On the maximum spreading of viscous droplets impacting flat solid surfaces
Bibliographic record
Abstract
We experimentally and theoretically examine the maximum spreading of viscous droplets impacting ultra-smooth solid surfaces, where viscosity plays a dominant role in governing droplet spreading. For low-viscosity droplets, viscous dissipation occurs mainly in a thin boundary layer near the liquid–solid interface, whereas for high-viscosity droplets, dissipation is expected to extend throughout the droplet bulk. Incorporating these dissipation mechanisms with energy conservation principles, two distinct theoretical scaling laws for the maximum spreading factor ( $\beta _m$ ) are derived: $\beta _m \sim ({\textit{We}}/ {\textit{Oh}})^{1/6}$ for low-viscosity regimes ( ${\textit{Oh}} \lesssim 0.1$ ) and $\beta _m \sim \textit{Re}^{1/5}$ for high-viscosity regimes ( ${\textit{Oh}} \gt 1$ ), where $\textit{We}$ , $\textit{Re}$ and $\textit{Oh}$ are the Weber, Reynolds and Ohnesorge numbers, respectively. Both scaling laws show good agreement with the experimental data for their respective validity ranges of $\textit{Oh}$ . Furthermore, to better model experimental data at vanishing $\textit{Re}$ , we introduce a semi-empirical scaling law, $\beta _m \sim (A + {\textit{We}}/ {\textit{Oh}})^{1/6}$ , where $A$ is a fitting parameter accounting for finite spreading ( $\beta _m \approx 1$ ) at negligible impact velocities. This semi-empirical law provides an effective description of $\beta _m$ for a broad experimental range of $10^{-3} \leqslant {\textit{Oh}} \leqslant 10^0$ and $10^1 \leqslant {\textit{We}} \leqslant 10^3$ .
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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) | 0.000 | 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".