Turbulent coagulation of colloidal particles

Abstract

Theoretical predictions for the coagulation rate induced by turbulent shear have often been based on the hypothesis that the turbulent velocity gradient is persistent (Saffman & Turner 1956) and that hydrodynamic and interparticle interactions (van der Waals attraction and electrostatic double-layer repulsion) between colloidal particles can be neglected. In the present work we consider the effects of interparticle forces on the turbulent coagulation rate, and we explore the response of the coagulation rate to changes in the Lagrangian velocity gradient correlation time (i.e. the characteristic evolution time for the velocity gradient in a reference frame following the fluid motion). Stokes equations of motion apply to the relative motion of the particles whose radii are much smaller than the lengthscales of turbulence (i.e. small particle Reynolds numbers). We express the fluid motion in the vicinity of a pair of particles as a locally linear flow with a temporally varying velocity gradient. The fluctuating velocity gradient is assumed to be isotropic and Gaussian with statistics taken from published direct numerical simulations of turbulence (DNS). Numerical calculations of particle trajectories are used to determine the rate of turbulent coagulation in the presence and absence of particle interactions. Results from the numerical simulations correctly reproduce calculated coagulation rates for the asymptotic limits of small and large total strain where total strain is a term used to describe the product of the characteristic strain rate and its correlation time. Recent DNS indicate that the correlation times for the fluctuating strain and rotation rate are of the same order as the Kolmogorov time (Pope 1990), suggesting theories that assume either small or large total strain may poorly approximate the turbulent coagulation rate. Indeed, simulations for isotropic random flows with intermediate total strain indicate that the coagulation rate in turbulence is significantly different from the analytical limits for large and small total strain. The turbulent coagulation rate constant for non-interacting monodisperse particles scaled with the Kolmogorov time and the particle radius is 8.62±0.02, whereas the commonly used model of Saffman & Turner (1956) predicts a value of 10.35 for non-rotational flows in the limit of persistent turbulent velocity gradients. Additional simulations incorporating hydrodynamic interactions and van der Waals attraction were used to estimate the actual rate of particle coagulation. For typical values of these parameters, particle interactions reduced the coagulation rate constant by at least 50%. In general, the collision efficiency (the ratio of coagulation with particle interactions to that without) decreased with increasing particle size and Kolmogorov shear rate.

Estimating the Collision Rate of Inertial Particles in a Turbulent Flow: Limitations of the "Ghost Collision" Approximation

“…in Wang et al, 2000;Zhou et al, 2001;Bec et al, 2005;Franklin et al, 2005;Bec et al, 2010). In a turbulent flow, however, the strain will not stay constant along the trajectory of a particle (Brunk et al, 1998) and it may be elliptic in some regions (Chong et al, 1990). Thus two particles can get close to each other, separate and come close again.…”

Clogging of microchannels by nano-particles due to hetero-coagulation in elongational flow

“…Greene et al [39] extended the results of Zeichner and Schowalter to different extensional flow fields, containing different amount of vorticity, and demonstrated that the aggregation efficiency is almost unaffected by the flow type, except for flow types similar to the simple shear. More recently Brunk et al [40] [41] modeled the particle aggregation in turbulent conditions and systematically investigated the role of rotation on the coagulation rate. They confirmed the results of Zeichner and Schowalter and showed that stability ratio (W) is proportional to 23 .…”

“…They developed a perturbation analysis of the pair probability equation for the relative motion of two particles with regard to diffusion and convection terms and came to the results that a little Brownian motion can either enhance or reduce the rate of rapid shear induced aggregation, depending on the flow number N f . Furthermore Brunk et al [40] pointed out that decreasing the shear rate while keeping N f constant, the aggregation efficiency for dilute suspensions with attractive interactions increases, since approaching particles will have many close encounters due to the Brownian motion, before they either collide or the flow carries them away from each other. On the other hand, at large total strain colliding particles have basically one opportunity for successful collision before the flow separates them.…”

A method for the computation of turbulent polymeric liquids including hydrodynamic interactions and chain entanglements

“…Notably, even in dilute systems, such turbulence-polymer couplings would most probably be entwined with both chain entanglement and hydrodynamic-interactions effects between chains in locally polymer-dense areas, as is the case, for example, in between turbulent coherent vortices where polymers might be expected to concentrate [14,24]. Similar ideas are also valid for turbulent flows in colloidal dispersions and aerosols [25], that feature particle aggregation/clustering phenomena [26,27] which require the capturing of hydrodynamic interactions between particles at high concentration areas, as is the case, for example, during rain initiation processes [28][29][30].…”

“…Notably, even in dilute systems, such turbulence-polymer couplings would most probably be entwined with both chain entanglement and hydrodynamic-interactions effects between chains in locally polymer-dense areas, as is the case, for example, in between turbulent coherent vortices where polymers might be expected to concentrate [14,24]. Similar ideas are also valid for turbulent flows in colloidal dispersions and aerosols [25], that feature particle aggregation/clustering phenomena [26,27] which require the capturing of hydrodynamic interactions between particles at high concentration areas, as is the case, for example, during rain initiation processes [28][29][30].Therefore, there is a need for mesoscopic, physical formulations/numerical methods that allow the direct computation of turbulent polymeric (and colloidal) fluids. Such methods have to overcome a number of challenges: (a) the forcing of the fluid by the particles is delta-function type (i.e., pointwise), hence standard methods for computational fluid dynamics need an extremely fine grid in order to capture the microscopic flow field in between the particles that corresponds to their hydrodynamic interactions, (b) efforts to average the forced Navier-Stokes equations in order to overcome this problem, lead, due to nonlinearity, to the appearance of subgrid scale stresses that need to be taken into account via some type of modeling, (c) the accompanying numerical method needs to handle the Brownian, i.e., stochastic motion of polymer and colloidal particles; this adds an additional level of complexity to standard computational methods for suspensions [31], (d) in polymeric liquids, the formulation and numerics need to describe the formation and dynamics of entanglements between macromolecular chains, in order for the approach to be applicable to arbitrary polymer volume fractions.…”