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And Inertia-Induced Inhomogeneity in the Distribution of Small Particles in fluid flows Julyan H

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And Inertia-Induced Inhomogeneity in the Distribution of Small Particles in fluid flows Julyan H CHAOS VOLUME 12, NUMBER 2 JUNE 2002 Noise- and inertia-induced inhomogeneity in the distribution of small particles in fluid flows Julyan H. E. Cartwrighta) Laboratorio de Estudios Cristalogra´ficos, CSIC, E-18071 Granada, Spain Marcelo O. Magnascob) Mathematical Physics Laboratory, Rockefeller University, P.O. Box 212, 1230 York Avenue, New York, New York 10021 Oreste Piroc) Institut Mediterrani d’Estudis Avanc¸ats, CSIC–UIB, E-07071 Palma de Mallorca, Spain ͑Received 2 November 2001; accepted 25 March 2002; published 20 May 2002͒ The dynamics of small spherical neutrally buoyant particulate impurities immersed in a two-dimensional fluid flow are known to lead to particle accumulation in the regions of the flow in which vorticity dominates over strain, provided that the Stokes number of the particles is sufficiently small. If the flow is viewed as a Hamiltonian dynamical system, it can be seen that the accumulations occur in the nonchaotic parts of the phase space: the Kolmogorov–Arnold–Moser tori. This has suggested a generalization of these dynamics to Hamiltonian maps, dubbed a bailout embedding. In this paper we use a bailout embedding of the standard map to mimic the dynamics of neutrally buoyant impurities subject not only to drag but also to fluctuating forces modeled as white noise. We find that the generation of inhomogeneities associated with the separation of particle from fluid trajectories is enhanced by the presence of noise, so that they appear in much broader ranges of the Stokes number than those allowing spontaneous separation. © 2002 American Institute of Physics. ͓DOI: 10.1063/1.1480441͔ Impurities suspended in a fluid flow are frequently ob- I. INTRODUCTION served to be distributed inhomogeneously. Even in very When impurities have a different density to the fluid, it is chaotic flows, particulate impurities arrange themselves intuitively clear that they will be expelled from rapidly ro- in extraordinarily structured distributions, in apparent tating regions of the flow—for heavy particles—or attracted contradiction to the high mixing efficiency expected from to the centers of these regions—for light particles—because the characteristics of the basic flow. To give just one ex- of centrifugal effects.7 However, it was recently demon- ample, in the particular instance of geophysical fluids, the strated that neutrally buoyant particles also tend to settle in filamentary structure, or patchiness, often displayed by the rotation ͑vorticity͒-dominated regions of a flow, but by a plankton populations in the oceans is a puzzling problem more subtle mechanism involving the separation between the currently under intense investigation.1–3 Several mecha- fluid and particle trajectories that can occur in the opposite 8 nisms to produce this type of inhomogeneity have been regions, i.e., in the areas of the flow dominated by strain. studied, and include dynamical aspects of the flow as well However, this mechanism is only relevant when the flow as the reactive properties of the considered impurities. gradients are of the order of the particle drag coefficient, a condition that may not be fulfilled in some physically inter- The basic idea in many of these mechanisms is that the esting situations. We show here by means of a minimal particle loss due either to the flow—in open flows—or to model that this condition may be relaxed if a small amount the chemical or population dynamics of the particles—in of noise is added to the forces acting on the impurity. closed flows—is minimized on some manifolds associated Our approach is qualitative, in the sense that instead of 4–6 with the hyperbolic character of the flow. In this paper considering a specific flow and the precise particle dynamics we explore an alternative purely dynamical mechanism induced by it, we describe the system with an iterative map for inhomogeneity with nonreactive particles in bounded whose evolution contains the basic features of both the fluid flows. We show that particle inertial effects combined flow and the particle dynamics: flow volume preservation, with fluctuating forces are capable of producing inhomo- together with particle separation in the hyperbolic regions. geneity even in cases in which the impurity and fluid den- The reason for moving to a discrete system is that in the map sities match exactly. the phenomena that we describe may be understood more intuitively, while translating the results back to the flow case is immediate. The strategy of attempting to understand fluid- ͒ dynamical phenomena by using iterated maps amenable to a Electronic mail: [email protected] b͒Electronic mail: [email protected] the powerful artillery of dynamical-systems theory has c͒Electronic mail: [email protected] proved successful on several different occasions. For ex- 1054-1500/2002/12(2)/489/7/$19.00489 © 2002 American Institute of Physics Downloaded 22 May 2002 to 130.206.33.86. Redistribution subject to AIP license or copyright, see http://ojps.aip.org/chaos/chocr.jsp 490 Chaos, Vol. 12, No. 2, 2002 Cartwright, Magnasco, and Piro ample, the structures of the chaotic advection induced by Here v represents the velocity of the particle, u that of the ␳ ␳ time-periodic three-dimensional incompressible flows were fluid, p the density of the particle, f the density of the fluid predicted by studying the qualitatively equivalent dynamics it displaces, ␯ the kinematic viscosity of the fluid, a, the of three-dimensional volume-preserving maps.9,10 Later, radius of the particle, and g, gravity. The terms on the right- these structures were confirmed in realistic flows.11–13 Other hand side of Eq. ͑1͒ represent, respectively, the force exerted examples are the treatment of the propagation of combustion by the undisturbed flow on the particle, buoyancy, Stokes fronts in laminar flows by a qualitative map approach,14 and drag, the added mass due to the boundary layer of fluid mov- the description of the formation of plankton population struc- ing with the particle,19,20 and the Basset–Boussinesq tures due to inhomogeneities of the nutrient sources.15 Re- force21,22 that depends on the history of the relative accelera- markably, in the instance we discuss in this paper, the pro- tions of particle and fluid. The terms in a2ٌ2u are the cedure is also useful from the point of view of dynamical- Faxe´n23 corrections. The Maxey–Riley equation is derived systems theory, as it has suggested a new technique—bailout under the assumptions that the particle radius and its Rey- embedding—for the control of Hamiltonian chaos.16 nolds number are small, as are the velocity gradients around The plan of the paper is as follows. First, we briefly the particle. review the classical model for the forces acting on a small First let us consider a minimal model for a neutrally ␳ ϭ␳ ͑ ͒ spherical particle moving relative to the fluid in which it is buoyant particle. For this we set p f in Eq. 1 . We con- immersed, and, concentrating on the case of a neutrally sider the Faxe´n corrections and the Basset–Boussinesq term buoyant impurity, we trace the construction of a minimal to be negligible. We now rescale space, time, and velocity by model that makes evident the separation of particle and fluid scale factors L, TϭL/U, and U, to arrive at trajectories in the regions in which the flow presents strong strain ͑Sec. II͒. On the basis of this model, we make a gen- dv Du Ϫ 1 dv Du ϭ ϪSt 1͑vϪu͒Ϫ ͩ Ϫ ͪ , ͑4͒ eralization that allows us to build a discrete mapping that dt Dt 2 dt Dt represents the Lagrangian evolution of the fluid parcels as ϭ 2 ␯ well as the dynamics of the particle ͑Sec. III͒. While this where St is the particle Stokes number St 2a U/(9 L) ϭ 2 map displays particle–fluid separation when the bailout pa- 2/9(a/L) Ref ,Ref being the fluid Reynolds number. The ͑ ͒ Ӷ rameter ␥, a function of the Stokes number, is relatively assumptions involved in deriving Eq. 1 require that St 1 ͑ ͒ small, we show in Sec. IV that a small amount of noise, in Eq. 4 . added to the dynamics of the particle to separate it continu- If we substitute the expressions for the derivatives in ͑ ͒ ͑ ͒ ͑ ͒ ally from the flow, enhances the impact of the hyperbolic Eqs. 2 and 3 into Eq. 4 , we obtain ␥ regions far beyond the values of required for separation in d the deterministic case. Section V extends and formalizes ͑ Ϫ ͒ϭϪ͑͑ Ϫ ͒" ͒ Ϫ␥͑ Ϫ ͒ ͑ ͒ v u v u “ u v u , 5 these ideas with analytical arguments. Our conclusions are to dt be found in Sec. VI. where we have written ␥ϭ2/3StϪ1. We may then write A ϭvϪu, whence II. MAXEY–RILEY EQUATIONS AND MINIMAL MODEL dA ϭϪ͑Jϩ␥I͒"A, ͑6͒ The equation of motion for a small, spherical particle in dt an incompressible fluid we term the Maxey–Riley equation,8,17,18 which may be written as where J is the velocity gradient matrix—we now concentrate on two-dimensional flows uϭ(u ,u )— dv Du 9␯␳ a2 x y ͪ ␳ ϭ␳ ϩ͑␳ Ϫ␳ ͒ Ϫ f ͩ Ϫ Ϫ ٌ2 ץ ץ p f p f g 2 v u u dt Dt 2a 6 xux yux Jϭͩ ͪ . ͑7͒ u ץ u ץ ␳ 2 f dv D a x y y y Ϫ ͩ Ϫ ͫuϩ ٌ2uͬͪ 2 dt Dt 10 If we diagonalize this matrix, and heuristically assume that the dependence on time of the diagonalizing transformation 9␳ ␯ t 1 d a2 Ϫ f ͱ ͵ Ϫ Ϫ ٌ2 ␨ is unimportant, we obtain ␲ ␨ ͩ v u uͪ d , 2a 0 ͱtϪ␨ d 6 ␭Ϫ␥ dAD 0 ͑1͒ ϭͩ ͪ "A , ͑8͒ dt 0 Ϫ␭Ϫ␥ D where the derivative Du/Dt is along the path of a fluid ele- ␭ Ͼ␥ ␭ ment so if Re( ) , AD may grow exponentially.
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