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The science
Surface tension gamma makes a curved interface carry a pressure jump (Young-Laplace): dP = gamma (1/R1 + 1/R2). In a tube of radius r with contact angle theta the pull 2 pi r gamma cos(theta) balances the weight of the raised column, giving Jurin's law h = 2 gamma cos(theta) / (rho g r); the liquid in the meniscus itself lowers this by about r/3. The program integrates the exact axisymmetric Young-Laplace equation (RK4 in arc length) and shoots on the apex curvature until the wall contact angle is right.
The scale is the capillary length lc = sqrt(gamma / (rho g)) (2.7 mm for water on Earth, 6.6 mm on the Moon); the Bond number is (r/lc)^2. The rise in time follows Lucas-Washburn with gravity, dh/dt = r^2 / (8 eta h) (2 gamma cos(theta)/r - rho g h). A pendant drop on a needle is the same equation with the liquid above the surface; its largest pinned volume is compared with Tate's estimate V = 2 pi r gamma / (rho g), which real drops reach only to about 60-75 %.
Simplifications: fixed contact angles (no hysteresis), pure liquids at 20 C, Poiseuille flow in the tube, no dynamic contact angle. All numbers use exactly the same physics code as the 3D page.
What to try
- Press Mercury is pushed down: contact angle 140 degrees gives a negative rise.
- Choose the Moon: the capillary length grows and the water climbs about six times higher.
- Switch to Pendant drop and slide the fill to 1: the drop is at its largest pinned volume and then falls.