Stagnation-Point Flow of Williamson Fluid: A Study on Dual Solutions and Stability Analysis
Bibliographic record
Abstract
Since non-Newtonian fluids are often encountered in engineering devices, the nonlinear boundary layer equations governing the flow and heat transfer properties of a non-Newtonian Williamson fluid over a stretching ( > 0) or shrinking ( < 0) sheet near the stagnation point are analyzed using two closely interrelated approaches.First, employing the shooting argument, it is proved that a unique solution exists when (-1, ) and second, using the BVP4C solver in MATLAB, two different solution branches are reported on the interval [ , -1], where is the bifurcation point.The values become more negative with increasing values of the Williamson parameter , marking the broadening of the solution range.Furthermore, the first solution branch continues for large positive values of , whereas the second branch seems to cease at (0) = 0 as -1.The smallest eigenvalue computed using temporal stability analysis of these solutions is found to be positive for the first branch, indicating that this branch is physically stable.These findings are relevant to various industrial processes involving non-Newtonian fluids, such as polymer processing and coating applications.Finally, an asymptotic expression is derived to provide insights into the behavior of large .
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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 | 0.000 |
| Meta-epidemiology (narrow) | 0.000 | 0.000 |
| Meta-epidemiology (broad) | 0.002 | 0.001 |
| Bibliometrics | 0.001 | 0.001 |
| Science and technology studies | 0.000 | 0.000 |
| Scholarly communication | 0.000 | 0.000 |
| Open science | 0.000 | 0.000 |
| Research integrity | 0.000 | 0.001 |
| 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".