PHY.9:5.2 - Separate pressure from the probe’s surface force

A large liquid reservoir maintains an unknown gauge pressure p during a small probe’s readout. A vertical circular capillary opens to the atmosphere. Let rho be liquid density, g gravitational acceleration, r the tube radius and h the meniscus height above the pressure reference. Assume hydrostatic equilibrium and a capillary regime in which the meniscus curvature is described by the wetting angle theta.

Hydrostatic pressure and the surface-pressure jump give

p=rho g h - s/r, where s=2 gamma cos(theta)

and gamma is surface tension. The height responds to the wanted pressure and to the capillary surface force.

If s is unknown but the same surface condition can be prepared in two narrow tubes of radii r_1 and r_2, use

rho g h_1=p+s/r_1, rho g h_2=p+s/r_2.

For distinct radii,

p=rho g (r_1 h_1-r_2 h_2)/(r_1-r_2).

Take rho g=10000 Pa/m, r_1=0.1 mm, r_2=0.2 mm, h_1=0.20 m and h_2=0.15 m. The inferred pressure is 1000 Pa and s=0.10 N/m. Interpreting the first height as p=rho g h_1 would give 2000 Pa. Both narrow radii must remain within the chosen capillary approximation.

Now suppose the two surfaces have different, unknown wetting conditions. The equations contain separate s_1 and s_2. The cancellation no longer determines p: two readings with three unknown quantities leave a consequential freedom. Restore the common surface condition, use useful bounds on the surface terms, or select another pressure-sensitive interaction.

The reservoir premise also matters. If filling the capillary changes the pressure whose earlier value was wanted, include the reservoir-probe volume and pressure relation through PHY.6 and C.16.MR. The maintained-pressure calculation cannot by itself recover that earlier value.