The Quarterly Journal of Mechanics and Applied Mathematics Advance Access originally published online on July 6, 2007
The Quarterly Journal of Mechanics and Applied Mathematics 2007 60(3):337-366; doi:10.1093/qjmam/hbm010
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Method of automorphic functions in the study of flow around a stack of porous cylinders
( Department of Mathematics, Louisiana State University, Baton Rouge, LA 70803, USA )
( Department of Mathematics, Gubkin Russian State University of Oil and Gas, Moscow 119991, Russia )
antipov{at}math.lsu.edu
v_silvestrov{at}mail.ru
Received 20 December 2006. Accepted 15 March 2007.
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Abstract |
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This paper studies the ideal flow around a stack of stationary cylinders with porous walls. The boundary conditions on the surfaces of the cylinders are nonlinear. For small values of the porosity parameters, by applying the asymptotic method, the boundary conditions are linearized. The use of Möbius transformations generating a symmetric Schottky group reduces the problem to a Riemann–Hilbert boundary-value problem for symmetric automorphic functions. Its solution is found in a series form in terms of a quasiautomorphic analogue of the Cauchy integral. The absolute and uniform convergence of the series is guaranteed when the associated flow domain symmetric Schottky group is a first class group. An example of a symmetric Schottky group of divergent type (not of the first class) is given. Formulae for the drag and lift forces acting on the cylinders are derived, and the dependence of the porosity parameters on the forces is studied. In particular, the drag force is zero for a single solid cylinder (d'Alambert's paradox), while for a cylinder with a porous surface this is not true.