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.

Inversion Transformation Geometric Inversion Stokes Flow Viscous Eddies Flow around a Cylinder Microflow
1. Introduction

Geometric inversion is a type of transformation of the Euclidean plane. This transformation preserves angles and map generalized circles into generalized circles, where a generalized circle means either a circle or a line (a circle with infinite radius). One of the main properties of this method is the transformation of a straight line to a circle. Many difficult problems in geometry become much more tractable when an inversion is applied.

In engineering, the geometric inversion could be very useful to solve complex problems. For example, in fluid mechanics, the equation of two-dimensional Stokes flow remains valid in the new coordinates system obtained by inversion. Thus two-dimensional Stokes flow around certain bodies presenting circular shape appears less difficult to calculate by inversion of flow in channels of parallel walls than by direct calculation using polar coordinates. Although this method is rather general, we will apply it to the case of cellular flows (recirculation flow) presenting viscous eddies. In fact these flows are characterized by the presence of dividing streamlines (separating streamlines) which also give by inversion in the new geometry dividing streamlines.

Much attention has been paid to the steady viscous flow between parallel plates at low Reynolds number (Stokes flow) because of its theoretical importance and also its engineering applications. The particular case of the flow with vanishing velocity to zero when (i.e. flow with mean rate equal zero) has been widely studied by authors motivated among others by separation phenomena. Thus, after the pioneering work of  who presented predictions of cellular motion in Stokes regime between parallel walls, theoretical works like those of  and  demonstrate the existence of such cellular flow. The corresponding authors showed that, independently of the motion source, any two-dimensional flow with mean rate null in a channel presented necessarily cellular motion composed by successive counter rotating eddies bounded by separating streamlines reattaching the walls. In order to examine the influence of the motion source, accurate computations for various motion sources have been performed [4-16]. Stokes flows and particularly cellular flows could be encountered in numerous applications in physics, biophysics, chemistry and MEMS (Micro and ElectroMechanical Systems) where microflows appear. The particularity of these applications is that they use microchannels [17-19]. Thus, several theoretical and numerical results are available. They could be useful to obtain by inversion transformation the structure and the features of Stokes flows around bodies of circular shape. This transformation is also useful to obtain flow around bodies with complex shape for which the direct calculation could be tiresome.

2. Geometric Inversion-Definitions and Properties

Inversion is the process of transforming points M to a corresponding set of points N known as their inverse points. Two points M and N drawn in Figure 1 are said to be inverses with respect to an inversion circle having inversion centre O and inversion radius R0 if N is the perpendicular foot of the altitude of the triangle OQM, where Q is a point on the circle such that OQ┴QM.

If M and N are inverse points, then the line L through M and perpendicular to OM is called a “polar” with respect to point N, known as the “inversion pole”. In addition, the curve to which a given curve is transformed under inversion is called its inverse curve (or more simply, its “inverse”).

From similar triangles, it immediately follows that the inverse points M and N obey to: or (1)

where the quantity is known as the circle power or inversion power .

The general equation for the inverse of the point relative to the inversion circle with inversion centre and inversion radius is given by , (2)

Note that a point on the circumference of the inversion circle is its own inverse point. In addition, any angle inverts to an opposite angle.

Treating lines as circles of infinite radius, all circles invert to circles. Furthermore, any two nonintersecting circles can be inverted into concentric circles by taking the inversion centre at one of the two so-called limiting points of the two circles , and any two circles can be inverted into themselves or into two equal circles. Or-

thogonal circles invert to orthogonal circles. The inversion circle itself, circles orthogonal to it, and lines through the inversion centre are invariant under inversion. Furthermore, inversion is a conformal map, so angles are preserved. Note that a point on the circumference of the inversion circle is its own inverse point. In addition, any angle inverts to an opposite angle.

The property that inversion transforms circles and lines to circles or lines (and that inversion is conformal) makes it an extremely important tool of plane analytic geometry. Figure 2 shows a simple example of application of geometric inversion.

The circle with dashed lines is the inversion circle of centre O and radius . Let take for example , the distance and the distance . Let’s make inversion of the straight lines L and L’ with centre O and power , we obtain the circles C and C’. The points N and Q are respectively the inverse images of the points M and P. The distances ON and OQ, calculated by using Equation (1), are and . Thus the radii of the circles C and C’ are respectively and .

In Figure 3 we present the classical example of inversion of a square relatively to a circle inscribed in this square. This inversion image becomes more complex if the number of squares is increased (see Mathematica Notebook presenting inversion of a grid).

Figure 4 illustrates the inversion of parabola. This transformation leads to a cardiod defined in Figure 4(c). It’s worth to note that the change of the inversion centre leads to other figures of inversion.

ReferencesH. K. Moffatt, “Viscous and Resistive Eddies near a Sharp Corner,” Journal of Fluid Mechanics, Vol. 18, No. 1, 1964, pp. 1-18.M. E. O’ Neill, “On Angles of Separation in Stokes Flow,” Journal of Fluid Mechanics, No. 133, 1983, pp. 427-442.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.C. Shen and J. M. Floryan, “Low Reynolds Number Flow over Cavi-ties,” Physics of Fluids, Vol. 28, No. 11, 1985, pp. 3191-3202.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.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.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.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.P. N. Shankar and M. D. Deshpande, “Fluid Mechanics in the Driven Cavity,” Annual Review of Fluid Mechanics, No. 32, 2000, pp. 93-136.J. T. Jeong, “Slow Viscous Flow in a Partitioned Channel,” Physics of Fluids, Vol. 13, No. 6, 2001, pp. 1577 -1582.M. Hellou, “Structures d'écoulements de Stokes dans une jonction bidimensionnelle de canaux,” Mécanique & Industries, Vol. 4, No. 5, 2003, pp. 575-583.C. Y. Wang, “Slow Viscous Flow between Hexagonal Cylinders,” Transport in Porous Media, Vol. 47, No. 1, 2002, pp. 67-80.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.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.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.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.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.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.C. Y. Wang, “Flow through a Finned Channel Filled with a Porous Medium,” Chemical Engineering Science, Vol. 65, No. 5, 2010, pp. 1826-1831.H. S. M. Coxeter, “Introduction to Geometry,” 2nd Edition, Wiley, New York, 1969, pp. 77-83.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.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.