Nondimensionalization rescaling

We are allowed to do this because we have expressed (3-77) in a form in which the various derivatives are guaranteed to be 0(1), i.e., independent of e. In this regard, we should note that rescaling from y (which is scaled by use of a) to ey is equivalent to nondimensionalizing the radial variable y using the gap width ae. 

Rather than starting with the original dimensional equations and searching for an appropriate characteristic length scale for the near-wall region, we can determine the correct form by simply rescaling the previously nondimensionalized equation, (4-21). To do this, let us introduce a new independent spatial variable,... 

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... 

We now seek a new nondimensionalization of the dimensional variable (z ) such that the viscous term remains in the problem as Re —> oo. To do this we introduce a rescaling of z by means of the dimensionless parameter Re, i.e.,... 

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... 

Certain features of the thermal problem can be ascertained by means of nondimensional-ization alone. An appropriate characteristic velocity is Uoo- Furthermore, it is convenient to introduce a dimensionless (i.e., rescaled) temperature,... 

As before, we recognize this rescaling as defining a new nondimensionalized radial variable, with a new characteristic length scale... 

Before discussing the solution of (9-173), it is perhaps useftd to review once more the rationale for rescaling and the choice m = 1/2 for the rescaling parameter. The change of variables (9 168) is just one way of changing the nondimensionalization of the radial variable r from the sphere radius a to the new length scale,... 

A second property of the Blasius solution is worth mentioning because it reflects a general property of the boundary-layer equations. The fact is that the Blasius solution is actually independent of the length of the plate. This may seem strange or even incorrect at first because we have obviously used the plate length L to nondimensionalize the equations leading to (10 59), and the plate length L also appears in the Reynolds number that was used to rescale the boundary-layer equations (10-64). However, we have shown that... 

The parameter K is the carrying capacity of the system, at which the growth rate vanishes. If we nondimensionalize the system by measuring the density in terms of the carrying capacity, p = p/K, and rescale time by the intrinsic growth rate, t = rt, then we obtain again KPP kinetics ... 

Figures 12 and 13 focus on the wall region of the jet. The profiles were nondimensionalized by the appropriate local peak values (denoted by subscript m) from Fig. 9, and rescaled by means of the boundary layer variable ...

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