One‐dimensional single rising bubble at low Reynolds numbers: solution of equation of motion by differential transformation method
Bibliographic record
Abstract
ABSTRACT The behavior of ascending bubbles is of fundamental importance for understanding many natural and artificial phenomena. In this study, the unsteady motion of a single spherical air bubble rising in a stationary viscous liquid has been studied for low Reynolds numbers. Bubble rising under the in?uence of gravitational force is one of the most common gas–liquid flow phenomena. Understanding the dynamic interaction between the phases is an important key for the design and operation of industrial applications. A particular instance of this situation occurs in gas–liquid column reactors. The equation of motion included an additional mass term and neglected the Basset term. An exact solution is derived for instantaneous velocity using a series‐based technique, called the differential transformation method (DTM), and the acceleration and position of the bubble were determined. A general solution of the equation was obtained, and also for several practical conditions with air as the gas phase and glycerin solutions of different concentrations as liquid phase. The results obtained demonstrate the effectiveness of using DTM and promote a new application of this powerful analytical tool for solving nonlinear problems in two‐phase flows. Copyright © 2010 Curtin University of Technology and 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 | 0.000 | 0.001 |
| 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 | 0.001 | 0.001 |
| Insufficient payload (model declined to judge) | 0.003 | 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".