Kelvin's circulation theorem

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In fluid mechanics, Kelvin's Circulation Theorem states "In an inviscid, barotropic flow with conservative body forces, the circulation around a closed curve moving with the fluid remains constant with time"[1]. The theorem was developed by William Thomson, 1st Baron Kelvin. Stated mathematically:

\frac{D\Gamma}{Dt} = 0

Where \frac{D}{Dt} is the substantive derivative (sometimes the Lagrangian derivative, material derivative or advective derivative), and Γ is the circulation. Stated more simply this theorem says that if one observes a closed contour at one instant, and follows the contour over time (by following the motion of all of its fluid elements), the circulation over the two locations of this contour are equal.

This theorem does not hold in cases with viscous stresses, nonconservative body forces (for example a coriolis force) or nonbarotropic pressure-density relations.

[edit] Mathematical Proof

Circulation around a closed contour C is defined as:

\Gamma = \oint_C u \cdot ds

Where u is the velocity vector, and ds is an element along the closed contour. Taking the material derivative of circulation gives:

\frac{D\Gamma}{Dt} = \oint_C \frac{Du}{Dt} \cdot ds

Using Stoke's theorem this becomes:

\oint_C \frac{Du}{Dt} \cdot ds = \int_A \nabla x \frac{Du}{Dt} \cdot dA

However,

\frac{Du}{Dt} = -\nabla(\omega + \Phi)

with Φ being the velocity potential. Since

\nabla x \nabla (\omega + \Phi) = 0

as is true for any function. Thus:

\frac{D\Gamma}{Dt} = 0

[edit] See Also

Helmholtz's theorems

[edit] References

  1. ^ Kundu, P and Cohen, I: "Fluid Mechanics", page 130. Academic Press 2002
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