TOPICS
Search

Vorticity

The vorticity omega of a velocity vector field u is its curl, omega=del xu. It measures the local rotation of the field. For a two-dimensional vector field u=(u_1,u_2), vorticity is commonly represented by the scalar field omega=partial_1u_2-partial_2u_1.

For a smooth solution of the three-dimensional incompressible Euler equations without external forcing, vorticity satisfies

(partialomega)/(partialt)+(u·del )omega=(omega·del )u
(1)
del ·u=0.
(2)

The right side is the vorticity-stretching term. It vanishes for two-dimensional incompressible flow. The Beale-Kato-Majda criterion (Beale et al. 1984) relates finite-time breakdown of a smooth three-dimensional Euler solution to a time integral of the maximum vorticity.

See also

Beale-Kato-Majda Criterion, Curl, Euler's Equations of Inviscid Motion, Vector Field

Explore with Wolfram|Alpha

References

Batchelor, G. K. An Introduction to Fluid Dynamics. Cambridge, England: Cambridge University Press, 1977.Beale, J. T.; Kato, T.; and Majda, A. "Remarks on the Breakdown of Smooth Solutions for the 3-D Euler Equations." Commun. Math. Phys. 94, 61-66, 1984. https://doi.org/10.1007/BF01212349.

Cite this as:

Weisstein, Eric W. "Vorticity." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/Vorticity.html

Subject classifications