The end regions

To show that this procedure is, in fact, acceptable, we need to follow the example of the previous subsection and consider the asymptotic problem in the end regions. For present purposes, it is sufficient to consider the end region at x = 0. The scaling (6-130) and the governing equations (6-131) are the same except for the body-force term -e3G that was retained in this section [see, e.g., Eq. (6-137)]. The boundary conditions atx = 0 and y = 0 are also identical to (6-132), but now the upper boundary is the interface. The boundary conditions there can be obtained by nondimensionalizing (6-10)-(6-12) using (6-130). The result is 

we can seek an asymptotic solution for this end region in the form 

The governing equations and boundary conditions at 9(e)define the problem for the 9(1) velocity components in the end region. The governing equations are 

the core solution was developed under the assumption that Ca/e2 1, and, for present purposes, it is convenient to express this condition in terms of the distinguished limit  

Thus we can consider the leading-order perturbation to h in the end region without having to solve the flow problem that was previously outlined. 

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In this assignment the assumption is made that the 22 helions of the mantle occupy the end regions of the highly deformed nucleus, as a result of their strong Coulomb repulsion. 

Considering the end regions of the column as well-mixed stages, with small but finite rates of mass transfer, component balance equations can be derived for end stage 0... 

Woodward and Hoffmann have first disclosed that the thermal (4M+2)-cyclization (and also the photochemical (4M)-cyclization) takes place via Type I process, and the photochemical (4m+2)-cyclization (and also the thermal (4m)-cyclization) via Type II process 51>. They called the former (Type I) process "disrotatory", while the latter (Type II) process was referred to as "conrotatory". They attributed this difference in selectivity to the symmetry of HO and SO MO in the ground-state and excited-state polyene molecules, respectively (Fig. 7.33). The former is symmetric with respect to the middle of the chain, and the latter antisymmetric, so that the intramolecular overlapping of the end regions having the same sign might lead to the Type I and Type II interactions, respectively. 

Distributions that fit the Sg distribution will also fit the log-normal distribution but differences will occur in the end regions. These differences arise since the log-normal distribution does not admit a... 

The active site in this kind of structure is a pocket located on one side of the hydrophobic core at the end region of the helices. 

The difference in scaling between the central core of the thin cavity (6-122) and the vicinity of the end walls (6-123) means that the asymptotic solution for s <dimensionless equations and a different form for the asymptotic expansion for e <[Pg.387]

At the leading order of approximation represented by (6-126)-(6 128), there is not a turning-flow contribution in the core region. All of this must occur in the end regions. Clearly (6-126) [and thus presumably also (6-127)] is not a uniformly valid first approximation to the solution of the full problem. In spite of the fact that it satisfies the zero-net-flux condition, (6-125), it does not satisfy the impermeability condition... 

Before we conclude this section, it is worthwhile to consider the formulation of the problem for flow in the end regions. Instead of obtaining the nondimensionalization and governing equations by rescaling the core equations, as we have done in the previous examples of matched asymptotic expansions, here we can directly apply the scaling (6-123) in the form... 

The latter comes from the matching requirement that the magnitude of the pressure in the end regions must be comparable with that in the core. The definitions (6-130) lead to the following nondimensionalized form of the Navier-Stokes equations for the end regions ... 

This completes formulation of the problem for a first approximation to flow in the end regions. However, in spite of the fact that this problem can be solved analytically, we do not pursue it here, as it has only a higher-order effect on the velocity and pressure fields in the core, and we do not seek any higher-order approximation beyond the solution (6 126) (6-128) in the core region. 

It will be noted that we have reverted, on the right hand side, to the general solution obtained for h in the core, without applying the boundary condition, (6-142), or the integral constraint, (6-143), which imply that the core solution for h can be extended all the way to the end walls. If we express the core solution in terms of the end-region variable x, we see that the matching condition becomes... 

As before, the latter conditions assume that we can extend the solution for h(x) in the core region all of the way to the end walls of the cavity. This is a valid assumption at the order of approximation that we consider, but we would need to explicitly consider the solutions in the end regions at higher orders in e. 

In fact, if the edge of the tool is applied flush with the conical portion already formed at the other end, the risk of such a deformation does not arise at all, and the end region easily makes a regular continuation of the cone... 

The major sources of errors associated with concentric cylinders are the end effects and turbulent flow in the end region. Princen (1986) proposed a modification to eliminate the end effects in a concentric cylinder rheometer. He proposed adding a pool of mercury at the bottom of the cup that essentially eliminates the torque exerted on the bottom of the inner cylinder and on the sample in the gap. However, the limitation of this modification is that the fluid to be tested must be considerably more viscous than mercury, and the angular velocity of the cup must be kept below the levels where centrifugal force or normal force effects start to significantly alter the shape of the sample/air and sample/mercury interfaces. 

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