Hostname: page-component-8448b6f56d-m8qmq Total loading time: 0 Render date: 2024-04-18T21:01:18.384Z Has data issue: false hasContentIssue false
  • English
  • Français

On the periodic fundamental solutions of the Stokes equations and their application to viscous flow past a cubic array of spheres

Published online by Cambridge University Press:  28 March 2006

H. Hasimoto
H. Hasimoto
Affiliation:
Department of Aeronautical Engineering, Faculty of Engineering, Kvoto University, Japan
Get access Rights & Permissions [Opens in a new window]

Abstract

Spatially periodic fundamental solutions of the Stokes equations of motion for a viscous fluid past a periodic array of obstacles are obtained by use of Fourier series. It is made clear that the divergence of the lattice sums pointed out by Burgers may be rescued by taking into account the presence of the mean pressure gradient. As an application of these solutions the force acting on any one of the small spheres forming a periodic array is considered. Cases for three special types of cubic lattice are investigated in detail. It is found that the ratios of the values of this force to that given by the Stokes formula for an isolated sphere are larger than 1 and do not differ so much among these three types provided that the volume concentration of the spheres is the same and small. The method is also applied to the two-dimensional flow past a square array of circular cylinders, and the drag on one of the cylinders is found to agree with that calculated by the use of elliptic functions.

Type
Research Article
Information
Journal of Fluid Mechanics , Volume 5 , Issue 2 , February 1959 , pp. 317 - 328
Copyright
© 1959 Cambridge University Press

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Born, M. & Misra, R. D. 1940 Proc. Camb. Phil. Soc. 36, 466.
Brinkman, H. C. 1947 Appl. Sci. Res., Hague, 1, A, 27.
Brinkman, H. C. 1948 Appl. Sci. Res., Hague, 1, A, 81.
Burgers, J. M. 1938 2nd Report on Viscosity and Plasticity, ch. III.
Burgers, J. M. 1941 Proc. K. Akad. Wet. Amst. 44, 1045, 1174.
Debye, P. & Bueche, A. M. 1948 J. Chem. Phys. 16, 573.
Emersleben, O. 1923 Phys. Z. 36, 173, 466.
Ewald, P. P. 1921 Ann. Phys. 64, 253.
Hasimoto, H. 1958a J. Phys. Soc. Japan, 13, 633.
Hasimoto, H. 1958b J. Phys. Soc. Japan (to be published).
Hobson, E. W. 1931 The Theory of Spherical and Ellipsoidal Harmonics. Cambridge University Press.
Kawaguchi, M. 1958 J. Phys. Soc. Japan, 13, 209.
Kuwabara, S. 1958 J. Phys. Soc. Japan (to be published).
Kynch, G. J. 1954 Brit. J. Appl. Phys. 3, 5.
Kynch, G. J. 1956 Proc. Roy. Soc. A, 237, 90.
Lamb, H. 1932 Hydrodynamics, 6th ed. Cambridge University Press.
Miyagi, T. 1958 J. Phys. Soc. Japan, 13, 493.
Tamada, K. & Fujikawa, H. 1957 Quart. J. Mech. Appl. Math. 10, 425.
Uchida, S. 1949 Rep. Inst. Sci. Technol. Tokyo, 3, 97.
Wigner, H. & Seitz, F. 1933 Phys. Rev. 43, 804.