Boundary layers region

In Boundary-Layer Region vs. z (After Pagano smd Pipes [4-15])... 

The volume fraction dependence of /cq/(4>) is plotted in Fig. 21b and shows that it increases strongly with 4>- Recall that this rate coefficient is independent of 4> if simple binary collision dynamics is assumed to govern the boundary layer region. The observed increase arises from the obstacle distribution in the vicinity of the catalytic sphere surface. When obstacles are present, a reactive... 

As suggested by Prandtl, the entire zone of motion can be subdivided into two regions a boundary layer region near the plate of thickness 6h = <5,(x), in... 

Stagnation flows can be viewed either as a similarity reduction of the flow equations in a boundary-layer region or as an exact reduction of the Navier-Stokes equations under certain simplifying assumptions. Depending on the circumstances of a particular problem of interest, one or the other view may be more natural. In either case, the same governing equations emerge, with the differences being in boundary conditions. The alternatives are explored in later sections, where particular problems and boundary conditions are discussed. 

Filtration is a physical separation whereby particles are removed from the fluid and retained by the filters. Three basic collection mechanisms involving fibers are inertial impaction, interception, and diffusion. In collection by inertial impaction, the particles with large inertia deviate from the gas streamlines around the fiber collector and collide with the fiber collector. In collection by interception, the particles with small inertia nearly follow the streamline around the fiber collector and are partially or completely immersed in the boundary layer region. Subsequently, the particle velocity decreases and the particles graze the barrier and stop on the surface of the collector. Collection by diffusion is very important for fine particles. In this collection mechanism, particles with a zig-zag Brownian motion in the immediate vicinity of the collector are collected on the surface of the collector. The efficiency of collection by diffusion increases with decreasing size of particles and suspension flow rate. There are also several other collection mechanisms such as gravitational sedimentation, induced electrostatic precipitation, and van der Waals deposition their contributions in filtration may also be important in some processes. 

In Eq. (12-27) it is assumed that the magnetic heating effects are confined to the boundary-layer region. Again, as in Sec. 5-6, we are able to write a cubic-parabola type of function for the temperature distribution so that... 

Tlie hypothetical line of u 0.99V divides the flow over a plate into two regions the boundary layer region, in which the viscous effects and the velocity changes are significant, and tlie irrotational flow region, in which the frictional effects are negligible and the velocity remains essentially constant. 

In most, but not all circumstances, the core gas temperature, T, is the natural reference temperature for the compressed gas because the highest temperature at the end of compression is responsible for the development of spontaneous ignition in the shortest time [88, 95]. Exceptionally, when the compression heats the reactants to temperatures that correspond to the region of ntc for that particular mixture, combustion may be initiated in the cooler boundary layer region. That is, gases which, at the end of compression, are colder than those in the adiabatic core control the duration of the ignition delay. This was demonstrated by Schreiber and coworkers by the simulation of alkane combustion, using various reduced kinetic schemes, in computational fluid dynamic calculations [102-104]. 

Let us now return to the solution of our problem for Rr 1. Although the arguments leading to (4-25) were complex, the resulting equation itself is simple compared with the original Bessel equation. Our objective here is an asymptotic approximation of the solution for the boundary-layer region. In general, we may expect an asymptotic expansion of the form... 

This small mismatch need not concern us at this stage in the solution scheme because we have so far considered only the first leading-order approximation for H in the boundary layer and in the core [see Eq. (4-18)]. With A =0, the final solution for H in the boundary-layer region can be expressed in the form... 

Thus, in the boundary-layer region, it follows from (4-1) and (4-2) that... 

To determine the appropriate scaling, and the correct form for the governing equation in this boundary layer region, we use the idea of rescaling that was introduced in Section A. First, for convenience, we follow the analysis from that section, and redefine the dimensionless radial variable as... 

To determine the appropriate characteristic length scale t for the boundary-layer region, we introduce a new dimensionless variable ... 

With this choice lor t, the governing equation in the boundary-layer region takes the form... 

Of course, the solution (4-181) is only the first approximation in the asymptotic series (4 175). In writing (4-177), we neglected certain smaller terms in the nondimensionalized equation, (4-170), because they were small compared with the terms that we kept. To obtain the governing equation for the second term in the boundary-layer region, we formally substitute the expansion, (4-175), into the governing equation, (4-170) ... 

Before this section is concluded, there is one key point that should be emphasized. The ultimate goal of the calculation was to calculate the effectiveness factor rjcat- In the present case, this came down to solving Eq. (4-170) for the concentration distribution in the boundary-layer near the particle surface. Because of all the assumptions that we made in the original problem setup, Eq. (4-170) is simple and easy to solve. However, even if the equation we achieved in the boundary-layer region had been much more complicated so that we could not solve it, the process of setting up the asymptotic framework, by means of nondimensionalization and rescaling, provides most of the important information about rjcat, and would do so even if we had not been able to solve (4-170). If we go back to the definition of rjcat in Eq. (4-191), we see that the effectiveness factor depends on (dc/dr) =. However, we see from fhe rescaling process that... 

Assuming m to be positive, we can see that the characteristic length scale in this boundary-layer region is much shorter than d. 

Once we specify m, this is the nondimensionalized form of the governing equation for/ but now scaled with the length scale t c that is appropriate to the boundary-layer region. The... 

Thus we see that the function F (and thus the velocity u) varies over a much shorter length scale in this boundary-layer region near the bottom surface of the disk than it does elsewhere in the domain. 

The main point here is that the solution procedure for this particular problem of a singular (or matched) asymptotic expansion follows a very generic routine. Given that there are two sub-domains in the solution domain, which overlap so that matching is possible (the sub-domains here are the core and the boundary-layer regions), the solution of a singular perturbation problem usually proceeds sequentially back and forth as we add higher order... 

I. Governing Equations and Rescaling in the Thermal Boundary-Layer Region... 

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