M099 · Fluid mechanics

Surface Tension & Capillary Laboratory

Compare five capillary bores, inspect real spherical menisci and switch imposed advancing or receding contact angles. Acquire volume-mean 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 $\text{Ready}$

GEOMETRY / PROBE / INFERENCE

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 is held at zero height. Five open bores share one liquid and imposed contact angle; radii are 0.6, 0.8, 1, 1.3 and 1.6 times the reference. Positive height means rise, negative height depression. Scene proportions are uniform; tube walls, clamps and the optical probe are schematic.

The spherical meniscus is a low-Bond approximation. Its volume-mean height balances the signed vertical surface force; centre and wall heights differ. Acquisition requires tube Bond number at most 0.1; larger tubes keep dashed candidate views. This screen is not an error bound or a gravity-deformed Young–Laplace solution.

Advancing and receding buttons impose the two contact-angle endpoints; they do not simulate motion or pinning. Optical heights are volume means referenced to the reservoir, not centre pixels. An affine height fit against inverse radius estimates wetting tension. Tension alone needs known angle; near neutral wetting it is withheld. Configuration changes clear readings; selecting another tube preserves them.

Static low-Bond spherical approximation with a maintained reservoir. Height data identify tension times the contact-angle cosine; tension alone needs independently known contact angle. No contact-line dynamics are solved.

Start with the essentials

Focus question: Which liquid height belongs in the force balance?
One-sentence intuition: Infer parameters from independent raw channels and check the approximation domain first.

Core mathematical model

Mean column force balance

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

Height is a signed volume mean relative to the maintained reservoir.

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 low-Bond spherical reference; rise and depression reverse its sign.

Gravity compared with capillarity

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

The spherical shape neglects gravity variation across the meniscus. Acquisition is screened at 0.1; no universal error tolerance is implied.

What the height record identifies

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

Infer wetting tension with a free height intercept. Dividing by a separately known contact-angle cosine gives tension, except near neutral wetting.

Imposed angle endpoints

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

The branch is receding, central or advancing. This is an imposed static comparison, not simulated pinning or wetting dynamics.

Common difficulties

Reference versus inference

Typical misconceptionThe estimate can be read without recording data.

Better mental modelClear the record and the estimate disappears. Sensor perturbations can produce nonphysical fits; estimates are not clipped or independently calibrated.

Run the experiment

01
Compare tube radii
Select Wetting rise; inspect the five menisci and the measurement-station view.
What to observe: Narrower tubes rise higher. The centre height differs from the volume mean.
02
Reverse wetting
Compare Neutral wetting and Nonwetting depression.
What to observe: Curvature, signed height and pressure jump change together.
03
Compare history endpoints
Select Angle hysteresis and switch advancing to receding.
What to observe: The same liquid and radius support different imposed static heights.
04
Acquire a tube array
Acquire eligible tubes, add height perturbations and export CSV.
What to observe: The affine inverse-radius fit estimates wetting tension; neutral height does not identify tension alone.