The stability of finite amplitude cellular convection and its relation to an extremum principle

Abstract

The stability of cellular convection flow in a layer heated from below is discussed for Rayleigh number R close to the critical value Rc. It is shown that in this region the stable stationary solution is determined by a minimum of the integral \[ \int_0^{H_0}R(H)\,dH, \] where R(H) is a functional of arbitrary convective velocity fields which satisfy the boundary conditions. For the stationary solutions R(H) is equal to the Rayleigh number. H0 is a given value of the convective heat transport. In a second part of the paper explicit results are derived for the convection problem with deviations from the Boussinesq approximation owing to the temperature dependence of the material properties.

Pattern Competition in Homogeneously Heated Fluid Layers

“…However, for cases where the breaking of isotropy is insignificant, rolls seem to be the preferred mode of evolution. Yet when isotropy is broken by the temperature dependence of material properties such as viscosity (Busse, 1967) then the stability of the rolls is reduced, as the magnitude of the isotropic breaking term increases. The cases described here, considered an additional heat flux that was applied homogeneously to a fluid layer with asymmetric boundary conditions.…”

“…Fourier space modes have long been used in the description and prediction of the formation of rolls, hexagons and squares in various forms of thermal convection. The technique, first introduced by Busse (1967), was applied recently in the analysis of rolls and hexagonal cells in an experimental study of convection in a so-called ferrofluid that was driven by two magnetic fields at right angles to one another (Groh et al, 2007). A new saddle node bifurcation was identified by Groh et al (2007), as a result of their analysis that bridges hexagons and stripes.…”

“…The study here is motivated partially by previous work (Busse, 1967;Generalis and Nagata, 2003), but primarily by the fact that the present problem has many important environmental and industrial applications. An industrial example is the case where such convection is driven by a homogeneous heat source that may be produced from fission product decay in a pool of molten fuel elements found at the lower head of a nuclear reactor that is undergoing a severe core damage accident (Asfia and Dhir, 1996).…”

Bifurcation and symmetry breaking in applied mathematics

“…This group is generated by the elements g = (1 2 3) and h = (1 2). It has three irreducible representations: Busse [7], motivated by an earlier work of Malkus, showed that there was a functional associated with the bifurcation problem and argued that this functional would be maximized by the physically correct solutions. Busse's functional is in fact the invariant tensor corresponding to the reduced bifurcation equations, so his method is equivalent to a linearized stability analysis of the bifurcating solutions.…”

Finite amplitude thermal convection with variable gravity

“…Here, the procedure to determine the growth rates σ 2 is similar to those used in [1] and [9]. Rather than repeating that procedure for deriving the eigenvalues σ 2 of (3.5), we refer the reader to these references for further details.…”