Asymptotic Solution for

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. 

In Chap. 3, we derived a general exact solution of this problem in terms of Bessel functions Jo for arbitrary R0J and then obtained an approximate form for Rr y 1 by approximating this solution. Instead, in the present section, let us suppose from the outset that R0J 1 and try to seek an approximate solution directly by approximating (4-1) and (4-2). 

Assuming that the scaling in (4-1) is correct, terms that do not contain R0J explicitly are independent of Rm. Thus, in the limit as Rm - 0, Eq. (4-1) reduces to the approximate form 

The solution of this equation subject to the boundary conditions, (4-2), is just 

in the limit R, - 0, the problem reduces to a quasi-steady Poiseuille flow with an instantaneous pressure gradient sin 7. In view of the analysis in Chap. 3, this result is not surprising, but we do note that the solution (4-4) was easier to obtain in this case in which we directly approximated the differential equation rather than first solving the exact problem and then approximating the solution. 

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Here the distortion (diagonal) and back coupling matrix elements in the two-level equations (section B2.2.8.4) are ignored so that = exp(ik.-R) remains an imdistorted plane wave. The asymptotic solution for ij-when compared with the asymptotic boundary condition then provides the Bom elastic ( =f) or inelastic scattering amplitudes... 

In the more general problem in which V (r) 0, the previous boundary condition is not applicable. Thus, B((a) 0 and the asymptotic solution for lttge values of r is given by [Eq. (5-148)]... 

The asymptotic solution for Pe -> oo embodied in Eq. (3-46) can be extended to finite Re in a similar way. The Oseen value for surface vorticity, Cs, predicts little effect of Re. However, the Proudman and Pearson expression for Cs Eq. (3-38), yields ... 

The surface velocities of Abdel-Alim and Hamielec (Al) can be used to obtain Sherwood numbers at intermediate k and Re. An equation which fits these calculated values, the numerical results for Sc oo, and the asymptotic solutions for K < 2 is... 

The following asymptotic solution for large values of (5 was developed by Scriven [6] ... 

LEUNG S ASYMPTOTIC SOLUTIONS FOR VAPOUR PRESSURE SYSTEMS... 

Quantitative Analysis of Spiral Flow (Combined Shear) and Asymptotic Solutions for a Power-Function Fluid... 

Let us first consider the asymptotic solution for this equation. During an accelerated phase of expansion, any physical length exits the Hubble radius at some stage during inflation the ratio k/H decreases with time. Therefore at late times, the evolution equation of the scalar perturbations becomes... 

The assumption of a steady-state profile in the oil laminates and small concentration drops in the water layers may be used to derive asymptotic solutions for the permeation problem. It may be shown that (2) for P P y and t[Pg.36]

This review concentrates on John Albery s work in the field of colloidal semiconductor photoelectrochemistry. John s major contributions to this area, as in so many others, have been through his astounding facility for generating useful asymptotic solutions for highly complex kinetic models of electrochemical systems. So as to put John s work in colloidal photoelectrochemistry into context. Sections 9.1-9.3 of this chapter provide a review of the more salient kinetic models of semiconductor photocatalysis developed over the last 20 years or so. Section 9.4 then concentrates on the Alberian view and presents, for the first time, John s model of the chronoamperometric behaviour of colloidal CdS. 

Laminar, spinning disk JcVdid, Nsh = D =0 8791V V IVfle < 104 [T] Asymptotic solution for large NSc. u = cojdisk/2, = rotational speed, rad/s. Rotating disks are often used in electrochemical research. [101] p. 60 [138] p. 240... 

Here, the asymptotic solution is a neutrally stable solution in Fig. 4.7-as one would expect. However, one also notices a constantly growing wave-front ahead of the asymptotic solution. For this case also, there are three modes with the leading mode neutrally stable and the other two modes are highly stable, given by oi = (0.3498239,0.0) 02 = (0.2149177,0.1454643) and 03 = (0.1604025,0.2593028). 

The tunneling mechanism only applies at low temperatures when the electrons and hole are immobile. The luminescence and LESR decay more rapidly above about 50 K. Hong, Noolandi and Street (1981) solved the complicated time-dependent diflfusion equation for geminate recombination when there is a distribution of thermalization distances and temperature-dependent multiple hopping of carriers. The asymptotic solution for the liuninescence intensity is... 

Monchick and Adrian [11] obtained an asymptotic solution for the polarization under the condition of slow STo mixing [12] ... 

Through Sects. (4.1.1) to (4.1.3) we have derived the asymptotic solutions for the respective models. In spite of the limiting appearance of these solu-... 

As discussed in Section 1, Rhee and Amundson [3] have shown that we can obtain an asymptotic solution for the concentration profile in the shock layer. This ... 

A. PULSATILE FLOW IN A CIRCULAR TUBE REVISITED - ASYMPTOTIC SOLUTIONS FOR HIGH AND LOW FREQUENCIES... 

We saw in the previous example lor R, <approximate solution, (4-4), which we obtained by taking the limit R, —> 0 in the exact equation, (4-1), was just the first term in an asymptotic solution for Rr approximate equation and solution (4-17) that we obtain by taking the limit Rm -> oo in (4-16) will also be the first term in a formal asymptotic expansion of H for Rm y>> 1. Assuming this expansion is regular, it will take the form... 

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