page_title: Vortex
section_title: Irrotational vortices
hierarchical_section_title: Vortex / Properties / Vortex types / Irrotational vortices
caption_attribution_description: English: Schematic illustration of particle trajectories in an idealized irrotational stationary vortex (where the linear velocity is constant and the angular velocity is inversely proportional to distance from the axis). The dashed line represents the axis of the vortex. The small spheres are sample particles, and the thin circles are their streamlines (or pathlines). Each niti transparent tube represents a surface of equal angular momentum; the distance r from the axis to each surface is 1, 2, 3 and 4 times some arbitrary unit. Note that this schematic animation is not entirely accurate. In a real vortex with curved axis, as shown here, the streamlines are not perfect concentric circles. Moreover, the pattern must break down in a core region close to the axis. This is one frame of an animation created on 2012-09-24 by J. Stolfi using POV-Ray. The code is available at http://www.ic.unicamp.br/~stolfi/EXPORT/videos/vortex-flow/2012-09-24-vortex-anim.tgz
context_page_description: In fluid dynamics, a vortex is a region in a fluid in which the flow revolves around an axis line, which may be straight or curved. Vortices form in stirred fluids, and may be observed in smoke rings, whirlpools in the wake of a boat, and the winds surrounding a tropical cyclone, tornado or dust devil.
  Vortices are a major component of turbulent flow. The distribution of velocity, vorticity, as well as the concept of circulation are used to characterize vortices. In most vortices, the fluid flow velocity is greatest next to its axis and decreases in inverse proportion to the distance from the axis.
  In the absence of external forces, viscous friction within the fluid tends to organize the flow into a collection of irrotational vortices, possibly superimposed to larger-scale flows, including larger-scale vortices. Once formed, vortices can move, stretch, twist, and interact in complex ways. A moving vortex carries with it some angular and linear momentum, energy, and mass.
context_section_description: In the absence of external forces, a vortex usually evolves fairly quickly toward the irrotational flow pattern, where the flow velocity u is inversely proportional to the distance r. Irrotational vortices are also called free vortices.
  For an irrotational vortex, the circulation is zero along any closed contour that does not enclose the vortex axis; and has a fixed value, Γ, for any contour that does enclose the axis once. The tangential component of the particle velocity is then . The angular momentum per unit mass relative to the vortex axis is therefore constant, .
  The ideal irrotational vortex flow in free space is not physically realizable, since it would imply that the particle speed (and hence the force needed to keep particles in their circular paths) would grow without bound as one approaches the vortex axis. Indeed, in real vortices there is always a core region surrounding the axis where the particle velocity stops increasing and then decreases to zero as r goes to zero. Within that region, the flow is no longer irrotational: the vorticity  becomes non-zero, with direction roughly parallel to the vortex axis. The Rankine vortex is a model that assumes a rigid-body rotational flow where r is less than a fixed distance r₀, and irrotational flow outside that core regions.
  In a viscous fluid, irrotational flow contains viscous dissipation everywhere, yet there are no net viscous forces, only viscous stresses. Due to the dissipation, this means that sustaining an irrotational viscous vortex requires continuous input of work at the core (for example, by steadily turning a cylinder at the core). In free space there is no energy input at the core, and thus the compact vorticity held in the core will naturally diffuse outwards, converting the core to a gradually-slowing and gradually-growing rigid-body flow, surrounded by the original irrotational flow. Such a decaying irrotational vortex has an exact solution of the viscous Navier–Stokes equations, known as a Lamb–Oseen vortex.
caption: Pathlines of fluid particles around the axis (dashed line) of an ideal irrotational vortex. (See animation)
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