Variational methods and maximal residual wall layers
Bibliographic record
Abstract
The problem of removing an unyielded visco-plastic material from the walls of a duct is considered. This forms a prototype problem for the mechanical removal of soft solids from the duct walls in both oilfield well construction and food processing. We consider two-layer axial flows for which both fluids may be characterized as Herschel–Bulkley fluids, a class of fluids including Newtonian, Bingham and power-law models. If the yield stress of the displaced fluid is sufficiently large, it is possible for static layers of fluid to persist at the walls of the duct, as has been shown in Allouche, Frigaard & Sona (2000). Following Prager (1954) two variational principles are derived for these flows. These may be loosely interpreted as rate-of-strain minimization and stress maximization principles. The rate-of-strain minimization principle leads directly to existence and uniqueness results for this class of flows. The stress maximization principle leads to a number of qualitative results. An adaptation of the stress maximization principle also allows us to define the concept of a maximal static wall layer in terms of minimization of a certain functional over the space of admissible fluid–fluid interfaces, i.e. a shape optimization problem. We present a number of geometrically simple examples that demonstrate the validity of this method.
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How this classification was reachedexpand
Full frame distilled prediction
Teacher imitationNot calibrated prevalence, not ground truth. Human validation pending. Learned from the 10,348 direct Codex labels and 10,348 direct Gemma labels. Candidate is the union of thresholded teacher heads; consensus is their intersection. These outputs are machine_predicted_unvalidated and are not human labels or direct frontier model labels.
Codex and Gemma teacher scores by category
| Category | Codex | Gemma |
|---|---|---|
| Metaresearch | 0.001 | 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 teacher head, 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".