop's circulation when a suitable surface exists, while Kelvin's circulation theorem tracks circulation along a moving material curve. The beauty is the precise link between a local derivative and a global line integral, with the domain deciding which conclusions are valid.

In an inviscid barotropic fluid subject to conservative body force, the circulation of a material loop is constant. For constant density, parameterize that loop by $\mathbf{r}(s,t)$ with ${\mathbf{r}}_t=\mathbf{u}$. Differentiating $\Gamma =\int \mathbf{u}\cdot {\mathbf{r}}_{s}ds$ gives an integral of $(D\mathbf{u}/Dt)\cdot d\mathbf{r}$ plus $\int {\partial }_s(|\mathbf{u}{|}^2/2)ds$. The latter vanishes on a closed loop, and the Euler equations make the former the integral of a gradient, also zero. For a barotropic density, $\nabla p/\rho$ is still a gradient of specific enthalpy. Constant circulation is consistent with changing local vorticity when a vortex tube changes its cross-sectional area.

In smooth barotropic inviscid flow with conservative body force, Kelvin's circulation theorem preserves circulation of material curves. Initially zero vorticity therefore gives zero circulation on contractible loops for all times. A loop winding around an excluded obstacle can have a nonzero constant circulation: the planar field $\mathbf{u}=\Gamma {\mathbf{e}}_{\theta }/(2\pi r)$ is irrotational outside the origin but has circulation $\Gamma$ on a positively oriented enclosing loop.

Use irrotational flow to identify where a velocity potential can describe the velocity field locally. An irrotational flow has $\mathbf{\omega }=\nabla \times \mathbf{u}=0$ 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. ^[ref]