The Behaviour of Fluid-Conveying Pipes, Supported at Both Ends, by the Complete Extensible Nonlinear Equations of Motion
Bibliographic record
Abstract
In this paper, the post-divergence behaviour of fluid-conveying pipes supported at both ends is studied using the complete extensible nonlinear equations of motion. The two coupled nonlinear partial differential equations are discretized via Galerkin’s method and the resulting set of ordinary differential equations is solved by Houbolt’s finite difference method and also using AUTO. Typically, the pipe is stable and retains its original static equilibrium position up to where it loses stability by a supercritical pitchfork bifurcation. By increasing the flow velocity, the amplitude of the buckled position increases, but no secondary instability can be observed thereafter, in agreement with Holmes’ results for his simplified model. The effect of different parameters on the behaviour of the pipe has been studied. By increasing the externally applied tension, or by increasing the gravity parameter, the critical flow velocity for the pitchfork bifurcation increases. The pitchfork bifurcation is subcritical if the nondimensional externally imposed tension, is greater than the nondimensional axial rigidity. The solution in the vicinity of the critical point for this case is confirmed to be subcritical, although the fold and the stable non-trivial solution thereafter could not be seen — perhaps because the model is correct to only third-order of magnitude. Dynamic instabilities may be possible for a pipe hinged at both ends but free to slide axially at the downstream end, according to preliminary results.
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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".