Boundary layers equations

In a boundary layer equation the mass center is considered with the help of the velocity (u, Uy, u ) and therefore a distribution of the velocity of the mass center is desirable. The diffusion velocity and diffusion factor are determined with regard to velocity v, giving a formula for Vax /x, but not for /ax - x useful approach is offered by Eq. (4.268c), using the artificial multiplication factor (v - ax /... 

To describe the velocity profile in laminar flow, let us consider a hemisphere of radius a, which is mounted on a cylindrical support as shown in Fig. 2 and is rotating in an otherwise undisturbed fluid about its symmetric axis. The fluid domain around the hemisphere may be specified by a set of spherical polar coordinates, r, 8, , where r is the radial distance from the center of the hemisphere, 0 is the meridional angle measured from the axis of rotation, and (j> is the azimuthal angle. The velocity components along the r, 8, and (j> directions, are designated by Vr, V9, and V. It is assumed that the fluid is incompressible with constant properties and the Reynolds number is sufficiently high to permit the application of boundary layer approximation [54], Under these conditions, the laminar boundary layer equations describing the steady-state axisymmetric fluid motion near the spherical surface may be written as ... 

There have been a number of direct solutions in the form of Equation (9.65) for convective combustion. These have been theoretical - exact or integral approximations to the boundary layer equations, or empirical - based on correlations to experimental data. Some examples are listed below ... 

For the case in which the Schmidt number is equal to 1, it can be shown [7] that the conservation equations [in terms of Cl see Eq. (6.17)] can be transposed into the form used for the momentum equation for the boundary layer. Indeed, the transformations are of the same form as the incompressible boundary layer equations developed and solved by Blasius [30], The important difference... 

Elzinga and Banchero (El) use Meksyn s boundary layer equation (M2) for flow around a rigid sphere, with the boundary condition that the interfacial velocity is not zero, to calculate a shift in the boundary-separation ring from an equivalent rigid-sphere location. Their calculated positions are slightly less than their observed shifts but confirm the thesis that these shifts are due to internal circulation. Similar quantitative results are reported by Garner and Tayeban (G7). 

The surface velocities of Haberman and Sayre (HI), when used in the thin concentration boundary layer equation for circulating spheres, Eq. (3-51), yield the mass transfer factors and X d shown in Fig. 9.7 for k <2. For a fluid sphere in creeping flow the relationship between the mass transfer factors is... 

The boundary layer equations for an axisymmetric body, Eqs. (1-55), (10-17), and (10-18) have been solved approximately for arbitrary Sc (L4). For Sc oo the mean value of Sh can be computed from Eq. (10-20). Solutions have also been obtained for Sc oo for some shapes without axial symmetry, e.g., inclined cylinders (S34). Data for nonspherical shapes are shown in Fig. 10.3 for large Rayleigh number. The characteristic length in Sh and Ra is analogous to that used in Chapters 4 and 6 ... 

Given the particular circumstances of the flow in a long, narrow channel, explain the reduction of the governing equations to a boundary-layer form that accommodates the momentum and species development length. Discuss the essential characteristics of the boundary-layer equations, including implications for computational solution. 

It is important to emphasize that the similarity behavior is not the result of approximations or assumptions where certain physical effects have been neglected. Instead, these are situations where the full two-dimensional behavior can be completely represented by a one-dimensional description for special sets of boundary conditions. Of course, in all finite-dimensional systems there are edge effects that violate that similarity behavior. By way of contrast, however, one may consider the difference between the governing equations used here and the boundary-layer equations for flow parallel to a solid surface. The boundary-layer equations are approximations in which certain terms are neglected because they are small compared to other terms. Thus terms are dropped even though they are not exactly zero, whereas here the mathematical reduction is accomplished because certain terms vanish naturally over nearly all of the domain (excluding edge effects). 

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. 

For the boundary-layer equations, where two-dimensional flow is retained, the continuity equation must retain both terms as order-one terms. Otherwise, a purely onedimensional flow would result. Certainly there are situations where one-dimensional flow... 

Given the scaling arguments in the previous sections, the axisymmetric channel-flow boundary-layer equations can be summarized as... 

The boundary-layer equations represent a coupled, nonlinear system of parabolic partial-differential equations. Boundary conditions are required at the channel inlet and at the extremeties of the y domain. (The inlet boundary conditions mathematically play the role of initial conditions, since in these parabolic equations x plays the role of the time-like independent variable.) At the inlet, profiles of the dependent variables (w(y), T(y), and Tt(y)) must be specified. The v(y) profile must also be specified, but as discussed in Section 7.6.1, v(y) cannot be specified independently. When heterogeneous chemistry occurs on a wall the initial species profile Yk (y) must be specified such that the gas-phase composition at the wall is consistent with the surface composition and temperature and the heterogeneous reaction mechanism. The inlet pressure must also be specified. 

The method of lines is a computational technique that is particularly suited for solving coupled systems of parabolic partial-differential equations (PDE). The boundary-layer equations can be solved by the method of lines (MOL), although the task is facilitated considerably by casting the problem in a differential-algebraic setting [13]. As an introductory illustration, consider the heat equation... 

An alternative to the standard-form representation is the differential-algebraic equation (DAE) representation, which is stated in a general form as g(r, y, y). The lower portion of Fig. 7.3 illustrates how the heat equation is cast into the DAE form. The boundary conditions can now appear as algebraic constraints (i.e., they have no time derivatives). For a problems as simple as the heat equations, this residual representation of the boundary conditions is not necessary. However, recall that implicit boundary-condition specification is an important aspect of solving boundary-layer equations. 

In the Von Mises form of the boundary-layer equations, it is r2 that must satisfy the consistent initial condition. Here, however, it is much easier to see that the initial profiles are not independent. Once the mesh in x/r has been set and the inlet profiles of u and T given, the consistent profiles in r2 follow easily from the solution of Eq. 7.60. 

We begin with the general boundary-layer equations (Section 7.2) where the stream function takes the form... 

Write a simulation program to solve the cylindrical-channel boundary-layer equations in primative form. 

Figure 7.9 shows a cylindrical duct whose radius varies as a function of position, R(z). As long as the radius varies smoothly and relatively smoothly, the channel flow may be treated as a boundary-layer problem. Discuss what, if any, changes must be made to the boundary-layer equations and the boundary-condition specifications to solve the variable-area boundary-layer problem. 

Write out the governing system of boundary-layer equations in physical coordinates. 

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. 

These relationships have been used by Spalding in the dimensionless presentation both of theoretical values obtained in his approximate solution of the boundary layer equations (58) and of the experimental data (51, 55, 60). Emmons (3), who has solved the problem of forced convection past a burning liquid plane surface in a more rigorous fashion, shows graphically the rather close correspondence between values obtained from his exact solution and that of Spalding, and between the calculated values for flat plates and the experimental values for spheres. 

C17. Curie, N., The Laminar Boundary Layer Equations. Oxford Univ. Press (Clarendon), London and New York, 1962. 

For the case of two-dimensional wavy film flow, Levich (L9) has shown that Eqs. (4) and (5) reduce to the familiar form of the boundary layer equations ... 

For controller-design purposes, it is practical to derive a state-space realization of the dynamics after the fastest boundary layer (Equation (5.20)) using a coordinate change (5.18) in which the control objectives appear directly. Thus, rather than expressing the dynamics of the system in terms of total holdups, we used... 

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