Young-Laplace Capillary Pressure & Jurin's Law Capillary Rise

Separate two different radii

For a single spherical droplet interface, the inside-minus-outside pressure is 2×surface tension/R, where R is the sphere radius. A circular tube has a different geometric input: its inner radius r. The meniscus pressure also depends on the contact angle.

In this workbook B3 is tube radius, B4 is contact angle measured through the liquid, and B7 is an independent droplet radius. Equal initial values for B3 and B7 do not make them interchangeable. The general Young–Laplace relation uses two principal curvatures; 2×tension/R assumes they are equal.

Calculate signed tube pressure

E2 computes gas-side pressure minus liquid-side pressure just across the meniscus: 2×tension×cos(theta)/r. The units are Pa when tension is in N/m and radius in metres.

At theta=0°, the initial inputs give +291.2 Pa. Above 90° the result is negative, corresponding to capillary depression under the stated convention. At exactly 90° the formula returns exactly zero, avoiding floating-point cosine residue.

=IF(COUNT(B2:B4)<>3,"Enter numeric tension,radius,angle",IF(OR(B2<0,B3<=0,B4<0,B4>180),"Require tension>=0,radius>0,angle 0..180",IF(B4=90,0,2*B2*COS(RADIANS(B4))/B3)))

Balance pressure with hydrostatic head

E3 uses h=E2/(rho×g) and converts metres to millimetres. Positive h means a rise above the large reservoir level; negative h means depression. Initially h rounds to +29.74 mm. At the illustrative angle 110°, it rounds to −10.17 mm.

This idealized equilibrium model assumes a narrow vertical circular tube, a large reservoir, constant tension and density, a prescribed contact angle and negligible gas density. It neglects the small meniscus-volume correction. It does not predict filling speed, contact-angle hysteresis or biological behavior.

=IF(NOT(ISNUMBER(E2)),E2,IF(COUNT(B5:B6)<>2,"Enter numeric density and gravity",IF(OR(B5<=0,B6<=0),"Require density and gravity>0",1000*E2/(B5*B6))))

Compare a separate spherical droplet

E4 uses B7 for a single spherical interface. It initially returns 291.2 Pa and is independent of the tube contact angle. Changing B4 to 110° changes E2 and E3, but must leave E4 unchanged.

This is not the tube pressure at arbitrary angle. A thin soap film with two interfaces has a different pressure relation and is not modeled by this output.

=IF(COUNT(B2,B7)<>2,"Enter numeric tension and sphere radius",IF(OR(B2<0,B7<=0),"Require tension>=0 and sphere radius>0",2*B2/B7))

Test units and boundaries

Enter 0.5 mm as 0.0005 m. The radius must be positive, density and gravity positive, and angle between 0° and 180°. Surface tension zero is accepted as a mathematical zero-capillarity limit. Missing and text inputs are reported.

Double the tube radius: the predicted pressure and height halve while droplet pressure stays unchanged. Double only the droplet radius: E4 halves while the tube outputs stay unchanged. The chosen 110° angle is a supplied example, not a universal contact angle for a named coating.

Further reading

• OpenStax — Surface Tension and Capillary Action
• Richard Fitzpatrick, University of Texas — Jurin’s Law
• MIT — Interfacial Phenomena lecture notes