Boundary layers asymptotic expansion

The comparison of flow conductivity coefficients obtained from Equation (5.76) with their counterparts, found assuming flat boundary surfaces in a thin-layer flow, provides a quantitative estimate for the error involved in ignoring the cui"vature of the layer. For highly viscous flows, the derived pressure potential equation should be solved in conjunction with an energy equation, obtained using an asymptotic expansion similar to the outlined procedure. This derivation is routine and to avoid repetition is not given here. 

Expansions (5.3.16) are expected to give an asymptotic representation of the solution to (5.3.13)-(5.3.14) everywhere, except for a boundary layer of thickness y/e, adjacent to x = 1, where an inner solution of the form ... 

Equations for each of the perturbation functions xu yh Xu Yl are derived by substituting the asymptotic expansions into the initial differential system, by matching terms with the same power in e, and finally by writing the proper initial and boundary layer conditions. The zeroth-order outer approximation is the solution to the system... 

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,... 

Let us now return to the solution of our problem for Rr 1. Although the arguments leading to (4-25) were complex, the resulting equation itself is simple compared with the original Bessel equation. Our objective here is an asymptotic approximation of the solution for the boundary-layer region. In general, we may expect an asymptotic expansion of 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. 

In the boundary-layer region, we also seek a solution in the form of an asymptotic expansion ... 

Of course, the solution (4-181) is only the first approximation in the asymptotic series (4 175). In writing (4-177), we neglected certain smaller terms in the nondimensionalized equation, (4-170), because they were small compared with the terms that we kept. To obtain the governing equation for the second term in the boundary-layer region, we formally substitute the expansion, (4-175), into the governing equation, (4-170) ... 

The main point here is that the solution procedure for this particular problem of a singular (or matched) asymptotic expansion follows a very generic routine. Given that there are two sub-domains in the solution domain, which overlap so that matching is possible (the sub-domains here are the core and the boundary-layer regions), the solution of a singular perturbation problem usually proceeds sequentially back and forth as we add higher order... 

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 should be remembered that (9-225) and its solution (9-240) represent only the first term in an asymptotic expansion for Pe - oo in the boundary-layer regime, and (9-203) and its solution (9-207) is the leading-order term in a corresponding expansion for the outer region. To obtain the next level of approximation, it is necessary to calculate an additional term in both of these expansions. We do not pursue this calculation here because of the... 

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... 

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 analysis in this chapter largely follows the original developments of J. D. Goddard and A. Acrivos, Asymptotic expansions for laminar forced-convection heat and mass transfer, Part 2, Boundary-layer flows, J. Fluid Mech. 24, 339-66 (1966). 

Diffusion boundary layer approximation. Now let us take into account the fact that common fluids are characterized by large Schmidt numbers Sc. Obviously, by substituting the leading term of the expansion of v as z -4 0 into (3.2.8) and (3.2.9), one can readily obtain the asymptotics of these formulas as Sc -4 oo. By using (3.2.5) and (3.2.8) and carrying out some transformation, we obtain the dimensionless concentration... 

In the mass exchange problem for a circular cylinder freely suspended in linear shear flow, no diffusion boundary layer is formed as Pe - oo near the surface of the cylinder. The concentration distribution is sought in the form of a regular asymptotic expansion (4.8.12) in negative powers of the Peclet number. The mean Sherwood number remains finite as Pe - oo. This is due to the fact that mass and heat transfer to the cylinder is blocked by the region of closed circulation. As a result, mass and heat transfer to the surface is mainly determined by molecular diffusion in the direction orthogonal to the streamlines. In this case, the concentration is constant on each streamline (but is different on different streamlines). 

The leading term in the comparable asymptotic expansion for large Peclet numbers, i.e., Pe -> 00, obtained by thermal boundary layer arguments (LlOa, Fll, Ala), is... 

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. 

Rewritting (6) under the dimensionless formulation, the small quantity 1/Re appears on the right hand-side, and the contribution of viscous stresses may be considered as a small perturbation to an ideal fluid motion (in ideal flows right hand-side of (6) is zero). The boundary layer equations are obtained as velocity asymptotic expansions which must satisfy the perturbed equation (Table 3). Physically, this is equivalent to say that the velocity gradient in the flow direction is very small compared with the normal one, and that the normal velocity component is much smaller than the axial one. It can be shown that the ratio of the transverse to longitudinal velocities is about Re and that the boundary layer thickness varies as Re" (Fig. 17). Such considerations may be applied to temperature and concentration profiles and lead to the so called Thermal boundary layer and Diffusion boundary layer . According to similitude laws, Pr, Le, Sc numbers allows a comparison to be made of these different layer thicknesses ... 

The zero order asymptotic solution (Eq. 6.53) agrees reasonably well with the exact solution over most of the domain [0,1] except close to the origin. The first order asymptotic expansion (Eq. 6.55), even though it agrees with the exact solution better, also breaks down near the origin. This suggests that a boundary layer solution must be constructed near the origin to take care of this nonuniformity. We will come back to this point later. 

Construction of an Asymptotic Expansion for the Parabolic Problem Other Problems with Corner Boundary Layers Nonisothermal Fast Chemical Reactions Contrast Structures in Partial Differential Equations A. Step-Type Solutions in the Noncritical Case Step-Type Solutions in the Critical Case Spike-Type Solutions Applications... 

The asymptotic expansion of the solution of problem (6.1), (6.2) consists of regular and boundary layer parts... 

The regular part of the asymptotic expansion does not generally satisfy the boundary condition (6.2). In the terminology of the paper of Vishik and Lyusternik [27], the regular terms of the asymptotics introduce a discrepancy into the boundary condition. The purpose of the boundary layer functions Il,(p, is to compensate for this discrepancy. The equality (6.5) shows that the boundary layer functions together with the regular terms must satisfy the boundary condition (6.2). 

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