Scaling Boundary Value Problems

It is instructive to study a much simpler mathematical equation that exhibits the essential features of boundary-layer behavior. There is a certain analogy between stiffness in initial-value problems and boundary-layer behavior in steady boundary-value problems. Stiffness occurs when a system of differential equations represents coupled phenomena with vastly different characteristic time scales. In the case of boundary layers, the governing equations involve multiple physical phenomena that occur on vastly different length scales. Consider, for example, the following contrived second-order, linear, boundary-value problem ... 

Next, suppose that for the particular boundary-value problem under consideration, the initial and boundary conditions are unchanged by scale change ... 

Matano developed a graphical method which, for certain classes of boundary value problems, relates the form of the diffusion profile with the concentration dependence of the interdiffusivity, D(c), introduced in Section 3.1.3 [5]. This method can determine D(c) from the diffusion profile in chemical concentration-gradient diffusion experiments where atomic volumes are sufficiently constant so that changes in overall specimen volume are insignificant and diffusion can be formulated in a F-frame. The method uses scaling, as discussed in Section 4.2.2. 

Solution. Yes. When D varies with concentration we have shown in Section 4.2.2 that the diffusion equation can be scaled (transformed) from zt-space to 77-space by using the variable rj = x/ /4Di (see Eq. 4.19). Also, under diffusion-limited conditions where fixed boundary conditions apply at the interfaces, the boundary conditions can also be transformed to 77-space, as we have also seen. Therefore, when D varies with concentration, the entire layer-growth boundary-value problem can be transformed into 77-space. Since the fixed boundary conditions at the interfaces require constant values of 77 at the interfaces, they will move parabolically. 

The pore scale boundary value problem is described by the advection-diffusion equation... 

With the scales by (3.3), one derives the following conjugation boundary-value problem that generalizes problem (3.6) ... 

In addition to the type of broad understanding that accompanies a description of phenomena in materials according to certain key scaling relations, much has been learned on the basis of the particular. Analytic and numerical solutions to boundary value problems as well as the advent of numerical simulation have all contributed to our understanding of material- or geometry- or mechanism-specific properties of material systems. 

The Dirichlet boundary value problem for a large scale region 2 (A.) has special interest for our discussions, as it seems to be a reasonable approximation to the free problem. To some extent, this is a simple problem, when some properties of 2 (A.) are supposed to be satisfied. We suppose that enlargement of A means extension of 2 (A) and any point of R3 belongs to some 2 (A) for a large enough A values. One may also suppose that for any A the distance between boundaries of 2 (A) and 2 (A + 8) is not less then KS for any 8 > 0 and some constant K. 

More sophisticated suction controlled tests which explore the behaviour of the bentonite under specific stress and suction paths are described in the references given above. Tests on small scale cells involving simultaneous hydration and heating were also performed. They are boundary value problems and their analysis may provide a refined evaluation of constitutive parameters. Some of the research groups participating in the reBEX benchmark test have used this information to their advantage. Their analysis is published elsewhere. 

The basis for this inequaUty is the separation of length scales indicated by equation 1.71, and a detailed discussion is available elsewhere (Whitaker, 1999). One should keep in mind that the boundary value problem given by equations 1.131—1.133 applies to aU N - 1 species and that the A - 1 concentration gradients are independent. This latter condition allows us to obtain... 

Thus, the critical measure of the applicability of a method to polyatomic reaction dynamics is the scaling of its computational cost with respect to the number of basis functions or degrees of freedom. Since the standard time-independent scattering methods solve boundary-value problems, they scale as with the number of basis functions N, and are thus difficult to extend to large systems. Until a few years ago, the reduced-dimensionality approach (RDA) [32, 33] provided the only means for treating the four-atom reactive scattering problem in which a four-atom reaction system is reduced to an effective atom-diatom system through the elimi-... 

Dimensional Analysis and Scaling of Boundary Value Problems... 

Krantz, W.B. and Sczechowski, J.G., Scaling initial and boundary value problems. Chemical Engineering Education, 28, 236, 1994. 

In the ensuring we assume that the interaction between species a and is reciprocal, i.e. = DijSap where S p is Kronecker s delta. For simplicity we treat only one species, and set the concentration as c = c , the source as p/ = ya where the superscript implies that this value is rapidly changed in a scale of the micro-domain as mentioned previously. Then, if we assume incompressibility of fluid (p = constant, 9vf/9v , = 0), we have the following initial-boundary value problem governing the advective-diffusive movement of the species ... 

The independent variation of ip and rj vanish at arbitrary temporal values t and in Hamilton s principle (3.3) yielding the following scaled boundary value problem ... 

The Chapman-Enskog approximation method leads, at all stages, to hydrod5mamic equations which are first order in the time and can therefore be solved subject to given initial conditions. This procedure by which the boundary-value problem is converted into an initial-value problem is, from the mathematical point of view, somewhat mysterious. It appears likely that the procedure will converge only for processes whose scale of variation is of the order of, or less than, the mean free path (or other characteristic length). The reduction to an initial-value problem would then be impossible for rapidly varying processes. 

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