On the origin of convection

Abstract

The method of investigating bifurcations developed in [1 and 2] is applicable to many hydrodynamic problems. In the present paper it is applied to investigate the origin of convection in a horizontal fluid layer heated from below. Secondary stationary flows are of particular interest in the convection problem since the loss of stability is associated with these flows: “the principle of the change in stability” is not only valid here but has been proved rigorously [3]. It has also been proved that secondary stationary flows are generated by branching off from the state of rest [4 and 5]. The problem under consideration is invariant relative to the group of motions of a horizontal plane. The single solution invariant relative to this whole group is the rest solution. When this solution is unstable, it is natural to expect the occurrence of solutions invariant relative to some subgroup of the group of motions. If the mentioned subgroup is generated by a pair of translations (in perpendicular directions), we arrive at doubly-periodic solutions (Section 1), and if invariance relative to rotation through a certain angle is required in addition, we arrive at solutions of hexagonal type (Section 2). As is known, precisely these latter are realized in convection experiments [6]. Deductions on the existence of doubly-pertodic convection flows are elucidated in Theorem 1.1, and the existence of solutions of hexagonal convection type is asserted in Theorem 1.2. The applied method has slight connection with the boundary conditions. Only for definiteness is it assumed that the boundaries of the layer are solid walls on each of which the temperature is specified.