Two experiments, one equation. Glass tubes lift a liquid to Jurin's height; a needle holds a hanging drop up to a largest volume. Both shapes come from integrating the Young-Laplace equation, and the capillary length sets the scale.
The science
Young-Laplace. The pressure jump across a liquid surface is Δp = γ (1/R1 + 1/R2). For an axisymmetric surface with arclength s and tangent angle ψ this becomes dψ/ds = 2/b ± z/ℓ2 - sin ψ / x with capillary length ℓ = √(γ / (ρ g)) (2.7 mm for water on Earth). The sign is + for a surface with the liquid below it (tube meniscus) and - for a hanging drop. The code integrates this with 4th-order Runge-Kutta.
Capillary rise. Jurin's law is h = 2 γ cos θ / (ρ g r). The exact apex height from the meniscus shape is slightly lower, h - r/3 for a narrow tube. Force balance: the liquid raised above the reservoir weighs ρ g (π r² H + Vmeniscus) = 2 π r γ cos θ. Mercury does not wet glass (θ about 140 degrees), so it is pushed down.
Dynamics. Lucas-Washburn with gravity: dh/dt = r²/(8 η h) (2 γ cos θ / r - ρ g h) (Poiseuille flow, inertia ignored). Early on h² = γ r cos θ t / (2 η). Glycerol (viscous) and narrow tubes take minutes; the time axis can be sped up.
Pendant drop. For a drop pinned on a needle of radius r the largest stable volume is found by scanning the apex curvature 2/b; the maximum of V(b) is the drop that falls. Tate's law gives the ideal weight m g = 2 π r γ; the exact static drop is smaller by a factor f (about 0.7 to 0.8 for these needles), and measured drops are another 4 to 8 percent lighter (Harkins-Brown) because they pinch off dynamically before the static limit. Weighing drops is a real method to measure γ.
Simplifications. Contact angles are fixed (no hysteresis), surface tension and density are 20 degree C literature values and are treated as typical, the dynamic rise ignores inertia and the entrance effect, and the drop leaves 10 % of the detached volume behind (a satellite is not simulated). The Blender models are generic lab objects.