# Complex Particulate Flows

## Concentration instabilities in sedimentation

**Research Assistant: Ramanathan Vishnampet (viscoelastic sedimentation)**

The sedimentation of small particles in a fluid at low Reynolds number is a complex problem of non-equilibrium statistical mechanics. Because of the very slow decay of hydrodynamic interactions in the Stokes flow regime, particles induce velocity fluctuations in the fluid at very large distances, and the calculation of statistical properties such as the mean sedimentation speed or the velocity variance typically leads to diverging integrals. Theoretical arguments, experiments and computer simulations all suggest that the case of anisotropic particles such as fibers or disks is yet more complex, as the velocity fluctuations in the fluid lead to a concentration instability by which particles aggregate into dense clusters. A similar instability also occurs for deformable particles which become anisotropic under flow, such as droplets, polymers, and elastic microcapsules. This clustering phenomenon has a strong influence on the sedimentation speed, which can become significantly greater that the maximum value for an isolated particle.

To investigate this instability and understand the wavenumber selection observed in experiments, we performed large-scale direct numerical simulations of the sedimentation of slender particles in Stokes flow, using two different levels of approximation: a detailed slender-body model based on an efficient smooth particle-mesh Ewald algorithm, and a point-particle method allowing the simulation of very large suspensions (up to 500,000 particles) in bounded geometries (see figure 1). These simulations showed very good qualitative and quantitative agreement with experiments, and in particular a wavenumber selection was observed in simulations for the first time. To elucidate the mechanism for wavenumber selection, a linear stability analysis that includes stratification was performed. In stably stratified suspensions, the stability analysis was shown to predict a wavenumber selection, and comparison of this prediction with simulations showed excellent agreement. It was also shown, in a different study, that the clustering instability can be controlled and suppressed by application of a vertical electric field, which has the effect of aligning the particles in the direction of the field.

**Figure 1:** Left: Simulation of the sedimentation of a suspension of rodlike particles. Right: Comparison to an experiment of B. Metzger, J. E. Butler, and E. Guazzelli.

More recently, we have also shown that a similar instability also arises in a dilute suspension of rigid spherical particles sedimenting in a viscoelastic fluid (but it does not occur in a Newtonian fluid). In the case of a second-order fluid, we performed a linear stability analysis and explained the instability as a result of the nonlinear coupling between the settling motion of a given particle and the mean-field flow driven by density fluctuations in the suspension. This instability was confirmed by weakly nonlinear numerical simulations, which hint at a pattern formation in the form of vertical columns, in agreement with experiments on polymeric fluids.

###### References

**Concentration instability of sedimenting spheres in a second-order fluid**

R. Vishnampet, D. Saintillan, *Phys. Fluids*, to appear (2012). [preprint]

**The effect of stratification on the wave number selection in the instability of sedimenting spheroids**

D. Saintillan, E. S. G. Shaqfeh, E. Darve, *Phys. Fluids*, **18** 121503 (2006). [reprint]

**The growth of concentration fluctuations in dilute dispersions of orientable and deformable particles under sedimentation**

D. Saintillan, E. S. G. Shaqfeh, E. Darve, *J. Fluid Mech.*, **553** 347 (2006). [reprint]

**A smooth particle-mesh Ewald algorithm for Stokes suspension simulations: The sedimentation of fibers**

D. Saintillan, E. Darve, E. S. G. Shaqfeh, *Phys. Fluids*, **17** 033301 (2005). [reprint]

## Gravity-driven particulate jets

**Collaborators: Elisabeth Guazzelli, Maxime Nicolas, and Florent Pignatel (CNRS, France)**

Gravity-driven suspension jets falling in viscous fluids at low Reynolds numbers are subject to an instability by which an initially cylindrical jet develops varicose fluctuations of its diameter. This instability, which has been observed in experiments, is qualitatively reminiscent of the classic Rayleigh-Plateau instability of an inertial jet of an immiscible fluid with surface tension, yet inertia is negligible and surface tension inexistent in the case of a suspension jet.

To understand the mechanism and dynamics of this instability, we have performed numerical simulations that capture far-field hydrodynamic interactions between particles based on a point-particle approximation, and have compared them to experiments performed by Guazzelli and coworkers. Snapshots of both an experiment and a simulation are shown in figure 2, where excellent qualitative agreement is observed. The jet is found to develop a mushroom-shaped head, and its diameter fluctuates in the vertical direction, with a characteristic wavelength of the order of a few diameters. While a complete understanding of the instability of still lacking, we obtain very good agreement between experiments and simulations for the magnitude and wavelength of the saturated instability, and for the jet diameter and mean velocity.

**Figure 2:** Gravity-driven suspension jet at zero Reynolds number: comparison between experiment (black and white) and simulation (red).

###### Reference

**Falling jets of particles in viscous fluids**

F. Pignatel, M. Nicolas, E. Guazzelli, D. Saintillan, *Phys. Fluids*, **21** 123303 (2009). [reprint]

(Also featured on the cover of *Physics of Fluids*)

## Weak inertial effects in particulate flows

**Collaborators: Elisabeth Guazzelli and Laurence Bergougnoux (CNRS, France)**

Complex particulate flows are ubiquitous in many environmental and industrial processes. Of particular relevance in civil and environmental engineering are gravity-driven flows, such as sediment flows in estuaries and turbidity currents, which are responsible for hydrocarbon well formation. While the vast majority of previous studies on suspensions flows have focused on the low-Reynolds-number highly viscous regime in which inertial effects are negligible and on the high-Reynolds-number turbulent flow regime, few studies have considered the intermediate case of moderate fluid inertia, which still occurs in many practical flows of interest.

In this work, we use a combination of numerical simulations and experiments (performed by Guazzelli and coworkers) to investigate the effects of weak inertia in suspension flows. We focus specifically on the case of a large-scale dilute suspension of spherical particles sedimenting under gravity, a system that has been studied in detail in the absence of inertia. The simulations use a full spectral solution of the Navier-Stokes equations with two-way coupling, based on a point-particle approximation, and a typical flow field in such a simulation is shown in figure 3. Preliminary results seem to indicate a decrease in the magnitude of velocity fluctuations with Reynolds number, which an apparent power-law as predicted by a theoretical argument due to Hinch. More detailed analysis and comparisons between experiments and simulations are currently underway.

**Figure 3:** Chaotic velocity field in the sedimentation of a dilute suspension of approximately 500,000 rigid spheres at a particle Reynolds number of 0.1 (simulation).

## Efficient algorithms for Stokes suspensions

The very slow decay of far-field hydrodynamic interactions between suspended particles in a viscous fluid at zero Reynolds number makes their calculation in computer simulations a very expensive task. Direct calculations scale with the square of the number of particles in the suspension, and are therefore typically limited to relatively small systems. There have been recents attempts to develop fast algorithms for the calculation of such interactions, most of which are inspired by existing methods from molecular dynamics, in which Coulombic interactions between charged particles have the same slow decay.

We have developed and implemented a Smooth Particle-Mesh Ewald (SPME) algorithm for the fast calculation of hydrodynamic interactions in periodic systems, after the existing algorithm for electrostatic interactions (Essmann et al., *J. Chem Phys.*, 1995). The method, which shares similarities with the Accelerated Stokesian Dynamics algorithm of Sierou and Brady (*J. Fluid Mech.*, 2001), is based on the decomposition of the slowly-decaying Green's function into two fast-converging sums: the first one involves the distribution of point forces and accounts for the singular short-range part of the interactions, while the second one is expressed in terms of the Fourier transform of the force distribution and accounts for the smooth and long-range part. Because of its smoothness the second sum is easily evaluated on a grid using cardinal B-spline interpolation and the Fast Fourier Transform algorithm. The total cost of the method scales in *N*.log(*N*) where *N* is the total number of point forces, which is a significant improvement and allows one to simulate much larger systems.

The method was successfully applied to simulate the sedimentation of rigid fibers in periodic boundary conditions, providing new insight into the mechanism of the concentration instability observed in such systems. More recently, we have extended it to Stokes dipole and potential quadrupole interactions, and we have applied it to calculate electrohydrodynamic interactions in large-scale suspensions of polarizable spheres in an electric field. We are currently working an modification of the algorithm to model confinement in doubly-periodic geometries.

###### References

**Dipolophoresis in large-scale suspensions of ideally polarizable spheres**

J. S. Park, D. Saintillan, *J. Fluid Mech.*, **662** 66 (2010). [reprint]

**A smooth particle-mesh Ewald algorithm for Stokes suspension simulations: The sedimentation of fibers**

D. Saintillan, E. Darve, E. S. G. Shaqfeh, *Phys. Fluids*, **17** 033301 (2005). [reprint]