Governing Equations and Rescaling in the Thermal Boundary-Layer Region

Governing Equations and Rescaling in the Thermal Boundary-Layer Region 

For the purposes of rescaling (9-24), it is convenient to redefine variables in the form 

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. 

Clearly, if we restrict our attention to a region (r - 1) = 0(Pe m), the magnitudes of both ur and ug in that region will depend on Pe. Indeed, if we introduce the rescaling (9-214) into (9-218) and (9 219) and restrict our attention to Y = (9(1), we confirm this fact  

Evidently, to leading order in Pe for Pe 1, we can neglect all but the first term in the Taylor series approximation for ur and ug. 

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