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## Kelvin's circulation theoremIn fluid mechanics, ## Additional recommended knowledgewhere Γ is the circulation around a material contour This theorem does not hold in cases with viscous stresses, nonconservative body forces (for example a coriolis force) or non-barotropic pressure-density relations. ## Mathematical ProofThe circulation Γ around a closed material contour where is an element along the closed contour.
dsThe governing equation for an inviscid fluid with a conservative body force is where D/D The condition of baratropicity implies that the density is a function only of the pressure, i.e. ρ = ρ(
For the first term, we substitute from the governing equation, and then apply Stoke's theorem, thus: The final equality arises since owing to baratropicity. For the second term, we note that evolution of the material line element is given by Hence The last equality is obtained by applying Stokes theorem. Since both terms are zero, we obtain the result ## See also## References**^**Kundu, P and Cohen, I: "Fluid Mechanics", page 130. Academic Press 2002
Categories: Equations of fluid dynamics | Fluid dynamics |

This article is licensed under the GNU Free Documentation License. It uses material from the Wikipedia article "Kelvin's_circulation_theorem". A list of authors is available in Wikipedia. |