B The Boundary-Layer Equations

The discussion in the previous section leads to the following qualitative description of the flow domain for Re y 1  

Because this inner boundary-layer region is infinitesimal in thickness relative to a, all curvature terms that appear when the equations of motion are expressed in curvilinear coordinates will drop out to first order in Re thus leaving boundary-layer equations that are effectively expressed in terms of a local Cartesian coordinate system. 

To illustrate this latter point, we first derive equations for the inner boundary-layer region for the specific problem of streaming flow past a circular cylinder, starting from the equations of motion expressed in a cylindrical coordinate system. These are 

To obtain the appropriate form of the equations of motion for the inner, boundary-layer region, we must rescale with a new characteristic length scale. Following the precedent of previous analyses, we do this by rescaling the radial variable y in the equations of motion and continuity, (10—21)—(10—23), according to 

approximating 1 + Y/Rea as 1, as is appropriate at the leading order of approximation, 

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

The boundary-layer equations may be solved by the technique outlined in Appendix B or by the integral method of Chap. 5. Eckert and Hartnett [3] have developed a comprehensive set Of solutions for the transpiration-cooling problem, and we present the results of their analysis without exploring the techniques employed for solution of the equations. 

J. B. Klemp and A. Acrivos, A Method for Integrating the Boundary-Layer Equations Through a Region of Reverse Flow, J. Fluid Mech., 53, pp. 177-191,1972. 

For convective crystal dissolution, the dissolution rate is u = (p/p )bD/8. For diffusive crystal dissolution, the dissolution rate is u = diffusive boundary layer thickness as 5 = (Df), the diffusive crystal dissolution rate can be written as u = aD/5, where a is positively related to b through Equation 4-100. Therefore, mass-transfer-controlled crystal dissolution rates (and crystal growth rates, discussed below) are controlled by three parameters the diffusion coefficient D, the boundary layer thickness 5, and the compositional parameter b. The variation and magnitude of these parameters are summarized below. 

We observe that the viscous-heating term in Eq. (B-18) contributes a particular solution to the equation. If there were no viscous heating, the adiabatic-wall solution would yield a uniform temperature profile throughout the boundary layer. We now assume that the temperature profile for the combined case of a heated wall and viscous dissipation can be represented by a linear combination of the solutions given in Eqs. (B-14) and (B-19). This assumption is justified in... 

Vtot in the above equation is a mass transfer velocity across the air-water interface. It has units of velocity (usually cm/s), and it is made up of two parts (a) the velocity of the compound through the boundary layer in the water to the interface (denoted by vw) and (b) the velocity of the compound through the boundary layer in the air as it leaves the air-water interface (denoted by va). The total mass transfer velocity is given by... 

In the case of thermal decomposition of a mineral, there is only the solid B on the left-hand side of equation (5.26). These thermal decompositions can also be treated by the same rate limiting steps as given previously. Although the product layer is often porous, it can produce a slower rate of either heat conduction or diffusion than the boundary layer. As a result fluid-solid reactions occur at a sharply defined reaction interface, at a position r within the particle of size R. The mass flux associated with boundary layer mass transfer is given by... 

Bradshaw et al. (B3) use Eqs. (40) to derive a differential equation for the turbulent shear stress t. The transport velocity Qa is taken as (Tmei/p), where Tm x is the maximum value of riy) in the boundary layer. G and I are prescribed as functions of the position across the boundary layer, and o is essentially taken as constant. Together with Eqs. (10a,b), Eq. (36) gives a closed set of equations for U, V, and t this system is of hyperbolic type, with three real characteristic lines. Bradshaw et al. construct a numerical solution using the method of characteristics it can also be done using small streamwise steps with an explicit difference scheme (Nl A. J. Wheeler and J. P. Johnston, private communications). There is a great physical appeal to the characteristics, especially since it is found that the solutions along the outward-going characteristic dominates the total solution. This... 

In the above equations n and N are the atom fractions of nitrogen-15 species in the gas phase and the liquid phase respectively c (moles/cc.), d (cm.2/sec.), b (cm.) for the gas phase are respectively the concentration of oxides of nitrogen, the diffusion coefficient, and the thickness of the boundary layer, while C, D, and B are the same quantities for the liquid phase k (cc./moles-sec.) is a rate constant for the exchange of oxides of nitrogen between the gas and liquid phase. The specific transfer rate kr (moles/sec.-cm. ) when multiplied by the interfacial area a (cm.2/cc.) in a 1 cm. length of column per cm. of cross-sectional area gives an interphase transfer rate fc a (moles/sec.-cc.). If chemical reaction is rate limiting, fc a will be determined by the first term of Equation 25, otherwise it will be determined by the diffusion terms. 

Experiments with Inulln and B-12 (open squares and x s) were not performed at sufficiently high volume fluxes to produce maxima in Robs Corrected values for R, calculated from observed values by using Equation 41, are entered on the graphs as shaded or closed circles, triangles, etc. If the boundary layer theory is valid, then according to Figure 1 for the Splegler-Kedem equation, each... 

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... 

The mass-transfer conditions prevailing in the boundary layer described so far correspond to diffusion of A through stagnant B in dilute solutions. Therefore, the flux can be written in terms of a -type mass-transfer coefficient, as described by equation (2-7), using the difference in surface and bulk concentrations as the driving force ... 

Equation (24-16) is applicable within the mass transfer boundary layer adjacent to the gas-liquid interface where 0 < x < MTBLTiiquid and Cb = Cb, at the interface where x = 0 (i.e., I b = 1 at = 0). At the other end of the stagnant liquid film, where the boundary layer meets the bulk liquid phase, either ... 

Figure 10.5.3 shows how the process in the liquid film of Fig. 10.5.1 is affected. As the dissolving substance A moves to the right in the boundary layer or film, a reactive component B diffuses to the left from the bulk of the liquid. We assume a rapid reaction between A and B. They will meet at some plane in the boundary layer and react with each other. The reaction plane is at some fraction x of the thickness of the film away from the interface. The equations of flux become... 

In order to derive the basic equation for a laminar or turbulent boundary layer, a small control volume in the boundary layer on a flat plate is used as shown in Fig. 3.10-5. The depth in the z direction is b. Flow is only through the surfacesand dj and also from the top curved surface at 8. An overall integral momentum balance using Eq. (2.8-8) and overall integral mass balance using Eq. (2.6-6) are applied to the control volume inside the boundary layer at steady state and the final integral expression by von Karman is (B2, S3)... 

When heat is transferred from one fluid through a solid boundary to another, fluid heat transfer coefficients are derived for the boundary layers on each side. The method for combining these coefficients is similar to the method for combining thermal resistances and an analogue of equation (4.7) will be obtained. The temperatures for the current example arc defined in figure 4.4(b),... 

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