Homogeneous Isotropic Particles

The calculation of the screened interaction tensors T- and the dispersion energy AE12 between two macroscopic particles 1 and 2 is greatly simplified if these particles are homogeneous and isotropic. Isotropy entails that the susceptibilities Xj(co) corresponding to the different positions j are scalars which can be separated from the interaction tensors 

Homogeneity enables interaction potentials rather than fields to be considered. Making use of Eq. (2.21) we find 

We are left with an integral over the surface of particle 1. However, a small spherical cavity around rj has to be excluded, since there is no interaction of dipole at r with itself. All induction fields result by summing over the remaining dipoles. This exclusion of a small spherical cavity is regularly required in microscopic theories on macroscopic susceptibilities. 

K particle 1 is a sphere with radius Ri and center r, we obtain according to Fig. 12 

The integration over cpi can be evaluated using the addition theorem for Legendre functions, Eq. (3.11(2)) in Ref. [2], 

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Let us consider a case when homogeneous isotropic particles with the dielectric constant C2 (or with the refractive index, or the index of refraction, n-i = y/TJ) are dispersed in a medium with ci =... 

The transmission boundary-value problem for homogeneous and isotropic particles has been formulated in Sect. 1.4 but we mention it in order for our analysis to be complete. We consider an homogeneous, isotropic particle occupying a domain D with boundary S and exterior (Fig. 2.1). The imit normal vector to S directed into is denoted by n. The exterior domain Ds is assumed to be homogeneous, isotropic, and nonabsorbing, and if t and jM. are the relative permittivity and permeability of the domain Ht, where t = s, i, we have s > 0 and ps > 0. The wave number in the domain Dt is kt = ko, /etPt, where ko is the wave number in the free space. The transmission boundary-value problem for a homogeneous and isotropic particle has the following formulation. 

Neal and Nader [260] considered diffusion in homogeneous isotropic medium composed of randomly placed impermeable spherical particles. They solved steady-state diffusion problems in a unit cell consisting of a spherical particle placed in a concentric shell and the exterior of the unit cell modeled as a homogeneous media characterized by one parameter, the porosity. By equating the fluxes in the unit cell and at the exterior and applying the definition of porosity, they obtained... 

It should be intuitively clear that the correlation between any two particles vanishes as ri — r21 oo. Therefore g -> 1 and /i 0 in this limit. For homogeneous fluids all positions are equivalent, and it follows that g(ri, r2) = gCfi — i 2)- For homogeneous-isotropic fluids g(ri, r2) = g( ri — r2 ), and similarly for h. In this case we refer to these functions as radial distribution functions. 

Zaichik, L. L, Simonin, O. Aupchenkov, V. M. 2003 Two statistical models for predicting collision rates of inertial particles in homogeneous isotropic turbulence. Physics of Fluids 15(10), 2995-3005. 

Grinding. An important aspect is often whether such a material can be ground, and how small then are the particles obtained. It can be derived from the theory that the thickness of the zone near a crack in which plastic deformation (yielding) occurs in a homogeneous isotropic material is given by... 

Though both parameters are equivalent for the description of polarised light, anisotropy is usually preferred. Following pulse excitation, the anisotropy of spherical particles in a homogeneous isotropic medium decays exponentially, given by ... 

Consider two conducting spherical particles with radii R and Rz, carrying charges qi and qz and placed in a uniform external electric field of strength Eo (Fig. 12.1). The angle between the center line of the particles and vector Eq is denoted by ft The space between the particles is occupied by a quiescent homogeneous isotropic dielectric with dielectric permittivity e the particles do not move relative to this dielectric medium. There are no free charges outside of the spheres, and the potential of the electric field in this area satisfies the Laplace equation ... 

The charges on the particles are unknown, but the total charge of both particles is considered to be specified and equal to Q. The particles are conducting, while the space around them is occupied by a homogeneous isotropic dielectric with dielectric permittivity s. The primary goal here is definition of the forces of interaction between the particles and of their charges. 

Although the expressions for and its relation to are originally derived for homogeneous, isotropic materials, it has been successfully utilized for particle-filled composites as follows ... 

So far, our discussion of length scales has focused on the smallest scales of motion. At the larger scales of motion, Taylor (1921) considered the turbulent dispersion of fluid particles by homogeneous isotropic turbulence in the absence of molecular diffusion. In his model, each fluid particle leaving a point source in a uniform velocity field is expected to deviate from the linear mean path in a random manner, depending on the local nature of the turbulence. The RMS deviation of the particle paths is observed as a continued divergence, spread, or dispersion as the particles are carried downstream. This eddy motion occurs even... 

The applicability of the present LBM to colliding point-particles snspended in homogeneous isotropic tnrbnlence has been recently demonstrated by Ernst and Sommerfeld [13], In this context, the direct numerical simulation of an nnsteady turbulence field is realised by applying a spectral forcing scheme. As a further step, flow-induced collision and agglomeration of resolved particles should be examined in order to understand the influence of the short-range hydrodynamic interaction between approaching particles at different particle Stokes and Reynolds numbers. 

Derksen JJ (2012) Direct numerical simulations of aggregation of monosized spherical particles in homogeneous isotropic turbulence. AIChE J 58 2589-2600... 

Ernst M, Sommerfeld M (2012) On the volume fraction effects of inertial colliding particles in homogeneous isotropic turbulence. J Fluids Eng 134 031302... 

As a simple illustrative application of dual Lanczos transformation theory, let us consider the problem of determining the Fourier transform a(]i, co) of the incoherent scattering function S(ic, t) associated with a particle in a spatially homogeneous, isotropic environment and executing motion described by Fokker-Planck dynamics. 

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