Infinite domains

One nice feature of the finite element method is the use of natural boundaiy conditions. It may be possible to solve the problem on a domain that is shorter than needed to reach some limiting condition (such as at an outflow boundaiy). The externally applied flux is still apphed at the shorter domain, and the solution inside the truncated domain is still vahd. Examples are given in Refs. 67 and 107. The effect of this is to allow solutions in domains that are smaller, thus saving computation time and permitting the solution in semi-infinite domains. 

Up to this point we have assumed a fluid occupying an infinite domain. For atmospheric applications a boundary at z = 0, the earth, is present. 

Unlike chemistry, mathematics often deals with infinite domains, and infinite axiom sets. If we allow the fact that two axioms infer the same conclusion to increase the truth value of that conclusion, we must choose some increment that reflects the importance of each individual axiom. If there are an infinite number of such axioms, then each axiom becomes infinitesimally important. Thus LT logic chooses to err on the side of conservatism, assuring that the conclusions will be valid, though perhaps less strong than they could actually be. 

Consider traveling wave soiutions u(x,t) = u ), where = x-Dt)l4e- The corresponding boundary vaiue probiem in the infinite domain takes the form... 

Here f f( ) is the electric field due to a single particle fc in an infinite domain and is the electric field induced by all other particles in the system (or external sources) at the surface of the particle k. Obviously, we expect that there should be no net electric force acting, in the absence of external sources, on a single charged particle in an infinite domain. We thus expect that... 

In these variables there is only one parameter, r,. Therefore, for a given value of the parameter, the equation can be solved once and for all. Furthermore the entire family of solutions can be determined as a function of the single parameter. Consequently the nondimensional formulation has offered a potentially enormous reduction of the problem. In Chapter 6 we make extensive use of this form of nondimensionalization for problems on semi-infinite domains. 

A principal assumption for similarity is that there exists a viscous boundary layer in which the temperature and species composition depend on only one independent variable. The velocity distribution, however, may be two- or even three-dimensional, although in a very special way that requires some scaled velocities to have only one-dimensional content. The fact that there is only one independent variable implies an infinite domain in directions orthogonal to the remaining independent variable. Of course, no real problems have infinite extent. Therefore to be of practical value, it is important that there be real situations for which the assumptions are sufficiently valid. Essentially the assumptions are valid in situations where the viscous boundary-layer thickness is small relative to the lateral extent of the problem. There will always be regions where edge effects interrupt the similarity. The following section provides some physical evidence that supports the notion that there are situations in which the stagnation-flow assumptions are valid. 

The steady-state stagnation-flow equations represent a boundary-value problem. The momentum, energy, and species equations are second order while the continuity equation is first order. Although the details of boundary-condition specification depend in the particular problem, there are some common characteristics. The second-order equations demand some independent information about V,W,T and Yk at both ends of the z domain. The first-order continuity equation requires information about u on one boundary. As developed in the following sections, we consider both finite and semi-infinite domains. In the case of a semi-infinite domain, the pressure term kr can be determined from an outer potential flow. In the case of a finite domain where u is known on both boundaries, Ar is determined as an eigenvalue of the problem. 

Fig. 6. 5 A stencil that illustrates the finite-difference discretization of the semi-infinite-domain axisymmetric stagnation flow problem.

Superposition of two displaced step-function initial conditions permits solutions that describe diffusion from an initially localized source into an infinite domain. The two step-function initial conditions in Fig. 4.4 have error-function solutions (Eq. 4.31), and their superposition is a localized source of width Ax. The two step functions are... 

Figure 4.5 One-dimensional diffusion into an infinite domain, (a) Point source diffusing into a line, (b) Line source diffusing into a plane, (c) Planar source diffusing into a volume.

The form of the solution for one-dimensional diffusion is illustrated in Fig. 5.3. The solution c(x,t) is symmetric about x = 0 (i.e., c(x,t) = c(—x,t)). Because the flux at this location always vanishes, no material passes from one side of the plane to the other and therefore the two sides of the solution are independent. Thus the general form of the solution for the infinite domain is also valid for the semi-infinite domain (0 < x < oo) with an initial thin source of diffusant at x = 0. However, in the semi-infinite case, the initial thin source diffuses into one side rather than two and the concentration is therefore larger by a factor of two, so that... 

The Laplace transform method is a powerful technique for solving a variety of partial-differential equations, particularly time-dependent boundary condition problems and problems on the semi-infinite domain. After a Laplace transform is performed on the original boundary-value problem, the transformed equation is often easily solved. The transformed solution is then back-transformed to obtain the desired solution. 

Solve the following boundary value problem on the semi-infinite domain with discontinuous initial conditions,... 

The balance equations used to model polymer processes have, for the most part, first order derivatives in time, related with transient problems, and first and second order derivatives in space, related with convection and diffusive problems, respectively. Let us take the heat equation over an infinite domain as... 

The coupled diffusion equations (6) together with the boundary conditions (9) and (10) can be solved in close form in the Laplace transform space, and numerically inverted to the time domain. At early time, the solution behaves according to the solution for a semi-infinite domain. At large time, the solution... 

In the strict, mathematical sense, fronts (or traveling waves) exist only in an infinite domain. In a finite domain, the moving interfaces are transient phenomena. However, for practical purposes, we can still identify moving interfaces as fronts as long as the domain is much larger than the width of the interface. 

One-dimensional (ID) heterogeneous non-reactive contaminant transport in a semi-infinite domain, described by the ADE is... 

An orthogonal collocation method for elliptic partial differential equations is presented and used to solve the equations resulting from a two-phase two-dimensional description of a packed bed. Comparisons are made between the computational results and experimental results obtained from earlier work. Some qualitative discrimination between rival correlations for the two-phase model parameters is possible on the basis of these comparisons. The validity of the numerical method is shown by applying it to a one-phase packed-bed model for which an analytical solution is available problems arising from a discontinuity in the wall boundary condition and from the semi-infinite domain of the differential operator are discussed. 

Transformation of the infinite domain. There are several ways of dealing with an infinite domain. Guertin et al. (21) chose perturbation solutions of the model as basis functions. 

The use of standard piecewise polynomials, together with an appropriate transformation of the infinite domain, overcomes these difficulties, provided some care is taken in the transformation. ... 

Calculating the first visit flux is thus relatively simple given the simple geometry of the system. Furthermore, charge and hydrodyanmic effects can be minimized by taking b, which is a computational parameter, to be large enough. The reaction probability is obtained from Brownian dynamics, but can be computed only for a finite domain. We discuss the methods to extrapolate the finite domain value to an infinite domain next. 

In the second method, we consider the balance of fluxes at the initiation surface (r=b) for a finite domain and an infinite domain to obtain the correction formula. In the case of an infinite domain, the average first visit flux to the initiation surface, is split into the average reaction flux, and... 

Finally, combining equation (33) with equation (35) and simplifying, we obtain the expression for the infinite domain reaction flux as... 

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