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.

Elapsed time $3.00\,\mathrm{s}$

Thinnest film $40.0\,\mathrm{\mu m}$

Peak surface speed $0.0600\,\mathrm{mm\,s^{-1}}$

Volume retained $100\,\%$

Pull on the surface

Spot temperature change

6

\Delta T\;(\mathrm K)

Dynamic viscosity

0.1

\mu\;(\mathrm{Pa\,s})

Spot position

4

x_H\;(\mathrm{mm})

Drag the glowing spot along the substrate. Change the sign of the temperature difference to gather liquid instead.

The warm patch is thinning as liquid flows outward.

Film model and limits

Spot width

0.800

w_H\;(\mathrm{mm})

The 8 mm periodic strip starts 50 micrometres thick. A prescribed temperature field sets both surface color and shear stress. The capillary pressure uses the reference tension, consistent with the leading-order small temperature-variation approximation.

Thermal transport, gravity, evaporation and surfactants are omitted. Integration stops below 10 micrometres or when the physical slope exceeds 0.22.

Read the dynamic-interface course

Bowen et al. (2005), Eq. (2.1)

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)

Minimum thickness through time

h_{\min}/H

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

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 misconceptionEvery layer must follow the outward motion at the surface.

Better mental modelDeformation 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

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.
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.
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.

Marangoni Surface Flow

The film remembers where the heat went

A temperature difference can pull a liquid surface

Core mathematical model

Temperature sets a tangential surface stress

Pressure and surface shear contribute to the same flux

The velocity vanishes at the solid wall

Common difficulties

The surface and the interior need not move together

Run the experiment

Watch a hot spot thin the film

Reverse the thermal pull

Move the source

Related experiments and articles

Curvature, films, thermal flow and liquid threads