surface-energy-gamma-area-cross-scan - Fluid Mechanics and Surface Tension Step by Step

A surface-energy cross-scan varies the surface area across rows at fixed gamma, showing how the energy E = gamma * A scales linearly row by row. The arithmetic is exact only for the stated gammas, areas, and rows in the ideal flat-surface model. Real interfaces add contamination, hysteresis, roughness, temperature dependence, meniscus geometry, calibration, and measurement uncertainty.

Surface energy is scanned across bound gamma and checked area. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Surface energy reuses a bound gamma source

The energy ledger is allowed to use gamma only by citing the checked line-force source. Area is the second independent input.

E=\gamma A=3\ \text{N/m}\cdot8\ \text{m}^{2}=24\ \text{J}

Changing gamma changes surface energy

Area is fixed in these three rows. The energy column changes only because the cited gamma source changes.

\begin{array}{c|c|c}\gamma&A&E\\2\ \text{N/m}&8\ \text{m}^{2}&16\ \text{J}\\3\ \text{N/m}&8\ \text{m}^{2}&24\ \text{J}\\4\ \text{N/m}&8\ \text{m}^{2}&32\ \text{J}\\\end{array}

Changing area also changes surface energy

Now the gamma source is fixed. The three area rows make the second input-output relationship visible.

\begin{array}{c|c|c}\gamma&A&E\\3\ \text{N/m}&4\ \text{m}^{2}&12\ \text{J}\\3\ \text{N/m}&8\ \text{m}^{2}&24\ \text{J}\\3\ \text{N/m}&12\ \text{m}^{2}&36\ \text{J}\\\end{array}