Multiple bifurcations in triple convection with non-ideal boundary conditions

Abstract

We investigate the behaviour associated with a multiple steady-oscillatory bifurcation in a triply diffusive fluid layer. We consider two-dimensional motion confined between two infinite horizontal planes under periodic lateral boundary conditions that confer O(2) symmetry upon the system. Previous investigations with “ideal” (stress-free, im-penetrable) horizontal boundaries have revealed a degenerate cubic-order normal form for the oscillatory bifurcation. This degeneracy does not arise when more realistic horizontal boundary conditions are applied, where at least one of the horizontal planes is rigid, and there is no slip of the fluid at a rigid boundary. For such (non-ideal) problems the normal form coefficients must be computed numerically. We perform these calculations for two triply diffusive fluid systems: heat-KCL-sucrose and KClNaCl-sucrose. Numerical integrations of the normal form equations for the multiple instability compare well with theoretical predictions of their behaviour. For various ranges of parameter values we find stable steady states, mixed steady-standing wave states, travelling waves and modulated travelling waves. We also find that a stable three-frequency flow (already known to exist for the ideal problem) can undergo period-doubling cascades, and can coexist with a branch of stable modulated travelling waves.