Equations with dimensional problems

Eq. (2.4.10b) there also occurs an exactly compensating change in 5 — 5. Thus, the value of S relative to S remains unaffected by the choice of units. However, the numerical values of S and S obviously do depend on the energy unit chosen for Cp and for R. Incidentally, on setting T=T and P = 1 bar it does appear as though Eq. (2.4.10b) were afflicted with dimensional problems so, it is best always to carry the P symbol along. Actually, statistical mechanics furnishes a relation for 5, which is cited as the Sackur-Tetrode equation in Section 1.12. 

In this chapter difference schemes for the simplest time-dependent equations are studied, namely, for the heat conduction equation with one or more spatial variables, the one-dimensional transfer equation and the equation of vibrations of a string. Two-layer and three-layer schemes are designed for the first, second and third boundary-value problems. Stability is investigated by different methods such as the method of separation of variables and the method of energy inequalities as well as by means of the maximum principle. Asymptotic stability of difference schemes is discovered for the heat conduction equation in ascertaining the viability of difference approximations. Finally, stability theory is being used, increasingly, to help us understand a variety of phenomena, so it seems worthwhile to discuss it in full details. 

In this section we consider the one-dimensional heat conduction equation with constant coefficients and difference schemes in order to develop various methods for designing the appropriate difference schemes in the case of time-dependent problems. 

LOS for equations with variable coefficients. One way of covering equations with variable coefficients is connected with possible constructions of locally one-dimensional schemes and the main ideas adopted for problem (15). It sufficies to point out only the necessary changes in the formulas for the operators Lc, and Aq., which will be used in the sequel, and then bear in mind that any locally one-dimensional scheme can always be written in the form (21)-(23). Several examples add interest and help in understanding. 

In principle, the task of solving a linear algebraic systems seems trivial, as with Gauss elimination a solution method exists which allows one to solve a problem of dimension N (i.e. N equations with N unknowns) at a cost of O(N ) elementary operations [85]. Such solution methods which, apart from roundoff errors and machine accuracy, produce an exact solution of an equation system after a predetermined number of operations, are called direct solvers. However, for problems related to the solution of partial differential equations, direct solvers are usually very inefficient Methods such as Gauss elimination do not exploit a special feature of the coefficient matrices of the corresponding linear systems, namely that most of the entries are zero. Such sparse matrices are characteristic of problems originating from the discretization of partial or ordinary differential equations. As an example, consider the discretization of the one-dimensional Poisson equation... 

For the flat one-dimensional problem with the sole coordinate x, the Poisson equation can be written as... 

In this overview we focus on the elastodynamical aspects of the transformation and intentionally exclude phase changes controlled by diffusion of heat or constituent. To emphasize ideas we use a one dimensional model which reduces to a nonlinear wave equation. Following Ericksen (1975) and James (1980), we interpret the behavior of transforming material as associated with the nonconvexity of elastic energy and demonstrate that a simplest initial value problem for the wave equation with a non-monotone stress-strain relation exhibits massive failure of uniqueness associated with the phenomena of nucleation and growth. 

Crystal growth Consider the case for crystal growth along one direction (hence a one-dimensional problem). Define the initial interface to be at x = 0 and the crystal is on the side with negative x (left-hand side) and the melt is on the positive side (Section 3.4.6). Due to crystal growth, the interface advances to the positive side. Define the interface position at time t to be at x = Xq, where Xq > 0 is a function of time. Let w be the mass fraction of the main equilibriumdetermining component then the diffusion equation in the melt is... 

Now consider the case of three-dimensional crystal dissolution. Let the radius of the crystal be a (which depends on time). In this case, the most often-used reference frame is fixed at the center of the crystal, i.e., lab-fixed reference frame (different from the case of one-dimensional crystal growth for which the reference frame is fixed at the interface) so that the problem has spherical symmetry. Ignore melt density variation. The crystal dissolution rate (u ) and melt growth rate at the interface (Ua) are related by the continuity equation with approximation of steady state ... 

The calculation was similar to that made for the problem of three-dimensional heat transmission and uniform heat generation in a rectangular parallelepiped (3) with the surfaces maintained at ambient temperature. By solving the parallelepiped equation with appropriate values, it can be shown that the temperature distribution over the sample cross section is approximately parabolic, with the maximum temperature at the center (7). (Figure 17 gives a graphical presentation of this solution.) This leads to the conclusion that the average temperature is approximately 63.5% of the maximum temperature. 

The solution of (2.3.69) is a purely mathematical problem well known in the theory of diffusion-controlled processes of classical particles. However, a particular form of writing down (2.3.69) allows us to use a certain mathematical analogy of this equation with quantum mechanics. Say, many-dimensional diffusion equation (2.3.69) is an analog to the Schrodinger equation for a system of N spinless particles B, interacting with the central particle A placed... 

A less rigorous approach is to imagine that we really have a one-dimensional problem, except that this one dimension is bent round to make a circle. Schrodinger s equation (with zero potential) can then be written... 

For K = 0, high temperatures, but weak disorder we adopt an alternative method by mapping the (classical) one-dimensional problem onto the Burgers equation with noise [26]. With this approach one can derive an effective correlation length given by... 

Consider a packet of emulsion phase being swept into contact with the heating surface for a certain period. During the contact, the heat is transferred by unsteady-state conduction at the surface until the packet is replaced by a fresh packet as a result of bed circulation, as shown in Fig. 12.6. The heat transfer rate depends on the rate of heating of the packets (or emulsion phase) and on the frequency of their replacement at the surface. To simplify the model, the packet of particles and interstitial gas can be regarded as having the uniform thermal properties of the quiescent bed. The simplest case is represented by the problem of one-dimensional unsteady thermal conduction in a semiinfinite medium. Thus, the governing equation with the boundary conditions and initial condition can be imposed as... 

Consider the infinitely long two-dimensional fin with convective boundary conditions, as shown in Fig. 4.2. The temperature of the surrounding fluid is T and the heat transfer coefficient is h. The x-axis is taken to be the axis of symmetry. The governing equations of the problem may be stated as below. 

The equation is linear if u is a known function. This may be considered to describe a time-dependent one-dimensional heat convection equation for a problem with a known flow field. For a fluid with constant properties in the temperature range considered, the momentum equation is decoupled from the energy equation. In the following example, the... 

We now wish to examine the applications of Fourier s law of heat conduction to calculation of heat flow in some simple one-dimensional systems. Several different physical shapes may fall in the category of one-dimensional systems cylindrical and spherical systems are one-dimensional when the temperature in the body is a function only of radial distance and is independent of azimuth angle or axial distance. In some two-dimensional problems the effect of a second-space coordinate may be so small as to justify its neglect, and the multidimensional heat-flow problem may be approximated with a one-dimensional analysis. In these cases the differential equations are simplified, and we are led to a much easier solution as a result of this simplification. 

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