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Creeping Motion Around Spheres at Rest in a Newtonian Fluid

  • Kolumban HutterEmail author
  • Yongqi Wang
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Part of the Advances in Geophysical and Environmental Mechanics and Mathematics book series (AGEM)

Abstract

This volume consists of 10 chapters and commences in this chapter with the determination of the creeping motion around spheres at rest in a Newtonian fluid. This is a classical problem of singular perturbations in the form of matched asymptotic expansions. For creeping flow the acceleration terms in Newton’s law can be ignored to approximately calculate flow around the sphere by this so-called Stokes approximation. It turns out that far away from the sphere the acceleration terms become larger than those in the Stokes solution, so that the latter solution violates the boundary conditions at infinity. This lowest order correction of the flow around the sphere is due to Oseen. In a systematic perturbation expansion the outer—Oseen—series and the inner—Stokes—series with the small Reynolds number as perturbation parameter must be matched together to determine all boundary and transition conditions of inner and outer expansions. This procedure is rather tricky, i.e., not easy to understand for beginners. This theory, originally due to Kaplun and Lagerström has been extended, and the drag coefficient for the sphere, which also can be measured, is expressible in terms of a series expansion of powers of the Reynolds number. However, for Reynolds numbers larger than unity, convergence to measured values is poor. In the 1990s of the last century a new mathematical approach was designed—the so-called Homotopy Analysis Method; it is based on an entirely different expansion technique, not restricted to small Reynolds numbers, and results for the drag coefficient lie much closer to the experimental values than values obtained with the ‘classical’ matched asymptotic expansion. Incidentally, the laminar flow of a viscous fluid around a cylinder can analogously be treated, but is not contained in this treatise.

Keywords

Creeping motion Stokes approximation StokesOseen expansion Drag coefficient for the sphere as a function of the Reynolds number Homotopy analysis 

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.c/o Versuchsanstalt für Wasserbau, Hydrologie und GlaziologieETH ZürichZürichSwitzerland
  2. 2.Department of Mechanical EngineeringTechnische Universität DarmstadtDarmstadtGermany

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