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Sandbox Physics
Dynamic interfaces

L3 · Chapter 5

How a temperature difference moves a liquid surface

Sandbox Physics ·

Heating a small patch usually sends liquid outward across a thin film. The warm surface has lower tension, so its cooler surroundings pull harder. As liquid leaves, the film thins and bends. The resulting pressure gradient can drive some of the liquid back below the surface.

The cooler surface pulls harder

Place a narrow heater beneath a liquid film. The patch above it warms before its surroundings. For a clean liquid with a negative temperature derivative of surface tension, the cool side pulls more strongly. The surface moves away from the heater. Flow driven by a gradient of surface tension is called Marangoni flow.

Let xx run right along a horizontal substrate and zz point upward. The wall is z=0z=0 and the surface is z=h(x,t)z=h(x,t). Temperature is TT, surface tension is γ\gamma and dynamic viscosity is μ\mu. Over a modest temperature range, approximate the tension by a straight line with negative slope.

γ(T)=γ0+γT(T−T0),γx=γTTx,γT<0\gamma(T)=\gamma_0+\gamma_T(T-T_0),\qquad \gamma_x=\gamma_TT_x,\qquad \gamma_T<0
Temperature increasing to the right gives tension decreasing to the right. The surface is pulled left.

Sources: Bowen · John W. M. Bush

Surface traction changes the upper boundary condition

The preceding chapter treated a surface with no imposed tangential stress. Now the surface pulls the liquid, and viscous shear must carry that traction. The solid wall still imposes no slip. Pressure remains approximately uniform through the thin layer.

Horizontal speed uu has units of metres per second and pressure pp has units of pascals. Tension has units of newtons per metre, so its horizontal derivative has units of pascals, matching the viscous stress. Negligible inertia and small physical slopes reduce the momentum equation to the form below.

μuzz=px,u(0)=0,μuz(h)=γx\mu u_{zz}=p_x,\qquad u(0)=0,\qquad \mu u_z(h)=\gamma_x
The wall fixes the velocity; the surface fixes its vertical derivative.
uz=pxμ(z−h)+γxμ,u=px2μ(z2−2hz)+γxμzu_z=\frac{p_x}{\mu}(z-h)+\frac{\gamma_x}{\mu},\qquad u=\frac{p_x}{2\mu}(z^2-2hz)+\frac{\gamma_x}{\mu}z
Integrate once and apply the top shear condition, then integrate again and apply wall no slip. Pressure contributes a parabola and surface traction a linear term.

Flux connects surface motion to film thickness

A velocity profile does not yet tell us how quickly the film rises or falls. Integrate the speed across its thickness to obtain volume flow per unit width. A short section becomes thinner when more liquid leaves than enters. This sequence, from velocity to flux to height, connects the temperature field to the interface.

At leading order in a weak variation of tension, use γ0\gamma_0 in the curvature pressure and retain the thermal variation in the shear. The Lab uses an 8 mm8\,\mathrm{mm} periodic strip initially 50 μm50\,\mathrm{\mu m} thick. Temperature is imposed externally. Modeling its transport by the flow and heat exchange with the substrate would require an additional heat equation.

q=∫0hu dz=−h33μpx+h22μγx,p=−γ0hxx,ht=−qxq=\int_0^h u\,dz=-\frac{h^3}{3\mu}p_x+\frac{h^2}{2\mu}\gamma_x,\qquad p=-\gamma_0h_{xx},\qquad h_t=-q_x
Periodic endpoints have equal flux, preserving total liquid volume. The displayed surface color and shear use the same temperature field.

Sources: Bowen

Calculate the first motion of a flat film

Choose h=50 μmh=50\,\mathrm{\mu m}, μ=0.10 Pa s\mu=0.10\,\mathrm{Pa\,s} and γT=−6.0×10−5 N m−1K−1\gamma_T=-6.0\times10^{-5}\,\mathrm{N\,m^{-1}K^{-1}}. Temperature rises by 4.0 K4.0\,\mathrm K from left to right over 2.0 mm2.0\,\mathrm{mm}. Initially the film is flat, giving zero curvature-pressure gradient. Divide the temperature difference by distance, then multiply by the tension derivative to obtain the surface stress.

The surface moves left, with progressively slower flow toward the wall. The mean speed is half the surface speed. Multiply it by film thickness to obtain the flux. Once the interface bends, pressure also contributes, and these initial speeds change.

Tx=2.0×103 K m−1,γx=−0.120 PaT_x=2.0\times10^3\,\mathrm{K\,m^{-1}},\qquad \gamma_x=-0.120\,\mathrm{Pa}
Warming points right and the surface traction points left.
u(h)=γxhμ=−0.0600 mm s−1,uˉ=−0.0300 mm s−1,q=−1.50×10−9 m2 s−1u(h)=\frac{\gamma_xh}{\mu}=-0.0600\,\mathrm{mm\,s^{-1}},\quad \bar u=-0.0300\,\mathrm{mm\,s^{-1}},\quad q=-1.50\times10^{-9}\,\mathrm{m^2\,s^{-1}}
These values describe a locally flat initial patch. On opposite sides of a hot spot the gradients have opposite signs, sending surface flow outward in both directions.

A moving surface can sit above a return flow

Continued surface flow deforms the film. Curvature pressure then pushes liquid through the lower layers. Consider a local instant with zero net flux: the surface still feels traction, but its integrated flow is canceled by the pressure contribution.

Insert that zero-flux condition into the velocity profile. The sign changes at z=2h/3z=2h/3. The upper third follows the surface traction and the lower part returns in the opposite direction. This is a local example of recirculation; a periodically heated film need not approach this condition everywhere.

q=0 ⟹ px=3γx2h,u(z)=γxμ(3z24h−z2)q=0\ \Longrightarrow\ p_x=\frac{3\gamma_x}{2h},\qquad u(z)=\frac{\gamma_x}{\mu}\left(\frac{3z^2}{4h}-\frac z2\right)
Surface traction carries the upper layer, and the pressure gradient can return liquid below.

Predict, change and explain

Predict the response to a hot or cold patch

  1. Choose A hot spot clears the film. Sketch the surface-flow direction on either side before comparing it with the streamlets and minimum-thickness history.
  2. Reverse only the temperature difference. Predict whether the center thickens or thins, then choose A cold spot gathers liquid to check.
  3. Drag the spot to one side. Notice the immediate velocity response and the later height response. Pause and compare upper and lower flow, then increase viscosity and explain the slower deformation.

Exercises and explanations

  1. Keep the temperature gradient and initial flat height fixed. What happens to surface speed and flux if viscosity triples?

    Read the explanation

    Both are inversely proportional to viscosity, so they fall to one third. At the same observation time, the liquid has had less time to develop a large height change.

  2. Double the initial flat-film thickness at fixed temperature gradient. By what factors do surface speed and flux change?

    Read the explanation

    With no initial pressure gradient, surface speed is proportional to thickness and doubles. Flux is proportional to thickness squared and increases fourfold. Later deformation requires including the pressure term as well.

  3. For the worked parameters, find the pressure gradient required for local zero flux and the resulting surface speed.

    Read the explanation

    Zero flux requires px=−3.60×103 Pa m−1p_x=-3.60\times10^3\,\mathrm{Pa\,m^{-1}}. Substitution gives surface speed −0.0150 mm s−1-0.0150\,\mathrm{mm\,s^{-1}}, one quarter of its initial pressure-free value. The lower return flow cancels the upper flux.

Sources and calculation scope

Examples and diagrams are newly authored. The paper comparison recalculates analytic relations; the research reading distinguishes those curves from experimental photographs and molecular simulations.

  1. John W. M. Bush, Surface Tension Module, MIT lecture notes

    Lecture 1, sections 1.1–1.3: interfacial energy, normal and tangential stress, and capillary scaling. Teaching source; the examples here are newly calculated.

  2. Bowen, Sur, Bertozzi & Behringer, Nonlinear dynamics of two-dimensional undercompressive shocks, Physica D 209, 36–48 (2005)

    Section 2, equation (2.1), identifies the thermal-stress, gravity and capillary fluxes. The horizontal, prescribed-temperature browser model retains the thermal and capillary terms. The paper’s inclined fronts and experimental images are not reproduced.

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