Boundary layer asymptotic limit

Indeed, the multi-layered model, applied to fiber reinforced composites, presented a basic inconsistency, as it appeared in previous publications17). This was its incompatibility with the assumption that the boundary layer, constituting the mesophase between inclusions and matrix, should extent to a thickness well defined by thermodynamic measurements, yielding jumps in the heat capacity values at the glass-transition temperature region of the composites. By leaving this layer in the first models to extent freely and tend, in an asymptotic manner, to its limiting value of Em, it was allowed to the mesophase layer to extend several times further, than the peel anticipated from thermodynamic measurements, fact which does not happen in its new versions. 

The asymptotic formulae, Eqs. (4-62) and (4-63), predict that K - 0 at the respective limits. However, Sh does not go to zero because the assumption of a thin concentration boundary layer breaks down for extreme values of E,... 

The exact solution of the problem leads to the same expression with a proportionality constant between 3 and 5, depending on the definition of the thickness of the boundary layer. In the following sections, the preceding evaluation procedure is applied to a large number of problems, particularly to complex cases for which limiting solutions can be obtained. As already noted in the introduction, the terms in the transport equations will be replaced by their evaluating expressions multiplied by constants. The undetermined constants will then be determined from solutions available for some asymptotic cases. 

In many respects, similar to the diffusion layer concept, there is that of the hydrodynamic boundary layer, <5H. The concept was due originally to Prandtl [16] and is defined as the region within which all velocity gradients occur. In practice, there has to be a compromise since all flow functions tend to asymptotic limits at infinite distance this is, to some extent, subjective. Thus for the rotating disc electrode, Levich [3] defines 5H as the distance where the radial and tangential velocity components are within 5% of their bulk values, whereas Riddiford [7] takes a figure of 10% (see below). It has been shown that... 

The analyses of Hunt, Liebovich and Richards, 1988 [287] and of Finnigan and Belcher, 2004 [189] divide the flow in the canopy and in the free boundary layer above into a series of layers with essentially different dynamics. The dominant terms in the momentum balance in each layer are determined by a scale analysis and the eventual solution to the flow held is achieved by asymptotically matching solutions for the flow in each layer. The model apphes in the limit that H/L 1. By adopting this limit, Hunt, Liebovich and Richards [287] were able to make the important simplification of calculating the leading order perturbation to the pressure held using potential how theory. This perturbation to the mean pressure, A p x, z), can then be taken to drive the leading order (i.e. 0(II/I.) ]) velocity and shear stress perturbations over the hill. 

Equation 17 gives the asymptotic limit of the boundary layer modulus in the far-downstream region of the fully developed region i.e., for a given value for the wall Pdcldt number, the modulus C /C eventually becomes independent of E and hence Z and... 

Figure 3. Asymptotic limit of boundary layer modulus in fully developed region fsee Equation 17)...

In this chapter, we discuss general concepts about asymptotic methods and illustrate a number of different types of asymptotic methods by considering relatively simple transport or flow problems. We do this by first considering pulsatile flow in a circular tube, for which we have already obtained a formal exact solution in Chap. 3, and show that we can obtain useful information about the high- and low-frequency limits more easily and with more physical insight by using asymptotic methods. Included in this is the concept of a boundary layer in the high-frequency limit. We then go on to consider problems for which no exact solution is available. The problems are chosen to illustrate important physical ideas and also to allow different types of asymptotic methods to be introduced ... 

The constant A cannot be determined from the boundary condition at the wall but must be obtained from the matching requirement that (4-27) reduce to the form of the core solution (4-17) in the region of overlap between the boundary layer and the interior region. Now, any arbitrarily large, but finite, value of Y will fall within the boundary-layer domain on the other hand, the corresponding value of y can be made arbitrarily small in the asymptotic limit R0J - oo. Thus the condition of matching is often expressed in the form... 

In the previous sections we have seen several examples of transport problems that are amenable to analysis by the method of regular perturbation theory. As we shall see later in this book, however, most transport problems require the use of singular-perturbation methods. The high-frequency limit of flow in a tube with a periodic pressure oscillation provided one example, which was illustrative of the most common type of singular-perturbation problem involving a boundary layer near the tube wall. Here we consider another example in which there is a boundary-layer structure that we can analyze by using the method of matched asymptotic expansions. 

This is known as the thermal boundary-layer equation for this problem. Because we have obtained it by taking the limit Pe -> oo in the full thermal energy equation (9-222) with m = 1/3, we recognize that it governs only the first term in an asymptotic expansion similar to (9-202) for this inner region. 

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... 

Although the dependence of the thermal boundary-layer thickness on the independent parameters Re and Pr (or Pe) remains to be determined, we may anticipate that the magnitudes of Re and Pe will determine the relative dimensions of the two boundary layers. If Pe yp Re yp 1, both the momentum and thermal layers will be thin, but it seems likely that the thermal layer will be much the thinner of the two. Likewise, if Pe Pe 1, we can guess that the momentum boundary layer will be thinner than the thermal layer. In the analysis that follows in later sections of this chapter, we consider both of the asymptotic limits Pr —> oc (Pe yy Re y> D and Pr 0 (Re yy> Pe p> 1). We shall see that the relative dimensions of the thermal and momentum layers, previously anticipated on purely heuristic grounds, will play an important and natural role in the theory. 

Apart from the special cases discussed in this section, the thermal boundary-layer equation (11-6) can only be solved analytically for the two asymptotic limits, Pr << I and Pr >> 1. This is the subject of the next two sections. 

In the high-T3 case, we obtain the leading-order term in an asymptotic expansion, for the part of the domain where the momentum boundary-layer scaling is applicable, by taking the limit Pr —oo in (11-6). The result is... 

The problem for the first term in an asymptotic solution for the temperature distribution 9 in the outer part of the thermal layer is thus to solve (11 66) subject to the conditions (11 67a), (11 67c), and (11 7c). Again, we see that the geometry of the body enters implicitly through the function ue (x) only. As in the high-Pr limit, a general solution of (11 66) is possible even for an arbitrary functional form for ue (x ). Before we move forward to obtain this solution, however, a few comments are probably useful about the solution (11 69) for the innermost part of the boundary layer immediately adjacent to the body surface. 

The limit Sc - oo implies that the normal velocity V(x,0) is asymptotically small regardless of the size of B. Thus the change in V(x, 0) is too small to affect the leading-order boundary-layer velocity distributions. Nevertheless, we shall see that the mass transferrate is still changed. The other two limits B -> oo or B - -1 both correspond to V(x, 0) —> oo, which means that the velocity profiles will change and there is an intimate coupling between the momentum and mass transfer equations. 

It is convenient to seek the asymptotics of the function Prandtl numbers starting from Eq. (3.3.3) with the extended variable 77 = C/Pr. As a result, we obtain the equation T"c + /((/Pr)T = 0. As Pr - 0, the argument of the function /(C/Pr) tends to infinity, which corresponds to a constant velocity inside the thermal boundary layer and f(j]) 77. In the other limit case as Pr —F 00, the argument of the function /(C/Pr) tends to zero, which corresponds to the linear approximation of the velocity inside the boundary layer and f rf) O.I66772. Substituting the above-mentioned leading terms of... 

The behavior of the alternate forms of eM/v in the near-wall region of a turbulent boundary layer is shown in Fig. 6.35. The classical Prandtl-Taylor model assumes a sudden change from laminar flow (eM/v = 0) to fully turbulent flow (Eq. 6.173) at y = 10.8. The von Kftrman model [88] allows for the buffer region and interposes Eq. 6.174 between these two regions. The continuous models depart from the fully laminar conditions of the sublayer around y+ = 5 and asymptotically approach limiting values represented by Eq. 6.173. In finite difference calculations, eM/v is allowed to increase until it reaches the value given by Eq. 6.158 and then is either kept constant at this value or diminished by an intermittency factor found experimentally by Klebanoff [92]. 

Numerical solution of the mass transfer equation begins at a small nonzero value of z = Zstart, uot at the inlet where Cp, x, y,z = 0) = Ca, miet for all values of x and y. This is achieved by invoking an asymptotically exact analytical solution for the molar density of reactant A from laminar mass transfer boundary layer theory in the limit of very large Schmidt and Peclet numbers. The boundary layer starting profile is valid under the following condition ... 

Limiting solutions based on pertubatlon methods have also been discussed in the literature. Goddard et al. ( ), Kreuzer and Hoofd (53). and Smith et al. (5j() all used matched asymptotic expansions to develop criteria for reactive boundary layer zones within facilitated transport membranes. These results can also be used to calculate solute fluxes. For systems of Interest, the reaction boundary layer will be negligible and an analysis of this detail is unnecessary. 

When the fluid and the immersed surface are at different temperatures, heat transfer will take place. If the heat transfer rate is small in relation to the thermal capacity of the flowing stream, its temperature will remain substantially constant. The surface may be maintained at a constant temperature, or the heat flux at the surface may be maintained constant or smface conditions may be intermediate between these two limits. Because the temperature gradient will be highest in the vicinity of the smface and the temperature of the fluid stream will be approached asymptotically, a thermal boundary layer may therefore be postulated which covers the region close to the surface and in which the whole of the temperature gradient is assmned to lie. 

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