Brinkman’s and Generalized Brinkman’s Equations and Associated Viscosities
Bibliographic record
Abstract
Under Poiseuelle flow, the effective viscosity associated with Brinkman’s equation is investigated and quantified for a high-porosity porous layer whose geometric factors are given by Ergun’s equation and the Kozney-Carmen equation. An expression for the relative viscosity is derived and expressed in terms of the geometric factors. Viscous and Darcy flow limits are quantified in terms of the relative viscosity and the geometric factors. This work also initiates investigations of the effective viscosity when the Brinkman flow is through a variable permeability porous layer and when the governing equation is the generalized Brinkman equation. When permeability is variable, it is shown that the relative viscosity is always less than unity when the Brinkman pressure gradient is the same as the corresponding Navier-Stokes pressure gradient. A condition on the pressure gradients under which the relative viscosity is greater than unity is derived. When the flow is governed by the generalized Brinkman’s equation, the problem of quantifying the effective viscosity is replaced by that of determining variations in viscosity due to pressure, and variations in pressure due to changes in the porous medium parameters.
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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".