Renormalized dimensionless variables

RENORMALIZATION OF THE DIMENSIONLESS VARIABLES REVEALS EXPLICIT DEPENDENCE OF g ON Re... 

The new set of dimensionless variables has transformed the equations of continuity and motion into two coupled partial differential equations for u and u with no explicit dependence on the Re5molds number because Re does not appear in either (12-21) or (12-22). Remember that is calculated from potential flow theory via (12-18c), exhibiting no dependence on Re, and x is the same as X. Hence, renormalization has no effect on the dimensionless dynamic pressure gradient in the x direction. One concludes from (12-21) and (12-22) that... 

The nonlinearity (or g ) contributes further to the renormalization of the temperature through the appropriate dimensionless variable u = KA/T )L (same as in Eq. (30). As in the previous section, the important flow equation is for this parameter u (upto constant factors). The renormalization of temperature acquires subtle d-dependence that... 

To describe the governing Eqs. (4.5) and (4.7) in dimensionless form, the variables are renormalized with dimensionless time x and other dimensionless variables denoted with tilde symbols as follows x = x/d, y = yld, % = Dt/d, where d is the characteristic length (10 m), and D is the diffusion coefficient. The temperature is rescaled to w = T/T. Then, the final governing equations are represented in dimensionless form ... 

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