Stagnation Point Flow of a Non-Newtonian Williamson Fluid over a Stretching/Shrinking Sheet: Existence Proofs, Dual Solutions, and a 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 (𝑐 > 0) or shrinking (𝑐 < 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 𝑐 ∈ (-1, ∞) and second, using the BVP4C solver in MATLAB, two different solution branches are reported on the interval [𝑐 𝑇 , -1], where 𝑐 𝑇 is the bifurcation point.The 𝑐 𝑇 values become more negative with increasing values of the Williamson parameter λ, marking the broadening of the solution range.Furthermore, the first solution branch continues for large positive values of 𝑐, whereas the second branch seems to cease at 𝐹′′(0) = 0 as 𝑐 → -1.The smallest eigenvalue computed using temporal stability analysis of these solutions is found 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 𝑐.
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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 | 0.003 |
| Meta-epidemiology (narrow) | 0.001 | 0.000 |
| Meta-epidemiology (broad) | 0.000 | 0.001 |
| Bibliometrics | 0.001 | 0.000 |
| Science and technology studies | 0.001 | 0.002 |
| Scholarly communication | 0.001 | 0.001 |
| Open science | 0.000 | 0.001 |
| Research integrity | 0.001 | 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".