Surface tension is the tendency of a liquid surface to behave like a stretched elastic skin.
It appears because molecules at the surface experience an imbalance of intermolecular forces.
Surface tension is important in capillary tubes, bubbles, droplets, wetting, biological fluids, and many engineering applications.
1. Surface tension force
Surface tension \(\gamma\) is force per unit length:
Surface-tension definition.
\[
\begin{aligned}
\gamma &= \frac{F}{L}.
\end{aligned}
\]
Therefore, the force along a wetted length is
\[
\begin{aligned}
F &= \gamma L.
\end{aligned}
\]
If a thin soap film has two surfaces, the force becomes
\[
\begin{aligned}
F &= 2\gamma L.
\end{aligned}
\]
2. Capillary rise and depression
In a narrow tube, surface tension acts around the contact perimeter of the liquid.
The vertical component of this force is
Vertical surface-tension force.
\[
\begin{aligned}
F_{\gamma,y}
&= 2\pi r\gamma\cos\theta.
\end{aligned}
\]
The weight of the raised liquid column is
\[
\begin{aligned}
W
&= \rho g(\pi r^2h).
\end{aligned}
\]
At equilibrium, the upward surface-tension component balances the column weight:
Capillary balance.
\[
\begin{aligned}
2\pi r\gamma\cos\theta
&= \rho g\pi r^2h.
\end{aligned}
\]
Solving for \(h\):
Capillary rise/depression formula.
\[
\begin{aligned}
h
&= \frac{2\gamma\cos\theta}{\rho g r}.
\end{aligned}
\]
If \(\theta<90^\circ\), then \(\cos\theta>0\), and the liquid rises.
If \(\theta>90^\circ\), then \(\cos\theta<0\), and the liquid is depressed.
3. Laplace pressure
A curved liquid surface creates a pressure difference.
For a spherical liquid drop with one interface:
\[
\begin{aligned}
\Delta P &= \frac{2\gamma}{r}.
\end{aligned}
\]
For a soap bubble, there are two surfaces, so
\[
\begin{aligned}
\Delta P &= \frac{4\gamma}{r}.
\end{aligned}
\]
More generally, this calculator uses
\[
\begin{aligned}
\Delta P &= k\frac{\gamma}{r},
\end{aligned}
\]
where \(k=2\) for a liquid drop and \(k=4\) for a soap bubble.
4. Siphon application
A siphon can be approximated with Bernoulli’s equation.
If the outlet is a vertical distance \(\Delta z\) below the reservoir surface, the ideal exit speed is
\[
\begin{aligned}
v_{\mathrm{ideal}}
&= \sqrt{2g\Delta z}.
\end{aligned}
\]
Real losses can be represented with a discharge coefficient:
\[
\begin{aligned}
v_{\mathrm{actual}}
&= C_d\sqrt{2g\Delta z}.
\end{aligned}
\]
The ideal maximum siphon crest height is limited by atmospheric pressure:
\[
\begin{aligned}
h_{\max}
&= \frac{P_{\mathrm{atm}}}{\rho g}.
\end{aligned}
\]
In real siphons, vapor pressure, dissolved gases, friction, and tube geometry reduce the practical limit.
5. Airplane lift preview
A simple Bernoulli-style lift estimate compares the air speed above and below a wing.
If the top flow is faster than the bottom flow, the pressure above the wing is lower.
Pressure difference from speed difference.
\[
\begin{aligned}
\Delta P
&= \frac{1}{2}\rho_{\mathrm{air}}
\left(v_{\mathrm{top}}^2-v_{\mathrm{bottom}}^2\right).
\end{aligned}
\]
The approximate lift force is
\[
\begin{aligned}
F_L
&= \Delta P A.
\end{aligned}
\]
This is only a simplified model. Real lift also depends on circulation, angle of attack, wing shape, viscosity, separation, and turbulence.
6. Blood-flow application
For a small cylindrical vessel with laminar flow, Poiseuille’s law estimates the volume flow rate:
Poiseuille law.
\[
\begin{aligned}
Q
&= \frac{\pi r^4\Delta P}{8\eta L}.
\end{aligned}
\]
This formula is very sensitive to vessel radius because \(Q\propto r^4\).
If the radius is reduced by a factor of 2, the ideal flow rate drops by a factor of 16.
The average speed is
\[
\begin{aligned}
\bar v
&= \frac{Q}{\pi r^2}.
\end{aligned}
\]
A Reynolds-number check is
\[
\begin{aligned}
Re
&= \frac{\rho \bar v D}{\eta}.
\end{aligned}
\]
7. Worked example: water in a glass capillary
Suppose water is in a clean glass capillary tube with radius \(0.3\ \mathrm{mm}\).
Use:
\[
\begin{aligned}
\gamma &= 72.8\times10^{-3}\ \mathrm{N/m},\\
\rho &= 998\ \mathrm{kg/m^3},\\
g &= 9.81\ \mathrm{m/s^2},\\
r &= 0.3\times10^{-3}\ \mathrm{m},\\
\theta &= 0^\circ.
\end{aligned}
\]
The capillary height is
Substitute into the capillary formula.
\[
\begin{aligned}
h
&= \frac{2\gamma\cos\theta}{\rho g r}\\
&= \frac{2(72.8\times10^{-3})\cos(0^\circ)}
{(998)(9.81)(0.3\times10^{-3})}\\
&\approx 4.96\times10^{-2}\ \mathrm{m}.
\end{aligned}
\]
So the water rises by about \(4.96\ \mathrm{cm}\).
8. Formula summary
| Application |
Formula |
Meaning |
| Surface-tension force |
\(F=n\gamma L\) |
Force along one or two liquid surfaces |
| Capillary rise/depression |
\(h=2\gamma\cos\theta/(\rho g r)\) |
Height change in a narrow tube |
| Liquid drop pressure |
\(\Delta P=2\gamma/r\) |
Pressure inside a spherical drop |
| Soap bubble pressure |
\(\Delta P=4\gamma/r\) |
Two surfaces contribute to pressure jump |
| Siphon speed |
\(v=C_d\sqrt{2g\Delta z}\) |
Approximate outlet speed |
| Siphon height limit |
\(h_{\max}=P_{\mathrm{atm}}/(\rho g)\) |
Ideal atmospheric-pressure limit |
| Lift estimate |
\(F_L=\frac{1}{2}\rho(v_{\mathrm{top}}^2-v_{\mathrm{bottom}}^2)A\) |
Simplified pressure-difference lift |
| Blood-flow preview |
\(Q=\pi r^4\Delta P/(8\eta L)\) |
Laminar cylindrical-vessel flow |
9. Assumptions and cautions
- Capillary formulas assume a clean tube, static equilibrium, and a stable contact angle.
- Surface contamination can strongly change surface tension.
- Laplace pressure assumes simple spherical curvature.
- Siphon estimates ignore friction, vapor pressure, bends, and turbulence.
- The lift estimate is a simplified Bernoulli preview, not a full aerodynamic model.
- The blood-flow preview assumes steady, laminar, Newtonian flow in a cylindrical vessel.
Key idea: surface tension controls interfaces, while pressure, viscosity, geometry, and flow speed control advanced fluid applications.
