Math 2.0

Fluid mechanics Definition

Vorticity equation

Builds on VorticityMaterial derivativeNavier–Stokes equationsCurl

Definition

Let uu be a smooth velocity field on an open set of ℝ3\R^3. Assume it solves the incompressible Navier–Stokes equations ∂u∂t+(u⋅∇)u=−∇p+νΔu−∇Φ,∇⋅u=0,\frac{\partial u}{\partial t} + (u \cdot \nabla) u = -\nabla p + \nu \Delta u - \nabla \Phi, \qquad \nabla \cdot u = 0, with kinematic viscosity ν≥0\nu \ge 0, pressure pp (divided by the constant density) and body force potential Φ\Phi. Let ω=∇×u\omega = \nabla \times u be the vorticity, the curl of uu. The vorticity equation is the equation DωDt=(ω⋅∇)u+νΔω,\frac{D\omega}{Dt} = (\omega \cdot \nabla) u + \nu \Delta \omega, where Dω/Dt=∂tω+(u⋅∇)ωD\omega/Dt = \partial_t \omega + (u \cdot \nabla) \omega is the material derivative. It holds for every such solution, as one sees by taking the curl of the momentum equation.

Remarks

The term (ω⋅∇)u(\omega \cdot \nabla) u describes stretching and tilting of vortex lines; νΔω\nu \Delta \omega describes viscous diffusion. For ν=0\nu = 0 it is the vorticity equation of the Euler equations.

Example: for a planar flow u=(u1(x,y,t),u2(x,y,t),0)u = (u_1(x, y, t), u_2(x, y, t), 0), the vorticity is ω=(0,0,ζ)\omega = (0, 0, \zeta) and (ω⋅∇)u=ζ ∂zu=0(\omega \cdot \nabla) u = \zeta \, \partial_z u = 0. So the equation reduces to the scalar equation Dζ/Dt=νΔζD\zeta/Dt = \nu \Delta \zeta.

Non-example: for a compressible fluid whose density is not a function of the pressure, the equation above is not satisfied in general. The general equation has the extra terms −ω(∇⋅u)-\omega (\nabla \cdot u) and the baroclinic term ∇ρ×∇p/ρ2\nabla \rho \times \nabla p / \rho^2, plus terms from viscous stresses.

Reading path

To understand this page from the ground up, read these 11 pages in order. The foundations come first.

Show the path as a graph
Euclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptMaterial derivative, concept, no evidence levelMaterial derivativeconceptNavier–Stokes equations, concept, no evidence levelNavier–Stokes equatio…conceptVorticity, concept, no evidence levelVorticityconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptCurl, concept, no evidence levelCurlconceptVorticity equation, definition, no evidence levelVorticity equationdefinition
Each page uses the pages above it. The colour of the bar on a box shows how strong the evidence is, and the box also gives it as text. The list has the same pages.
  • no evidence level
  1. Euclidean space object
  2. Function concept
  3. Material derivative concept
  4. Navier–Stokes equations concept
  5. Vorticity concept
  6. Function of several variables concept
  7. Vector-valued function concept
  8. Partial derivative concept
  9. Vector field concept
  10. Curl concept
  11. Vorticity equation definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Vorticity_equation
About this page
TagWVVDKKM9D2
URIm2://WVVDKKM9D2
Kinddefinition
WritingW0 stub
MSC76B47, 76D17, 76D05
DefinesVorticity equation
UsesVorticity, Material derivative, Navier–Stokes equations, Curl
Changed3 November 2026

Markdown · JSON · Edit on GitHub

What this page needs for W1

  • an inventory row: the row of the inventory that the node accounts for (0 of 1)
  • a source with a locator (0 of 1)

The writing levels say how complete the text of a page is. They do not measure its truth.

What links here

No page links to this page yet.