Download Equi-angular packings for 3D vortex systems

Equi-angular packings for 3D vortex systems
Geert C. Dijkhuis
Eindhoven University of Technology, The Netherlands
Recent experiments in superconducting nanostructures reveal spontaneous formation of symmetric
vortex-antivortex patterns conserving square or triangular sample symmetry [1]. Similar
symmetry-induced vortex patterns are expected in superfluid helium and laser-stirred BoseEinstein condensates confined by boundaries with well-defined discrete symmetries. Their ideal
flow patterns have analytic forms for potential and streamlines around quantized vortex-antivortex
tubes alternating on square, triangular or hexagonal grids [2]. Analogous 3D vortex crystals
achieve uniform and minimal dissipation with equal angles between tangent tubes. Ref. 2 obtains
tetragonal, octahedral and rhombohedral vortex lattices with four equi-angular axis directions
parallel to the body diagonals of a cube, or normal to the octahedral faces at common angle
arccos(1/3)= 70.50. Here we construct analogous 3D packings of straight vortex tubes with axis
directions parallel to the six body diagonals of an icosahedron, or normal to the dodecahedral
faces. Their common angle arccos(1/√5)=63.40, or common distance in elliptic geometry [3], is
smaller than for cube diagonals, reducing shear and dissipation in resulting vortex lattices.
Fig. 1(a) Dodecahedral cell
with six equi-angular tubes.
(b) Cubic cell with
six equi-angular tubes.
(c) Rhombohedral cell with
four equi-angular tubes.
In the dodecahedral cell of Fig. 1a six equi-angular tubes tubes connect edges of opposite faces at
right angles to their axes. Its scaling extension smoothly connects vortex tubes at common edges.
With g=½(1+√5)≅1.618 for the Golden Section, we find that its fractal dimension g+2log20≅2.33
approaches the Von Karman constant viewed as a threshold dimension for transience or
recurrence of random walks [4]. On cube faces as in Fig. 1b six equi-angular tubes give irrational
slopes equal to g precluding their smooth extension into a periodic lattice. But upon removal of
two out of six tubes the remaining four tubes align with face diagonals in a specific cuboidal unit
cell smoothly packing the tubes into a periodic lattice. A second periodic lattice results when four
tubes align with the edges and the shortest body diagonal in the rhombohedral unit cell shown in
Fig. 1c. This cell has a Golden Rhombus as base, and differs from the obtuse Golden
Rhombohedron only by its height. A third periodic lattice results with tubes connecting edges of
octahedral faces shaped as isosceles triangles instead of equilateral triangles defining a regular
octahedron. These three lattice types pack cylinders with two different radii and twofold
symmetry, lower than the 3 and 4-fold symmetry of corresponding equi-angular and equi-radial
vortex crystals presented in ref. 2.
1 V.L. Moshchalkov, Europhysics News, Vol. 22, pp. 176-177, 2001.
2 G.C. Dijkhuis, Proceedings, 5th International Symposium on Aerothermodynamics of Internal Flows (ISAIF),
Gdansk, Poland, September 2001 (in press).
3 J.H. van Lint and J.J. Seidel, in: Geometry and Combinatorics, pp. 3-16, Academic Press, 1991.
4 G.C. Dijkhuis, Physica B, Vol. 228, pp.144-152 (1996).
Abstract for 3rd AMIF Conference, Lissabon, Portugal, April 17-20, 2002
Eindhoven, 1 Nov 2001.