Analytic approximations asymptotic solutions

When considering analytic description, asymptotically optimal estimates are of importance. Asymptotically optimal estimates assume infinite duration of the observation process for fjv —> oo. For these estimates an additional condition for amplitude of a leap is superimposed The amplitude is assumed to be equal to the difference between asymptotic and initial values of approximating function a = <2(0, xo) — <2(oc,Xq). The only moment of abrupt change of the function should be determined. In such an approach the required quantity may be obtained by the solution of a system of linear equations and represents a linear estimate of a parameter of the evolution of the process. 

The earliest studies of heat-loss effects in premixed flames were based on analytical approximations to the solution of the equation for energy conservation [35] [39]. Two such approximations that have been sufficiently popular to be presented in books are those of Spalding (see [40]) and of von Karman (see [5]). Later work involved numerical integrations [41]-[43] and, more recently, activation-energy asymptotics [44]-[46]. 

Inside the drop, we require that the velocity and pressure fields be bounded at the origin [which is a singular point for the spherical coordinate system that we will use to solve (7 199)]. Finally, at the drop surface, we must apply the general boundary conditions at a fluid interface from Section L of Chap. 2. However, a complication in using these boundary conditions is that the drop shape is actually unknown (and, thus, so too are the unit normal and tangent vectors n and t and the interface curvature V n). As already noted, we can expect to solve this problem analytically only in circumstances when the shape of the drop is approximately (or exactly) spherical, and, in this case, we can use the method of domain perturbations that was first introduced in Chap. 4. In this procedure, we assume that the shape is nearly spherical, and develop an asymptotic solution that has the solution for a sphere as the first approximation. An obvious question in this case is this When may we expect the shape to actually be approximately spherical ... 

It has been stated repeatedly that the boundary-layer and potential-flow equations apply to only the leading term in an asymptotic expansion of the solution for Re F> 1. This is clear from the fact that we derived both in their respective domains of validity by simply taking the limit Re -= oc in the appropriately nondimensionalized Navier-Stokes equations. Frequently, in the analysis of laminar flow at high Reynolds number, we do not proceed beyond these leading-order approximations because they already contain the most important information a prediction of whether or not the flow will separate and, if not, an analytic approximation for the drag. Nevertheless, the reader may be interested in how we would proceed to the next level of approximation, and this is described briefly in the remainder of this section.13... 

The aim of this chapter is to present the fundamentals of adsorption at liquid interfaces and a selection of techniques, for their experimental investigation. The chapter will summarise the theoretical models that describe the dynamics of adsorption of surfactants, surfactant mixtures, polymers and polymer/surfactant mixtures. Besides analytical solutions, which are in part very complex and difficult to apply, approximate and asymptotic solutions are given and their range of application is demonstrated. For methods like the dynamic drop volume method, the maximum bubble pressure method, and harmonic or transient relaxation methods, specific initial and boundary conditions have to be considered in the theories. The chapter will end with the description of the background of several experimental technique and the discussion of data obtained with different methods. 

Next, we consider the two limiting cases in order to derive compact analytical expressions for Ux. The first is the case that the two pairs of transition points (/ /4) and (t2, t3) are well isolated from each other along the real axis and can be treated separately. Roughly speaking, this corresponds to b2 1 in both the LZ and NT cases. The connection matrix L which connects the coefficients of the asymptotic solutions are z - and is directly related to scattering matrix can be well approximated by... 

The aim of this chapter is to present the fundamentals of adsorption kinetics of surfactants at liquid interfaces. Theoretical models will be summarised to describe the process of adsorption of surfactants and surfactant mixtures. As analytical solutions are either scarcely available or very complex and difficult to apply, also approximate and asymptotic solutions are given and their ranges of application demonstrated. For particular experimental methods specific initial and boundary conditions have to be considered in these theories. In particular for relaxation theories the experimental conditions have to be met in order to quantitatively understand the obtained results. In respect to micellar solutions and the impact of micelles on the adsorption layer dynamics a detailed description on the theoretical basis as well as a selection of representative experiments will follow in Chapter 5. 

FIG. 3.20. Asymptotic and approximate analytical solutions for the maximum temperature rise. — Exact solution,-Asymptotic solution (a < 0.1), - - O - - Approxi-... 

We note that for the spherical case of d = 3, the only approximation in Eq. [139] is use of Eq. [140], Thus, the solution developed below will be almost exact for a sphere, numerically as well as analytically, and we will see that the asymptotic form of the solution exactly matches the spherical Debye-Hiickel potential. For the cylindrical case, however, Eq. [139] can be justified only on numerical grounds since the asymptotic solution should behave as a sum of modified Bessel functions. To put Eq. [139] into a form more amenable to solution, we temporarily change variables from r to <[>0 and introduce the following substitutions... 

So the MCT does not provide an analytical form of the memory function, as other theoretical models do, but it defines a general hierarchical way of building it. Clearly this definition of the memory introduces a self-coupling phenomena in the correlation d5mamics. These coupled equations, (2.47) and (2.48), can be solved using a few asymptotic approximations. The solution of these equations provides an analytical description of the density dynamics, called asymptotic results. 

The cumulative uptake can easily be calculated by numerical integration of Eq. (29). Comparison of the numerical and analytical solutions for a range of nutrient parameters showed very close agreement, indicating that the asymptotic approximations were valid. 

Exact analytical solutions of the coupled equations for simultaneous mass transfer, heat transfer, and chemical reaction cannot be obtained. However, various authors have employed linear approximations (56-57), perturbation techniques (58), or asymptotic approaches (59) to obtain approximate analytical solutions to these equations. Numerical solutions have also been obtained (60-61). Once the solution for the concentration profile has been determined, equation 12.3.98 may be used to determine the temperature profile. The effectiveness factor may also be determined from the concentration profile, using the approach we have... 

The concentration and temperature Tg will, for example, be conditions of reactant concentration and temperature in the bulk gas at some point within a catalytic reactor. Because both c g and Tg will vary with position in a reactor in which there is significant conversion, eqns. (1) and (15) have to be coupled with equations describing the reactor environment (see Sect. 6) for the purpose of commerical reactor design. Because of the nonlinearity of the equations, the problem can only be solved in this form by numerical techniques [5, 6]. However, an approximation may be made which gives an asymptotically exact solution [7] or, alternatively, the exponential function of temperature may be expanded to give equations which can be solved analytically [8, 9]. A convenient solution to the problem may be presented in the form of families of curves for the effectiveness factor as a function of the Thiele modulus. Figure 3 shows these curves for the case of a first-order irreversible reaction occurring in spherical catalyst particles. Two additional independent dimensionless paramters are introduced into the problem and these are defined as... 

Incorporation of the superposition approximation leads inevitably to a closed set of several non-linear integro-differential equations. Their nonlinearity excludes the use of analytical methods, except for several cases of asymptotical automodel-like solutions at long reaction time. The kinetic equations derived are solved mainly by means of computers and this imposes limits on the approximations used. For instance, we could derive the kinetic equations for the A + B — C reaction employing the higher-order superposition approximation with mo = 3,4,... rather than mo = 2 for the Kirkwood one. (How to realize this for the simple reaction A + B —> B will be shown in Chapter 6.) However, even computer calculations involve great practical difficulties due to numerous coordinate variables entering these non-linear partial equations. 

Therefore, analytical estimates of the solution of a set of non-linear kinetic equations based on the superposition approximation confirm reduction of the asymptotic exponent. 

Since the analytic solutions have been obtained, it is interesting to examine the asymptotic behavior of particle motion for the case of near contact between the cylinder and the plane (X 1). The approximate expression for the electrophoretic velocity is given by... 

The mathematical structure of the models is their unifying background systems of nonlinear coupled differential equations with eventually nonlocal terms. Approximate analytic solutions have been calculated for linearized or reduced models, and their asymptotic behaviors have been determined, while various numerical simulations have been performed for the complete models. The structure of the fixed points and their values and stability have been analyzed, and some preliminary correspondence between fixed points and morphological classes of galaxies is evident—for example, the parallelism between low and high gas content with elliptical and spiral galaxies, respectively. 

In order to demonstrate the physical significance of asymjjtotic nonadiabatic transitions and especially the aiialj-tical theory developed an application is made to the resonant collisional excitation transfer between atoms. This presents a basic physical problem in the optical line broadening [25]. The theoretical considerations were mad( b( for< [25, 27, 28, 29, 25. 30] and their basic id( a has bec n verified experimentally [31]. These theoretical treatments assumed the impact parameter method and dealt with the time-dependent coupled differenticil equations imder the common nuclear trajectory approximation. At that time the authors could not find any analytical solutions and solved the coupled differential equations numerically. The results of calculations for the various cross sections agree well with each other and also with experiments, confirming the physical significance of the asymptotic type of transitions by the dipole-dipole interaction. 

Introductory note Most transport and/or fluids problems are not amenable to analysis by classical methods for linear differential equations, either because the equations are nonlinear (or simply too comphcated in the case of the thermal energy equation, which is linear in temperature if natural convection effects can be neglected), or because the solution domain is complicated in shape (or in the case of problems involving a fluid interface having a shape that is a priori unknown). Analytic results can then be achieved only by means of approximations. One approach is to simply discretize the equations in some way and turn on the computer. Another is to use the family of approximations methods known as asymptotic approximations that lead to useful concepts such as boundary layers, etc. This course is about the latter approach. However, it is not just a... 

In the present case, in which the basic governing equation is linear, the asymptotic analysis serves only to simplify the solution procedure, for example, by avoiding the need to deal with Bessel s equation when Rn> 1. Later, however, we shall see that the same basic methods may often allow approximate analytic solutions to be obtained for nonlinear problems, even when no exact solution is possible. 

With a first-order reaction, the governing equation is linear and could thus be solved without any use of scaling or asymptotic methods. However, we could just as easily assume that the reaction rate is second order in c or add other complications that do not so easily allow an exact analytic solution. The point here is to illustrate the idea of the asymptotic approximation technique, which is easily generalizable to all of these problems. 

If the boundaries of the flow domain are not parallel, the magnitude of the primary velocity component must vary as a function of distance in the flow direction. This not only introduces a number of new physical phenomena, as we shall see, but it also means that the Navier-Stokes equations cannot be simplified following the unidirectional flow assumptions of Chap. 3, and exact analytical solutions are no longer possible. In this chapter, we thus consider only a special limiting case, known as the thin-gap limit, in which the distance between the boundaries is small compared with the lateral gap width. In this case, we shall see that we can obtain approximate analytical solutions by using the asymptotic and scaling techniques that were introduced in the preceding chapter. 

Now, the general problem of (5—5)—(5—10) is highly nonlinear and, for an arbitrary occentric cylinder geometry, it can only be solved numerically - i.e., for arbitrary e and X in the range 0 < X < 1.3 However, for Re = 0, an exact analytic solution can be obtained by a coordinate transformation. In addition, for Re / 0, there are two limiting cases for which we can use asymptotic methods to obtain approximate analytic solutions. These are slight eccentricity... 

Ca <very small, and we shall see that we can obtain an approximate analytic solution using the asymptotic method of domain perturbations. 

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