^{1}, Christel Métivier

^{1}, Jean-Michel Piau

^{1}, Albert Magnin

^{1}and Ahmed Abdelali

^{1}

### Abstract

The influence of rheological and interfacial properties of yield stress fluids is investigated on the onset of the Rayleigh-Bénard convection. Different Carbopol® (B.F. Goodrich) gels are used in a circular cell for Rayleigh-Bénard experimental setup. The influence of the boundary conditions is also investigated by controlling either slip or no-slip conditions. The onset of thermoconvection is shown by measuring temperature differences and also by using shadowgraph flow visualization. Experimental results show that convection occurs in the range of our experiments. Considering Carbopol gels as elasto-plastic materials with a yield stress τ y , a generalized Rayleigh number is obtained: Ra g = Y −1, with Y the yield number, which represents the balance between the yield stress of the gel and the buoyancy effects. The results show that the Rayleigh number is proportional to d, the height of the setup, and that the control parameter is the yield number at the onset of convection. Critical values of Y −1 have been determined for slip conditions as well as for no-slip conditions . It highlights that the change in surface conditions affect significantly the critical conditions.

This work was supported by the French National Research Agency (ANR), via the grant called “ThiM.”

I. INTRODUCTION

II. SCIENTIFIC CONTEXT

III. EXPERIMENTAL METHODS

A. Rayleigh-Bénard experimental setup

B. Material and rheological properties

C. Control of interface conditions

IV. RESULTS AND DISCUSSION

V. CONCLUSION

### Key Topics

- Gels
- 51.0
- Yield stress
- 38.0
- Convection
- 31.0
- Shear rate dependent viscosity
- 19.0
- Microgels
- 14.0

##### F15D

## Figures

Rayleigh-Bénard setup.

Rayleigh-Bénard setup.

Rheological properties of Carbopol gels at T = 293 K. (a) Steady-state rheometry: Shear stress as a function of shear rate. (Black diamonds: τ y = 0.0047 Pa, diamonds: τ y = 0.006 Pa, plus: τ y = 0.01 Pa, crosses: τ y = 0.009 Pa, black circles: τ y = 0.031 Pa, triangles: τ y = 0.045 Pa, black squares: τ y = 0.104 Pa, dashed lines: Herschel-Bulkley model). (b) Oscillatory rheometry: Variation of G ′ (black diamonds) and G ′ ′ (white diamonds) as function of the strain for a Carbopol gel τ y = 0.006 Pa, and a frequency f = 0.1 Hz.

Rheological properties of Carbopol gels at T = 293 K. (a) Steady-state rheometry: Shear stress as a function of shear rate. (Black diamonds: τ y = 0.0047 Pa, diamonds: τ y = 0.006 Pa, plus: τ y = 0.01 Pa, crosses: τ y = 0.009 Pa, black circles: τ y = 0.031 Pa, triangles: τ y = 0.045 Pa, black squares: τ y = 0.104 Pa, dashed lines: Herschel-Bulkley model). (b) Oscillatory rheometry: Variation of G ′ (black diamonds) and G ′ ′ (white diamonds) as function of the strain for a Carbopol gel τ y = 0.006 Pa, and a frequency f = 0.1 Hz.

Steady-state response of a Carbopol gel (τ y = 0.031 Pa) when different lower surfaces are used at T = 293 K. In the case of rough or treated surfaces (black circles: sandpaper, white circles: treated PMMA) the abscissa represents the shear rate and the ordinate axis the shear stress. In the case of smooth surfaces or non-treated surface (black diamonds: glass, black triangles: copper alloy, white diamonds: raw PMMA), the results represent an average of the shear stress vs an apparent shear rate.

Steady-state response of a Carbopol gel (τ y = 0.031 Pa) when different lower surfaces are used at T = 293 K. In the case of rough or treated surfaces (black circles: sandpaper, white circles: treated PMMA) the abscissa represents the shear rate and the ordinate axis the shear stress. In the case of smooth surfaces or non-treated surface (black diamonds: glass, black triangles: copper alloy, white diamonds: raw PMMA), the results represent an average of the shear stress vs an apparent shear rate.

Temperature difference ΔT as a function of the total heat input Q t , for different yield stress values of Carbopol gels. (Black diamonds: τ y = 0.0047 Pa, diamonds: τ y = 0.006 Pa, plus: τ y = 0.01 Pa, crosses: τ y = 0.009 Pa, black circles: τ y = 0.031 Pa, triangles: τ y = 0.045 Pa, black squares: τ y = 0.104 Pa ; PMMA surfaces in the cavity-circles with crosses: treated PMMA and τ y = 0.031 Pa, white circles: raw PMMA and τ y = 0.031 Pa). (a) d = 0.017 m; (b) d = 0.03 m.

Temperature difference ΔT as a function of the total heat input Q t , for different yield stress values of Carbopol gels. (Black diamonds: τ y = 0.0047 Pa, diamonds: τ y = 0.006 Pa, plus: τ y = 0.01 Pa, crosses: τ y = 0.009 Pa, black circles: τ y = 0.031 Pa, triangles: τ y = 0.045 Pa, black squares: τ y = 0.104 Pa ; PMMA surfaces in the cavity-circles with crosses: treated PMMA and τ y = 0.031 Pa, white circles: raw PMMA and τ y = 0.031 Pa). (a) d = 0.017 m; (b) d = 0.03 m.

Temperature difference ΔT as a function of the total heat input Q t , for different values of d, τ y , and slip condition (untreated copper alloy, glass, and PMMA surfaces). The black (resp. white) symbols represent the results obtained by increasing (resp. decreasing) Q t . (a) d = 0.01 m, τ y = 0.006 Pa; (b): d = 0.017 m, τ y = 0.01 Pa.

Temperature difference ΔT as a function of the total heat input Q t , for different values of d, τ y , and slip condition (untreated copper alloy, glass, and PMMA surfaces). The black (resp. white) symbols represent the results obtained by increasing (resp. decreasing) Q t . (a) d = 0.01 m, τ y = 0.006 Pa; (b): d = 0.017 m, τ y = 0.01 Pa.

Temperature difference ΔT as a function of the total heat input Q t , for the value of d = 0.01 m, τ y = 0.031 Pa, and no-slip conditions (treated PMMA surfaces at walls). The black (resp. white) symbols represent the results obtained by increasing (resp. decreasing) Q t .

Temperature difference ΔT as a function of the total heat input Q t , for the value of d = 0.01 m, τ y = 0.031 Pa, and no-slip conditions (treated PMMA surfaces at walls). The black (resp. white) symbols represent the results obtained by increasing (resp. decreasing) Q t .

(a) Variation of βΔT c as a function of d for different Carbopol gels and surface conditions (white diamonds: τ y = 0.006 Pa, slip surfaces (glass and copper alloy); crosses: τ y = 0.009 Pa, slip surfaces (glass and copper alloy), black circles: τ y = 0.031 Pa, slip surfaces (glass and copper alloy); white circles, white circles with crosses: τ y = 0.031 Pa and, respectively, slip surfaces (raw PMMA), no-slip surfaces (treated PMMA)). (b) Variation of βΔT c /τ y as a function of d for one Carbopol gel (τ y = 0.031 Pa) for the reference case: no-slip surfaces (treated PMMA).

(a) Variation of βΔT c as a function of d for different Carbopol gels and surface conditions (white diamonds: τ y = 0.006 Pa, slip surfaces (glass and copper alloy); crosses: τ y = 0.009 Pa, slip surfaces (glass and copper alloy), black circles: τ y = 0.031 Pa, slip surfaces (glass and copper alloy); white circles, white circles with crosses: τ y = 0.031 Pa and, respectively, slip surfaces (raw PMMA), no-slip surfaces (treated PMMA)). (b) Variation of βΔT c /τ y as a function of d for one Carbopol gel (τ y = 0.031 Pa) for the reference case: no-slip surfaces (treated PMMA).

Evolution of the Nusselt number Nu as a function of the inverse of the yield number 1/Y for one Carbopol gel τ y = 0.031 Pa, d = 0.017 m (white circles with crosses: treated PMMA, white circles: raw PMMA).

Evolution of the Nusselt number Nu as a function of the inverse of the yield number 1/Y for one Carbopol gel τ y = 0.031 Pa, d = 0.017 m (white circles with crosses: treated PMMA, white circles: raw PMMA).

Shadowgraph visualizations for one Carbopol (τ y = 0.031 Pa) at different Nu values, d = 0.017 m. (a) Nu = 0.98, 1/Y = 12; (b) Nu = 1.12, 1/Y = 59; (c) Nu = 1.39, 1/Y = 60; (d) Nu = 1.84, 1/Y = 63.

Shadowgraph visualizations for one Carbopol (τ y = 0.031 Pa) at different Nu values, d = 0.017 m. (a) Nu = 0.98, 1/Y = 12; (b) Nu = 1.12, 1/Y = 59; (c) Nu = 1.39, 1/Y = 60; (d) Nu = 1.84, 1/Y = 63.

## Tables

Identification of the gels coefficients and the values of yield stress obtained by both methods: Flow and oscillatory measurements at T = 293 K.

Identification of the gels coefficients and the values of yield stress obtained by both methods: Flow and oscillatory measurements at T = 293 K.

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