Sinusoidal oscillations radiating from a cylindrical source in thermal conduction or groundwater flow: Closed‐form solutions
Bibliographic record
Abstract
Abstract The diffusion equation governs thermal conduction and groundwater flow phenomena. In this paper, we study the two‐dimensional radial propagation of a sinusoidal perturbation radiating from a cylindrical source within an infinite slab of homogeneous material. The solution of this problem has several applications. For instance, it can be used to determine the hydraulic diffusivity of the subsurface based on measurements of the hydraulic head around a vertical well during its development. For thermal problems, it can be used to determine the thermal diffusivity based on measurements of the temperature distribution around a cylindrical heat source generating a sinusoidal power per unit length. In this paper, we present a comprehensive analytical solution of this problem and we compare these solutions with numerical solutions. Two approximate analytical solutions, which can be relevant in practice, are also presented. Finally, we give an upper bound for the survival time of the transient part of the solution and we provide an estimate of the radius of influence of the sinusoidal solicitation. Copyright © 2010 John Wiley & Sons, Ltd.
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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 | 0.001 | 0.005 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.001 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.000 | 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) | 0.005 | 0.001 |
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".