Partial differential equations similarity solutions

The methods of calculation of numerical solutions discussed in Chapter 10 for a single component can easily be extended to the case of multicomponent mixtures. In this case, we have n partial differential equations similar to Eqs. 10.78,10.80, or 10.82 (f = 1,2, , )... 

Burns and Curtiss (1972) and Burns et al. (1984) have used the Facsimile program developed at AERE, Harwell to obtain a numerical solution of simultaneous partial differential equations of diffusion kinetics (see Eq. 7.1). In this procedure, the changes in the number of reactant species in concentric shells (spherical or cylindrical) by diffusion and reaction are calculated by a march of steps method. A very similar procedure has been adopted by Pimblott and La Verne (1990 La Verne and Pimblott, 1991). Later, Pimblott et al. (1996) analyzed carefully the relationship between the electron scavenging yield and the time dependence of eh yield through the Laplace transform, an idea first suggested by Balkas et al. (1970). These authors corrected for the artifactual effects of the experiments on eh decay and took into account the more recent data of Chernovitz and Jonah (1988). Their analysis raises the yield of eh at 100 ps to 4.8, in conformity with the value of Sumiyoshi et al. (1985). They also conclude that the time dependence of the eh yield and the yield of electron scavenging conform to each other through Laplace transform, but that neither is predicted correctly by the diffusion-kinetic model of water radiolysis. 

They are in line with the traditional Lie approach to the reduction of partial differential equations, since they exploit symmetry properties of the equation under study in order to construct its invariant solutions. And again, any deviation from the standard Lie approach requires solving overdetermined system of nonlinear determining equations. A more profound analysis of similarities and differences between these approaches can be found elsewhere [33,56,64]. 

More recently Perry and Pigford (P2) made a similar calculation for the absorption of a solute accompanied by a slow second-order reaction, A + B 2C. The problem, which involves the simultaneous solution of three coupled, second-order, partial-differential equations, was worked out by means of an electronic digital computer. 

L. Dresner, Similarity Solutions of Nonlinear Partial Differential Equations, Pitman, Boston, 1983. 

This is a linear parabolic partial differential equation that can be readily solved as soon as boundary conditions are specified. There is a symmetry condition at the centerline, and it is presumed that the mass fraction Yk vanishes at the wall, Yk = 0. It is important to note that it has been implicitly assumed that the velocity profile has been fully developed, such that the similarity solution / is valid. This assumption is analogous to that used in the Graetz problem (Section 4.10). 

We therefore advise that the reader should consult a recent series of papers published by Galvez et al. [171, 172] encompassing all the mechanisms mentioned in Sect. 7.1, elaborated for both d.c. and pulse polarography. The principles of the Galvez method are clearly outlined in the first part of the series [171]. It is similar to the dimensionless parameter method of Koutecky [161], which enables the series solutions for the auxiliary concentration functions cP and cQ exp (kt) and

combined directly with the partial differential equations of the type of eqn. (203). In some of the treatments, the sphericity of the DME is also accounted for. The results are usually visualized by means of predicted polarograms, some examples of which are reproduced in Fig. 38. Naturally, the numerical description of the surface concentrations at fixed potential are also immediately available, in terms of the postulated power series, and the recurrent relationships obtained for the coefficients of these series. 

At present, we see only one route to complete solution of the problem for t as r, for example, up to t = 5r or lOr, we solve the partial differential equations with a given pressure law at the piston continuing the solution for t > r in asymptotic (self-similar) form, we determine the constants A and B from the condition that the two solutions coincide at the extreme point to which the calculation is carried. 

For systems that exhibit slow anomalous transport, the incorporation of external fields is in complete analogy to the existing Brownian framework which itself is included in the fractional formulation for the limit a —> 1 The FFPE (19) combines the linear competition of drift and diffusion of the classical Fokker-Planck equation with the prevalence of a new relaxation pattern. As we are going to show, also the solution methods for fractional equations are similar to the known methods from standard partial differential equations. However, the temporal behavior of systems ruled by fractional dynamics mirrors the self-similar nature of its nonlocal formulation, manifested in the Mittag-Leffler pattern dominating the system equilibration. 

The equilibrium equations of a hyperboloid of revolution used for cooling towers derived by using membrane theory under an arbitrary static normal load are reduced to a single partial differential equation with constant coefficients. The problem of finding displacements is reduced to a similar type of equation so that the solution for this problem becomes straightforward. 11 refs, cited. 

The single-column process (Figure 1) is similar to that of Jones et al. (1). This process is useful for bulk separations. It produces a high pressure product enriched in light components. Local equilibrium models of this process have been described by Turnock and Kadlec (2), Flores Fernandez and Kenney (3), and Hill (4). Various approaches were used including direct numerical solution of partial differential equations, use of a cell model, and use of the method of characteristics. Flores Fernandez and Kenney s work was reported to employ a cell model but no details were given. Equilibrium models predict... 

Thus 1 is the total heat energy required to bring a solid from an initial temperature Tso to Tm and to melt it at that temperature. Sundstrom and Young (33) solved this set of equations numerically after converting the partial differential equations into ordinary differential equations by similarity techniques. Pearson (35) used the same technique to obtain a number of useful solutions to simplified cases. He also used dimensionless variables, which aid in the physical interpretation of the results, as shown below ... 

The above equation is often converted to dimensionless variables and solved. The solution of this partial differential equation is recorded in the literature [Otake and Kunigata, Kagaku Kogaku, 22 144 (1958)]. The plots of E(tr) versus /. are bell-shaped, similar to the response for a series of n CSTRs model (Fig. 19-7). A relation between a2(tr), n, and Pe (for the closed-ends condition) is... 

In the preceding sections, the solution for boundary layer flow over a flat plate wav obtained by reducing the governing set of partial differential equations to a pair of ordinary differential equations. This was possible because the velocity and temperature profiles were similar in the sense that at all values of x, (u u ) and (Tw - T)f(Tw - T > were functions of a single variable, 17, alone. Now, for flow over a flat plate, the freestream velocity, u, is independent of x. The present section is concerned with a discussion of whether there are any flow situations in which the freestream velocity, u 1, varies with Jr and for which similarity solutions can still be found [1],[10]. 

Similarity and integral equation methods for solving the boundary layer equations have been discussed in the previous sections. In the similarity method, it will be recalled, the governing partial differential equations are reduced to a set of ordinary differential equations by means of a suitable transformation. Such solutions can only be obtained for a very limited range of problems. The integral equation method can, basically, be applied to any flow situation. However, the approximations inherent in the method give rise to errors of uncertain magnitude. Many attempts have been made to reduce these errors but this can only be done at the expense of a considerable increase in complexity, and, therefore, in the computational effort required to obtain the solution. 

The fact that the original partial differential equations have been reduced to a pair of ordinary differential equations confirms the assumption that similarity solutions do in fact exist. 

There is an obvious similarity between the equations of the crossflow model and those of the modified mixing-cell model. With suitable redefinition of parameters, it can easily be shown that these partial differential equations are mathematically identical. Thus, the solution for the modified mixing-cell model is identical to the solution for the crossflow model, for the same set of boundary conditions. 

The conditions specified by Eq. (6.206) provide the conditions required to design the model, also called similarity requirements or modeling laws. The same analysis could be carried out for the governing differential equations or the partial differential equation system that characterize the evolution of the phenomenon (the conservation and transfer equations for the momentum). In this case the basic theorem of the similitude can be stipulated as A phenomenon or a group of phenomena which characterizes one process evolution, presents the same time and spatial state for all different scales of the plant only if, in the case of identical dimensionless initial state and boundary conditions, the solution of the dimensionless characteristic equations shows the same values for the internal dimensionless parameters as well as for the dimensionless process exits . 

C What is a similarity variable, aud what is it used for For what kinds of functions can we expect a similarity solution for a set of partial differential equations to exist ... 

A similarity transformation may be used when the solution to a parabolic partial differential equation, written in terms of two independent variables, can be expressed in terms of a new independent variable that is a combination of the original independent variables. The success of this transformation requires that ... 

Solution The method for solving partial differential equations generally involves finding a method to express them as coupled ordinary differential equations. A similarity transformation is possible if c,- can be expressed as a function of only a new variable. This requirement implies that equation (2.48) can be expressed as a function of only the new variable, and that the three conditions (2.49) in time and position can collapse into two conditions in the new variable. 

The basis of the solution of complex heat conduction problems, which go beyond the simple case of steady-state, one-dimensional conduction first mentioned in section 1.1.2, is the differential equation for the temperature field in a quiescent medium. It is known as the equation of conduction of heat or the heat conduction equation. In the following section we will explain how it is derived taking into account the temperature dependence of the material properties and the influence of heat sources. The assumption of constant material properties leads to linear partial differential equations, which will be obtained for different geometries. After an extensive discussion of the boundary conditions, which have to be set and fulfilled in order to solve the heat conduction equation, we will investigate the possibilities for solving the equation with material properties that change with temperature. In the last section we will turn our attention to dimensional analysis or similarity theory, which leads to the definition of the dimensionless numbers relevant for heat conduction. 

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. 

Parabolic partial differential equations are solved using the similarity solution technique in this section. This method involves combining the two independent variables (x and t) as one (rj). For this purpose, the original initial and boundary conditions should become two boundary conditions in the new combined variable (rj). The methodology involves converting the governing equation (PDF) to an ordinary differential equation (ODE) in the combined variable (rj). This variable transformation is very difficult to do by hand. In this chapter, we will show how... 

Similarity Solution Technique for Elliptic Partial Differential Equations... 

Similarity Solution Technique for Nonlinear Partial Differential Equations 341... 

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