Isotropic media, homogeneous

1) Fullerene C6o is perfectly spherical but for of rotational motions. Moreover, its 

For isotropic motions in an isotropic medium, the values of the instantaneous and steady-state emission anisotropies are linked to the rotational diffusion coefficient Dr by the following relations (see Chapter 5)  

The changes in correlation time upon an external perturbation (e.g. temperature, pressure, additive, etc.) reflects well the changes in fluidity of a medium. It should again be emphasized that any microviscosity value that could be calculated from the Stokes-Einstein relation would be questionable and thus useless. 

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For flow through homogeneous isotropic media (K = Ky = K ), Equation 1.28 reduces to the Laplace equation... 

The hydrodynamic approaches assume instantaneous reaction, and apart from a dependence on density, the simplest theories assume detonation parameters to be invariant for a substance and applicable to propagation in infinite, homogeneous (isotropic) media. They give no information on the effect of size or crystal orientation, or on the detailed mechanism by which a detonation propagates. Several theories developed by Jones in the United Kingdom and by Eyring, Wood, and Cook in the United States related detonation velocity to reaction-zone length and explosive diameter, but experimental problems severely limited their validation and application to azides. 

The parameters of the Hamiltonian (3), i.e. the frequencies and the number of oscillators (more specifically, the strength of oscillators) are determined by the imaginary part of a complex dielectric function a(k, co) which characterizes the dielectric losses for polarization fluctuations in a medium. This model is, strictly speaking, applicable to homogeneous isotropic media in which the spatial correlations of polarization fluctuations SP r)dP r ), which determine the dependence of s k, co) on the wave vector k, depend on the difference of coordinates r—r only. 

Since P represents the induced electric dipole moment per unit volume and M represents the induced magnetic dipole moment per unit volume, in homogeneous isotropic media these quantities may therefore be expressed as... 

All of the above pressure equations are complicated and given that boundary conditions are typically prescribed on awkward near- and farfield boundaries, it is no wonder that recourse to numerical models is often made. Analytical approaches typically stop at the classical logarithmic solution for pressure, which is restricted to purely radial flows, and progress no further. However, two simple solutions, introduced here, can be leveraged to produce large classes of solutions for flows past fractures and shales. In most books, the simple radial flow model for liquids in homogeneous, isotropic media is discussed. It is based on the parabolic and elliptic equations... 

Properties of the simple streamfunction. Let us consider first the two-dimensional, steady, constant density flow of a liquid in homogeneous, isotropic media. The Darcy flow, in this case, satisfies Laplace s equation... 

Flows of gases in isotropic media. Although we have considered liquids (with m = 0) exclusively, complex variables ideas readily extend to steady-state compressible gases. Consider the flow of a gas in homogeneous, isotropic media, following Discussion 4-2. Equations 4-17 and 4-18 suggest that... 

Example 6-2. Simple front tracking for liquids in homogeneous, isotropic media. ... 

As in Example 6-1, we consider three botmdary value problems for steady gas flows in homogeneous, isotropic media. Two are easily posed and solved, but the third requires nonlinear iteration. 

Single-phase flow pressure equations. Fluid flows are governed by partial differential equations. For example, single-phase flows of constant density liquids in homogeneous, isotropic media satisfy Laplace s equation for pressure. 

Example 6-3. Steady-state gas flows in homogeneous, isotropic media, 111... 

The permeation rate of ions across membranes can be estimated using a continuum dielectric model of a water-membrane system. In this model, both water and membrane are represented as homogeneous, isotropic media, characterized by dielectric constants and ej, respectively, and separated by a sharp planar boundary. If the ion is represented as a point charge q located at the center of a cavity of radius a, the change in the excess chemical potential associated with the transfer of the ion from bulk water to the center of the membrane (the free energy barrier), is expressed in this model as [58,59] ... 

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