Brownian diffusion coefficients

For displacements shorter than the mean pore dimension, (z2) < a, where flow velocities tend to be spatially constant and homogeneously distributed, Brownian diffusion is the only incoherent transport phenomenon that contributes to the hydrodynamic dispersion coefficient. As a direct consequence, the dispersion coefficient approaches the ordinary Brownian diffusion coefficient,... 

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. 

The simple form in Eq. (7) can be maintained by replacing the Brownian diffusion coefficient in the expression kc = /-An /5 by the shear-induced hydrodynamic diffusion coefficient for the particles, Ds. Shear-induced hydrodynamic diffusion of particles is driven by random displacements from the streamlines in a shear flow as the particles interact with each other. For particle volume fractions between 20 and 45%, Ds has been related to... 

To obtain the mean free path Xp, we recall that in Section 9.1, using kinetic theory, we connected the mean free path of a gas to measured macroscopic transport properties of the gas such as its binary diffusivity. A similar procedure can be used to obtain a particle mean free path A,p from the Brownian diffusion coefficient and an appropriate kinetic theory expression for the diffusion flux. Following an argument identical to that in Section 9.1, diffusion of aerosol particles can be viewed as a mean free path phenomenon so that... 

On the other hand, the two individual Brownian diffusion coefficients are defined by (dr2) = 6D dt and (dr ) = 6D2dt. Consequently, we find that... 

D,i Brownian diffusion coefficient describing relative motion of two ... 

To determine the value of the diffusivity that connects the two approaches, we follow Einstein s thermodynamic arguments given in Section 5.2 for evaluating the translational Brownian diffusion coefficient. The basis for this is the random Brownian motion of the monomer units in the gel, which translates into the gel osmotic pressure. If, as above, the flow through the gel is assumed to follow Darcy s law (Eq. 4.7.7), then we may write the applied hydrodynamic force per mole of solution flowing through the gel as... 

Here, n represents the local number density of the particles of radius a2, and D,2 is the Brownian diffusion coefficient describing the relative motion of the two particles. However, since the motion of the particles is assumed to be independent, it is readily shown that D,2 = D, -F Dj (Friedlander 1977). 

The diffusion of NPs in blood flow can be due to (1) Brownian diffusion caused by the bombardment of fluid molecules and (2) shear-induced diffusion due to the presence of red blood cells (RBCs) in shear flow. The Brownian diffusion coefficient, Dpr, can be calculated using the Stokes-Einstein equation ... 

At the point of zero charge, there is no repulsive electrical force on the particles and so the full adhesion between the grains is developed. If this adhesion is strong, then each Brownian collision between singlet particles will produce a doublet. This was the case considered by Smoluchowski in 1917 and extended by Fuchs in 1934. The theory was based on the idea that colloidal particles behave like molecules which can react to form abimolecular compound. Thus the rate of appearance of doublets dAf /df is proportional to the square of singlet concentration N per unit volume by the law of mass action, and is limited by the Brownian diffusion coefficient to give... 

Capillary and extrudate diameter, respectively Droplet deformability in extensional flow Rotational Brownian diffusion coefficient Steady-state tensile compliance Volume-to-surface average particle diameter Elasticity of the interphase Interaction energy Tensile, or Young s, modulus Electron... 

Brownian motion is the random thermal motion of a particle suspended in a fluid. This motion results from collisions between fluid molecules and suspended particles. For time intervals At much larger than the particle inertial response time, the dynamics of Brownian motion are independent of inertial parameters such as particle and fluid density. The Brownian diffusion coefficient D is given by the Stokes-Einstein equation as... 

The particle size and temperature dependence of this collection mechanism is introduced via the Brownian diffusion coefficient, Dbm (T/ds). 

For rigid spheres the rotary Brownian diffusion coefficient is... 

The relaxation time A. is inversely proportional to Dr, the rotary Brownian diffusion coefficient, eq. 10.3.4. 

WS7 Brownian diffusion coefficient for aqua sols (i.e., DOC, colloids, etc.) in the bed pore-waters WS8 Bioturbation driven bio-diffusion coefficient for particle and pore-water transport within the bed WS9 Particle settling and deposition into the deep ocean... 

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