Bubble & Pendant Drop Interface Calculator

An engineering tool to calculate the analytical interface shapes, volumes, and detachment thresholds of fluid bubbles and pendant drops on flat surfaces.

This toolkit implements the Adams-Bashforth integration method to solve the Young-Laplace equation across varying wetting states (pinned vs. spreading/moving contact lines), validated against classical literature experimental datasets.

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Mathematical Background

The profile shapes are governed by the non-dimensionalized Young-Laplace equation under hydrostatic pressure. The interface arc-length parameter ($s$) is solved numerically using an explicit Adams-Bashforth integration routine according to:

$$\frac{d\phi}{ds} = \frac{2}{R_t} - \frac{z}{\lambda^2} - \frac{\sin\phi}{r}$$

Where:

• $R_t$ is the radius of curvature at the apex/top.
• $\lambda$ is the capillary length ($\lambda = \sqrt{\gamma / \Delta \rho g}$).
• $\phi$ is the tangent angle made with the horizontal plane.

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Project Structure

├── bubble.py          # Core integration algorithms & profile loop re-orderers
├── run.py             # Main entry point to execute sweeps and profile tasks
├── plot.py            # Comprehensive Matplotlib script for manuscript-quality figures
├── simData/           # Output directory for generated integration text profiles
├── plots/             # Output folder for generated plots and diagrams (PDF format)
└── exptData/           # Experimental verification benchmarks (Demirkir24, Allred21, etc.)

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Installation & Requirements

Ensure you have a Python 3 environment with standard scientific computing libraries installed.

pip install numpy scipy matplotlib

Note: The plotting script uses LaTeX rendering (text.usetex: True). Ensure you have a functioning LaTeX distribution installed on your system path (e.g., TeX Live, MiKTeX) to avoid pipeline plotting errors.

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How to Run

1. Execute Simulations and Analysis

Run the primary execution pipeline to trigger profiling routines and process structural spatial data maps:

python run.py

2. Custom Scripting & Core Module Usage

Import the simulation module inside your scripts to manually compute properties for unique fluid interfaces:

from bubble import AdamsBashforthProfile

# Parameters: Capillary Length, Apex Radius, file path destination
volume, radius, height, centroid, final_psi = AdamsBashforthProfile(
    capLen=1.0,
    RadTop=0.5,
    fname="simData/my_bubble_profile.txt"
)

print(f"Calculated Drop Volume: {volume:.4f} λ³")

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Outputs & Experimental Validation

Calculated analytical profiles are verified natively against established experimental benchmarks located inside the /exptData directory:

File	Reference
demirkir24life.txt	Demirkır, Ç., Wood, J. A., Lohse, D., and Krug, D. (2024). "Life beyond Fritz: On the Detachment of Electrolytic Bubbles." Langmuir, 40(39), 20474–20484. https://doi.org/10.1021/acs.langmuir.4c01963
allred21role.txt	Allred, T. P., Weibel, J. A., and Garimella, S. V. (2021). "The Role of Dynamic Wetting Behavior during Bubble Growth and Departure from a Solid Surface." Int. J. Heat Mass Transf., 172, 121167. https://doi.org/10.1016/j.ijheatmasstransfer.2021.121167
huang25effects.txt	Huang, J. and Li, R. (2026). "Effects of Surface Wettability on Bubble Dynamics and Induced Liquid Flow: Finite-difference Analysis of Two-Phase Particle Image Velocimetry." Phys. Rev. Fluids, 11(2), 023603. https://doi.org/10.1103/jvxz-8mzv
gunde01measurement.txt	Gunde, R., Kumar, A., Lehnert-Batar, S., Mäder, R., and Windhab, E. J. (2001). "Measurement of the Surface and Interfacial Tension from Maximum Volume of a Pendant Drop." J. Colloid Interface Sci., 244(1), 113–122. https://doi.org/10.1006/jcis.2001.7916
sasetty23stability.txt	Sasetty, S. and Ward, T. (2023). "Stability and Critical Volume of a Suspended Pendant Drop in Air via Experiments and Eigenvalue Analysis." Colloids Surf. A, 666, 131346. https://doi.org/10.1016/j.colsurfa.2023.131346
LesageVolVsContRadSq.txt	Lesage, F. J. and Marois, F. (2013). "Experimental and Numerical Analysis of Quasi-Static Bubble Size and Shape Characteristics at Detachment." Int. J. Heat Mass Transf., 64, 53–69. https://doi.org/10.1016/j.ijheatmasstransfer.2013.04.019
MoriVolByContCubeVsContSqByCapSq.txt	Mori, B. K. and Baines, W. D. (2001). "Bubble Departure from Cavities." Int. J. Heat Mass Transf., 44(4), 771–783. https://doi.org/10.1016/s0017-9310(00)00133-2

The output pipeline generates automated technical reports saved inside /plots, containing capillary curves, maximum volume vs. detachment limits, and shape evolution configurations.