Instability in gravity-driven flow over uneven surfaces
Bibliographic record
Abstract
We consider the gravity-driven laminar flow of a shallow fluid layer down an uneven incline with the principal objective of investigating the effect of bottom topography and surface tension on the stability of the flow. The equations of motion are approximations to the Navier–Stokes equations which exploit the assumed relative shallowness of the fluid layer. Included in these equations are diffusive terms that are second order relative to the shallowness parameter. These terms, while small in magnitude, represent an important dependence of the flow dynamics on the variation in bottom topography and play a significant role in theoretically capturing important aspects of the flow. Some of the second-order terms include normal shear contributions, while others lead to a nonhydrostatic pressure distribution. The explicit dependence on the cross-stream coordinate is eliminated from the equations of motion by means of a weighted residual approach. The resulting mathematical formulation constitutes an extension of the modified integral-boundary-layer equations proposed by Ruyer-Quil and Manneville [Eur. Phys. J. B 15, 357 (2000)] for flows over even surfaces to flows over variable topography. A linear stability analysis of the steady flow is carried out by taking advantage of Floquet–Bloch theory. A numerical scheme is devised for solving the nonlinear governing equations and is used to calculate the evolution of the perturbed equilibrium flow. The simulations are used to confirm the analytical predictions and to investigate the interfacial wave structure. The bottom profile considered in this investigation corresponds to periodic undulations characterized by measures of wavelength and amplitude. Conclusions are drawn on the combined effect of bottom topography and surface tension.
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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 | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.000 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 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".