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Dynamics of Finite-Size Particles in Chaotic Fluid Flows

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Dynamics of Finite-Size Particles in Chaotic Fluid Flows Dynamics of Finite-Size Particles in Chaotic Fluid Flows Julyan H.E. Cartwright, Ulrike Feudel, György Károlyi, Alessandro de Moura, Oreste Piro, and Tamás Tél Abstract We review recent advances on the dynamics of finite-size particles advected by chaotic fluid flows, focusing on the phenomena caused by the inertia of finite-size particles which have no counterpart in traditionally studied passive tracers. Particle inertia enlarges the phase space and makes the advection dynamics much richer than the passive tracer dynamics, because particles’ trajectories can diverge from the trajectories of fluid parcels. We cover both confined and open flow regimes, and we also discuss the dynamics of interacting particles, which can undergo fragmentation and coagulation. 1 Introduction and Overview A correct formulation of the problem of the motion of finite-size particles in fluid flows has presented difficult challenges for generations of fluid dynamicists. Although in principle this problem is “just” another application of the Navier– Stokes equation, with moving boundary conditions, a direct solution of the fluid dynamical equations is not only very difficult, but also not very illuminating. So from the nineteenth century onwards efforts were made to find the appropriate approximations which allow one to write the equations of motion of small rigid particles in a given flow in the form of ordinary differential equations, regarding the flow’s velocity field as given. Some very subtle issues are involved in making the right kinds of approximations and assumptions in a self-consistent way, and a number of incorrect results appeared in the early literature. The issue was finally resolved when Maxey and Riley [1] wrote down the equations of motion for a small spherical rigid particle advected by a (smooth) flow and Auton et al. [2], following Taylor’s work [3], corrected the form of the added-mass term. A. de Moura (B) Institute of Complex Systems and Mathematical Biology, University of Aberdeen, King’s College, Meston Building, AB24 3UE Aberdeen, United Kingdom e-mail: [email protected] M. Thiel et al. (eds.), Nonlinear Dynamics and Chaos: Advances and Perspectives, 51 Understanding Complex Systems, DOI 10.1007/978-3-642-04629-2_4, C Springer-Verlag Berlin Heidelberg 2010 52 J.H.E. Cartwright et al. The Maxey-Riley equations allow the global dynamics of a single advected finite- size particle to be investigated with the techniques of dynamical systems theory. The dynamics of advected particles in fluid flows have been a favourite subject of investigation of chaos and related complex dynamical regimes since the pioneering work of Aref [4], but all the early works assumed that the particles’ size and their inertia could be neglected – the passive tracer assumption. Finite size results in inertia, which introduces a new richness to the dynamics, since finite-size particles are no longer enslaved to the motion of the flow surrounding them – they have their own dynamics, distinct from that of the fluid. A whole new world of challenges and possibilities opens up to dynamicists once finite size and inertia are considered. This gives this subject a great importance from the theoretical point of view alone; not to mention its practical importance: polluting particles suspended in the atmosphere and plankton organisms floating in the ocean are just a few of the systems whose understanding involves the theory of the dynamics of finite-size particles in complex flows. This chapter presents an overview of the subject of the dynamics of finite-size particles in chaotic flows, focusing on a few chosen topics of current research. The choice of topics reflects the authors’ own research interests; we make no apologies for that: we in no way claim this to be an exhaustive review on this area. But we do think the topics we cover here give the reader a good idea of what is going on in this exciting area of research. We first introduce the Maxey-Riley equation in Sect. 2. Its assumptions and range of validity are discussed, as well as some of its basic consequences; but we do not show the derivations. The dynamics of finite-size particles can often be understood by using the simpler dynamics of passive tracers as a starting point. In Sect. 3 we discuss the chaotic advection of non-inertial tracers, and also introduce some of the flows which will be used as examples in later sections. The dynamics of non-interacting finite-size particles is the subject of Sects. 4 and 5. Section 4 deals with confined flows, whereas Sect. 5 focuses on open flows. These two kinds of flow have very different long-time behaviours, which lead to quite distinct particle dynamics. In these sections, we focus especially on the new phenomena caused by the particles’ inertia, which are not present in the case of passive tracers. The challenging and very important topic of interacting finite-size particles is covered by Sect. 6, which reviews recent results on the processes of fragmentation and coagulation of finite-size particles. Finally, in Sect. 7, we make some remarks on the future directions of this research area. 2 Motion of Finite-Size Particles in Fluid Flows When studying the motion of particles advected by fluid flows, it is commonly assumed that one can consider the particles as passive tracers, with negligible mass and size. This amounts to neglecting the particles’ inertia, and assuming that they take on the velocity of the surrounding fluid, instantaneously adapting to any changes in the fluid velocity. If u is the (possibly time-dependent) fluid’s velocity Dynamics of Finite-Size Particles in Chaotic Fluid Flows 53 field, and denoting by r(t) the position of a particle, the passive tracer assumption implies that r satisfies the differential equation: r˙(t) = u(r(t),t). (1) The passive tracer assumption is extensively used in fluid dynamics [5, 6], and it is a good approximation in a number of cases. There are many situations, however, where it does not apply, and we need to take into account the fact that particles have finite sizes and masses (for reviews see [7–9]). Finite-size particles are not able to adjust their velocities instantaneously to that of the fluid, and in addition their density may be different from that of the fluid. Therefore, in general the particle velocity differs from the fluid velocity. This means that the dynamics of finite-size particles is far richer and more complex than that of passive tracers. 2.1 The Maxey-Riley Equation In order to study the dynamics of finite-size particles advected by chaotic flows, we need to have a simple formulation of the equations of motion of the advected parti- cles. The problem is that finite-size particles are actually extended objects with their own boundaries. The rigorous way to analyse their dynamics would involve solving the Navier–Stokes equation for moving boundaries, with all the complications this implies. The partial differential equations resulting from this approach would be very difficult to solve and analyse; and as dynamicists, we would like to have the particle’s motion described by ordinary differential equations, similar to Eq. (1). Fortunately, an approximate differential equation for the motion of small spherical particles in flows may be written down [1, 2]. If a particle has radius a and mass mp, its motion is given to a good approximation by the Maxey-Riley equation: D 1 D 1 2 2 mpv˙ = mf u(r(t),t) − mf v˙ − u(r(t),t) + a ∇ u(r(t),t) Dt 2 Dt 10 t τ / τ 2 dq( ) d −6πaρf νq(t) + (mp − mf )g − 6πa ρf ν dτ √ ,(2) 0 πν(t − τ) where 1 q(t) ≡ v(t) − u(r(t),t) − a2∇2u. 6 Here r(t) and v(t) ≡ dr(t)/dt are the position and velocity of the particle, respec- tively, and u(r,t) is the undisturbed flow field at the location of the particle. mf denotes the mass of the fluid displaced by the particle, and ν is the kinematic vis- cosity of the fluid of density ρf ; g is the gravitational acceleration. The derivative Du ∂u = + (u ·∇)u (3) Dt ∂t 54 J.H.E. Cartwright et al. is the total hydrodynamical derivative, taken along the path of a fluid element, whereas du ∂u = + (v ·∇)u (4) dt ∂t is taken along the particle’s trajectory. The first term on the right-hand side is the acceleration of the fluid element in position r(t) at time t and represents the force exerted on the particle by the undis- turbed fluid. The second term represents the added-mass effect, which accounts for the fact that when the particle moves relative to the fluid, it displaces a certain amount of fluid with it; the result is that the particle behaves as if it had additional mass. The third and fourth terms represent the Stokes drag caused by the fluid’s viscosity and the buoyancy force, respectively. The integral is called the Basset- Boussinesq history term, and arises from the fact that the vorticity diffuses away from the particle due to viscosity [10, 11]. The terms involving the factor a2∇2u are the so-called Faxén corrections [94], and they account for the spatial variation of the flow field across the particle. Equation (2) is valid for small particles at low particle Reynolds numbers Rep. This Reynolds number is calculated by using the particle size as the length scale, and the relative velocity between particle and neighbouring fluid as the velocity scale: Rep = a|v − u|/ν. This implies that for Eq. (2) to be a valid approximation, the initial velocity difference between particle and fluid must be small [1].
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