# Transport and mixing of finite-size particles

Inertial (or finite-sized) particles in fluid flows are commonly encountered in natural phenomena and industrial processes. Examples of inertial particles include dust, impurities, droplets, and air bubbles, with applications in pollutant transport in the ocean and atmosphere, rain initiation, coexistence between plankton species in the hydrosphere, and even planet formation by dust accretion in the solar system. Finite-size or inertial particle dynamics in fluid flows can differ markedly from infinitesimal particle dynamics: both clustering and dispersion are well-documented phenomena in inertial particle motion, while they are absent in the incompressible motion of infinitesimal particles.

*Systems that can be modeled as finite-size particles in fluid flows. *

**Reduced order dynamics of finite-size particles**

We study the transport and mixing properties of finite-size particles which using tools of geometrical singular perturbation theory. In particular we derive a modified flow field (slow invariant manifold), which governs the motion of these particles in general, flow fields [Read more...]. Using this modified flow field we achieve an efficient way to study and simulate mixing and transport properties of finite size particles in generic fluid flows. In addition we are able to perform source inversion (i.e. solve the dynamical equations backward in time) which is, in general, a computationally intractable task due to the unstable character of the original Maxey-Riley equations when one tries to solve backward in time.

In the following movie we present the slow invariant manifold that governs the velocity field of finite-size particles as a blue surface. We release two set of particles: the red particles evolve under the full dynamics, while the black are evolving through the reduced order inertial equation. Trying to invert the position of the particles in time reveals the unstable character of the full equation. On the other hand the reduced order inertial equation is able to recover the original position of the particles.

*Source inversion for finite-size particles using slow manifold reduction*

**Dispersion of finite size particles due to finite size**

Slow manifold reduction of the full equation that governs the dynamics of small particles reveals the existence of a vector field (slightly perturbed from the fluid flow field) that governs the kinematics of finite-size particles. For the special case of neutrally buoyant particles this vector field is exactly the velocity field of the flow. This is always the case for sufficiently small particles. For larger particles, however, a stability analysis of the normal dynamics of this manifold reveals zones of instabilities where particles velocity is not consistent with the predicted flow field anymore – a mechanism that may result in chaotic dispersion of particles even in laminar flows. By performing a stability argument normal to the slow manifold we are able to derive an analytical criterion that predicts spatial locations of the flow where these instabilities will take place. In the movie shown below the slow manifold is colored according to the analytical criterion. Red zones indicate locations where particles velocity diverges from the kinematics predicted using slow manifold reduction [Read more...].

*Zones of transverse instabilities on the slow manifold lead to divergence of the dynamics from the inertial equation.*

**Clutering of finite size particles**

The analytical form of the vector field that governs the motion of finite-size particles allows for the derivation of analytical criteria that describe and predict their clustering properties [Read more...]. Using ergodicity theory we are able to derive criteria that need only a single trajectory in order to predict clustering surfaces for finite-size particles. Our criteria are applicable to two dimensional unsteady and three dimensional steady flow fields. On the following figures we present clustering regions for various cases of fluid flows.

**Comparison of theory with experiments and representation of errors through Lagrangian stochastic models**

We study the validity of various models that describe the motion of finite-sized particles in fluids by means of a direct comparison between theory and experimentally measured trajectories. In the left panel we present the tiny particles (black dots) used to measure the 2D chaotic flow (shown in the right) as well as a small number of larger particles (red circles) which are used to compare with theory. Our analysis indicates that finite-sized particles follow the predicted particle dynamics given by the Maxey-Riley equation, except for random correlated fluctuations that are not captured by deterministic terms in the equations of motion, such as the Basset-Boussinesq term or the lift force. We describe the fluctuations via spectral methods and we propose three different Lagrangian stochastic models to account for them. These Lagrangian models are stochastic generalizations of the Maxey-Riley equation with coefficients calibrated to the experimental data [Read more...].

**Modeling of predator-prey interaction in jellyfish feeding **

We have applied the developed tools for finite-sized particles to study predator-prey interactions in jellyfish feeding. We have explored the effect of prey mass on its dynamics and how this couples with the prey self-propulsion as well as the predator flow field. One of the biological insights from our study is that the model demonstrates, from a physical and mathematical perspective that prey selection is dependent on a combination of many factors, including characteristics of flow induced by the predator, prey size, self-propulsion, escape strategies, etc. One of the advantages of the model is that compared with empirical studies it is able to isolate the effect of these parameters and potentially determine the dominant factor [Read more...].

**Lagrangian Coherent Structures for weather balloons in hurricane Isabel**

In this work we have identified inertial Lagrangian coherent structures (ILCS) that govern the motion of inertial particles in a numerical model of Hurricane Isabel. Using the reduced order inertial equation we have calculated the backward and forward-time finite time Lyapunov exponent fields on the slow manifold. The ridges of these scalar fields mark the location of attracting and repelling ILCS, respectively. Additionally, we have determined the critical size of inertial particles below which ILCS derived from the inertial equation govern the asymptotic particle motion in forward and backward time [Read more...]

**Publications
**

`T. Sapsis, N. Ouellette, J. Gollub, & G. Haller, `**Neutrally buoyant particle dynamics in fluid flows: Comparison of Experiments with Lagrangian stochastic models**, *Phys. Fluids , ***23** (2011) 093304.[pdf]

`G. Haller & T. Sapsis, `**Lagrangian Coherent Structures and the Smallest Finite-Time Lyapunov Exponent**, *Chaos, ***21** (2011) 023115. [pdf]

`T. Sapsis, J. Peng, & G. Haller, `**Instabilities on prey dynamics in jellyfish feeding**, *Bulletin of Mathematical Biology, ***73** (2011) 1841-1856. [pdf]

`T. Sapsis & G. Haller, `**Clustering Criterion for Inertial Particles in 2D Time-Periodic and 3D Steady Flows**,* **Chaos*, **20** (2010) 017515. [pdf]

`G. Haller & T. Sapsis, `**Localized instability and attraction along invariant manifolds**, *SIAM J. of Appl. Dynamical Systems*, **9** (2010) 611-633. [pdf]

`T. Sapsis & G. Haller, `**Inertial particle dynamics in a hurricane**,* J. Atmosph. Sci.*, **66** (2009) 2481-2492.[pdf]

`T. Sapsis & G. Haller, `**Instabilities in the dynamics of neutrally buoyant particles**.* Phys. Fluids*, **20**(2008) 017102. [pdf]

`G. Haller & T. Sapsis, `**Where do inertial particles go in fluid flows? ***Physica D*, **237** (2008) 573-583.[pdf]