Capillary Rise in Three Tubes

What is modelled

A wetting liquid has a curved meniscus at the top of a thin tube. The surface tension pulls the meniscus up with a pressure jump; the column weight and viscous drag in the tube oppose it.

Equations

• Capillary pressure pc = 2 γ cosθ / r; hydrostatic pressure ph = ρ g h.
• Jurin height (equilibrium) hJ = 2 γ cosθ / (ρ g r).
• Poiseuille resistance 8 μ (h + he) / r2 with a small entrance length h_e, giving dh/dt = (pc - ph) r2 / (8 μ (h + he)).
• Without gravity the solution is Washburn-like: (h + he)2 = he2 + γ r cosθ t / (2 μ).

Checks run in the page

• The simulated final height of each tube against the Jurin height.
• The simulation with gravity switched off against the closed Washburn solution.
• Force balance: surface-tension force 2π r γ cosθ against the weight of the lifted column.
• Scaling: h_J r is the same in all three tubes.

Honest simplifications

Quasi-steady Poiseuille flow, constant contact angle, no dynamic contact-angle effect, no inertia at very early time, no evaporation. Tube widths are drawn enlarged for visibility.

• Pick Water: a thin tube lifts the liquid much higher than a wide one but needs a few seconds.
• Pick Oil: viscosity makes the climb slow although the final height is set only by surface tension and density.
• Lower gravity to the Moon value and the equilibrium height grows by the same factor.