Introduction to Vortex Filaments in Equilibrium by Timothy D. Andersen & Chjan C. Lim

Author:Timothy D. Andersen & Chjan C. Lim
Language: eng
Format: epub
Publisher: Springer New York, New York, NY

(5.29)

where δ = ε∕R and κ = 1∕R is the curvature. The contribution is the non-local induction from the Biot–Savart integral. In the LIA, the characteristic wavelength of the filament is much smaller than the average radius of curvature, L ≪ R, and non-local effects become small, vanishing as R → ∞—the perfectly straight case. The LIA can then be written as

(5.30)

where is the tangential vector. This uses the approximation

(5.31)

where is a small parameter [78].

As with the non-local induction equations, C 0 must be measured indirectly by measuring the motion of the filament and deriving its value from the equations.

If the filament is nearly parallel, then , the basis vector in the z-direction in Cartesian coordinates (x, y, z). The resulting vector then approximately points in the x-y plane and z ≈ s if scaled appropriately. Under this assumption, is two-dimensional and the LIA is

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