Differential equations, boundary layer

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. 

Heat and Mass Transfer Differential Equations in the Boundary Layer and the Corresponding Analogy 131... 

The rate of thickening of the boundary layer is then obtained by differentiating equation 11.17 ... 

The basic assumption for a mass transport limited model is that diffusion of water vapor thorugh air provides the major resistance to moisture sorption on hygroscopic materials. The boundary conditions for the mass transport limited sorption model are that at the surface of the condensed film the partial pressure of water is given by the vapor pressure above a saturated solution of the salt (Ps) and at the edge of the diffusion boundary layer the vapor pressure is experimentally fixed to be Pc. The problem involves setting up a mass balance and solving the differential equation according to the boundary conditions (see Fig. 10). 

The term in the bracket can be regarded as an equivalent heat of combustion for the more complete problem. If this effect is followed through in the stagnant layer solution of the ordinary differential equations with the more complete boundary condition given by... 

In the modeling of the electrode reaction (2.230) proceeding in a thin-layer cell or within a thin film on the electrode surface, the differential equation (1.2) and (1.3), together with the following boundary conditions (2.231) to (2.233) have to be considered ... 

Equation (E4.4.1) is a nonlinear partial differential equation, because of the velocity u that appears in front of the velocity gradient du/dx. The boundary layer thickness is generally defined as the distance from the plate where the momentum reaches 99% of the free-stream momentum. We will assign (Blasius, 1908)... 

It is apparent from the above sections that the understanding of electrophoretic mobility involves both the phenomena of fluid flow as discussed in Chapter 4 and the double-layer potential as discussed in Chapter 11. In both places we see that theoretical results are dependent on the geometry chosen to describe the boundary conditions of the system under consideration. This continues to be true in discussing electrophoresis, for which these two topics are combined. As was the case in Chapters 4 and 11, solutions to the various differential equations that arise are possible only for rather simple geometries, of which the sphere is preeminent. 

Hiemenz (in 1911) first recognized that the relatively simple analysis for the inviscid flow approaching a stagnation plane could be extended to include a viscous boundary layer [429]. An essential feature of the Hiemenz analysis is that the inviscid flow is relatively unaffected by the viscous interactions near the surface. As far as the inviscid flow is concerned, the thin viscous boundary layer changes the apparent position of the surface. Other than that, the inviscid flow is essentially unperturbed. Thus knowledge of the inviscid-flow solution, which is quite simple, provides boundary conditions for the viscous boundary layer. The inviscid and viscous behavior can be knitted together in a way that reduces the Navier-Stokes equations to a system of ordinary differential equations. 

When the boundary-layer approximations are applicable, the characteristics of the steady-state governing equations change from elliptic to parabolic. This is a huge simplification, leading to efficient computational algorithms. After finite-difference or finite-volume discretization, the resulting problem may be solved numerically by the method of lines as a differential-algebraic system. 

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. 

Following the very brief introduction to the method of lines and differential-algebraic equations, we return to solving the boundary-layer problem for nonreacting flow in a channel (Section 7.4). From the DAE-form discretization illustrated in Fig. 7.4, there are several important things to note. The residual vector F is structured as a two-dimensional matrix (e.g., Fuj represents the residual of the momentum equation at mesh point j). This organizational structure helps with the eventual software implementation. In the Fuj residual note that there are two timelike derivatives, u and p (the prime indicates the timelike z derivative). As anticipated from the earlier discussion, all the boundary conditions are handled as constraints and one is implicit. That is, the Fpj residual does not involve p itself. 

An important issue in the boundary-layer problem, and in differential-algebraic equations generally, is the specification of consistent initial conditions. We think first of the physical problem (not in Von Mises form), since the inlet profiles of u, v, and T, as well as pressure p, must be specified. However, all the initial conditions are not independent, as they would be for a system of standard-form ordinary differential equations. So assuming that the axial velocity u and temperature T profiles are specified, the radial velocity must be required to satisfy certain constraints. 

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 ... 

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