Peclet number Brownian diffusion

In this section, we consider flow-induced aggregation without diffusion, i.e., when the Peclet number, Pe = VLID, where V and L are the characteristic velocity and length and D is the Brownian diffusion coefficient, is much greater than unity. For simplicity, we neglect the hydrodynamic interactions of the clusters and highlight the effects of advection on the evolution of the cluster size distribution and the formation of fractal structures. 

A rheological measurement is a useful tool for probing the microstructural properties of a sample. If we are able to perform experiments at low stresses or strains the spatial arrangement of the particles and molecules that make up the system are only slightly perturbed by the measurement. We can assume that the response is characteristic of the microstructure in quiescent conditions. Here our convective motion due to the applied deformation is less than that of Brownian diffusion. The ratio of these terms is the Peclet number and is much less than unity. In Equation (5.1) we have written the Peclet number in terms of stresses ... 

The Peclet number compares the effect of imposed shear (known as the convective effect) with the effect of diffusion of the particles. The imposed shear has the effect of altering the local distribution of the particles, whereas the diffusion (or Brownian motion) of the particles tries to restore the equilibrium structure. In a quiescent colloidal dispersion the particles move continuously in a random manner due to Brownian motion. The thermal motion establishes an equilibrium statistical distribution that depends on the volume fraction and interparticle potentials. Using the Einstein-Smoluchowski relation for the time scale of the motion, with the Stokes-Einstein equation for the diffusion coefficient, one can write the time taken for a particle to diffuse a distance equal to its radius R, as... 

Consider a sphere of radius a held fixed in a creeping flow field with approach velocity U. The fluid contains Brownian particles having a diffusion coefficient D. Should the Peclet number 2aUiDUj have a value much greater than unity, the diffusion boundary layer will be sufficiently thin so that curvature effects and tangential diffusion are negligible. Under these conditions the convective-diffusion equation assumes the following form ... 

The external electric field is in the direction of the pore axis. The particle is driven to move by the imposed electric field, the electroosmotic flow, and the Brownian force due to thermal fluctuation of the solvent molecules. Unlike the usual electroosmotic flow in an open slit, the fluid velocity profile is no longer uniform because a pressure gradient is built up due to the presence of the closed end. The probability of the particle position is obtained by solving the Fokker-Planck equation. The penetration depth is found to be dependent upon the Peclet number, which is a measure of significance of the convective electroosmotic flow relative to the Brownian diffusion, and the Damkohler number, which is a ratio of the characteristic diffusion-to-deposition times. 

Changes in microstructure of the suspension become important when the diffusion time fj becomes long compared to the characteristic time of the process, fp. This number hcis been discussed earlier as the De number. The importance of convection relative to diffusion is compared in the Peclet number Pe (in which u is the fluid velocity). The importance of convection forces relative to the dispersion force is compared in Nf just as the dispersion force compared to the Brownian force. The electrical force compared to the dispersion or Brownian force is given by N. The particle size compared to the range of the electrical force is compared in UK. 

The first group of terms on the right-hand-side of Eq. 4 describes particle transport to a collector surface by Brownian diffusion. NPe is the Peclet number, a ratio of particle transport by fluid advection to transport by molecular or viscous processes. The term As is introduced to account for the effects of neighboring collectors or media grains on the fluid flow around a collector of interest. The results here assume Happel s model (Happel, 1958) for flow around a sphere in a packed bed 4S depends only on the porosity of the bed (Table 1). The derivation for diffusive transport is based on the early work of Levich (1962). 

Suspensions, even in Newtonian liquids, may show elasticity. Hinch and Leal [1972] derived relations expressing the particle stresses in dilute suspensions with small Peclet number, Pe = y/D 1 (D is the rotary diffusion coef-hcient) and small aspect ratio. The origin of elastic effect lies in the anisometry of particles or their aggregates. Rotation of asymmetric entities provides a mechanism for energy storage. Brownian motion for its recovery. Eor suspensions of spheres, this mechanism does not exist. 

A measure of the importance of Brownian motion is given by the ratio of the Brownian diffusion time to the convection time. The diffusion time may be interpreted as the time taken for a particle to diffuse a distance equal to its radius, which is the characteristic time given by the reciprocal of This time characterizes the time taken for the restoration of the equilibrium microstructure from a disturbance caused, for example, by viscous convection. The characteristic convection time is simply given by the reciprocal of the shear rate. We denote the ratio of these two times by the Peclet number symbol, since they measure viscous convection to Brownian diffusion, and we write... 

We treat first the capture by a collector of submicrometer-size particles undergoing Brownian motion in a low-speed flow of velocity U. The collector is taken to be a sphere of radius a and is assumed to be ideal in that all of the particles that impinge on its surface stick to it (Fig. 8.3.1). Because the Brownian particle diffusivities D - kTIGiruap, where is the particle radius, are typically about a thousand times smaller than the molecular diffusivities, the diffusion Peclet number (Ua/D) is generally very large compared with unity. The diffusive flux of the particles to the surface is therefore governed by the steady, convective, diffusion boundary layer equation, with the particles treated as diffusing points. ... 

This is exactly the Peclet number defined by Eq. (5.3.25), which measures the characteristic rotational Brownian diffusion time to the time scale defined by the reciprocal of the shear rate. It is the same measure found for dilute polymer solutions with the particle radius here replacing the Flory radius for the polymer. 

The majority of publications on Brownian motion are limited to the cases Pen 1 (purely Brownian diffusion) or Pep 1 (particle interaction without taking into account Brownian diffusion). The case of arbitrary Peclet number is considered in [28]. 

We assume that deposition on the sphere is ideal, that is, each collision of a particle with the sphere results in the particle being captured. The factor of Brownian diffusion Dj,r = kTwhere Oj, is the particle s radius, is much smaller than the factor of molecular diffusion, therefore the Peclet diffusion number is Peo = Ua/Dhr 1- By virtue of this inequality (see Section 6.5), the diffusion flux of particles toward the sphere can be found by solving the stationary equation of convective diffusion with a condition corresponding to a thick or thin diffusion-boundary layer. Particles may then be considered as point-like, and the diffusion equation will become ... 

When dealing with the sedimentation of colloidal particles, it is principally necessary to regard the Brownian motion of the particles, which results in diffusive particle transport and, thus, acts against the migration in the gravitational or centrifugal field. The relevance of the Brownian motion can be roughly estimated by means of a Peclet-number ... 

Frank et al. [13] investigated particle migration in concentrated Brownian suspensions both by experiment and by modeling of flow in a mixer. The flow rate was quantified by the dimensionless Peclet number, which, conceptually, is the ratio of the time required for Brownian diffusion to move a particle by its own size, a /D = cP j kTl itrjci), to the time required for shear flow to move it by the same distance, y (where y is the shear rate of the surrounding flow field), yielding... 

The hydrodynamic forces are always proportional to the viscosity of the medium. Therefore, suspension viscosities are scaled with the viscosity of the suspending medium, meaning that relative viscosities are used. As for dilute systems, the balance between Brownian motion and flow can be expressed by a Peclet number. Here the translational diffusivity D, has to be used, but that does not change the functionality (for spheres. Dr is proportional to D,). A dimensionless number is obtained by taking the ratio of the time scales for diffusion (D,) and convective motion (y). This is again a Peclet number ... 

Before discussing theoretical models for the rheology of fiber suspensions and its connection to fiber orientation, there are three topics that must be discussed Brownian motion, concentration regimes, and fiber flexibility. Brownian motion refers to the random movement of any sufficiently small particle as a result of the momentum transfer from suspending medium molecules. The relative effect that Brownian motion may have on orientation of anisotropic particles in a dynamic system can be estimated using the rotary Peclet number, Pe s y Dm, where y is the shear rate and Ao is the rotary diffusivity, which defines the ratio of the thermal energy in the system to the resistance to rotation. Doi and Edwards (1988) estimated the rotary diffusivity, Ao, to be... 

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