line-force-gamma-length-cross-scan - Fluid Mechanics and Surface Tension Step by Step

A line-force cross-scan varies the contact-line length across rows at fixed gamma, showing how the pull F = gamma * L scales linearly row by row. The arithmetic is exact only for the stated gammas, line lengths, and rows in the ideal wetting model. Real interfaces add contamination, hysteresis, roughness, temperature dependence, meniscus geometry, calibration, and measurement uncertainty.

Line force is scanned across gamma and contact length with exact rows. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Line force starts from one checked contact edge

The contact edge supplies the measured line length. Gamma and length are both shown before the force number is accepted.

F=n\gamma L=1\cdot3\ \text{N/m}\cdot4\ \text{m}=12\ \text{N}

Changing gamma changes force one for one

Hold the surface count and contact length fixed. Three gamma rows show the force scaling without needing a paragraph.

\begin{array}{c|c|c|c}n&\gamma&L&F\\1&2\ \text{N/m}&4\ \text{m}&8\ \text{N}\\1&3\ \text{N/m}&4\ \text{m}&12\ \text{N}\\1&4\ \text{N/m}&4\ \text{m}&16\ \text{N}\\\end{array}

Changing line length also changes force one for one

The second scan changes a different input. The same gamma source now pulls harder only because the checked edge is longer.

\begin{array}{c|c|c|c}n&\gamma&L&F\\1&3\ \text{N/m}&2\ \text{m}&6\ \text{N}\\1&3\ \text{N/m}&4\ \text{m}&12\ \text{N}\\1&3\ \text{N/m}&6\ \text{m}&18\ \text{N}\\\end{array}