Diffusion equation regularization

Spinodal decomposition (SD) driven by the chemical reaction proceeds isothermally, but the quench depth AT (expressing the temperature difference between LCST and the reaction temperature), increases with time. This situation is quite different from the familiar SD under isoquench depth, where after a temperature jump (or drop) SD proceeds isothermally, and the AT is constant. However, the regular morphology is also obtained in the kinetically driven SD, as in the iso-quench SD. This observation was confirmed by the computer simulation using the Cahn-Hilliard non-linear diffusion equation [Ohnaga et al., 1994]. This should also be the case for the solution casting, described in preceding section and the shear-dependent decomposition in next section. 

Kopell and Howard have written several nice expository papers on travelling wave solutions of reaction-diffusion equations (e.g. Kopell and Howard, 1974, 1975) One result which is particularly worth mertioning is that one can construct axisymmetric periodic travelling waves which are completely regular at the origin (see also Greenberg, 1975). That is. 

Prandtl modified the Reynolds analogy by writing the regular molecular diffusion equation for the viscous sublayer and a Reynolds-analogy equation for the turbulent core region. Then since these processes are in series, these equations were combined to produce an overall equation (Gl). The results also are poor for fluids where the Prandtl and Schmidt numbers differ from 1.0. 

Many materials are used in very demanding conditions. Submitted to deformation, corrosion, irradiation, etc., their defect populations acquire complex behaviors, well described by reaction diffusion equations, and may therefore become organized in very regular structures that affect their physical properties. It is also clear now that instabilities and patterns occur all the time in materials science. They affect the properties of the materials, hence they need to be understood and controlled. 

Figure 2 An example of a regular spatial grid used in the simulation of the reaction-diffusion equation. The grid spacing is Ax. The heavy hnes connecting grid points show the coupling used in the evaluation of the diffusion term.

In order to model pattern formation in chemical systems at the macroscopic level, we must be able to solve the reaction-diffusion equation [1]. The simplest numerical method that one can use to solve this equation is an Euler scheme, where space is divided into a regular grid of points with separation Ax. In Figure 2, we show such a grid for a two-dimensional system. Each grid point is labeled by r = (f, ). Time is also divided into small segments At. 

In order to construct mesoscopic models, we again begin by partitioning the system into cells located at the nodes of a regular lattice, but now the cells are assumed to contain some small number of molecules. We cannot use a continuum description of the dynamics in a cell as we did for the reaction-diffusion equation. Instead, we describe the reactions and motions of molecules using stochastic rules that mimic the dynamics of these processes on meso-scales. The stochastic element arises because we do not take into account the detailed motions of all solvent species or the dynamics on microscopic scales. Nevertheless, because the number of molecules in a cell may be small, we must account for the fact that this number can change by random reactive events and random motions of molecules that take them into and out of a... 

This equation is the first term of an infinite series which appears in the rigorous solution of the quasi-diffusion. This equation describes the regular process of quasi-diffusion. For the low values of the Fourier number (irregular quasi-diffusion) it is necessary to use Eq. (5.1) or Boyd-Barrer approximation [105, 106] for the first term in Eq. (5.1)... 

If (1) the diffusion coefficient D of the polymer in the used solvent, (2) the specific volume v of the dissolved polymer, and (3) the density of the solvent solvent known, one can determine the molecular weight of the dissolved polymer according to the above equation by measuring the sedimentation coefficient (by measuring the maximum of the concentration gradient at regular time intervals). 

Fick s second law states the conservation of the diffusing species i no i is produced (or annihilated) in the diffusion zone by chemical reaction. If, however, production (annihilation) occurs, we have to add a (local) reaction term r, to the generalized version of Fick s second law c, = —Vjj + fj. In Section 1.3.1, we introduced the kinetics of point defect production if regular SE s are thermally activated to become irregular SE s (i.e., point defects). These concepts and rate equations can immediately be used to formulate electron-hole formation and annihilation... 

Here, we use the example of the Taylor dispersion problem discussed Section III to illustrate the regularization procedure. For simplicity, we illustrate this only for the case of PeT —> oo (negligible axial diffusion). In this case, the global equation is given by... 

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