In a sufficiently narrow (i.e., low Bond number) tube of circular cross-section (radius a), the interface between two fluids forms a meniscus that is a portion of the surface of a sphere with radius R. The pressure jump across this surface is related to the radius and the surface tension γ by
This may be shown by writing the Young–Laplace equation in spherical form with a contact angle boundary condition and also a prescribed height boundary condition at, say, the bottom of the meniscus. The solution is a portion of a sphere, and the solution will exist only for the pressure difference shown above. This is significant because there isn't another equation or law to specify the pressure difference; existence of solution for one specific value of the pressure difference prescribes it.
The radius of the sphere will be a function only of the contact angle, θ, which in turn depends on the exact properties of the fluids and the container material with which the fluids in question are contacting/interfacing:
so that the pressure difference may be written as:
In order to maintain hydrostatic equilibrium, the induced capillary pressure is balanced by a change in height, h, which can be positive or negative, depending on whether the wetting angle is less than or greater than 90°. For a fluid of density ρ:
where g is the gravitational acceleration. This is sometimes known as the Jurin's law or Jurin height after James Jurin who studied the effect in 1718.
For a water-filled glass tube in air at sea level:
*γ = 0.0728 J/m at 20 °C
*θ = 20° (0.35 rad)
*ρ = 1000 kg/m
*g = 9.8 m/s
and so the height of the water column is given by:
Thus for a 2 mm wide (1 mm radius) tube, the water would rise 14 mm. However, for a capillary tube with radius 0.1 mm, the water would rise 14 cm (about 6 inches).