Similarity solutions nonlinear diffusion equations

B. Self-Similar Solutions of Nonlinear Diffusion Equations... 

The first thing to notice about (6-77) is that it cannot be cast into the form of the nonlinear diffusion equation that was discussed in Section B. Hence, unlike the preceding case, here we cannotapply the solution ofthe diffusion equation from Subsection C.2. However, we can apply the same method of analysis in the hope of finding a self-similar form for the solution of the (/-dimensional symmetric version of (6-77). We begin by adopting the notation of (6-37) and/or (6 19) to express (6-77) in the form... 

When the fast reactions occurring in the system have stoichiometries different from the simple one shown by Eq. (5.78), analytical solutions of the diffusion equations are difficult to obtain. Nevertheless, numerical solutions can be obtained by iterative routines, and the results are conceptually similar to those described. The additional complications introduced by non-steady-state diffusion and nonlinear concentration gradients can be similarly handled. 

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... 

Transient heat conduction or mass transfer in solids with constant physical properties (diffusion coefficient, thermal diffusivity, thermal conductivity, etc.) is usually represented by a parabolic partial differential equation. For steady state heat or mass transfer in solids, potential distribution in electrochemical cells is usually represented by elliptic partial differential equations. In this chapter, we describe how one can arrive at the analytical solutions for linear parabolic partial differential equations and elliptic partial differential equations in semi-infinite domains using the Laplace transform technique, a similarity solution technique and Maple. In addition, we describe how numerical similarity solutions can be obtained for nonlinear partial differential equations in semi-infinite domains. 

In this equation the ) (17) function assumes a similar role to the f(x,y, 17) function in the previous example. The reader is referred back to Section 12.5 for a discussion of this equation and the one dimensional time dependent solution. In the present chapter in Section 13.9 an example of linear diffusion into a two dimensional surface was presented. For that example, a triangular mesh array was developed and shown in Figure 13.15. The present example combines the nonlinear diffusion model of Section 12.5 with the FE mesh of Figure 13.15 to demonstrate a second nonlinear PDE solution using the FE approach. The reader should review this previous material as this seetion builds upon that material. 

You saw how the equations governing energy transfer, mass transfer, and fluid flow were similar, and examples were given for one-drmensional problems. Examples included heat conduction, both steady and transient, reaction and diffusion in a catalyst pellet, flow in pipes and between flat plates of Newtonian or non-Newtonian fluids. The last two examples illustrated an adsorption column, in one case with a linear isotherm and slow mass transfer and in the other case with a nonlinear isotherm and fast mass transfer. Specific techniques you demonstrated included parametric solutions when the solution was desired for several values of one parameter, and the use of artificial diffusion to smooth time-dependent solutions which had steep fronts and large gradients. 

This case has also been considered by Kim and coworkers/ who derived a nonlinear partial differential equation of a form similar to that outlined in Eqn. 101. The problem was also considered by Hermans many years ago. The method of solution proposed by Bartlett and Gardner is perhaps the most comprehensive suggested to date, and it is based on the Neumann analysis of the phase transformation problem in a semiinfinite diffusion space described in the classic text written by Carlsaw and Jaeger. The reader is referred to the original paper for full details of the analysis. We note therefore that the process of dopant transport and reaction in a polymer matrix can be complex. 

One important feature of reaction-diffusion fields, not shared by fluid dynamical systems as another representative class of nonlinear fields, is worth mentioning. This is the fact that the total system can be viewed as an assembly of a large number of identical local systems which are coupled (i.e., diffusion-coupled) to each other. Here the local systems are defined as those obeying the diffusionless part of the equations. Take for instance a chemical solution of some oscillating reaction, the best known of which would be the Belousov-Zhabotinsky reaction (Tyson, 1976). Let a small element of the solution be isolated in some way from the bulk medium. Then, it is clear that in this small part a limit cycle oscillation persists. Thus, the total system may be imagined as forming a diffusion-coupled field of similar limit cycle oscillators. 

Analytical solutions (e.g., obtained by eigenfunction expansion, Fourier transform, similarity transform, perturbation methods, and the solution of ordinary differential equations for one-dimensional problems) to the conservation equations are of great interest, of course, but they can be obtained only under restricted conditions. When the equations can be rendered linear (e.g., when transport of the conserved quantities of interest is dominated by diffusion rather than convection) analytical solutions are often possible, provided the geometry of the domain and the boundary conditions are not too complicated. When the equations are nonlinear, analytical solutions are sometimes possible, again provided the boundary conditions and geometry are relatively simple. Even when the problem is dominated by diffusive transport and the geometry and boundary conditions are simple, nonlinear constitutive behavior can eliminate the possibility of analytical solution. 

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