Skip to main content
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 415
  • Cited by
    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Bose, Sujit K. and Dey, Subhasish 2016. Circular Far-Wake Flow behind a Sphere: Solutions to the Second Order. Journal of Engineering Mechanics, Vol. 142, Issue. 1, p. 06015005.

    Fukada, Toshiaki Takeuchi, Shintaro and Kajishima, Takeo 2016. Interaction force and residual stress models for volume-averaged momentum equation for flow laden with particles of comparable diameter to computational grid width. International Journal of Multiphase Flow, Vol. 85, p. 298.

    Hutter, Kolumban and Wang, Yongqi 2016. Fluid and Thermodynamics.

    Kishore, Nanda and Ramteke, Rahul Ramdas 2016. Forced convective heat transfer from spheres to Newtonian fluids in steady axisymmetric flow regime with velocity slip at fluid–solid interface. International Journal of Thermal Sciences, Vol. 105, p. 206.

    Kundu, Pijush K. Cohen, Ira M. and Dowling, David R. 2016. Fluid Mechanics.

    Wagner, Caroline E. and McKinley, Gareth H. 2016. The importance of flow history in mixed shear and extensional flows. Journal of Non-Newtonian Fluid Mechanics, Vol. 233, p. 133.

    Amador, G. J. Mao, W. DeMercurio, P. Montero, C. Clewis, J. Alexeev, A. and Hu, D. L. 2015. Eyelashes divert airflow to protect the eye. Journal of The Royal Society Interface, Vol. 12, Issue. 105, p. 20141294.

    Einarsson, J. Candelier, F. Lundell, F. Angilella, J. R. and Mehlig, B. 2015. Effect of weak fluid inertia upon Jeffery orbits. Physical Review E, Vol. 91, Issue. 4,

    Fanghui Yin, Farzaneh, Masoud and Xingliang Jiang, 2015. 2015 IEEE Conference on Electrical Insulation and Dielectric Phenomena (CEIDP). p. 427.

    Guillod, Julien and Wittwer, Peter 2015. Asymptotic behaviour of solutions to the stationary Navier–Stokes equations in two-dimensional exterior domains with zero velocity at infinity. Mathematical Models and Methods in Applied Sciences, Vol. 25, Issue. 02, p. 229.

    Lamtyugova, S. N. and Sidorov, M. V. 2015. Numerical analysis of the external slow flows of a viscous fluid using the R-function method. Journal of Engineering Mathematics, Vol. 91, Issue. 1, p. 59.

    Mohajer, Behzad Aliakbar, Vahid Shams, Mehrzad and Moshfegh, Abouzar 2015. Heat Transfer Analysis of a Microspherical Particle in the Slip Flow Regime by Considering Variable Properties. Heat Transfer Engineering, Vol. 36, Issue. 6, p. 596.

    Srivastava, Deepak Kumar Yadav, Raja Ram and Yadav, Supriya 2015. Creeping flow past rotating axi-symmetric isolated body-class of deformed sphere. Journal of the Brazilian Society of Mechanical Sciences and Engineering, Vol. 37, Issue. 4, p. 1199.

    Villadsen, Naja Andreasen, Daniel Ø. Hagelskjær, Jesper Thøgersen, Jan Imparato, Alberto and Keiding, Søren Rud 2015. Pushing the limit: investigation of hydrodynamic forces on a trapped particle kicked by a laser pulse. Optics Express, Vol. 23, Issue. 10, p. 13141.

    Henry, Christophe and Minier, Jean-Pierre 2014. Progress in particle resuspension from rough surfaces by turbulent flows. Progress in Energy and Combustion Science, Vol. 45, p. 1.

    Hormozi, Sarah and Ward, Michael J. 2014. A hybrid asymptotic-numerical method for calculating drag coefficients in 2-D low Reynolds number flows. Journal of Engineering Mathematics,

    Khair, Aditya S. and Chisholm, Nicholas G. 2014. Expansions at small Reynolds numbers for the locomotion of a spherical squirmer. Physics of Fluids, Vol. 26, Issue. 1, p. 011902.

    Li, Dandan Li, Shichen Xue, Yahui Yang, Yantao Su, Weidong Xia, Zhenhua Shi, Yipeng Lin, Hao and Duan, Huiling 2014. The effect of slip distribution on flow past a circular cylinder. Journal of Fluids and Structures, Vol. 51, p. 211.

    Victor Nunes de Sousa, João Roberto Lins de Macêdo, Antônio Ferreira de Amorim Junior, Wanderley and Gilson Barbosa de Lima, Antonio 2014. Numerical Analysis of Turbulent Fluid Flow and Drag Coefficient for Optimizing the AUV Hull Design. Open Journal of Fluid Dynamics, Vol. 04, Issue. 03, p. 263.

    2014. Flows and Chemical Reactions in Heterogeneous Mixtures.


Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder

  • Ian Proudman (a1) and J. R. A. Pearson (a1)
  • DOI:
  • Published online: 01 March 2006

This paper is concerned with the problem of obtaining higher approximations to the flow past a sphere and a circular cylinder than those represented by the well-known solutions of Stokes and Oseen. Since the perturbation theory arising from the consideration of small non-zero Reynolds numbers is a singular one, the problem is largely that of devising suitable techniques for taking this singularity into account when expanding the solution for small Reynolds numbers.

The technique adopted is as follows. Separate, locally valid (in general), expansions of the stream function are developed for the regions close to, and far from, the obstacle. Reasons are presented for believing that these ‘Stokes’ and ‘Oseen’ expansions are, respectively, of the forms $\Sigma \;f_n(R) \psi_n(r, \theta)$ and $\Sigma \; F_n(R) \Psi_n(R_r, \theta)$ where (r, θ) are spherical or cylindrical polar coordinates made dimensionless with the radius of the obstacle, R is the Reynolds number, and $f_{(n+1)}|f_n$ and $F_{n+1}|F_n$ vanish with R. Substitution of these expansions in the Navier-Stokes equation then yields a set of differential equations for the coefficients ψn and Ψn, but only one set of physical boundary conditions is applicable to each expansion (the no-slip conditions for the Stokes expansion, and the uniform-stream condition for the Oseen expansion) so that unique solutions cannot be derived immediately. However, the fact that the two expansions are (in principle) both derived from the same exact solution leads to a ‘matching’ procedure which yields further boundary conditions for each expansion. It is thus possible to determine alternately successive terms in each expansion.

The leading terms of the expansions are shown to be closely related to the original solutions of Stokes and Oseen, and detailed results for some further terms are obtained.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
Please enter your name
Please enter a valid email address
Who would you like to send this to? *