Diffusion equation scaling

This complex Ginzburg-Landau equation describes the space and time variations of the amplitude A on long distance and time scales detennined by the parameter distance from the Hopf bifurcation point. The parameters a and (5 can be detennined from a knowledge of the parameter set p and the diffusion coefficients of the reaction-diffusion equation. For example, for the FitzHugh-Nagumo equation we have a = (D - P... 

At small driving forces a completely fiat interface cannot move at a constant speed. This is basically a result of the inherent scaling property of the diffusion equation, which scales lengths proportional to the square-root of time, so an advancing interface would slow down with time. 

Computer simulation and analytical methods have both been used, based on diffusion equation, partition function and scaling theory approaches. There are a number of parameters which are common to most of these theories some of these are also relevant to theories of polymer solutions, i.e. 

As discussed in Section 4.3, the linear-eddy model solves a one-dimensional reaction-diffusion equation for all length scales. Inertial-range fluid-particle interactions are accounted for by a random rearrangement process. This leads to significant computational inefficiency since step (3) is not the rate-controlling step. Simplifications have thus been introduced to avoid this problem (Baldyga and Bourne 1989). 

Regarding the electrochemical method, the generalized forms of the Cottrell relation and the Randles-Sevcik relation were theoretically derived from the analytical solutions to the generalized diffusion equation involving a fractional derivative operator under diffusion-controlled constraints and these are useful in to determining the surface fractal dimension. It is noted that ionic diffusion towards self-affine fractal electrode should be described in terms of the apparent self-similar fractal dimension rather than the self-affine fractal dimension. This means the fractal dimension determined by using the diffusion-limited electrochemical method is the self-similar fractal dimension irrespective of the surface scaling property. 

If Da = 1 is defined as the transition between diffusionally controlled and kinetically controlled regimes, an inverse relationship is observed between the particle diameter and the system pressure and temperature for a fixed Da. Thus, for a system to be kinetically controlled, combustion temperatures need to be low (or the particle size has to be very small, so that the diffusive time scales are short relative to the kinetic time scale). Often for small particle diameters, the particle loses so much heat, so rapidly, that extinction occurs. Thus, the particle temperature is nearly the same as the gas temperature and to maintain a steady-state burning rate in the kinetically controlled regime, the ambient temperatures need to be high enough to sustain reaction. The above equation also shows that large particles at high pressure likely experience diffusion-controlled combustion, and small particles at low pressures often lead to kinetically controlled combustion. 

According to measurements made in the atmosphere, the Lagrangian time scale is of the order of 100 sec (Csanady, 1973). Using a characteristic particle velocity of 5 m sec", the above conditions are 100 sec and L > 500 m. Since one primary concern is to examine diffusion from point sources such as industrial stacks, which are generally characterized by small T and L, it is apparent that either one (but particularly the second one) or both of the above constraints cannot be satisfied, at least locally, in the vicinity of the point-like source. Therefore, in these situations, it is important to assess the error incurred by the use of the atmospheric diffusion equation. 

Because of the way it was derived, the last term in the diffusion equation due to the gradient of <1> has the same scaling laws as the first one. Therefore it does not... 

Constraints may be introduced either into the classical mechanical equations of motion (i.e., Newton s or Hamilton s equations, or the corresponding inertial Langevin equations), which attempt to resolve the ballistic motion observed over short time scales, or into a theory of Brownian motion, which describes only the diffusive motion observed over longer time scales. We focus here on the latter case, in which constraints are introduced directly into the theory of Brownian motion, as described by either a diffusion equation or an inertialess stochastic differential equation. Although the analysis given here is phrased in quite general terms, it is motivated primarily by the use of constrained mechanical models to describe the dynamics of polymers in solution, for which the slowest internal motions are accurately described by a purely diffusive dynamical model. 

Before we turn to this issue, we would like to substantiate the above discussion of basic features of nonlinear diffusion with some examples based upon the well-known similarity solutions of the Cauchy problems for the relevant diffusion equations. Similarity solutions are particularly instructive because they express the intrinsic symmetry features of the equation [6], [28], [29], Recall that those are the shape-preserving solutions in the sense that they are composed of some function of time only, multiplied by another function of a product of some powers of the time and space coordinates, termed the similarity variable. This latter can usually be constructed from dimensional arguments. Accordingly, a similarity solution may only be available when the Cauchy problem under consideration lacks an explicit length scale. Thus, the two types of initial conditions compatible with the similarity requirement are those corresponding to an instantaneous point source and to a piecewise constant initial profile, respectively, of the form... 

However, for (r0 — R) < a, Noyes [269] developed a different form of j30. Since such motion must be on a molecular scale a < R) or even less, the author is doubtful that such approximations are more valid than that of the diffusion equation. To estimate a, it may be noted that Northrup and Hynes [103] have found that... 

The dissociation of a molecule in solution and the approach to an equilibrium distribution of molecules and radicals has been treated by Berg [278]. His detailed and careful analysis uses the diffusion equation exclusively to describe microscopic motion. During molecular dissociation on a microscopic scale (i.e. involving only a few molecules), molecules dissociate, recombine, dissociate etc. many times. The global rate of dissociation is much less than that of an individual molecule, indeed smaller by a factor of (1 + kACijAiiRD), that is an average number of times the molecule dissociates and recombines. For reactions which do not go to completion... 

In this chapter, the motion of solute and solvent molecules is considered in rather more detail. Previously, it has been emphasised that this motion approximates to diffusion only over times which are long compared with the velocity relaxation time (see Chap. 8, Sect. 2.1). At times comparable with or a little longer than the velocity relaxation time, the diffusion equation does not provide a satisfactory description of molecular motion. An alternative approach must be sought. This introduces considerable complications to a theoretical analysis of very fast reactions in solution. To develop an understanding of chemical reactions occurring over very short time intervals, several points need to be discussed. Which reactions might be of interest and over what time scale What is known of the molecular motion of solute and solvent molecules Why does the Markovian (hydrodynamic) continuum analysis fail and what needs to be done to develop a better theory These points will be considered in further detail in this chapter. 

Another derivation has been given by Resibois and De Leener. In principle, eqn. (287) can be applied to describe chemical reactions in solution and it should provide a better description than the diffusion (or Smoluchowski) equation [3]. Reaction would be described by a spatial- and velocity-dependent term on the right-hand side, — i(r, u) W Sitarski has followed such an analysis, but a major difficulty appears [446]. Not only is the spatial dependence of the reactive sink term unknown (see Chap. 8, Sect. 2,4), but the velocity dependence is also unknown. Nevertheless, small but significant effects are observed. Harris [523a] has developed a solution of the Fokker—Planck equation to describe reaction between Brownian particles. He found that the rate coefficient was substantially less than that predicted from the diffusion equation for aerosol particles, but substantially the same as predicted by the diffusion equation for molecular-scale reactive Brownian particles. 

At times t < f0 w [where f0 ° is an infinitesimal amount less than f0 ], the density is zero. Only after the pair is formed can there be any probability of its existence [499]. This is cause and effect, but strictly only applicable at a macroscopic level. On a microscopic scale, time reversal symmetry would allow us to investigate the behaviour of the pair at time and so it reflects the inappropriateness of the diffusion equation to truly microscopic phenomena. The irreversible nature of diffusion on a macroscopic scale results from the increase of entropy, and should be related to microscopic events described by the Sturm—Liouville equation (for instance) and appropriately averaged. 

Then i) is invariant under the scaling corresponding to Eq. 4.19 and c becomes a function of the single variable, r). The diffusion equation becomes an ordinary differential equation (i.e., d — d). 

Using the scaling parameter 77 = x/V41, the diffusion equation becomes... 

To solve the diffusion equation in the principal coordinate system (i.e., Eq. 4.61), the Cartesian space can now be stretched or contracted along the principal axes by scaling ... 

This scaling conserves the volume. Using Eq. 4.64, the diffusion equation can now be written in terms of the ... 

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