TITLE:
Study of bifurcations and stability in Rayleigh-B\'enard convection

AUTHORS
D. Puigjaner (1), C. Sim\'o (2), F.Giralt (3)

(1) Dept. Enginyeria Inform\`atica i Matem\`atiques, ETSE, 
Universitat Rovira i Virgili, 
Ctra. de Salou s/n, 43006 Tarragona (Spain).

(2) Dept. Matem\`atica Aplicada i An\`alisi, Fac. Matem\`atiques, 
Universitat de Barcelona, 
Gran Via, 585, 08007 Barcelona (Spain).

(3) Dept. Enginyeria Qu\'{\i}mica, ETSEQ, 
Universitat Rovira i Virgili, 
Ctra. de Salou s/n, 43006 Tarragona (Spain).

ABSTRACT
A path-continuation Galerkin method is proposed to determine the bifurcations 
and stability of steady convective flows in cavities. It is based on a 
complete, divergent-free set of basis functions satisfying all boundary 
conditions. The method is applied to the Rayleigh--B\'enard flow in a cubical 
cavity. Three bifurcations from the conductive state are identified for 
Ra<8000. At the first bifurcation a stable x--roll and an unstable 
diagonal--roll are formed. The second bifurcation yields an initially 
unstable four--rolls structure which becomes stable later on, while the 
third bifurcation results in a highly unstable structure.  
