Modified perturbation solutions for Stefan problems with convective boundary conditions at high Stefan numbers
Bibliographic record
Abstract
The classical Stefan problem is one of the formulations to represent the moving boundary problems, such as solidification and melting processes. The nonlinearity of the differential equation that governs the moving boundary, i.e., the Stefan condition, makes finding the exact solutions a difficult task. Hence, perturbation theory is often applied to generate approximate analytical solutions by assuming a small Stefan number, i.e., Ste ≤ 0 . 01 , which indicates the ratio of the sensible heat over latent heat in phase change processes. This assumption, however, limits the thermal engineering application of the approximate solution. The present study introduces a modified perturbation solution by adding a correction term after the leading-order solution that extends the validity to a wider range of Stefan numbers (i.e., 0.01 ≤ Ste ≤ 1). Specifically, the Stefan problem is first formulated in Cartesian, cylindrical, and spherical coordinates subject to a realist Robin boundary condition. Then, the leading-order perturbation solution is calculated and a correction term is added by using the Monte-Carlo statistical method and a multi-variant regression analysis. Results indicate that the correction term changes linearly with the Stefan number and is not significantly influenced by the Biot number. The proposed modified solution represents a rapid and precise method to predict the nonlinear moving boundary and temperature profiles in phase change processes.
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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 | 0.000 | 0.000 |
| 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".