non-Boussinesq Conditions Research Articles

Stability of mixed-convection flow in a tall vertical channel under non-boussinesq conditions

We have examined the linear stability of the fully developed mixed-convection flow in a differentially heated tall vertical channel under non-Boussinesq conditions. The Three-dimensional analysis of the stability problem was reduced to an equivalent two-dimensional one by the use of Squire's transformation. The resulting eigenvalue problem was solved using an integral Chebyshev pseudo-spectral method. Although Squire's theorem cannot be proved analytically, two-dimensional disturbances are found to be the most unstable in all cases. The influence of the non-Boussinesq effects on the stability was studied. We have investigated the dependence of the critical Grashof and Reynolds numbers on the temperature difference. The results show that four different modes of instability are possible, two of which are new and due entirely to non-Boussinesq effects.

Stability of natural convection flow in a tall vertical enclosure under non-Boussinesq conditions

We have examined the linear stability of the fully developed natural convection flow in a differentially heated tall vertical enclosure under non-Boussinesq conditions. The three-dimensional analysis of the stability problem was reduced to a two-dimensional one by the use of Squire's theorem. The resulting eigenvalue problem was solved using an integral Chebyshev collocation method. The influence of non-Boussinesq effects on the stability was studied. We have investigated the dependence of the critical Rayleigh number on the temperature difference. The results show that two different modes of instability are possible, one of which is new and due entirely to non-Boussinesq effects. Both types of instability are oscillatory, and the critical disturbance wave speed is zero only in the Boussinesq limit.

Stability of unstably stratified shear flow in a channel under non-Boussinesq conditions

We investigate the linear stability of unstably stratified Poiseuille flow between two horizontal parallel plates under non-Boussinesq conditions. It is shown that Squire's transformation can be used to reduce the three-dimensional stability problem to an equivalent two-dimensional one. The eigenvalue problem, consisting of the generalized Orr-Sommerfeld equations, is solved numerically using an integral Chebyshev pseudo-spectral method. The influence of the non-Boussinesq effects on stability is studied. The dependence of the critical Rayleigh number on the Reynolds number and temperature difference parameter is obtained. As in the Boussinesq case, results show that the most unstable mode is that of longitudinal rolls. However, in contrast to the Boussinesq case, the rolls are highly distorted for large temperature differences. In addition, the critical Rayleigh number increases with the increase of the temperature difference and is independent of the Reynolds number.

Transition from hexagons to rolls in convection in fluids under non-Boussinesq conditions

The transition between hexagons and rolls in convective patterns has been studied. The transition thresholds and changes in the Nusselt number are discussed theoretically in terms of calculations made by Busse (1967a) and with amplitude equations. Experiments have been done in a shallow layer of pure water under non-Boussinesq conditions using complementary techniques: shadowgraph (qualitative), optical (based on the deflections of a laser beam) and calorimetric. The experimental values of the critical Rayleigh number Rc and the critical wavenumber kc are in agreement with theoretical ones. However, theory and experiments show some discrepancies in the slopes of the non-dimensional convective heat flow curve and in the thresholds of the hysteretic hexagons-rolls transition. These discrepancies are discussed in terms of lateral effects and of the presence of defects in the pattern.

Defects in roll-hexagon competition.

The defects of a system where hexagons and rolls are both stable solutions are considered. On the basis of topological arguments we show that the unstable phase is present in the core of the defects. This means that a roll is present in the penta-hepta defect of hexagons and that a hexagon is found in the core of a grain boundary connecting rolls with different orientations. These results are verified in an experiment of thermal convection under non-Boussinesq conditions.

Competition between different symmetries in convective patterns.

The transition between hexagons and rolls in convective patterns has been studied in pure water under non-Boussinesq conditions. The transition is characterized qualitatively by shadowgraph images and quantitatively by heat-flow measurements and a local technique based on the deflections of a laser beam. We also show that with this local technique one can recover global quantities such as the convective heat flow.