Geometric Inversion of Two-Dimensional Stokes Flows – Application to the Flow between Parallel Planes

Mustapha Hellou

.

DOI: 10.4236/eng.2010.210100                

Abstract

Geometric inversion is applied to two-dimensional Stokes flow in view to find new Stokes flow solutions. The principle of this method and the relations between the reference and inverse fluid velocity fields are presented. They are followed by applications to the flow between two parallel plates induced by a rotating or a translating cylinder. Thus hydrodynamic characteristics of flow around circular bodies obtained by inversion of the plates are thus deduced. Typically fluid flow patterns around two circular cylinders in contact placed in the centre of a rotating or a translating circular cylinder are illustrated.

Keywords

Inversion Transformation, Geometric Inversion, Stokes Flow, Viscous Eddies, Flow around a Cylinder, Microflow

Conflicts of Interest

The authors declare no conflicts of interest.

References

[1]	H. K. Moffatt, “Viscous and Resistive Eddies near a Sharp Corner,” Journal of Fluid Mechanics, Vol. 18, No. 1, 1964, pp. 1-18.
[2]	M. E. O’ Neill, “On Angles of Separation in Stokes Flow,” Journal of Fluid Mechanics, No. 133, 1983, pp. 427-442.
[3]	J. M. Bourot, “Sur la structure cellulaire des écoulements plans de Stokes, à debit moyen nul, en canal indéfini à parois parallèles,” Comptes rendus de l'Académie des sciences, Vol. 298, Serie II, 1984, pp. 161-164.
[4]	C. Shen and J. M. Floryan, “Low Reynolds Number Flow over Cavi-ties,” Physics of Fluids, Vol. 28, No. 11, 1985, pp. 3191-3202.
[5]	J. M. Bourot and F. Moreau, “Sur l’utilisation de la série cellulaire pour le calcul d’écoulements plans de Stokes en canal indéfini: Application au cas d’un cylindre circulaire en translation,” Mechanics Research Communications, Vol. 14, No. 3, 1987, pp. 187-197.
[6]	P. Carbonaro and E. B. Hansen, “Transient Stokes Flow in a Channel Driven by Moving Sleeves,” ASME Journal of Applied Mechanics, Vol. 57, No. 4, 1990, pp. 1061 -1065.
[7]	M. Hellou and M. Coutanceau, “Cellular Stokes Flow Induced by Rotation of a Cylinder in a Closed Channel,” Journal of Fluid Mechanics, No. 236, 1992, pp. 557-577.
[8]	F. Moreau and J. M. Bourot, “Ecoulements cellulaires de Stokes produits en canal plan illimité par la rotation de deux cylindres,” Journal of Applied Mathematics and Physics, Vol. 44, No. 5, 1993, pp. 777-798.
[9]	P. N. Shankar and M. D. Deshpande, “Fluid Mechanics in the Driven Cavity,” Annual Review of Fluid Mechanics, No. 32, 2000, pp. 93-136.
[10]	J. T. Jeong, “Slow Viscous Flow in a Partitioned Channel,” Physics of Fluids, Vol. 13, No. 6, 2001, pp. 1577 -1582.
[11]	M. Hellou, “Structures d'écoulements de Stokes dans une jonction bidimensionnelle de canaux,” Mécanique & Industries, Vol. 4, No. 5, 2003, pp. 575-583.
[12]	C. Y. Wang, “Slow Viscous Flow between Hexagonal Cylinders,” Transport in Porous Media, Vol. 47, No. 1, 2002, pp. 67-80.
[13]	C. Y. Wang, “The Recirculating Flow due to a Moving Lid on a Cavity Containing a Darcy–Brinkman Medium,” Applied Mathematical Modelling, Vol. 33, No. 4, 2009, pp. 2054-2061.
[14]	A. H. Abd El Naby and M. F. Abd El Hakeem, “The Flow Separation through Peristaltic Motion of Power- Law Fluid in Uniform Tube,” Applied Mathematics Sciences, Vol. 1, No. 26, 2007, pp. 1249-1263.
[15]	D. van der Woude, H. J. H. Clercx, G. J. F. van Heijst and V. V. Meleshko, “Stokes Flow in a Rectangular Cavity by Rotlet Forcing,” Physics of Fluids, Vol. 19, No. 8, 2007, pp. 083602-083602-19.
[16]	M. Zabarankin, “Asymetric Three-diemensional Stokes Flows about Two Fused Equal Spheres,” Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, Vol. 463, No. 2085, 2007, pp. 2329 -2350.
[17]	J. P. Brody, P. Yager, R. E. Goldstein and R. H. Austin, “Biotechnology at Low Reynolds Numbers,” Biophysical Journal, Vol. 71, No. 6, 1996, pp. 3430-3441.
[18]	J. Yeom, D. D. Agonafer, J.-H. Han and M. A. Shannon, “Low Reynolds Number Flow across an Array of Cylindrical Microposts in a Microchannel and Figure-of-Merit Analysis of Micropost-Filled Microreactors,” Journal of Mi-cromechanics Microengineering, Vol. 19, No. 6, 2009.
[19]	C. Y. Wang, “Flow through a Finned Channel Filled with a Porous Medium,” Chemical Engineering Science, Vol. 65, No. 5, 2010, pp. 1826-1831.
[20]	H. S. M. Coxeter, “Introduction to Geometry,” 2nd Edition, Wiley, New York, 1969, pp. 77-83.
[21]	M. A. Laurentiev and B. V. Chabat, “Les méthodes de la théorie des fonctions de la variable complexe,” traduit du russe par Damadian H., Ed., Mir 1972, 1977.
[22]	R. Bouard and M. Coutanceau, “Etude théorique et expérimentale de l’écoulement engendré par un cylindre en translation uniforme dans un fluide visqueux en régime de Stokes,” Journal of Applied Mathematics and Physics, Vol. 37, No. 5, 1986, pp. 673-684.