Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Hilbert formulas for $ r$-analytic functions and the Stokes flow about a biconvex lens

Authors: Michael Zabarankin and Andrei F. Ulitko
Journal: Quart. Appl. Math. 64 (2006), 663-693
MSC (2000): Primary 30E20, 35Q15, 35Q30, 76D07
DOI: https://doi.org/10.1090/S0033-569X-06-01011-7
Published electronically: September 14, 2006
MathSciNet review: 2284465
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Abstract: The so-called $ r$-analytic functions are a subclass of $ p$-analytic functions and are defined by the generalized Cauchy-Riemann system with $ p(r,z)=r$. In the system of toroidal coordinates, the real and imaginary parts of an $ r$-analytic function are represented by Mehler-Fock integrals with densities, which are assumed to be meromorphic functions. Hilbert formulas, establishing relationships between those functions, are derived for the domain exterior to the contour of a biconvex lens in the meridional cross-section plane. The derivation extends the framework of the theory of Riemann boundary-value problems, suggested in our previous work, to solving the three-contour problem for the case of meromorphic functions with a finite number of simple poles. For numerical calculations, Mehler-Fock integrals with Hilbert formulas reduce to the form of regular integrals. The 3D problem of the axially symmetric steady motion of a rigid biconvex lens-shaped body in a Stokes fluid is solved, and the Hilbert formula for the real part of an $ r$-analytic function is used to express the pressure in the fluid via the vorticity analytically. As an illustration, streamlines and isobars about the body, the vorticity and pressure at the contour of the body and the drag force exerted on the body by the fluid are calculated.

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Additional Information

Michael Zabarankin
Affiliation: Department of Mathematical Sciences, Stevens Institute of Technology, Castle Point on Hudson, Hoboken, New Jersey 07030
Email: mzabaran@stevens.edu

Andrei F. Ulitko
Affiliation: Department of Mechanics and Mathematics, National Taras Shevchenko University of Kiev, 7 Academic Glushkov Prospect, Kiev, Ukraine

DOI: https://doi.org/10.1090/S0033-569X-06-01011-7
Keywords: $r$-analytic function, generalized Cauchy-Riemann system, Hilbert formula, Riemann boundary-value problem, analytic function, biconvex lens, toroidal coordinates, Mehler-Fock integral transform, Stokes model, pressure, vorticity, drag force
Received by editor(s): October 24, 2005
Published electronically: September 14, 2006
Article copyright: © Copyright 2006 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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