Rescaling variables

In fig. 3.20, we show this distribution for several different temperatures. For ease of plotting, we have rescaled variables so that the Fermi-Dirac distribution is written in the form... 

The question now is what value to choose for the scaling coefficient m. To determine this, we substitute the rescaled variable Y = IT into (4-166). The result is... 

With the correct choice for m, this is the length scale characteristic of the inner (or boundary-layer) region. In the rescaled variables, the change in 0 from 0 = 1 to approximately the free-stream value 0 = 0 will occur over an increment A Y = 0(1) so that 30/37 = 0(1) independent of Pe. For m > 0, an increment A 7 = 0(1) clearly corresponds to a very small increment in the radial distance Ay, scaled with respect to a. In particular, the thickness of the so-called thermal boundary layer is only 0(Pe m) relative to the sphere radius a. 

To determine a and the appropriate form of the equations of motion, (10 21) and (10 22), in the inner region, we substitute the rescaled variables (10-24) and (10 27) into these equations. The result for the tangential component (10-21) is... 

We proceed formally. Thus we introduce the rescaled variable... 

The left-hand side of this condition contains, at leading order, just the solution (11-57). It is convenient to express this solution in terms of the rescaled variables for the outer part of the thermal boundary layer, namely,... 

We recall that the profiles presented in Fig. 11 were obtained from the numerical solution of Eqs. (55) and (56), including the effect of small ions and excluded volume. The scaling relations are verified by plotting in Fig. 12 the same sets of data as in Fig. 11, using rescaled variables as defined in Eqs. (61) and (62). That is, the rescaled electrostatic potential x)/-fs and polymer concentration Cm(x)/cM Cm x)a ys are plotted as functions of the rescaled dis-tancex/D xf l ys l / a. The different curves roughly collapse on the same curve. 

We obtain the following dimensionless equation by substituting the explicit form of H S into Eq. (1) and by rescaling variables in the equation [11] ... 

Namely, one decomposes u x) in a smooth contribution U(x), plus a fufiction (p x/r ) which depends on the rescaled variable x/rj. The function (p x/rj) is only non negligible in the inner region corresponding to the set of reactor cell points where the solution displays a small-scale behavior characterized by Uxx (see Figure 16). The outer region is the complementary set... 

The saddle equilibrium states are the saddle fixed points of the shift map, and respectively, their separatrices are the invariant manifolds. Returning to the original (non-rescaled) variables we find that the fixed points must lie apart from the origin at some distance of order e. If the third iteration (10.6.2) of the map (10.6.1) were the shift map of the reduced system (10.6.5), then the above theorem would follow from our arguments because the fixed points Oi, O2,03 of the third iterations correspond to the cycle of period three of the original map. 

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