^{1}, Alexander Oron

^{2,a)}and Alexander Alexeev

^{1,b)}

### Abstract

Using long wave theory and direct numerical solutions of the Navier–Stokes equations, we investigate thermocapillary flows arising in a thin liquid film covering a heated solid substrate with non-uniform temperature in the form of traveling thermal waves. Our results indicate that unidirectionally propagating interfacial waves are formed in the liquid film. The interfacial waves transport liquid, thereby creating a net pumping effect. We show that the frequency of thermal waves leading to the most efficient pumping is defined by their wave length and weakly depends on other system parameters. The results are useful for designing new methods for transporting liquids in open microfluidic devices.

Funding from United States–Israel Binational Science Foundation under Award No. 2008038 is gratefully acknowledged. The research was also partially supported by IRSES Grant No. 269207 (to A.O.) and by NSF travel Award CBET-1028778 (to A.A.).

I. INTRODUCTION

II. PROBLEM STATEMENT

III. METHODOLOGY

A. Volume-of-fluid method

B. Long-wave approximation

IV. RESULTS AND DISCUSSION

A. Stationary thermal waves

B. Traveling thermal waves

C. Fluid pumping

V. SUMMARY

### Key Topics

- Liquid thin films
- 21.0
- Thermocapillary flows
- 13.0
- Gas liquid interfaces
- 10.0
- Navier Stokes equations
- 9.0
- Viscosity
- 8.0

##### B81B

## Figures

(a) Velocity and temperature distributions in a liquid film heated from below by a stationary thermal wave at the bottom wall with L = 25, a = 0.1, M = 100, δ = 3, H = 0.0152. The thick curve indicates the position of the liquid-gas interface in the steady state. (b) The temperature profile at the bottom wall. The vertical dashed line indicates the border line between the counter-rotating vortices in the liquid film, namely, counterclockwise near the ends of the periodic domain and clockwise in the center of the domain. The bright and dark shades correspond to high and low temperatures, respectively.

(a) Velocity and temperature distributions in a liquid film heated from below by a stationary thermal wave at the bottom wall with L = 25, a = 0.1, M = 100, δ = 3, H = 0.0152. The thick curve indicates the position of the liquid-gas interface in the steady state. (b) The temperature profile at the bottom wall. The vertical dashed line indicates the border line between the counter-rotating vortices in the liquid film, namely, counterclockwise near the ends of the periodic domain and clockwise in the center of the domain. The bright and dark shades correspond to high and low temperatures, respectively.

(a) Time evolution of the film deformation. (b) Film interface deformation at the steady state. The dashed and solid curves denote the results obtained from LW theory and VOF simulation, respectively. The parameter set is L = 25, a = 0.1, M = 100, δ = 3, H = 0.0152.

(a) Time evolution of the film deformation. (b) Film interface deformation at the steady state. The dashed and solid curves denote the results obtained from LW theory and VOF simulation, respectively. The parameter set is L = 25, a = 0.1, M = 100, δ = 3, H = 0.0152.

The case of a stationary temperature wave at the solid-liquid boundary. (a) Film interface deformation Δh 0 = h max − h min as a function of Marangoni number M for different wave periods L and a fixed temperature amplitude a = 0.1. (b) Film deformation Δh 0 as a function of M for different wave amplitudes a and a fixed wave period L = 10. Empty and filled symbols represent the results obtained from LW theory and VOF simulations, respectively. All parameter sets shown in the figure correspond to linearly stable films with h = 1 in the case of a uniform temperature with a = 0 at the solid-liquid boundary. Curves serve to guide the reader's eye.

The case of a stationary temperature wave at the solid-liquid boundary. (a) Film interface deformation Δh 0 = h max − h min as a function of Marangoni number M for different wave periods L and a fixed temperature amplitude a = 0.1. (b) Film deformation Δh 0 as a function of M for different wave amplitudes a and a fixed wave period L = 10. Empty and filled symbols represent the results obtained from LW theory and VOF simulations, respectively. All parameter sets shown in the figure correspond to linearly stable films with h = 1 in the case of a uniform temperature with a = 0 at the solid-liquid boundary. Curves serve to guide the reader's eye.

Snapshot of the velocity and temperature fields for a traveling thermal wave at the solid-liquid boundary for L = 25, a = 0.1, M = 150, ω = 0.0136, δ = 3, H = 0.0152. The bottom panel shows the instantaneous temperature profile at the solid-liquid boundary. Phase lag Δϕ denotes the phase difference between the thermal wave and the film interface. The vertical dashed lines indicate the border line between the counter-rotating vortices, namely counterclockwise is on the left of the center line and clockwise is on the right. The bright and dark shades correspond to high and low temperatures, respectively.

Snapshot of the velocity and temperature fields for a traveling thermal wave at the solid-liquid boundary for L = 25, a = 0.1, M = 150, ω = 0.0136, δ = 3, H = 0.0152. The bottom panel shows the instantaneous temperature profile at the solid-liquid boundary. Phase lag Δϕ denotes the phase difference between the thermal wave and the film interface. The vertical dashed lines indicate the border line between the counter-rotating vortices, namely counterclockwise is on the left of the center line and clockwise is on the right. The bright and dark shades correspond to high and low temperatures, respectively.

Evolution of the liquid film in the case of L = 25, a = 0.1, M = 150, δ = 3, H = 0.0152, and different values of the wave frequency ω of propagating thermal wave. (a) Interface profile of the film at the saturated (steady for ω = 0) state. The inset shows the maximum and the minimum thickness of the film, and the interface deformation as a function of the wave frequency ω. (b) Variation of the interface velocity at the saturated state. The inset shows the magnitudes of positive and negative interface velocities, and the difference between them as a function of wave frequency ω. In both panels, the dashed and solid curves correspond to the results obtained from LW and VOF, respectively. In the insets, the empty and filled symbols correspond to the results obtained with LW and VOF, respectively.

Evolution of the liquid film in the case of L = 25, a = 0.1, M = 150, δ = 3, H = 0.0152, and different values of the wave frequency ω of propagating thermal wave. (a) Interface profile of the film at the saturated (steady for ω = 0) state. The inset shows the maximum and the minimum thickness of the film, and the interface deformation as a function of the wave frequency ω. (b) Variation of the interface velocity at the saturated state. The inset shows the magnitudes of positive and negative interface velocities, and the difference between them as a function of wave frequency ω. In both panels, the dashed and solid curves correspond to the results obtained from LW and VOF, respectively. In the insets, the empty and filled symbols correspond to the results obtained with LW and VOF, respectively.

Film evolution for L = 25, a = 0.1, δ = 3, H = 0.0152 with different wave frequencies ω. (a) Effect of the Marangoni number M on the flow rate q averaged over the periodic domain. (b) Phase difference between the traveling thermal wave and the interfacial wave. (c) Film interface deformation relative to the case of a static thermal wave with the same M. Empty and filled symbols correspond to the results obtained from LW theory and VOF simulations, respectively. Curves serve to guide the reader's eye.

Film evolution for L = 25, a = 0.1, δ = 3, H = 0.0152 with different wave frequencies ω. (a) Effect of the Marangoni number M on the flow rate q averaged over the periodic domain. (b) Phase difference between the traveling thermal wave and the interfacial wave. (c) Film interface deformation relative to the case of a static thermal wave with the same M. Empty and filled symbols correspond to the results obtained from LW theory and VOF simulations, respectively. Curves serve to guide the reader's eye.

Film evolution for L = 10, M = 400, δ = 3, H = 0.0152 with different wave frequencies ω. (a) Effect of the wave amplitude on the flow rate. (b) Phase difference between the traveling thermal wave and the interfacial wave. (c) Film interface deformation relative to the case of a static thermal wave. Empty and filled symbols correspond to LW and VOF, respectively. Curves serve to guide the reader's eye.

Film evolution for L = 10, M = 400, δ = 3, H = 0.0152 with different wave frequencies ω. (a) Effect of the wave amplitude on the flow rate. (b) Phase difference between the traveling thermal wave and the interfacial wave. (c) Film interface deformation relative to the case of a static thermal wave. Empty and filled symbols correspond to LW and VOF, respectively. Curves serve to guide the reader's eye.

Film evolution for a = 0.1, M = 150, δ = 3, H = 0.0152 with different wave frequencies ω. Note that the horizontal axis is in logarithmic scale. (a) Effect of the thermal wave period L on the flow rate q. (b) Phase difference between the traveling thermal wave and the interfacial wave. (c) Variation of the interface deformation relative to the case of a static thermal wave. Empty and filled symbols correspond to LW and VOF, respectively. Curves serve to guide the reader's eye.

Film evolution for a = 0.1, M = 150, δ = 3, H = 0.0152 with different wave frequencies ω. Note that the horizontal axis is in logarithmic scale. (a) Effect of the thermal wave period L on the flow rate q. (b) Phase difference between the traveling thermal wave and the interfacial wave. (c) Variation of the interface deformation relative to the case of a static thermal wave. Empty and filled symbols correspond to LW and VOF, respectively. Curves serve to guide the reader's eye.

Optimal wave frequency as a function of the wave period L. Circle and square symbols represent LW theory and VOF simulations, respectively. The range of parameters is 0.1 ⩽ a ⩽ 0.3 and 100 ⩽ M ⩽ 1500.

Optimal wave frequency as a function of the wave period L. Circle and square symbols represent LW theory and VOF simulations, respectively. The range of parameters is 0.1 ⩽ a ⩽ 0.3 and 100 ⩽ M ⩽ 1500.

## Tables

Dimensionless parameters.

Dimensionless parameters.

Material properties for the bilayer liquid-gas system.

Material properties for the bilayer liquid-gas system.

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