Initial and boundary condition problem

In a surfactant mixture the initial conditions, for each component, are equivalent to those for a single surfactant system. When solving the given initial and boundary condition problem the result is Eq. (4.1). The derivation of the solution was performed using Green s functions (Ward Tordai 1946, Petrov Miller 1977) or by the Laplace operator method (Hansen 1961). Appendix 4E demonstrates the application of the operator method for solving such types of transport problems. 

Despite claims often made by one source or another, there is no well-agreed explanation for how this phenomenon occurs. Actually, up to now explanations have just been guesswork about what is happening during the Mpemba effect with focusing mainly on extrinsic factors. Little attention [14, 15] has yet been paid to the nature of the heat source or mechanism behind the entire source-path-drain cycling system. A combination of formulization of measurements and a solution to the one-dimensional, nonlinear Fourier s initial and boundary condition problem using the finite element method with examination of all possible parameters enabled the solution to this mystery. 

The equilibrium problem for a plate is formulated as some variational inequality. In this case equations (3.92)-(3.94) hold, generally speaking, only in the distribution sense. Alongside (3.95), other boundary conditions hold on the boundary F the form of these conditions is clarified in Section 3.3.3. To derive them, we require the existence of a smooth solution to the variational inequality in question. On the other hand, if we assume that a solution to (3.92)-(3.94) is sufficiently smooth, then the variational inequality is a consequence of equations (3.92)-(3.94) and the initial and boundary conditions. All these questions are discussed in Section 3.3.3. In Section 3.3.2 we prove an existence theorem for a solution to the variational equation and in Section 3.3.4 we establish some enhanced regularity properties for the solution near F. ... 

It is noteworthy that the original equilibrium problem for a plate with a crack can be stated twofold. On the one hand, it may be formulated as variational inequality (3.98). In this case all the above-derived boundary conditions are formal consequences of such a statement under the supposition of sufficient smoothness of a solution. On the other hand, the problem may be formulated as equations (3.92)-(3.94) given initial and boundary conditions (3.95)-(3.97) and (3.118)-(3.122). Furthermore, if we assume that a solution is sufficiently smooth then from (3.92)-(3.97) and (3.118)-(3.122) we can derive variational inequality (3.98). 

Similar equations may be developed for other geometries such as spheres and cylinders. To complete the mathematical representation of a problem, initial and boundary conditions are specified. 

Optimization of a distributed parameter system can be posed in various ways. An example is a packed, tubular reactor with radial diffusion. Assume a single reversible reaction takes place. To set up the problem as a nonlinear programming problem, write the appropriate balances (constraints) including initial and boundary conditions using the following notation ... 

For a given mass transfer problem, the above conservation equations must be complemented with the applicable initial and boundary conditions. The problem of finding the mathematical function that represents the behaviour of the system (defined by the conservation equations and the appropriate set of initial and boundary conditions), is known as a boundary value problem . The boundary conditions specifically depend on the nature of the physicochemical processes in which the considered component is involved. Various classes of boundary conditions, resulting from various types of interfacial processes, will appear in the remainder of this chapter and Chapters 4 and 10. Here, we will discuss some simple boundary conditions using examples of the diffusion of a certain species taken up by an organism ... 

The diffusion-reaction problem is set by the following partial derivative equations accompanied by a set of initial and boundary conditions. 

Because of the possible wide differences among properties and characteristics of solid phases and the varied chemical compositions of contaminants or a contaminant leachate, field measurement variables present average properties over a large volume/area. The problem which complicates the picture is that ideal models are applied to a material or space which is highly non-ideal, non-uniform, and does not permit easy specification or identification of both initial and boundary conditions. To avoid this discrepancy, field and laboratory methods should be developed or modified to complement one another. Thus, ideal theory needs to be supported with physical evidence if rational applications to field studies and environmental simulation are desired. 

Figure 2. Description of the initial and boundary conditions for the hydrogen diffusion problem in the pipeline. The parameter / denotes hydrogen flux and C,(P) is normal interstitial lattice site hydrogen concentration at the inner wall-surface of the pipeline in equilibrium with the hydrogen gas pressure P as it increases to 15 MPa in 1 sec. At time zero, the material is hydrogen free,...

Figure 5. Description of (a) boundary conditions for the elastoplastic problem and (b) initial and boundary conditions for the hydrogen diffusion problem at the blunting crack tip in the MBL formulation. The parameter bCl denotes the crack tip opening displacement in the absence of hydrogen. The parameter C, (P) denotes NILS hydrogen concentration on the crack face in equilibrium with hydrogen gas pressure P. and / is hydrogen flux.

Equation 3-10 is the most basic diffusion equation to be solved, and has been solved analytically for many different initial and boundary conditions. Many other more complicated diffusion problems (such as three-dimensional diffusion with spherical symmetry, diffusion for time-dependent diffusivity, and... 

The above derivation has not made use of the initial and boundary conditions yet, and shows only that A may take any constant value. The value of A can be constrained by boundary conditions to be discrete Ai, A2,..., as can be seen in the specific problem below. Because each function corresponding to given A is a solution to the diffusion equation, based on the principle of superposition, any linear combination of these functions is also a solution. Hence, the general solution for the given boundary conditions is... 

Because D increases with increasing temperature (the Arrhenius equation 1-73), time-dependent D is often encountered in geology because an igneous rock may have cooled down from a high temperature, or metamorphic rock may have experienced a complicated thermal history. If the initial and boundary conditions are simple and if D depends only on time, the diffusion problem is easy to deal with. Because D is independent of x. Equation 3-9 can be written as... 

Many diffusion problems cannot be solved anal3dically, such as concentration-dependent D, complicated initial and boundary conditions, and irregular boundary shape. In these cases, numerical methods can be used to solve the diffusion equation (Press et al., 1992). There are many different numerical algorithms to solve a diffusion equation. This section gives a very brief introduction to the finite difference method. In this method, the differentials are replaced by the finite differences ... 

The solution for the zero initial and boundary conditions may be found from Carslaw and Jaeger (1959, p. 242). The nonzero initial condition may be treated the same way as for the case of plane sheets. The procedure for simplifying the problem is as follows. First, let w = rC then Equation 5-95a becomes... 

In order to complete the problem, the initial and boundary conditions must be given. The temperature and degree of cine or crystallinity must initially (at time zero) be specified at every point inside the composite and the mandrel. For the latter only the temperature is required. As boundary conditions, the temperatures or heat fluxes at the composite outside diameter and mandrel inner diameter must be specified. 

However, most CFD software programs available to date for simulation of transport phenomena require the user to define the model equations and parameters and specify the initial and boundary conditions in accordance with the program s language and code, often highly specialized. A practical interim solution to the computational problem presented by Equation (46) and its non-Newtonian counterparts is at hand now in the form of software developed by Visimix Ltd. (74) VisiMix 2000 Laminar... 

State the appropriate initial and boundary conditions that are needed to solve the problem in nondimensional form. 

Eqn. (3.4-76) has been solved for a number of initial- and boundary conditions using a variety of techniques [32]. Application to dispersion problems provides information on the axial Peclet number, defined as ... 

Equation (11) is also applicable as a good, or reasonably good, approximation to a number of techniques classified as d.c. voltammetry , in which the response to a perturbation is measured after a fixed time interval, tm. The diffusion layer thickness, 5/, will be a function of D, and tm and the nature of this function has to be deduced from the rigorous solution of the diffusion problem in combination with the appropriate initial and boundary conditions [21—23]. The best known example is d.c. polarography [11], where the d.c. current is measured at the dropping mercury electrode at a fixed time, tm, after the birth of a new drop as a function of the applied d.c. potential. The expressions for 5 pertaining to this and some other techniques are given in Table 1. 

Let us discuss briefly the solution to the stress induced transport problem for short times after loading one side of the specimen at = 0 with species i (q( = 0)). Under the given initial and boundary conditions, the non-local transport term in... 

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

The following example illustrates the method. Consider a one-dimensional diffusion problem with the initial and boundary conditions for the domain 0 < x < L ... 

For the grain-boundary grooving problem, the initial and boundary conditions derive from the initial shape of the surface and Young s equation at the groove notch ... 

Except for some special cases, the presence of a free boundary introduces a nonlinearity, so that only a few exact solutions are known. These are in all cases self-similar solutions, which means that the differential equation and associated initial and boundary conditions can all be expressed in terms of a single independent variable. The problem is thereby reduced... 

---