We develop a model of the forces on a spherical particle suspended in flow through a curved duct under the assumption that the particle Reynolds number is small. This extends an asymptotic model of inertial lift force previously developed to study inertial migration in straight ducts. Of particular interest is the existence and location of stable equilibria within the cross-sectional plane towards which particles migrates. The Navier-Stokes equations determine the hydrodynamic forces acting on a particle. A leading order model of the forces within the cross-sectional plane is obtained through the use of a rotating coordinate system and a perturbation expansion in the particle Reynolds number of the disturbance flow. We predict the behaviour of neutrally buoyant particles at low flow rates and examine the variation in focusing position with respect to particle size and bend radius, independent of the flow rate. In this regime, the lateral focusing position of particles approximately collapses with respect to a dimensionless parameter dependent on three length scales, specifically the particle radius, duct height, and duct bend radius. Additionally, a trapezoidal shaped cross-section is considered in order to demonstrate how changes in the cross-section design influence the dynamics of particles. †

Abstract. We present a general theory of fractal transformations and show how it leads to new type of method for filtering and transforming digital images. This work substantially generalizes earlier work on fractal tops. The approach involves fractal geometry, chaotic dynamics, and an interplay between discrete and continuous representations. The underlying mathematics is established and applications to digital imaging are described and exemplified.Key words. Iterated function systems, dynamical systems, fractal transformations.AMS subject classifications. 37B10, 54H20, 68U101. Introduction. Fractal transformations are mappings between pairs of attractors of iterated function systems. They are defined with the aid of code space structures, and can be quite simple to handle and compute. They can be applied to digital images when the attractors are rectangular subsets of R 2 . They are termed "fractal" because they can change the box-counting, Hausdorff, and other dimensions of sets and measures upon which they act. In this paper we substantially generalize and develop the theory and we illustrate how it may be applied to digital imaging. Previous work was restricted to fractal transformations defined using fractal tops.Fractal tops were introduced in [2] and further developed in [5,6,7,11]. The main idea is this: given an iterated function system with a coding map and an attractor, a section of the coding map, called a tops function, can be defined using the "top" addresses of points on the attractor. Given two iterated function systems each with an attractor, a coding map, and a common code space, a mapping from one attractor to the other can be constructed by composing the tops function, for the first iterated function system, with the coding map for the second system. Under various conditions the composed map, from one attractor to the other, is continuous or a homeomorphism. In the cases of affine and projective iterated function systems, practical methods based on the chaos game algorithm [8] are feasible for the approximate digital computation of such transformations. Fractal tops have applications to information theory and to computer graphics. They have been applied to the production of artwork, as discussed for example in [4], and to real-time image synthesis [18]. In the present paper we extend the theory and applications.Much of the material in this paper is new. The underlying new idea is that diverse sections of a coding map may be defined quite generally, but specifically enough to be useful, by associating certain dynamical systems with the iterated function system. These sections provide novel collections of fractal transformations; by their means we generalize the theory and applications of fractal tops. We establish properties of fractal transformations, including conditions under which they are continuous. The properties are illustrated by examples related to digital imaging.A notable result, Theorem 5.3, states the existence of nontrivial fractal homeomorphisms between attractors of some affi...

The term special overlapping refers to a certain simple type of piecewise continuous function from the unit interval to itself and also to a simple type of iterated function system (IFS) on the unit interval. A correspondence between these two classes of objects is used (1) to find a necessary and sufficient condition for a fractal transformation from the attractor of one special overlapping IFS to the attractor of another special overlapping IFS to be a homeomorphism and (2) to find a formula for the topological entropy of the dynamical system associated with a special overlapping function.

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