Boundary-Layer Behavior

The general notion of a boundary later is found in many aspects of modeling physical systems. Recognizing boundary-layer behavior can very often lead to important simplifications in the analysis and modeling of such systems. Certainly the analysis and study of fluid mechanics is greatly facilitated by the exploitation of boundary-layer approximations. 

Scaling arguments are used to establish the circumstances where the boundary-layer behavior is valid. These arguments, which are usually made for external flows over surfaces, may be found in many texts on fluid mechanics (e.g., [350]). The essential feature of the boundary-layer approximation is that there is a principal flow direction in which the convective effects significantly dominate the diffusive behavior. As a result the flow-wise diffusion may be neglected, while the cross-flow diffusion and convection are retained. Mathematically this reduction causes the boundary-layer equations to have essentially parabolic characteristics, whereas the Navier-Stokes equations have essentially elliptic characteristics. As a result the computational simulation of the boundary-layer equations is much simpler and more efficient. 

While our primary interest in this text is internal flow, there are certain similarities with the classic aerodynamics-motivated external flows. Broadly speaking, the stagnation flows discussed in Chapter 6 are classified as boundary layers where the outer flow that establishes the stagnation flow has a principal flow direction that is normal to the solid surface. Outside the boundary layer, there is typically an outer region in which viscous effects are negligible. Even in confined flows (e.g., a stagnation-flow chemical-vapor-deposition reactor), it is the existence of an inviscid outer region that is responsible for some of the relatively simple correlations of diffusive behavior in the boundary layer, like heat and mass transfer to the deposition surface. 

It is instructive to study a much simpler mathematical equation that exhibits the essential features of boundary-layer behavior. There is a certain analogy between stiffness in initial-value problems and boundary-layer behavior in steady boundary-value problems. Stiffness occurs when a system of differential equations represents coupled phenomena with vastly different characteristic time scales. In the case of boundary layers, the governing equations involve multiple physical phenomena that occur on vastly different length scales. Consider, for example, the following contrived second-order, linear, boundary-value problem  

As e - 0, the right-hand side of Eq. C.5 is neglected to get the reduced form 

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Boundary-layer behavior is one of several potential simplifications that facilitate channel-flow modeling. Others include plug flow or one-dimensional axial flow. The boundary-layer equations, however, are the ones that require the most insight and effort to derive and to establish the ranges of validity. The boundary-layer equations retain a full two-dimensional representation of all the field variables as well as all the nonlinear behavior of Navier-Stokes equations. Nevertheless, when applicable, they provide a very significant simplification that can be used to great benefit in modeling. 

Fig. C.l Exact solutions for the mathematical prototype for boundary-layer behavior. The solution shown is for varying values of the parameter e and for a = 0.4. Also shown is the solution to the reduced outer equation that does not satisfy the boundary condition at x=0.

Fig. C.2 Comparison of the inner solution with the exact solution for the mathematical prototype equation for boundary-layer behavior.

From a fluid mechanical point of view, we concentrate on viscous behavior in boundary layers. It is often the boundary-layer behavior near a surface that is responsible for important outcomes, like uniform thin-film growth. Quite often the analysis of boundary-layer flows can take advantage of some major mathematical simplifications of the general flow equations. Moreover, and perhaps more important, it is the characteristics of certain boundary layers that are responsible for desirable properties of the process. Unlike much fluid-mechanical literature, which con-... 

Sakiadis, B. C., Boundary-layer behavior on continuous solid surfaces. 2. Boundary layer on a continuous flat surface, AIChE J., Vol. 7, No. 2, pp. 221-225, 1961. 

Uniform Surface Temperature, Air as Coolant. When the boundary layer and coolant gases are the same, the equations controlling boundary layer behavior are Eqs. 6.6-6.8. The mass injection at the surface simply alters the boundary conditions (Eq. 6.9) at the wall to be... 

J. Vleggaar, Laminar Boundary-Layer Behavior on Continuous Accelerating Surfaces," Chem. Eng. Sci., 32, pp. 1517-1525,1977. 

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