Averaged diffusion equation

Two applications of the flucUiathig diffusion equation are made here to illustrate tlie additional infonnation the flucUiations provide over and beyond the detenninistic behaviour. Consider an infinite volume with an initial concentration, c, that is constant, Cq, everywhere. The solution to the averaged diffusion equation is then simply (c) = Cq for all t. However, the two-time correlation fiinction may be shown [26] to be... 

Turbulent flux of a scalar quantity averaged diffusion equation... 

Figure 9.2 shows the geometrical characteristics of the evolution over time of the travel and growth of the spot. We denote by x t) the middle position of the spot at time t, and by D(t) its thickness. Because of [9.4], the space and time evolution of the average concentration c(x,t) inside the reactor depend only on time and on the X coordinate. It is governed by the averaged diffusion equation (equation [8.31] of Chapter 8) ... 

This example illustrates how the Onsager theory may be applied at the macroscopic level in a self-consistent maimer. The ingredients are the averaged regression equations and the entropy. Together, these quantities pennit the calculation of the fluctuating force correlation matrix, Q. Diffusion is used here to illustrate the procedure in detail because diffiision is the simplest known case exlribiting continuous variables. 

The shear viscosity is a tensor quantity, with components T] y, t],cz, T)yx> Vyz> Vzx> Vzy If property of the whole sample rather than of individual atoms and so cannot be calculat< with the same accuracy as the self-diffusion coefficient. For a homogeneous fluid the cor ponents of the shear viscosity should all be equal and so the statistical error can be reducf by averaging over the six components. An estimate of the precision of the calculation c then be determined by evaluating the standard deviation of these components from tl average. Unfortunately, Equation (7.89) cannot be directly used in periodic systems, evi if the positions have been unfolded, because the unfolded distance between two particl may not correspond to the distance of the minimum image that is used to calculate the fore For this reason alternative approaches are required. 

If average diffusion coefficients are used, then the finite difference equation is as follows. 

Similar convection-diffusion equations to the Navier-Stokes equation can be formulated for enthalpy or species concentration. In all of these formulations there is always a superposition of diffusive and convective transport of a field quantity, supplemented by source terms describing creation or destruction of the transported quantity. There are two fundamental assumptions on which the Navier-Stokes and other convection-diffusion equations are based. The first and most fundamental is the continuum hypothesis it is assumed that the fluid can be described by a scalar or vector field, such as density or velocity. In fact, the field quantities have to be regarded as local averages over a large number of particles contained in a volume element embracing the point of interest. The second hypothesis relates to the local statistical distribution of the particles in phase space the standard convection-diffusion equations rely on the assumption of local thermal equilibrium. For gas flow, this means that a Maxwell-Boltzmann distribution is assumed for the velocity of the particles in the frame-of-reference co-moving with the fluid. Especially the second assumption may break dovm when gas flow at high temperature or low pressure in micro channels is considered, as will be discussed below. 

The first and second integrals have their coordinate systems centered on the catalytic C and noncatalytic N spheres, respectively. The local nonequilibrium average microscopic density field for species a is pa(r) = [Y = 5(r - ( )) The solution of the diffusion equation can be used to estimate this nonequilibrium density, and thus the velocity of the nanodimer can be computed. The simple model yields results in qualitative accord with the MPC dynamics simulations and shows how the nonequilibrium density field produced by reaction, in combination with the different interactions of the B particles with the noncatalytic sphere, leads to directed motion [117],... 

Since turbulent fluctuations not only occur in the velocity (and pressure) field but also in species concentrations and temperature, the convection diffusion equations for heat and species transport under turbulent-flow conditions also comprise cross-correlation terms, obtained by properly averaging products of... 

An equivalent dehnition of the drift velocity E" may be obtained by using the diffusion equation alone to calculate the average flux velocity in a statishcal ensemble characterized by a probability distribution... 

Two approaches can be used for the analysis of turbulent mass transfer near a liquid-fluid interface. One has the time-averaged convective diffusion equation as the starting point. For obtaining in that procedure an equation for... 

Substantially the same result is obtained using the solutions of the two-dimensional diffusion equation given by Adam and Delbriick.1 We suppose that at time ( = 0 an IgG antibody molecule binds to a lipid hapten with one site. What is the average rate at which a second hapten diffuses up to the second combining site From (18) of Adam and Delbriick1 this is... 

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