Long-wave equations and direct simulations for the breakup of a viscous fluid thread surrounded by an immiscible viscous fluid
Bibliographic record
Abstract
We consider capillary driven breakup of a viscous liquid thread in a second immiscible viscous liquid of infinite extent and arbitrary viscosity. Postulating the existence of long-wave dynamics, we use the slenderness parameter ε ≪ 1 (proportional to the interfacial slope, for example) to construct consistent asymptotic theories using matched asymptotic expansions at arbitrary viscosity ratios. Three canonical models are found, two of which hold for asymptotically small values of the inner to outer viscosity ratio λ ∼ ε2 and λ ∼ 1/ln(1/ε), respectively, and the third valid for large λ ∼ 1/(ε2 ln(1/ε)). The smallest and largest λ produce appropriate limits of models described in the literature when the inner or outer fluid is air, respectively. The intermediate λ model is found to be ill-posed and a technique is described to regularize it by considering terms arising from the asymptotic forms of the λ ∼ ε2 model as its scaled viscosity ratio becomes large, and from the asymptotic forms of the λ ∼ 1/(ε2 ln(1/ε)) as its scaled viscosity becomes small. Time-dependent direct numerical simulations based on boundary integral methods are also used to predict the dynamics for a large range of viscosity ratios (0.001 ≤ λ ≤ 20). Intermediate values of λ indicate that the dynamics is not long-wave, consistent with the asymptotic analysis, but are self-similar in a pinch-off region with order one aspect ratio. Simulations at large and small values of λ produce intricate dynamics near pinching and in particular the formation of necks, threads and bulges. The direct simulations and the asymptotic models valid at λ ∼ ε2 and λ ∼ 1/(ε2 ln(1/ε)) complement each other and the former confirm the validity of the long-wave assumptions.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.001 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.001 | 0.001 |
| Research integrity | 0.002 | 0.001 |
| Insufficient payload (model declined to judge) | 0.002 | 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".