---
title: "Marangoni Surface Flow Interactive Lab | Sandbox Physics"
description: "Why does a warm spot push a thin liquid film aside? Surface tension is usually weaker in the warmer patch."
canonical_url: "https://sandboxphysics.com/labs/marangoni-flow/"
language: "en"
page_type: "lab"
alternate_languages:
  zh-CN: "https://sandboxphysics.com/zh/labs/marangoni-flow/"
last_modified: "2026-10-28"
---

FR18 · A surface pulled by temperature

# Marangoni Surface Flow

**Why does a warm spot push a thin liquid film aside?** Surface tension is usually weaker in the warmer patch. The cooler surface pulls liquid outward, thinning the film over the hot spot. As the surface bends, curvature pressure drives a competing flow below it.

Drag a warm or cold spot across a liquid film. Watch the surface pull toward stronger tension, the film reshape and pressure build a return flow.

> Interactive simulation: open this page in a web browser to change the parameters and run the model.

Model assumptions

A one-dimensional, small-slope Newtonian film evolves on a periodic substrate. Temperature is prescribed and surface tension falls linearly with temperature; its small variation drives shear, while curvature pressure uses the reference tension. The model omits thermal transport, evaporation, surfactants, gravity and contact-line motion. The run stops before excessive thinning or slope. Height is enlarged in the cross-section.

## The film remembers where the heat went

Cyan: saved film thickness. Amber: imposed temperature. The time series comes from the same evolving film shown above; these are simulated data.

Thickness and temperature along the strip $h/H,\quad (T-T_0)/(8\,\mathrm K)$

> Computed film thickness and prescribed temperature.

Minimum thickness through time $h_{\min}/H$

> Minimum thickness extracted from saved simulated height records.

Surface work and viscosity

With no thermal forcing, capillary surface energy decreases. Heating supplies surface work, so surface energy need not decrease during a driven run. The export contains the energy, work rate, dissipation and all saved height samples.

$$
\frac{d}{dt}\int\frac{h_x^2}{2}\,dx=\mathcal P_{\mathrm{surface}}-\mathcal D
$$

Physics tutorial

## A temperature difference can pull a liquid surface

**Background** A clean liquid surface usually pulls harder when it is cooler. Heat one small patch of a thin film and the surrounding surface pulls liquid away from it.

**Why it matters** A film can move even on a horizontal substrate. The driving force acts along its free surface, so the direction of the temperature gradient matters.

Start with the essentials

- **Focus question**: Why does a warm spot push a thin liquid film aside?
- **One-sentence intuition**: The surface moves toward stronger surface tension, and curvature pressure responds to the deformation that follows.

### Core mathematical model

#### Temperature sets a tangential surface stress

$$
\gamma(T)=\gamma_0+\gamma_T(T-T_0),\qquad \mu\,\partial_z u\big|_{z=h}=\partial_x\gamma,\qquad \gamma_T<0
$$

The coordinate x runs along the horizontal substrate and z points upward. Surface tension is gamma, dynamic viscosity is mu, and u is horizontal speed. A negative temperature derivative means the cold side pulls harder.

#### Pressure and surface shear contribute to the same flux

$$
p=-\gamma_0\partial_{xx}h,\qquad q=-\frac{h^3}{3\mu}\partial_xp+\frac{h^2}{2\mu}\partial_x\gamma,\qquad \partial_t h=-\partial_xq
$$

Film height is h and q is volume flow per unit width. The first flux term responds to curvature pressure; the second carries liquid toward increasing surface tension. Periodic boundaries keep the total liquid volume fixed.

#### The velocity vanishes at the solid wall

$$
u(x,z)=\frac{\partial_xp}{2\mu}(z^2-2hz)+\frac{\partial_x\gamma}{\mu}z,\qquad 0\le z\le h
$$

Integrating this profile over the film gives the displayed flux. The solid wall imposes no slip, and the top slope of the velocity profile supplies the prescribed surface stress.

### Common difficulties

#### The surface and the interior need not move together

**Typical misconception** Every layer must follow the outward motion at the surface.

**Better mental model** Deformation creates a pressure gradient. Its parabolic velocity contribution can oppose the surface-driven motion and produce a return flow in the lower film.

### Run the experiment

1. 01

   #### Watch a hot spot thin the film

   Choose A hot spot clears the film and watch the bright surface streamlets.

   **What to observe:** The surface moves away from the warm center, where the film becomes thinner. Total liquid volume stays fixed.

2. 02

   #### Reverse the thermal pull

   Choose A cold spot gathers liquid and compare the surface-flow direction.

   **What to observe:** The sign change reverses the surface stress and gathers liquid under the cold patch.

3. 03

   #### Move the source

   Drag the glowing spot to one side, then pause to compare the surface and the lower film.

   **What to observe:** The velocity responds immediately to the new temperature gradient. The height profile changes over the subsequent viscous flow time.

Physical conclusion

Temperature sets the surface shear; conservation of liquid turns that flow into a changing film thickness.

## Related experiments and articles

Dynamic interfaces · Reading path

### Curvature, films, thermal flow and liquid threads

Articles on this path that use this experiment, followed by the other experiments they suggest.

1. [Article **How a temperature difference moves a liquid surface**](https://sandboxphysics.com/stories/dynamic-interfaces/thermal-surface-flow/)

Motion & dynamics · Starter path

## From a falling ball to an orbit

Start by measuring gravity and splitting motion into components. Then add contact forces and collisions, and follow the same laws all the way to an orbit.

[Start here → **Free Fall & Measuring Gravity** Capture a few drops and fit the camera positions to estimate gravitational acceleration. Then add air damping and watch the fit drift from the truth. Observe · 14 min](https://sandboxphysics.com/labs/free-fall/)

1. [1 Free Fall & Measuring Gravity](https://sandboxphysics.com/labs/free-fall/)
2. [2 Projectile Flight Designer](https://sandboxphysics.com/labs/projectile-flight/)
3. [3 Inclined Plane & Friction](https://sandboxphysics.com/labs/inclined-plane/)
4. [4 One-Dimensional Collision Bench](https://sandboxphysics.com/labs/collision-bench/)
5. [5 Mass–Spring Oscillator](https://sandboxphysics.com/labs/mass-spring/)
6. [6 How Orbits Take Shape](https://sandboxphysics.com/labs/orbit/)

[View the full path →](https://sandboxphysics.com/topics/mechanics/#starter-path)