Bifurcation in steady laminar mixed convection flow in uniformly heated inclined tubes

The fully developed laminar mixed convection flow in inclined tubes subject to axially and circumferentially uniform heat flux has been studied numeri cally for a Boussinesq fluid. Dual solutions characterized by a two- and a four-vortex secondary flow structure in a cross-section normal to the tube' s longitudinal axis have been found for different combinations of the Grash of number Gr and of the tube inclination alpha for all Prandtl numbers betw een 0.7 and 7 In the two-parameter space defined by Gr and alpha dual solut ions occur: at a given alpha, if the Grashof number exceeds a critical valu e Gr(l) (for horizontal tubes Gr(l) is approximately 5.5 x 10(5), 1.7 x 10( 5) and 1.7 x 10(4) respectively for Pr = 0.7, 7 and 70); at a given Gr, if the tube inclination is below a critical value alpha(c) (for Gr = 10(6) thi s critical angle is approximately 62.5 degrees and 83.5 degrees respectivel y for Pr = 0.7 and 7). Numerical experiments carried out for developing flo ws indicate that the two vortex solution is the only stable flow structure.