Finite-amplitude Rayleigh–Bénard convection and pattern selection for viscoelastic fluids
Bibliographic record
Abstract
The influence of inertia and elasticity on the onset and stability of Rayleigh–Bénard thermal convection is examined for highly elastic polymeric solutions with constant viscosity. These solutions are known as Boger fluids, and their rheology is approximated by the Oldroyd-B constitutive equation. The Galerkin projection method is used to obtain the departure from the conduction state. The solution is capable of displaying complex dynamical behaviour for viscoelastic fluids in the elastic and inertio-elastic ranges, which correspond to ${\it Ra} \,{<}\, {\it Ra}_c^s$ and ${\it Ra} \,{>}\, {\it Ra}_c^s $ , respectively, ${\it Ra}_c^s $ being the critical Rayleigh number at which stationary thermal convection emerges. This behaviour is reminiscent of that observed experimentally for viscoelastic Taylor–Couette flow. For a given ${\it Ra}$ in the pre-critical range, finite-amplitude periodic oscillatory convection emerges when the elasticity number, $E$ , exceeds a threshold. Periodicity is lost as $E$ increases, leading to a $T^{2}$ quasi-periodic behaviour, and the breakup of the torus as $E$ increases further. Although no experimental data are available for direct comparison, this scenario is reminiscent of the flow sequence observed by Muller et al. (1993) in the Taylor–Couette flow of a Boger fluid. Stationary thermal convection emerges, via a supercritical bifurcation, when ${\it Ra}$ exceeds ${\it Ra}_c^s $ . The amplitude of motion is found to be little influenced by fluid elasticity or retardation time, especially as the Rayleigh number increases. However, the range of stability of the stationary thermal convection narrows considerably for viscoelastic fluids. In this case, oscillatory thermal convection is favoured. The onset and the stability of other steady convective patterns, namely hexagons and squares, are studied in the inertio-elastic range by using an amplitude equation approach. The range of stability of each pattern is examined, simultaneously allowing the validation of the two-dimensional picture.
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How this classification was reachedexpand
Full frame machine prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. The Gemma side is a direct model label for every work in the frame, read from the title-only record. The Codex side is a classifier learned from the 10,348 direct Codex labels and calibrated to design-weighted sample rates; fields without enough sample support carry no Codex call. Candidate is the union of the two sides; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels.
Distilled classifier scores by category (both heads)
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.000 | 0.001 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.000 |
| Bibliometrics | 0.000 | 0.000 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.000 |
| Insufficient payload (model declined to judge) | 0.000 | 0.000 |
Machine scores (provisional)
The two teacher heads of the student model, read on this work. A score orders the frame for review; it never asserts a category, and the validation status ships verbatim with every row.
Baseline scores from an immature model (maturity gate not passed, 7 training rounds). Scores rank; they never assert a category.
score_only:v0-immature-baseline · verbatim from the scoring run: score_only means the number may rank works, and no category label ships from itClassification
machine, unvalidatedMachine predicted; a candidate call from one source (direct Gemma or distilled Codex), not a consensus.
How this classification was reached, model by model and score by score, is at the end of the page under "How this classification was reached".