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The Quarterly Journal of Mechanics and Applied Mathematics Advance Access published online on March 21, 2007
The Quarterly Journal of Mechanics and Applied Mathematics, doi:10.1093/qjmam/hbl026
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© The author 2007. Published by Oxford University Press; all rights reserved. For Permissions, please email: journals.permissions@oxfordjournals.org

GENERALIZED ANALYTIC FUNCTIONS IN 3D STOKES FLOWS

M Zabarankin{dagger}

( Department of Mathematical Sciences, Stevens Institute of Technology, Castle Point on Hudson, Hoboken, NJ 07030, USA )

P Krokhmal{ddagger}

( Department of Mechanical and Industrial Engineering, The University of Iowa, 2403 Seamans Center, Iowa City, IA 52242, USA )

{dagger} mzabaran{at}stevens.edu

{ddagger} krokhmal{at}engineering.uiowa.edu

Received 15 December 2006.
  Abstract

A class of generalized analytic functions, defined by a special case of the Carleman system that arises in three-dimensional asymmetric problems of hydrodynamics of Stokes flows, stationary electromagnetic fields in conductive materials, etc., has been considered. Hilbert formulae, establishing relationships between the real and imaginary parts of a generalized analytic function from this class, have been derived for the domains exterior to the contour of bi-spheres and torus in the meridional cross-section plane. This special case of the Carleman system has been reduced to an infinite three-diagonal system of algebraic equations with respect to the coefficients in series representations of the real and imaginary parts. For torus, the system has been solved by means of the Fourier transform, while for bi-spheres, it has been solved by an algebraic method. As examples, analytical expressions for the pressure in the problems of the steady axially symmetric motion of rigid bi-spheres and torus in a Stokes fluid have been derived based on the corresponding Hilbert formulae.


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