Surface Tension and Capillarity

Surface tension $\gamma$ (N/m) arises because molecules at a fluid surface have fewer neighbours than those in the bulk, creating a net inward force. The surface behaves like an elastic membrane that resists stretching.

Capillary Rise

A liquid in a narrow tube rises (or falls) due to the balance between surface tension and gravity. The equilibrium height is:

$h = \frac{2\gamma\cos\theta}{\rho g r}$

where:

• $\gamma$ — surface tension (N/m)
• $\theta$ — contact angle between the liquid and the tube wall
• $\rho$ — liquid density (kg/m³)
• $g = 9.81$ m/s²
• $r$ — tube radius (m)

Water on glass has $\theta \approx 0°$ ($\cos\theta = 1$), so it rises. Mercury on glass has $\theta \approx 140°$ ($\cos\theta < 0$), so it is depressed.

Pressure Inside Curved Surfaces

Surface tension creates a pressure jump across a curved interface (Young-Laplace equation). For a spherical bubble in a liquid (two surfaces):

$\Delta P = \frac{4\gamma}{r}$

For a spherical drop in air (one surface):

$\Delta P = \frac{2\gamma}{r}$

Your Task

Implement:

• capillary_rise(gamma, theta_deg, rho, r) — capillary rise height $h$ (m). Accept $\theta$ in degrees. Use $g = 9.81$ m/s² inside the function.
• bubble_pressure(gamma, r) — excess pressure inside a bubble $\Delta P$ (Pa)
• drop_pressure(gamma, r) — excess pressure inside a drop $\Delta P$ (Pa)