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https://hdl.handle.net/11147/2392
Title: | Vortex images and q-elementary functions | Authors: | Pashaev, Oktay Yılmaz, Oğuz |
Keywords: | Group theory Special functions Boundary-value problems Electrostatics Vortex dynamics |
Publisher: | IOP Publishing Ltd. | Source: | Pashaev, O., and Yılmaz, O. (2008). Vortex images and q-elementary functions. Journal of Physics A: Mathematical and Theoretical, 41(13). doi:10.1088/1751-8113/41/13/135207 | Abstract: | In the present paper, the problem of vortex images in the annular domain between two coaxial cylinders is solved by the q-elementary functions. We show that all images are determined completely as poles of the q-logarithmic function, and are located at sites of the q-lattice, where a dimensionless parameter q = r 2 2/r 2 1 is given by the square ratio of the cylinder radii. The resulting solution for the complex potential is represented in terms of the Jackson q-exponential function. Our approach in this paper provides an efficient path to rediscover known solutions for the vortex-cylinder pair problem and yields new solutions as well. By composing pairs of q-exponents to the first Jacobi theta function and conformal mapping to a rectangular domain we show that our solution coincides with the known one, obtained before by elliptic functions. The Schottky-Klein prime function for the annular domain is factorized explicitly in terms of q-exponents. The Hamiltonian, the Kirchhoff-Routh and the Green functions are constructed. As a new application of our approach, the uniformly rotating exact N-vortex polygon solutions with the rotation frequency expressed in terms of q-logarithms at Nth roots of unity are found. In particular, we show that a single vortex orbits the cylinders with constant angular velocity, given as the q-harmonic series. Vortex images in two particular geometries with only one cylinder as the q → ∞ limit are studied in detail. | URI: | http://doi.org/10.1088/1751-8113/41/13/135207 http://hdl.handle.net/11147/2392 |
ISSN: | 1751-8113 1751-8113 1751-8121 |
Appears in Collections: | Mathematics / Matematik Scopus İndeksli Yayınlar Koleksiyonu / Scopus Indexed Publications Collection WoS İndeksli Yayınlar Koleksiyonu / WoS Indexed Publications Collection |
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