CoMPhy Lab
Drop impact · Weber–Ohnesorge scaling
SL theory Drop-impact calculator

Unifying theory of scaling in drop impact: forces and maximum spreading diameter.

Enter the Weber number \(We\) and Ohnesorge number \(Oh\) of an impacting drop. The page returns the Reynolds number, the dissipation regime on the \(We\)–\(Oh\) map, and the predicted maximum spreading factor \(\beta\) from the theory of Sanjay and Lohse (2025).

Read the paper Open access on arXiv Watch the talk
Inputs
\(We\), \(Oh\) Two dimensionless groups
Outputs
\(Re\), regime, \(\beta\) Rounded to two decimals
Validity
\(1 \leq We \leq 10^3\) \(10^{-3} \leq Oh \leq 10^2\)
Overview

What the theory predicts

Drop impact on a rigid surface depends on the drop velocity, size and material properties. Two dimensionless groups organise the scaling theory: the Weber number \(We\), which compares inertia with surface tension, and the Ohnesorge number \(Oh\), which measures viscous effects.

The theory is built from direct numerical simulations spanning \(1 \leq We \leq 10^3\) and \(10^{-3} \leq Oh \leq 10^2\). It separates the dominant dissipation mechanisms across the impact process and predicts the maximum spreading diameter, showing why viscous dissipation remains important even at low \(Oh\).

Inputs

Dimensionless groups

Weber number

\(We = \dfrac{\rho V_0^2 D_0}{\gamma}\)

Inertia relative to surface tension. Larger values mean stronger inertial forcing of the spreading lamella.

Ohnesorge number

\(Oh = \dfrac{\mu}{\sqrt{\rho \gamma D_0}}\)

Viscous relative to inertio-capillary effects. Larger values mean stronger viscous damping during impact.

Phase diagram

Weber–Ohnesorge parameter space

Phase diagram of the four impact regimes in Weber–Ohnesorge parameter space
Regimes I–IV as used by the classifier, with boundaries at \(Oh = We^{-2}\), \(Oh = We^{-3/4}\) and \(Oh = We^{1/2}\). Valid inputs in the calculator are marked on the map.
Talk

Video overview

Open on YouTube
Batch

CSV upload

Upload a CSV with We and Oh header columns. The model fills in beta as the third column for every row and returns the completed file. Extra columns are preserved unchanged.

First row must be a header containing We and Oh. Rows outside the theory range are marked error. Uploads are limited to 1 MB.

Citation

Cite this work

Phys. Rev. Lett. · 2025 arXiv · 2408.12714

Vatsal Sanjay and Detlef Lohse, “Unifying Theory of Scaling in Drop Impact: Forces and Maximum Spreading Diameter,” Physical Review Letters 134(10), 104003 (2025).

DOI 10.1103/PhysRevLett.134.104003

BibTeX entry
Scholar-style export 
@article{PhysRevLett.134.104003,
  title = {Unifying Theory of Scaling in Drop Impact: Forces and Maximum Spreading Diameter},
  author = {Sanjay, Vatsal and Lohse, Detlef},
  journal = {Phys. Rev. Lett.},
  volume = {134},
  issue = {10},
  pages = {104003},
  numpages = {9},
  year = {2025},
  month = {Mar},
  publisher = {American Physical Society},
  doi = {10.1103/PhysRevLett.134.104003},
  url = {https://link.aps.org/doi/10.1103/PhysRevLett.134.104003}
}