Self-diffusion of bimodal suspensions of hydrodynamically interacting spherical particles in shearing flow

1994 ◽  
Vol 281 ◽  
pp. 51-80 ◽  
Author(s):  
Chingyi Chang ◽  
Robert L. Powell
Keyword(s):  
Diffusion Coefficients ◽  
Cluster Size ◽  
Near Field ◽  
Spherical Particles ◽  
Random Structure ◽  
Shearing Flow ◽  
Self Diffusion ◽  
Long Time ◽  
Mobility Matrix ◽  
Many Body Interactions

We study the average mobilities and long-time self-diffusion coefficients of a suspension of bimodally distributed spherical particles. Stokesian dynamics is used to calculate the particle trajectories for a monolayer of bimodal-sized spheres. Hydrodynamic forces only are considered and they are calculated using the inverse of the grand mobility matrix for far-field many-body interactions and lubrication formulae for near-field effects. We determine both the detailed microstructure (e.g. the pair-connectedness function and cluster formation) and the macroscopic properties (e.g. viscosity and self-diffusion coefficients). The flow of an ‘infinite’ suspension is simulated by considering 25, 49, 64 and 100 particles to be one ‘cell’ of a periodic array. Effects of both the size ratio and the relative fractions of the different-sized particles are examined. For the microstructures, the pair-connectedness function shows that the particles form clusters in simple shearing flow due to lubrication forces. The nearly symmetric angular structures imply the absence of normal stress differences for a suspension with purely hydrodynamic interactions between spheres. For average mobilities at infinite Péclet number, Ds0, our simulation results suggest that the reduction of Ds0 as concentration increases is directly linked to the influence of particle size distribution on the average cluster size. For long-time self-diffusion coefficients, Ds∞, we found good agreement between simulation and experiment (Leighton & Acrovos 1987 a; Phan and Leighton 1993) for monodispersed suspensions. For bimodal suspensions, the magnitude of Ds∞, and the time to reach the asymptotic diffusive behaviour depend on the cluster size formed in the system, or the viscosity of the suspension. We also consider the effect of the initial configuration by letting the spheres be both organized (size segregated) and randomly placed. We find that it takes a longer time for a suspension with an initially organized structure to achieve steady state than one with a random structure.

Self-diffusion in sheared suspensions

1996 ◽  
Vol 312 ◽  
pp. 223-252 ◽  
Author(s):  
Jeffrey F. Morris ◽  
John F. Brady
Keyword(s):  
Volume Fraction ◽  
Diffusion Tensor ◽  
Hard Spheres ◽  
Strain Tensor ◽  
The Self ◽  
Hydrodynamic Interactions ◽  
Spherical Particles ◽  
Tagged Particle ◽  
Self Diffusion ◽  
Long Time

Self-diffusion in a suspension of spherical particles in steady linear shear flow is investigated by following the time evolution of the correlation of number density fluctuations. Expressions are presented for the evaluation of the self-diffusivity in a suspension which is either raacroscopically quiescent or in linear flow at arbitrary Peclet number $Pe = \dot{\gamma}a^2/2D$, where $\dot{\gamma}$ is the shear rate, a is the particle radius, and D = kBT/6πa is the diffusion coefficient of an isolated particle. Here, kB is Boltzmann's constant, T is the absolute temperature, and η is the viscosity of the suspending fluid. The short-time self-diffusion tensor is given by kBT times the microstructural average of the hydrodynamic mobility of a particle, and depends on the volume fraction $\phi = \frac{4}{3}\pi a^3n$ and Pe only when hydrodynamic interactions are considered. As a tagged particle moves through the suspension, it perturbs the average microstructure, and the long-time self-diffusion tensor, D∞s, is given by the sum of D0s and the correlation of the flux of a tagged particle with this perturbation. In a flowing suspension both D0s and D∞ are anisotropic, in general, with the anisotropy of D0s due solely to that of the steady microstructure. The influence of flow upon D∞s is more involved, having three parts: the first is due to the non-equilibrium microstructure, the second is due to the perturbation to the microstructure caused by the motion of a tagged particle, and the third is by providing a mechanism for diffusion that is absent in a quiescent suspension through correlation of hydrodynamic velocity fluctuations.The self-diffusivity in a simply sheared suspension of identical hard spheres is determined to O(øPe3/2) for Pe ≤ 1 and ø ≤ 1, both with and without hydro-dynamic interactions between the particles. The leading dependence upon flow of D0s is 0.22DøPeÊ, where Ê is the rate-of-strain tensor made dimensionless with $\dot{\gamma}$. Regardless of whether or not the particles interact hydrodynamically, flow influences D∞s at O(øPe) and O(øPe3/2). In the absence of hydrodynamics, the leading correction is proportional to øPeDÊ. The correction of O(øPe3/2), which results from a singular advection-diffusion problem, is proportional, in the absence of hydrodynamic interactions, to øPe3/2DI; when hydrodynamics are included, the correction is given by two terms, one proportional to Ê, and the second a non-isotropic tensor.At high ø a scaling theory based on the approach of Brady (1994) is used to approximate D∞s. For weak flows the long-time self-diffusivity factors into the product of the long-time self-diffusivity in the absence of flow and a non-dimensional function of $\bar{P}e = \dot{\gamma}a^2/2D^s_0(\phi)$. At small $\bar{P}e$ the dependence on $\bar{P}e$ is the same as at low ø.

Long-time self-diffusion coefficients of suspensions

1992 ◽  
Vol 46 (8) ◽  
pp. 5012-5019 ◽  
Author(s):  
Grzegorz Szamel ◽  
Jan A. Leegwater
Keyword(s):  
Diffusion Coefficients ◽  
Self Diffusion ◽  
Long Time

The Spread of Pollutants from Harbour Sludge Depots

1984 ◽  
Vol 16 (3-4) ◽  
pp. 623-633
Author(s):  
M Loxham ◽  
F Weststrate
Keyword(s):  
Time Scales ◽  
Molecular Diffusion ◽  
Near Field ◽  
Parameter Analysis ◽  
Migration Patterns ◽  
Long Time ◽  
Dilution Effects ◽  
Short Time ◽  
Harbour Sludge

It is generally agreed that both the landfill option, or the civil techniques option for the final disposal of contaminated harbour sludge involves the isolation of the sludge from the environment. For short time scales, engineered barriers such as a bentonite screen, plastic sheets, pumping strategies etc. can be used. However for long time scales the effectiveness of such measures cannot be counted upon. It is thus necessary to be able to predict the long term environmenttal spread of contaminants from a mature landfill. A model is presented that considers diffusion and adsorption in the landfill site and convection and adsorption in the underlaying aquifer. From a parameter analysis starting form practical values it is shown that the adsorption behaviour and the molecular diffusion coefficient of the sludge, are the key parameters involved in the near field. The dilution effects of the far field migration patterns are also illustrated.

Far field modeling of the Mamala Bay outfalls

1998 ◽  
Vol 38 (10) ◽  
pp. 323-330
Author(s):  
Philip J. W. Roberts
Keyword(s):  
Field Model ◽  
Near Field ◽  
Far Field ◽  
Field Modeling ◽  
Visitation Frequency ◽  
Statistical Variation ◽  
Long Time ◽  
Harmonic Average ◽  
Ocean Outfall ◽  
Acoustic Doppler Current Profilers

The results of far field modeling of the wastefield formed by the Sand Island, Honolulu, ocean outfall are presented. A far field model, FRFIELD, was coupled to a near field model, NRFIELD. The input data for the models were long time series of oceanographic observations over the whole water column including currents measured by Acoustic Doppler Current Profilers and density stratification measured by thermistor strings. Thousands of simulations were made to predict the statistical variation of wastefield properties around the diffuser. It was shown that the visitation frequency of the wastefield decreases rapidly with distance from the diffuser. The spatial variation of minimum and harmonic average dilutions was also predicted. Average dilution increases rapidly with distance. It is concluded that any impact of the discharge will be confined to a relatively small area around the diffuser and beach impacts are not likely to be significant.

scholarly journals Theoretical Modelling of Ion Exchange Processes in Glass: Advances and Challenges

2021 ◽  
Vol 11 (11) ◽  
pp. 5070
Author(s):  
Xesús Prieto-Blanco ◽  
Carlos Montero-Orille
Keyword(s):  
Ion Exchange ◽  
Diffusion Coefficients ◽  
Molten Salts ◽  
Theoretical Modelling ◽  
Copper Films ◽  
Theoretical Frameworks ◽  
Self Diffusion ◽  
Exchange Processes ◽  
The One ◽  
Made In

In the last few years, some advances have been made in the theoretical modelling of ion exchange processes in glass. On the one hand, the equations that describe the evolution of the cation concentration were rewritten in a more rigorous manner. This was made into two theoretical frameworks. In the first one, the self-diffusion coefficients were assumed to be constant, whereas, in the second one, a more realistic cation behaviour was considered by taking into account the so-called mixed ion effect. Along with these equations, the boundary conditions for the usual ion exchange processes from molten salts, silver and copper films and metallic cathodes were accordingly established. On the other hand, the modelling of some ion exchange processes that have attracted a great deal of attention in recent years, including glass poling, electro-diffusion of multivalent metals and the formation/dissolution of silver nanoparticles, has been addressed. In such processes, the usual approximations that are made in ion exchange modelling are not always valid. An overview of the progress made and the remaining challenges in the modelling of these unique processes is provided at the end of this review.

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