Geometric approach to laminar convection

The analytical approach to the solution of a steady-state convection problem for Hagen-Poiseuille flow in tubes of arbitrary cross section was analyzed. The problem was studied in the view of isoperimetric inequalities, that is of inequalities holding for domain functionals, provided that equality is attained. The formal analogy with continuum solid mechanics allowed to develop a semiqualitative theory of laminar convection in tubes of arbitrary cross section. The heat transfer rate per unit length across the wall can be expressed in terms of a geometric parameter for a fluid of constant shear viscosity. The heat transfer rate could be calculated in terms of overall quantities without requiring pointwise solution of the Hagen-Poiseuille problem.