Regular perturbation

Flow of trains of surfactant-laden gas bubbles through capillaries is an important ingredient of foam transport in porous media. To understand the role of surfactants in bubble flow, we present a regular perturbation expansion in large adsorption rates within the low capillary-number, singular perturbation hydrodynamic theory of Bretherton. Upon addition of soluble surfactant to the continuous liquid phase, the pressure drop across the bubble increases with the elasticity number while the deposited thin film thickness decreases slightly with the elasticity number. Both pressure drop and thin film thickness retain their 2/3 power dependence on the capillary number found by Bretherton for surfactant-free bubbles. Comparison of the proposed theory to available and new experimental... 

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) ... 

The main results of our first-order regular perturbation analysis are the expressions for the constant thin film thickness, hQ, and for the total hydrodynamic pressure drop across the entire... 

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. 

The free propagator in (1.8) determines other building blocks and the form of a two-body equation equivalent to the BS equation, and the regular perturbation theory formulae in this case were obtained in [9, 10]. 

Even though the total kernel in (1.23) is unambiguously defined, we still have freedom to choose the zero-order kernel Kq at our convenience, in order to obtain a solvable lowest order approximation. It is not difficult to obtain a regular perturbation theory series for the corrections to the zero-order ap>-proximation corresponding to the difference between the zero-order kernel Kq and the exact kernel Kq + 6K... 

B, is regular in the neighborhood of /c = 0 for some fixed l l0. It should be noted that direct definition of regularity of IIy. is impossible, for IIy is not assumed to be bounded. Also we speak of regular perturbation when Hy is considered as a, perturbed operator to the unperturbed one H0. In particular if Il% belongs to B, the above definition is easily seen to coincide with that given in 1. 2. 

Some examples. In the preceding chapter we have shown that p.m. is really convergent in the case of the regular perturbation. As we already pointed out in Introduction, however, the assumption of regularity is rather restrictive, and the theory is not able to comprise all important applications of p.m., especially in problems... 

This is in remarkable contrast with p.m. of the first kind discussed in Chap. I. There the convergence of the formal series was established under the assumption of regular perturbation, but this is not the case with p.m. of the second kind now under consideration. Even if we assume the simplest case of regular perturbation where... 

Thus the power expansion (16. 9) seems to be valid practically only in the case of bounded perturbation. So we are obliged to regard it as asymptotic expansion even in the case of regular perturbation. In this section we shall assume (17.1) and examine under what conditions the first several coefficients of (16. 9) are significant, and then, discuss the validity of the asymptotic expansion. 

Summing up, the series (16. 9) Is valid when interpreted as an asymptotic expansion at least to the 0-th, order for arbitrary initial state fo, to the fkst order if second order if special case of regular perturbation (17.1) is assumed. 

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. 

In this case, the model of the process with recycle is a regular perturbation of the nominal (no recycle) series system. In light of the concepts introduced in Chapter 2, we can expect that the presence of the (small) material recycle stream will not have a significant impact on the dynamics of the process. 

Moreover, the fact that, at the unit level, the presence of flow rates of vastly different magnitudes is modeled as a regular perturbation, while at the process... 

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. 

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