projected-pull-is-smaller-than-line-force - Fluid Mechanics and Surface Tension Step by Step

Surface tension pulls along the contact line with force gamma per unit length, but only the component projected onto the surface normal does the lifting, so the effective pull is gamma * cos(theta). The arithmetic is exact only for the stated gamma, line length, and contact angle in the ideal wetting model. Real interfaces add contamination, hysteresis, roughness, temperature dependence, meniscus geometry, calibration, and measurement/source uncertainty.

A rational projection factor reduces a checked line force. Exact arithmetic here means exact results for the stated model inputs; measured inputs still carry uncertainty and significant-figure limits.

highlighted = computed this step

Projection begins from a checked line force

The projected pull is allowed only because it cites the existing line-force source.

F=24\ \text{N}\qquad p=\frac{1}{2}

Allowed projection factors reduce the pull

The training surface only allows zero, one half, and one. The projected force follows that bounded factor exactly.

\begin{array}{c|c|c}p&F&F_{\parallel}\\0&24\ \text{N}&0\ \text{N}\\\frac{1}{2}&24\ \text{N}&12\ \text{N}\\1&24\ \text{N}&24\ \text{N}\\\end{array}

A projection factor can only reduce the line force

The projection is a rational multiplier on a checked line-force source, not a real contact-angle law.

F_{\parallel}=\frac{1}{2}\cdot24\ \text{N}=12\ \text{N}