curvature-pressure-radius-factor-cross-scan - Fluid Mechanics and Surface Tension Step by Step

A curvature-pressure cross-scan varies the droplet radius across rows, showing how the pressure jump Delta-P = 2 * gamma / R scales as 1/R through the Young-Laplace law. The arithmetic is exact only for the stated gammas, radii, and rows in the ideal spherical-interface model. Real interfaces add contamination, hysteresis, roughness, temperature dependence, meniscus geometry, calibration, and measurement uncertainty.

Curvature pressure is scanned across inverse radius and direct factor. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Curvature pressure has a radius input and a surface-count factor

The radius marker is visible in the diagram. The factor is a checked integer that distinguishes a droplet-style surface from a two-sided film surface.

\Delta P=k{\gamma\over R}=2\cdot6\ \text{N/m}/3\ \text{m}=4\ \text{Pa}

Increasing radius lowers the pressure jump

Gamma and factor are fixed here. The three radius rows expose the inverse relationship without decimal approximations.

\begin{array}{c|c|c|c}k&\gamma&R&\Delta P\\2&6\ \text{N/m}&2\ \text{m}&6\ \text{Pa}\\2&6\ \text{N/m}&3\ \text{m}&4\ \text{Pa}\\2&6\ \text{N/m}&6\ \text{m}&2\ \text{Pa}\\\end{array}

Increasing the curvature factor raises the pressure jump

The factor rows keep radius fixed. A two-sided film factor gives a larger pressure jump than a one-sided training case.

\begin{array}{c|c|c|c}k&\gamma&R&\Delta P\\1&6\ \text{N/m}&3\ \text{m}&2\ \text{Pa}\\2&6\ \text{N/m}&3\ \text{m}&4\ \text{Pa}\\4&6\ \text{N/m}&3\ \text{m}&8\ \text{Pa}\\\end{array}