Dimensionless variables for free convection, the dimensionless variables in Eqs 11.4-39 to 43 can be obtained by simple arguments. The form of Θ is dictated by the boundary conditions and that of ζ is suggested by the geometry. The remaining dimensionless variables may be found as follows: 

(a) Set η = y/y₀, Φ₂ = v₂/v_z0, and Φ_y = v_y/v_y0, the subscript-zero quantities being constants, then the differential equations in Eqs. 11.4-33 to 35 become with the boundary conditions given in Eqs. 11.4-47 to 49.

(b) Choose appropriate values of v_z0, v_y0, and y₀ to convert the equations in (a) into Eqs. 11.4-44 to 46, and show that the definitions in Eqs. 11.4-41 to 43 follow directly. 

(c) Why is the choice of variables developed in (b) preferable to that obtained by setting the three dimensionless groups in Eqs, 11B.15-1 and 2 equal to unity?

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