Homogeneous and Isotropic Particles

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. 

This case can also be approached using Kolmogoroff s (K9, H15) theory of local isotropic turbulence to predict the velocity of suspended particles relative to a homogeneous and isotropic turbulent flow. By examining this situation for spherical particles moving with a constant relative velocity, varying randomly in direction, Levich, (L3) has demonstrated that... 

As mentioned in the introduction, the discussion starts from a macroscopic system composed of 1 identical particles. Among the particles there exists an attractive interaction that is responsible for the formation of condensed phases. The particles on the other hand possess a certain degree of freedom of motion in any direction within the system, as required by the liquid state. Because no preferred distribution of particles can be assumed, the system seems on average to be totally homogeneous and isotropic. This leads to an essential simplification of the problem. 

The macroscopic transport of particles requires a departure 5f(u) from the homogeneous and isotropic thermal equilibrium distribution in the velocity space fo r,u,t). The net motion of species in the entire velocity space no longer vanishes ... 

For homogeneous and isotropic systems, the two-particle correlation function depends only on the distance between the centers of the particles, r = ri — r2, i.e.,... 

Thus, for homogeneous and isotropic systems, g(r) is the autocorrelation function of the local particle-concentration. 

In the general case, the scattering fiom a homogeneous and isotropic dispersion of spherical particles (pores) can be written as the product of,... 

Consider a homogeneous and isotropic system of classical noninteracting charged particles under an external, position-independent electric field Exit) in the x direction. In this case... 

In this book, we shall only be interested in homogeneous and isotropic fluids. In such a case, there is a redundancy in specifying the full configuration of the pair of particles by 12 coordinates (X, X"). It is clear that for any configuration of the pair X, X", the correlation g(X, X ) is invariant to translation and rotation of the pair as a unit, keeping the relative configuration of one particle toward the other fixed. Therefore, we can reduce to six the number of independent variables necessary for the full description of the pair correlation function. For instance, we may choose the location of one particle at the origin of the coordinate system, R = 0, and fix its orientation, say, at (j) = O = t// = 0. Hence, the pair correlation function is a function only of the six variables X" = R", S2". 

Most micromechanical theories treat composites where the thermoelastic properties of the matrix and of each filler particle are assumed to be homogeneous and isotropic within each phase domain. Under this simplifying assumption, the elastic properties of the matrix phase and of the filler particles are each described by two independent quantities, usually the Young s modulus E and Poisson s ratio v. The thermal expansion behavior of each constituent of the composite is described by its linear thermal expansion coefficient (3. It is far more complicated to treat composites where the properties of some of the individual components (such as high-modulus aromatic polyamide fibers) are themselves inhomogeneous and/or anisotropic within the individual phase domains, at a level of theory that accounts for the internal inhomogeneities and/or anisotropies of these phase domains. Consequently, there are very few analytical models that can treat such very complicated but not uncommon systems truly adequately. 

The properties of a two-phase system consisting of a continuous "matrix" phase and a discontinuous "filler" phase are calculated in tenns of the component properties and volume fractions. It is assumed that the thennoelastic properties within each phase domain are homogeneous and isotropic, and that there is perfect adhesion between adjacent phase domains. The shapes of the filler particles are assumed to have biaxial symmetry. If a filler particle is anisotropic (as in fibers or platelets), it is oriented uniaxially at this stage of the calculation. Particle shape is described by the aspect ratio Af (defined as the ratio of the largest dimension of the filler divided by its smallest dimension), and if Af l then also by... 

The pea coal was spherical particles which was homogeneous and isotropic ... 

Throughout this chapter we consider a homogeneous and isotropic fluid composed of N rigid molecules (or spherical particles) in a volume V at the inverse temperature 0 = 1/ksT, and the thermodynamic limit with density p = N/V is implied. 

Recently, Pozorski and Apte [39] performed a systematic study of the direct effect of subgrid scale velocity on particle motion for particle-laden forced isotropic turbulence using a stochastic model based on filtered particle tracking (FPT). The FPT approach statistically reconstructs the imresolved carrier-phase velocity along particle trajectories. A reasonable assumption for LES is to consider the residual turbulent motion as locally homogeneous and isotropic. Then, the fluid velocity... 

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... 

Much of the treatment contained within this volume is limited by the assumptions that the adhesives, and usually the adherends, are linear elastic, homogenous, and isotropic. For bulk adhesives, the assumption of isotropy is usually justified, although instances do arise where preferred orientation of filler particles or crystalline regions can lead to anisotropic behavior. Common adherends such as fiber-reinforced composites, wood, and cold-drawn metals often exhibit anisotropic behavior that can significantly affect Joint behavior. 

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