^{1}and Jean-Pierre Delville

^{1}

### Abstract

The deformation of a fluid-fluidinterface due to the thermocapillary stress induced by a continuous Gaussian laser wave is investigated analytically. We show that the direction of deformation of the liquidinterface strongly depends on the viscosities and the thicknesses of the involved liquid layers. We first investigate the case of an interface separating two different liquid layers while a second part is dedicated to a thin film squeezed by two external layers of same liquid. These results are predictive for applications fields where localized thermocapillary stresses are used to produce flows or to deform interfaces in presence of confinement, such as optofluidics.

I. INTRODUCTION

II. TWO-FLUID THEORETICAL MODEL

A. Heat equations

B. Fluid equations of motion

C. Velocity field

D. Pressure field

E. Interface deflection

III. RESULTS AND DISCUSSION

A. Double layer system

1. Flow pattern

2. Influence of the heating length

3. Influence of the liquid layer thickness ratio

4. Influence of the viscosity ratio

B. Triple layer system

1. Influence of the film thickness

2. Inducing a dimple

3. Inducing noses

IV. CONCLUSION

### Key Topics

- Viscosity
- 20.0
- Liquid thin films
- 19.0
- Thermocapillary flows
- 18.0
- Laser heating
- 16.0
- Liquid liquid interfaces
- 15.0

## Figures

Schematic representation of the liquid-liquid system heated by a laser beam. See text for other notations. σ^{+} indicated that the interfacial tension increases with temperature. In this configuration, we arbitrarily consider ∂σ/∂*T* > 0, a situation often encountered in microfluidics due to the presence of large amount of surfactant.

Schematic representation of the liquid-liquid system heated by a laser beam. See text for other notations. σ^{+} indicated that the interfacial tension increases with temperature. In this configuration, we arbitrarily consider ∂σ/∂*T* > 0, a situation often encountered in microfluidics due to the presence of large amount of surfactant.

Steady flow pattern in a double layer configuration for *H* _{1} = 5*H* _{2} = 5*w*, α = 1, and η_{2} = η_{1}.

Steady flow pattern in a double layer configuration for *H* _{1} = 5*H* _{2} = 5*w*, α = 1, and η_{2} = η_{1}.

Reduced radial velocity *u* _{ r }/*u* _{0} (top left), reduced axial velocity *u* _{ z }/*u* _{0} (top right), and interface deformation (bottom) for *H* _{1} = 5*H* _{2} = 5*w*, α = 1, and η_{2} = η_{1}.

Reduced radial velocity *u* _{ r }/*u* _{0} (top left), reduced axial velocity *u* _{ z }/*u* _{0} (top right), and interface deformation (bottom) for *H* _{1} = 5*H* _{2} = 5*w*, α = 1, and η_{2} = η_{1}.

Variations of the radial (continuous) and the axial (dashed with symbols) velocities versus the fluid layer reduced thicknesses *H* _{1}/*w* (*H* _{1} = *H* _{2} and *r* = 0.1 *w*). The right inset shows the reduced location *z* _{0}/*w* of null radial velocity and maximum axial velocity (*z* _{0} = *z*(*u* _{ r } = 0) = *z*(*u* _{ zmax })). α = 1 and η_{2} = η_{1}. The left inset shows the reduced deformation height *h*/*w* as a function of *H* _{1}/*w* for η_{2} = 10η_{1}.

Variations of the radial (continuous) and the axial (dashed with symbols) velocities versus the fluid layer reduced thicknesses *H* _{1}/*w* (*H* _{1} = *H* _{2} and *r* = 0.1 *w*). The right inset shows the reduced location *z* _{0}/*w* of null radial velocity and maximum axial velocity (*z* _{0} = *z*(*u* _{ r } = 0) = *z*(*u* _{ zmax })). α = 1 and η_{2} = η_{1}. The left inset shows the reduced deformation height *h*/*w* as a function of *H* _{1}/*w* for η_{2} = 10η_{1}.

Ratio of maximum axial velocities in each fluid *u* _{ z1}/*u* _{ z2} as a function of *H* _{1}/*H* _{2} in a double layer system. Inset shows the reduced deformation height *h*/*H* _{1} as a function of *H* _{1}/*H* _{2}. *H* _{1} = *w*, *r* = 0.1*w* and η_{2} = η_{1}.

Ratio of maximum axial velocities in each fluid *u* _{ z1}/*u* _{ z2} as a function of *H* _{1}/*H* _{2} in a double layer system. Inset shows the reduced deformation height *h*/*H* _{1} as a function of *H* _{1}/*H* _{2}. *H* _{1} = *w*, *r* = 0.1*w* and η_{2} = η_{1}.

Reduced maximum velocity *u* _{ max }/*u* _{0} as a function of viscosity ratio η_{2}/η_{1} with η_{1} = *cste*. The inset shows the reduced deformation height *h*/*w* as a function of 1/(1 + η_{2}/η_{1}). *H* _{1} = *H* _{2}.

Reduced maximum velocity *u* _{ max }/*u* _{0} as a function of viscosity ratio η_{2}/η_{1} with η_{1} = *cste*. The inset shows the reduced deformation height *h*/*w* as a function of 1/(1 + η_{2}/η_{1}). *H* _{1} = *H* _{2}.

Steady flow pattern in a triple layer configuration for *H* _{1} = 0.2*w*, *H* _{2} = *w*, α = 1, and η_{2} = η_{1}.

Steady flow pattern in a triple layer configuration for *H* _{1} = 0.2*w*, *H* _{2} = *w*, α = 1, and η_{2} = η_{1}.

Reduced maximum velocity *u* _{ r1max }/*u* _{ r }(*z* = 0) as a function of *H* _{1}/*H* _{2} in a triple layer system. Inset shows the reduced deformation height *h*/*H* _{1} as a function of *H* _{1}/*H* _{2}. *H* _{1} = *w* and η_{2} = η_{1}.

Reduced maximum velocity *u* _{ r1max }/*u* _{ r }(*z* = 0) as a function of *H* _{1}/*H* _{2} in a triple layer system. Inset shows the reduced deformation height *h*/*H* _{1} as a function of *H* _{1}/*H* _{2}. *H* _{1} = *w* and η_{2} = η_{1}.

Reduced radial velocity *u* _{ r }/*u* _{0} (top left), reduced axial velocity *u* _{ z }/*u* _{0} (top right), and interface deformation (bottom) in a three-layer system for *H* _{1} = 0.1*w*, *H* _{2} = 2*w*, α = 0.1, and η_{2} = η_{1}.

Reduced radial velocity *u* _{ r }/*u* _{0} (top left), reduced axial velocity *u* _{ z }/*u* _{0} (top right), and interface deformation (bottom) in a three-layer system for *H* _{1} = 0.1*w*, *H* _{2} = 2*w*, α = 0.1, and η_{2} = η_{1}.

Reduced radial velocity *u* _{ r }/*u* _{0} (top left), reduced axial velocity *u* _{ z }/*u* _{0} (top right), and interface deformation (bottom) in a three-layer system for *H* _{1} = 0.1*w*, *H* _{2} = 2*w*, α = −0.075, and η_{2} = η_{1}.

Reduced radial velocity *u* _{ r }/*u* _{0} (top left), reduced axial velocity *u* _{ z }/*u* _{0} (top right), and interface deformation (bottom) in a three-layer system for *H* _{1} = 0.1*w*, *H* _{2} = 2*w*, α = −0.075, and η_{2} = η_{1}.

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