Math 2.0

Fluid mechanics Definition

Circulation

Builds on Line integral of a vector fieldVector fieldVorticityStokes' theorem

Definition

Let Ω⊆ℝn\Omega \subseteq \R^n be open, with n=2n = 2 or n=3n = 3, and let 𝐮 ⁣:Ω→ℝn\mathbf{u} \colon \Omega \to \R^n be a continuous velocity field. Let CC be a closed, oriented, piecewise continuously differentiable curve in Ω\Omega. The circulation of 𝐮\mathbf{u} around CC is the line integral ΓC=∮C𝐮⋅d𝐥=∫ab𝐮(γ(s))⋅γ′(s) ds,\Gamma_C = \oint_C \mathbf{u} \cdot d\mathbf{l} = \int_a^b \mathbf{u}(\gamma(s)) \cdot \gamma'(s) \, ds, where γ ⁣:[a,b]→Ω\gamma \colon [a, b] \to \Omega is a parametrisation of CC with γ(a)=γ(b)\gamma(a) = \gamma(b).

Remarks

If 𝐮\mathbf{u} is continuously differentiable and CC bounds an oriented surface SS in Ω\Omega, then Stokes' theorem gives ΓC=∬S𝝎⋅𝐧 dA\Gamma_C = \iint_S \boldsymbol{\omega} \cdot \mathbf{n} \, dA, where 𝝎\boldsymbol{\omega} is the vorticity. Example: for 𝐮=(−y,x)\mathbf{u} = (-y, x) and the unit circle γ(s)=(cos⁡s,sin⁡s)\gamma(s) = (\cos s, \sin s), 0≤s≤2π0 \le s \le 2\pi, the integrand is sin⁡2s+cos⁡2s=1\sin^2 s + \cos^2 s = 1, so Γ=2π\Gamma = 2\pi; this equals the vorticity 22 times the area π\pi. Non-example: for the uniform field 𝐮=(1,0)\mathbf{u} = (1, 0), the integral along the upper half of the circle is −2-2. This is not a circulation, because the curve is not closed; around the full circle the circulation is 00. Reversing the orientation of CC changes the sign of ΓC\Gamma_C.

Reading path

To understand this page from the ground up, read these 63 pages in order. The foundations come first.

  1. Euclidean space object
  2. Function concept
  3. Function composition concept
  4. Infimum concept
  5. Supremum concept
  6. Vorticity concept
  7. Fréchet derivative definition
  8. Total derivative concept
  9. Function of several variables concept
  10. Vector-valued function concept
  11. Jordan measure definition
  12. Limit of a function of several variables concept
  13. Differentiable function of several variables concept
  14. Parametrised curve concept
  15. Vector field concept
  16. Iterated integral definition
  17. Partial derivative concept
  18. Continuity in R^n definition
  19. Differentiable function of several variables definition
  20. Partial derivative definition
  21. Line integral of a vector field concept
  22. Gradient concept
  23. Chain rule in several variables statement
  24. Differentiable on an open set definition
  25. Double integral definition
  26. Gradient definition
  27. Higher partial derivatives definition
  28. Jacobian matrix definition
  29. Iterated derivative definition
  30. Chain rule for a composition with a curve statement
  31. Fubini's theorem statement
  32. Triple integral definition
  33. C^k function characterised by continuous derivatives statement
  34. Differential form definition
  35. Parametrised curve definition
  36. Vector field definition
  37. Wedge product of forms definition
  38. Closed curve definition
  39. Parametrised surface definition
  40. Rectifiable curve definition
  41. Conservative vector field definition
  42. Curl definition
  43. Tangent vector of a curve definition
  44. Divergence definition
  45. Line integral of a scalar field definition
  46. Line integral of a vector field definition
  47. Tangent plane and normal vector of a surface definition
  48. Regular curve definition
  49. Green's theorem statement
  50. One-form definition
  51. Surface area definition
  52. Arc length parametrisation definition
  53. Surface integral definition
  54. Line integral as an integral over the parameter interval statement
  55. Exterior derivative definition
  56. Divergence theorem on a rectangle in the plane statement
  57. Orientable surface definition
  58. Gradient theorem statement
  59. Flux through a surface definition
  60. Divergence theorem statement
  61. Generalised Stokes theorem statement
  62. Stokes' theorem statement
  63. Circulation definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Circulation_(physics)
About this page
Tag7ZT5QQ8MHC
URIm2://7ZT5QQ8MHC
Kinddefinition
WritingW0 stub
MSC76A02, 76B47
DefinesCirculation
UsesLine integral of a vector field, Vector field, Vorticity, Stokes' theorem
Changed3 November 2026

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