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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.

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 [

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 _{0} 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 [

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.

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.

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