M099 · Fluid mechanics

Surface Tension & Capillary Laboratory

Can the height a liquid climbs in a thin tube tell surface tension apart from wetting? Not from the height alone. The rise in a narrow tube depends only on surface tension times the cosine of the contact angle, a combination called the wetting tension. A liquid with weak tension that wets well and one with strong tension that wets poorly can climb to the same height.

Compare capillary tubes of five different bores, look at the actual spherical meniscus in each and switch the imposed contact angle between advancing and receding. Measure the volume-averaged heights to infer the wetting tension.

Volume-mean height $\text{Ready}$

Meniscus centre height $\text{Ready}$

Contact-line height $\text{Ready}$

Imposed branch angle $\text{Ready}$

Liquid-minus-gas pressure jump $\text{Ready}$

Tube Bond number $\text{Ready}$

Signed vertical tension force $\text{Ready}$

Force closure residual $\text{Ready}$

Wetting tension inferred from heights $\text{Ready}$

Tension with independently known angle $\text{Ready}$

Acquired stations $\text{Ready}$

Raw-coordinate fit residual $\text{Ready}$

Inference status $\text{Ready}$

Model domain $\text{Ready}$

Set the geometry, then infer from readings

What does the liquid level reveal?

Reference tube radius

r_0

0.50\,mm

Surface tension

\gamma

72.80\,mN/m

Central contact angle

\theta_0

25.00\,{}^\circ

Imposed angle interval

\Delta\theta

0.00\,{}^\circ

Branch: receding, central, advancing

b

0.00\,

Density

\rho

1000.00\,kg/m^3

Gravity

g

9.81\,m/s^2

Selected tube fraction

\xi_p

0.50\,

Height perturbation gain

\epsilon

0.00\,

The reservoir surface is held at zero height. Five open tubes share one liquid and one set contact angle, with radii of 0.6, 0.8, 1, 1.3 and 1.6 times the reference. A positive height means the liquid rises; a negative height means it is pushed down. The scene is drawn to one uniform scale, but the tube walls, clamps and optical probe are schematic.

Treating the meniscus as part of a sphere is an approximation for small Bond number. Its volume-mean height is set by balancing the vertical surface force, which can point up or down; the heights at the centre and at the wall differ. Acquisition needs a tube Bond number of at most 0.1, and wider tubes show only dashed candidate views. This cutoff is not an error bound, and the model does not solve the Young–Laplace equation with gravity deforming the surface.

The advancing and receding buttons set the contact angle to either end of its range; they do not simulate the contact line moving or pinning. Optical heights are volume averages measured from the reservoir surface, not the height at the centre pixel. A straight-line fit of height against one over the radius estimates the wetting tension. Finding the surface tension alone needs a known angle, and near neutral wetting no value is given. Changing the setup clears the readings; selecting another tube keeps them.

The model is static: at low Bond number the meniscus is taken as spherical, and the reservoir level is held fixed. Height data determine the surface tension times the cosine of the contact angle; finding the tension alone needs a contact angle known independently. Contact-line dynamics are not solved.

Start with the essentials

Focus question: Which liquid height belongs in the force balance?
One-sentence intuition: Check first that the approximation applies, then infer parameters from independent raw readings.

Core mathematical model

Force balance on the average column

\rho g\pi r^2\overline h=2\pi r\gamma\cos\theta

The height is the volume-averaged height measured from the fixed reservoir level, and it is negative below that level.

Spherical meniscus

\begin{aligned}\overline h&=\frac{2\gamma\cos\theta}{\rho gr}\\\Delta p&=p_l-p_g=-\frac{2\gamma\cos\theta}{r}\end{aligned}

The pressure jump is the reference value for a spherical meniscus at low Bond number. Its sign is opposite for a rising and a depressed column.

Gravity compared with capillarity

\mathrm{Bo}=\rho gr^2/\gamma

A spherical shape ignores how gravity varies across the meniscus. Tubes are measured only when this number is at most 0.1; that cutoff is not a universal error tolerance.

What the heights can determine

\begin{aligned}\overline h_i&=b+S/r_i\\\widehat{\gamma\cos\theta}&=\rho gS/2\end{aligned}

Fit the heights with a free intercept to infer the wetting tension. Dividing by a separately known cosine of the contact angle gives the tension, except near neutral wetting.

The imposed contact angle

\theta=\operatorname{clip}\!\left(\theta_0+b\Delta\theta/2,0^\circ,180^\circ\right)

The angle takes its receding, central or advancing value. These are static cases set by hand; pinning and wetting dynamics are not simulated.

Common difficulties

A reference curve is not an estimate

Typical misconceptionYou can read off the estimate without recording any data.

Better mental modelClear the recorded data and the estimate disappears. Sensor perturbations can produce unphysical fits; the estimates are neither clipped nor independently calibrated.

Run the experiment

01
Compare tube radii
Select Wetting rise, then look at the five menisci and the measurement-station view.
What to observe: Liquid rises higher in narrower tubes. The height at the centre differs from the volume-averaged height.
02
Reverse wetting
Compare Neutral wetting and Nonwetting depression.
What to observe: The curvature, the signed height and the pressure jump all change together.
03
Compare advancing and receding angles
Select Angle hysteresis and switch advancing to receding.
What to observe: With the same liquid and the same radius, each imposed angle holds the column at a different static height.
04
Measure the tube array
Acquire eligible tubes, add height perturbations and export CSV.
What to observe: A straight-line fit of height against inverse radius estimates the wetting tension. At neutral wetting the height cannot determine the tension by itself.

Surface Tension & Capillary Laboratory

A meniscus as a force balance

Core mathematical model

Force balance on the average column

Spherical meniscus

Gravity compared with capillarity

What the heights can determine

The imposed contact angle

Common difficulties

A reference curve is not an estimate

Run the experiment

Compare tube radii

Reverse wetting

Compare advancing and receding angles

Measure the tube array

Related experiments and articles

Curvature, films, thermal flow and liquid threads