Method asymptotic analogies

The Method of Asymptotic Analogies in Theory of Mass and Heat Transfer... 

The basic idea of the method of asymptotic analogies is to use the expression (4.1.5) (or (4.1.6)) to approximate similar characteristics for a wider class of problems describing qualitatively similar phenomena or processes. Specifically, after the relation (4.1.5) has been constructed with the help of (4.1.1) for some specific (say, the simplest) case, we can evaluate w for other problems of this class by finding the asymptotics wo (as r —> 0) and wx (as r -4 oo) and then by substituting these asymptotics into (4.1.5). The approximate formulas thus obtained are asymptotically sharp in both limit cases r -4 0 and r -4 oo. 

To approximate the dependence of the bulk temperature on time, we use the method of asymptotic analogies. The simplest original problem is taken to be the one-dimensional (with respect to spatial coordinates) heat exchange problem for a sphere of radius a. The solution of this problem is well known [277] and results in the following expression for bulk temperature ... 

Following the method of asymptotic analogies, we shall use formula (4.2.7) for the calculation of bulk temperature for nonspherical bodies. To this end, for a body of a given shape, we must first calculate the asymptotics of bulk temperature for small and large f and then substitute these asymptotics into (4.2.7). 

Particles and bubbles. Using the method of asymptotic analogies, we shall derive formulas for the calculation of the Sherwood number in a laminar flow past spherical particles, drops, and bubbles for an arbitrary structure of the nonperturbed flow at infinity. We assume that closed streamlines are lacking in the flow. 

The method of asymptotic analogies (see Section 4.1) permits one to generalize formulas (5.4.2)—(5.4.4) to cavities of an arbitrary shape. In the special case of a first-order volume reaction, we obtain the formula... 

The conceptually simplest approach to solve for the -matrix elements is to require the wavefimction to have the fonn of equation (B3.4.4). supplemented by a bound function which vanishes in the asymptote [32, 33, 34 and 35] This approach is analogous to the fiill configuration-mteraction (Cl) expansion in electronic structure calculations, except that now one is expanding the nuclear wavefimction. While successfiti for intennediate size problems, the resulting matrices are not very sparse because of the use of multiple coordinate systems, so that this type of method is prohibitively expensive for diatom-diatom reactions at high energies. 

Analogous results are obtained for this system, see Table 12. Also in this case the SCF-MI and MCSCF-MI results show a greater stability for what concerns the basis set if compared with the SCF method. The difference between the force constants relative to asymptotic and equilibrium geometries (AK) is positive. The AK for the NH3-HCI complex is greater than that of the H2O-HCI complex. 

To perform the minimization to the free energy (Eq. (3.1)) let us use the variational method with the trial function (2.5). After the calculations, which are analogous to those performed in the Onsager paper7, we arrive at the following results for the boundaries of the region of phase separation in the asymptotic limit... 

Near the point where the two streams first meet the chemical reaction rate is small and a self-similar frozen-flow solution for Yp applies. This frozen solution has been used as the first term in a series expansion [62] or as the first approximation in an iterative approach [64]. An integral method also has been developed [62], in which ordinary differential equations are solved for the streamwise evolution of parameters that characterize profile shapes. The problem also is well suited for application of activation-energy asymptotics, as may be seen by analogy with [65]. The boundary-layer approximation fails in the downstream region of flame spreading unless the burning velocity is small compared with u it may also fail near the point where the temperature bulge develops because of the rapid onset of heat release there,... 

Here lli is the wave function of the < uasiniolcculc when a valence electron is located near the first core (an electron is connected with the first nucleus), and 2 corresponds to electron location near the second nucleus, H is the Hamiltonian of electrons. Note that for an accurate evaluation of this interaction it is necessary to use the accurate wave functions of the (luasimolccule which take into account interaction of a valence electron located between the cores with both cores simultaneously. We assume this to be fulfilled within the framework of the asymptotic theory. Using a general method of calculation of the exchange interaction potential A(i ) by analogy with that for the case a of Hund coupling [3, 17, 20, 21], we... 

We will consider three such parameters, namely the c-availability and the two c-intensities. They correspond to analogous characteristics for two-state systems. The first parameter, denoted A(c, t,z), is the probability that at time t the network state at a point z is greater or equal to c. The remaining two, denoted A+(c,r,z) and A (c,r,z), are the intensities with which the network state crosses the level c from below or above respectively. They will be referred to as transition intensities, where a transition takes place between the states >c and reliability parameters of repairable systems can be derived from the asymptotic intensities. The presented method is illustrated by the calculation of A(c,r,z) for the exemplary network. 

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