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Stokes’ paradox: creeping flow past a two-dimensional cylinder in an infinite domain

Published online by Cambridge University Press:  21 March 2017

Arzhang Khalili Open the ORCID record for Arzhang Khalili [Opens in a new window]  and
Bo Liu
Arzhang Khalili*
Affiliation:
Department of Biogeochemistry, Max Planck Institute for Marine Microbiology, 28359 Bremen, Germany Department of Physics and Earth Sciences, Jacobs University Bremen, 28759 Bremen, Germany
Bo Liu
Affiliation:
Department of Biogeochemistry, Max Planck Institute for Marine Microbiology, 28359 Bremen, Germany Geochemistry and Isotope Biogeochemistry Group, Marine Geology Department, Leibniz Institute for Baltic Sea Research (IOW), 18119 Warnemünde, Germany
*
†Email address for correspondence: akhalili@mpi-bremen.de
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Abstract

Finite container sizes in experiments and computer simulations impose artificial boundaries which do not exist when they are meant to mimic ambient fluid of infinite extent. We show here that this is the case with flows past an infinite cylinder placed in an infinite ambient fluid (Stokes’ paradox). Using a highly efficient and stable numerical method that is capable of handling computational domains several orders of magnitude larger than in previous studies, we provide a criterion for the minimum necessary extent around an object in order to provide accurate velocity and pressure fields, which are prerequisites for correct calculation of secondary quantities such as drag coefficient. The careful and extensive simulations performed suggest an improved relation for the drag coefficient as a function of Reynolds number, and identify the most suitable experimental data available in the literature.

JFM classification

Low-Reynolds-number flows: Low-Reynolds-number flows Low-Reynolds-number flows: Stokesian dynamics
Type
Papers
Information
Journal of Fluid Mechanics , Volume 817 , 25 April 2017 , pp. 374 - 387
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Copyright
© 2017 Cambridge University Press 

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References

Bairstow, L. & Cave, B. M. 1923 The resistance of a cylinder moving in a viscous fluid. Phil. Trans. R. Soc. Lond. A 223, 383–432.Google Scholar
Bönisch, S., Heuveline, V. & Wittwer, P. 2005 Adaptive boundary conditions for exterior flow problems. J. Math. Fluid Mech. 7, 85–107.CrossRefGoogle Scholar
Bönisch, S., Heuveline, V. & Wittwer, P. 2008 Second order adaptive boundary conditions for exterior flow problems: non-symmetric stationary flows in two dimensions. J. Math. Fluid Mech. 10, 45–70.CrossRefGoogle Scholar
Chen, S. & Doolen, G. D. 1998 Lattice Boltzmann method for fluid flows. Annu. Rev. Fluid Mech. 30, 329–364.CrossRefGoogle Scholar
Chopard, B. & Droz, M. 1998 Cellular Automata Modeling of Physical Systems. Cambridge University Press.CrossRefGoogle Scholar
van Dyke, M. 1970 Extension of Goldstein’s series for the Oseen drag of a sphere. J. Fluid Mech. 44, 365–372.CrossRefGoogle Scholar
van Dyke, M. 1975 Perturbation Methods in Fluid Mechanics. Parabolic.Google Scholar
Filippova, O. & Hänel, D. 1998 Grid refinement for lattice–BGK models. J. Comput. Phys. 147, 219–228.CrossRefGoogle Scholar
Finn, R. K. 1953 Determination of the drag on a cylinder at low Reynolds number. J. Appl. Phys. 24, 771–773.Google Scholar
Goldstein, S. 1957 Lectures on Fluid Mechanics. Interscience.Google Scholar
Greene, D. F. & Johnson, E. A. 1990 The aerodynamics of plumed seeds. Funct. Ecol. 4, 117–125.CrossRefGoogle Scholar
Huner, B. & Hussey, R. G. 1977 Cylinder drag at low Reynolds number. Phys. Fluids 20, 1211–1218.Google Scholar
Imai, I. 1954 A new method of solving Oseen’s equations and its application to the flow past an inclined elliptic cylinder. Proc. R. Soc. Lond. A 224, 141–160.Google Scholar
Jayaweera, K. O. L. F. & Mason, B. J. 1965 The behaviour of freely falling cylinders and cones in a viscous fluid. J. Fluid Mech. 22, 709–720.Google Scholar
Kaplun, S. 1957 Low Reynolds number flow past a circular cylinder. J. Math. Mech. 6, 595–603.Google Scholar
Karp-Boss, L. E. & Jumars, P. 1996 Nutrient fluxes to planktonic osmotrophs in the presence of fluid motion. Oceanogr. Mar. Biol.: Annu. Rev. 34, 71–107.Google Scholar
Keller, J. B. & Ward, M. J. 1996 Asymptotics beyond all orders for a low Reynolds number flow. J. Math. Mech. 30, 253–265.Google Scholar
Kida, T. & Take, T. 1992a Asymptotic expansions for low Reynolds number flow past a cylindrical body. JSME Intl J. 35, 144–150.Google Scholar
Kida, T. & Take, T. 1992b Integral approach of asymptotic expansions for low Reynolds number flow past an arbitrary cylinder. JSME Intl J. 35, 138–143.Google Scholar
Lamb, H. 1911 On the uniform motion of a sphere through a viscous fluid. Phil. Mag. 6 (21), 112–121.Google Scholar
Lamb, Sir H. 1932 Hydrodynamics. Cambridge University Press.Google Scholar
Landau, L. D. & Lifshitz, E. M. 1993 Fluid Mechanics. Pergamon.Google Scholar
Litchman, E., Klausmeier, C. A. & Yoshiyama, K. 2009 Contrasting size evolution in marine and freshwater diatoms. Proc. Natl Acad. Sci. USA 106, 2665–2670.CrossRefGoogle ScholarPubMed
Liu, B. & Khalili, A. 2008 Acceleration of steady-state lattice Boltzmann simulation for exterior flows. Phys. Rev. E 78, 056701.Google Scholar
Liu, B. & Khalili, A. 2009 Lattice Boltzmann model for exterior flows with an annealing preconditioning method. Phys. Rev. E 79, 066701.CrossRefGoogle ScholarPubMed
Munk, W. H. & Riley, G. A. 1952 Absorption of nutrients by aquatic plants. J. Mar. Res. 11, 215–240.Google Scholar
Oseen, C. W. 1910 Über die Stokessche Formel und über die verwandte Aufgabe in der Hydrodynamik. Ark. Mat. Astron. Fys. 6 (29), 143–152.Google Scholar
Oseen, C. W. 1913 Über den Gültigkeitsbereich der Stokesschen Widerstandsformel. Ark. Mat. Astron. Fys. 9, 1–15.Google Scholar
Proudman, I. & Pearson, J. R. A. 1957 Expansion at small Reynolds number for the flow past a sphere and a circular cylinder. J. Fluid Mech. 2, 237–262.Google Scholar
Qian, Y. H., D’Humières, D. & Lallemand, P. 1992 Lattice BGK models for Navier–Stokes equation. Europhys. Lett. 17 (6), 479–484.CrossRefGoogle Scholar
Qian, Y. H. & Orszag, S. A. 1993 Lattice BGK models for the Navier–Stokes equation: nonlinear deviation in compressible regimes. Europhys. Lett. 21 (3), 255–259.Google Scholar
Relf, E. F.1914 Discussion of the results of measurements of the resistance of wires, with some additional tests on the resistance of wires of small diameter. British A.R.C. Rep. Mem. 102.Google Scholar
Rheinländer, M. 2005 A consistent grid coupling method for lattice–Boltzmann schemes. J. Stat. Phys. 121, 49–74.CrossRefGoogle Scholar
Riebesell, U., Körtzinger, A. & Oschlies, A. 2009 Sensitivities of marine carbon fluxes to ocean change. Proc. Natl Acad. Sci. USA 106, 20602–20609.Google Scholar
Schiller, L. 1932 Fallversuche mit Kugeln und Scheiben, Hydro- und Aerodynamik. Handbuch der Experimentalphysik 4, 337–387.Google Scholar
Schulte, K., Gojny, F. H., Wichmann, M. H. G. & Fiedler, B. 2005 Influence of different carbon nanotubes on the mechanical properties of epoxy matrix composites – a comparative study. Compos. Sci. Technol. 65, 2300–2313.Google Scholar
Skinner, L. A. 1975 Generalized expansions for slow flow past a cylinder. Q. J. Mech. Appl. Maths 28, 333–340.Google Scholar
Sohm, J. A., Webb, E. A. & Capone, D. G. 2011 Emerging patterns of marine nitrogen fixation. Nat. Rev. Microbiol. 9, 499–508.Google Scholar
Stokes, G. G. 1851 On the effect of the internal friction of fluids on the motion of pendulums. Trans. Camb. Phil. Soc. 9, 8–106.Google Scholar
Succi, S. 2001 The Lattice Boltzmann Equation for Fluid Dynamics and Beyond. Oxford University Press.Google Scholar
Suzuki, H., Sasaki, H. & Fukuchi, M. 2003 Loss processes of sinking fecal pellets of zooplankton in the mesopelagic layers of the Antarctic marginal ice zone. J. Oceanogr. 59, 809–818.Google Scholar
Taneda, S. 1956 Experimental investigation of the wakes behind cylinder and plates at low Reynolds number. J. Phys. Soc. Japan 11, 302–307.CrossRefGoogle Scholar
Tölke, J., Freudiger, S. & Krafczyk, Manfred 2006 An adaptive scheme using hierarchical grids for lattice Boltzmann multi-phase flow simulations. Comput. Fluids 35, 820–830.Google Scholar
Tomotika, S. & Aoi, T. 1950 The steady flow of viscous fluid past a sphere and circular cylinder at small Reynolds numbers. Q. J. Mech. Appl. Maths 3, 140–161.Google Scholar
Tritton, D. J. 1959 Experiments on the flow past a circular cylinder at low Reynolds number. J. Fluid Mech. 6, 547–567.Google Scholar
Veysey, J. & Goldenfeld, N. 2007 Singular perturbations in simple low Reynolds number flows: from boundary layers to the renormalization group. Rev. Mod. Phys. 79, 883–927.Google Scholar
Whitehead, A. N. 1889 Second approximations to viscous fluid motion. Q. J. Pure Appl. Maths 23, 143–152.Google Scholar
Wieselsberger, A. N. 1921 Neuere Feststellungen über die Gesetze des Flüssigkeits- und Luftwiderstandes. Z. Phys. 22, 321–328.Google Scholar