Regular perturbation solution

Regular Perturbation Solution. To effect an analytical expression for the bubble-flow resistance, we consider fast sorption kinetics or equivalently, small deviations from equilibrium surfactant coverage making 0 large. Hence, a regular perturbation expansion is performed in 1/0 about the constant-tension case. The resulting equations for and rf are to zero and first order in 1/0 (21) ... 

As usual in asymptotic problems, we first seek a regular perturbation solution. This means that we seek a solution in the form... 

Can this be done by a regular perturbation solution, or is it necessary to use the method of matched asymptotic expansions Calculate the torque to include the first inertial correction. 

The asymptotic solution (6.15) differs only slightly from the base solution, which is the solution when = 0. This is the regular perturbation solution. The asymptotic solution (6.20) deviates substantially from the base solution it is the singular perturbation solution. 

The first term in both Equations 17 and 18 is the constant surface-tension contribution and the second term gives the first-order contribution resulting from the presence of a soluble surfactant with finite sorption kinetics. A linear dependence on the surfactant elasticity number arises because only the first-order term in the regular perturbation expansion has been evaluated. The thin film thickness deviates negatively by only one percent from the constant-tension solution when E = 1, whereas the pressure drop across the bubble is significantly greater than the constant-tension value when E - 1. 

Figure 8 reveals that the few data available for surfactant-laden bubbles do confirm the capillary-number dependence of the proposed theory in Equation 18. Careful examination of Figure 8, however, reveals that the regular perturbation analysis carried out to the linear dependence on the elasticity number is not adequate. More significant deviations are evident that cannot be predicted using only the linear term, especially for the SDBS surfactant. Clearly, more data are needed over wide ranges of capillary number and tube radius and for several more surfactant systems. Further, it will be necessary to obtain independent measurements of the surfactant properties that constitute the elasticity number before an adequate test of theory can be made. Finally, it is quite apparent that a more general solution of Equations 6 and 7 is needed, which is not restricted to small deviations of surfactant adsorption from equilibrium. 

Equation (2.2) is said to be a regular perturbation problem. Notice that in the limiting case, as e —> 0, the regular perturbation problem reduces to the original problem (2.1). Intuitively, the solution of the regular perturbation problem should not differ significantly from that of the unperturbed problem. For example, for n = 1, m = 0, the solution of Equation (2.2) is of the form... 

The solution (2.3) is known as a regular perturbation expansion, Xio(f) is the solution of the original problem (2.1), and the higher-order terms xi i(t),... are determined successively by substituting the regular expansion (2.3) into the original differential equation (2.1) (Haberman 1998). 

The above equation is in the form of Equation (2.2), with the presence of the leak constituting a regular perturbation to the system dynamics. It is therefore to be expected that the solutions in the two cases differ by a small, 0(e) quantity. 

Here, we explain the origin of the term singular. It is used in contrast with the expression regular. In general, regular perturbation theory presupposes that the solution obtained by setting s = 0 resembles the one for a small and positive E. However, for Eq. (4), there exist no solutions at all for arbitrary initial conditions when we set s = 0. 

We could obtain a general solution of Eq. (4-94). However, there is no obvious way to apply the boundary condition at the channel wall, at least in the general form (4-95) or (4-96). The method of domain perturbations provides an approximate way to solve this problem for e << 1. The basic idea is to replace the exact boundary condition, (4-96), with an approximate boundary condition that is asymptotically equivalent for e 1 but now applied at the coordinate surface y = d/2. The method of domain perturbations leads to a regular perturbation expansion in the small parameter s. 

Unlike the case of a regular perturbation expansion, we cannot know the gauge functions a priori, and these must be determined as a part of the solution. To obtain governing equations for the unknown concentration functions co and c, we substitute (4-173) into the governing equation, (4-158) (remembering that Da = e ). Now, recalling from Section B that... 

In the analysis of stability that follows, we seek an approximate asymptotic solution of this equation in which the small parameter is the dimensionless magnitude of the initial departure from the equilibrium radius RE. Hence we seek a solution in the form of a regular perturbation expansion ... 

Write the governing equations and boundary conditions in dimensionless form. Solve for the velocity profile in the cone-and-plate flow as a regular perturbation problem for small Reynolds numbers. The solution will be very similar to that of the parallel plate problem. 

Hence we seek an asymptotic solution for the leading-order terms in the thin-film approximation, i.e., u<(>>p<(l>w<(>>, as a regular perturbation in 8, i.e.,... 

Let us suppose, then, that an asymptotic solution exists for Pe <regular perturbation expansion ... 

If fi (Pe) = Pe, as assumed in the regular perturbation expansion (9 25), then, of course, the governing equation for 0 is still (9 26), and the general solution for 0 is still (9 27). However, we do not know a priori what formal and the othcr/ (/L) in (9 32) should take this must be determined as part of the solution of the problem. 

Unlike the regular perturbation expansion discussed earlier, the method of matched asymptotic expansions often leads to a sequence of gauge functions that contain terms like Pe2 In Pe or Pe3 In Pe that are intermediate to simple powers of Pe. Thus, unlike the regular perturbation case, for which the form of the sequence of gauge functions can be anticipated in advance, this is not generally possible when the asymptotic limit is singular In the latter case, the sequence of gauge functions must be determined as a part of the matched asymptotic-solution procedure. 

Now, let us return to the fluid mechanics problem of streaming flow past a solid sphere at small, but nonzero, Reynolds numbers. The objective is to see how the steady-state creeping-flow solution is modified if we consider the nonlinear terms in (7-2) for Re < 1. As we shall see, there is essentially the same problem of trying to obtain a solution of (7-2) in the form of a regular perturbation expansion as was encountered when we tried to obtain such a solution of the thermal problem for IP

[Pg.616]

Following Whitehead s approach, we seek a solution of (9-75) and (9-76) by means of a regular perturbation expansion of the form... 

A similarity solution is available for Eqs. 9.93 to 9.95 subject to negligible macroscopic inertial and viscous forces, that is, small permeabilities and constant DL [115,116]. The inertial and viscous forces are included by Kaviany [117] through the regular perturbation of the similarity solution for plain media, that is, the Nusselt solution [113]. The perturbation parameter used is... 

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