Boundary layers exact solutions

The determination of the constants Cs, t, and is a complicated problem requiring a complete solution of the Boltzmann equation, including the kinetic boundary layer. Exact solutions have been found only for certain modeled Boltzmann equations, like the BGK equation, in flows with a simple geometry (e-g- stationary shear flow along a flat plate in a semi-infinite space, the so-called Kramers problem). Approximate results have been obtained by using variational methods and moment expansions. ... 

In finite boundary conditions the solute molecule is surrounded by a finite layer of explicit solvent. The missing bulk solvent is modeled by some form of boundary potential at the vacuum/solvent interface. A host of such potentials have been proposed, from the simple spherical half-harmonic potential, which models a hydrophobic container [22], to stochastic boundary conditions [23], which surround the finite system with shells of particles obeying simplified dynamics, and finally to the Beglov and Roux spherical solvent boundary potential [24], which approximates the exact potential of mean force due to the bulk solvent by a superposition of physically motivated tenns. 

Let us recapitulate. We have achieved a solution to boundary-layer-like burning of a steady liquid-like fuel. A thin flame or fast chemistry relative to the mixing of fuel and oxygen is assumed. All effects of radiation have been ignored - a serious omission for flames of any considerable thickness. This radiation issue cannot easily be resolved exactly, but we will return to a way to include its effects approximately. 

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 ... 

Figure 5. Exact (numerical solution, continuous line) and linearised (equation (24), dotted line) velocity profile (i.e. vy of the fluid at different distances x from the surface) at y = 10-5 m in the case of laminar flow parallel to an active plane (Section 4.1). Parameters Dt = 10 9m2 s-1, v = 10-3ms-1, and v = 10-6m2s-1. The hydrodynamic boundary layer thickness (<50 = 5 x 10 4 m), equation (26), where 99% of v is reached is shown with a horizontal double arrow line. For comparison, the normalised concentration profile of species i, ct/ithe linear profile of the diffusion layer approach (continuous line) and its thickness (<5, = 3 x 10 5m, equation (34)) have been added. Notice that the linearisation of the exact velocity profile requires that <5,

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. 

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.

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. 

PL Donoughe and N B Livingood. Exact Solutions of Laminar Boundary-layer Equations with Constant Property Values for Porous Wall with Variable Temperature. NASA Technical Report 1229,1958. 

This is the momentum equation of the laminar boundary layer with constant properties. The equation may be solved exactly for many boundary conditions, and the reader is referred to the treatise by Schlichting ll] for details of the various methods employed in the solutions. In Appendix B we have included the classical method for obtaining an exact solution to Eq. (5-13) for laminar flow over a flat plate. For the development in this chapter we shall be satisfied with an approximate analysis which furnishes an easier solution without a loss in physical understanding of the processes involved. The approximate method is due to von Karman [2],... 

The exact solution of the boundary-layer equations as given in Appendix B yields... 

EXACT SOLUTIONS OF LAMINAR-BOUNDARY-LAYER EQUATIONS... 

Boundary-layer theory has been applied to solve the heat-transfer problem in forced convection laminar flow along a heated plate. The method is described in detail in numerous textbooks (El, G5, S3). Some exact solutions and approximate solutions are also obtained (B2, S3). 

It will further be shown that the exact solution of the boundary layer equation yields a value of 0.332 which is only very slightly different from the value 0.331 obtained here. 

The boundary-layer theory used here to correct observed rejection coefficients is an improvement over thin-film theory, but it ag ears limited to filtrate velocities, J, below about 0.5 X 10 cm/sec for highly rejected solutes. Xn exact theory for incomplete rejection by hollow fibers is needed to define the validity of Equation 23, over the range of conditions of the experiments. 

The parameter, D, calculated is probably strongly related to a true protein diffusivity, but an exact boundary layer theory for protein solutions is needed to accurately establish that relationship. However, irrespective of a theoretical explanation, the observed Independence of graphs of J versus P from axial velocity and fiber length is a new region of Xhe protein ultrafiltration process that should be investigated further. 

With a few exceptions, the fluid flow must be simulated before the mass-transfer simulations can be rigorously performed. Nevertheless, here are several important situations, such as that at a rotating disk electrode, where the fluid flow is known analytically or from an exact, numerical solution. Thus there exists a body of work that was done before CFD was a readily available tool (for example, see Refs. 34-37). In many of these studies, a boundary-layer analysis, based on a Lighthill transformation (Ref. 1, Chapter 17), is employed. 

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 ... 

Hartree18 also obtained a family of solutions for f3 between 0 and —0.1988 that were physically acceptable in the sense that 1 from below as i] —> oo. Several such profiles are sketched in Fig. 10-7. These correspond to the boundary layer downstream of the corner in Fig. 10-6(b) (assuming that the upstream surface is either a slip surface or is short enough that one can neglect any boundary layer that forms on this surface). It should be noted that solutions of the Falkner-Skan equation exist for (l < -0.1988, but these are unacceptable on the physical ground that f —> 1 from above as r] —> oo, and this would correspond to velocities within the boundary layer that exceed the outer potential-flow value at the same streamwise position, x. It may be noted from Fig. 10-7 that the shear stress at the surface (r] = 0) decreases monotonically as (l is decreased from 0. Finally, at /3 = -0.1988, the shear stress is exactly equal to zero, i.e., /"(0) = 0. It will be noted from (10-113) that the pressure gradient... 

We have considered the thermal boundary-layer problem in this chapter for an arbitrary 2D body with no-slip boundary conditions for Re 1 and Pr (or Sc) either arbitrarily large or small. If we assume that we have a body of the exact same shape, but the surface of which is a slip surface (e.g., it is an interface, so that the surface tangential velocity is not zero), the form of the correlation between Nusselt number and Pr will change for Pr 1. Solve this problem, i.e., derive the governing boundary-layer equation, and show that it has a similarity solution. What is the resulting form of the heat transfer correlation among Nu, Re, and Pr ... 

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