^{1,a)}, Olivier Crumeyrolle

^{1}and Innocent Mutabazi

^{1}

### Abstract

The thermal convection driven by the dielectrophoretic force is investigated in annular geometry under microgravity conditions. A radial temperature gradient and a radial alternating electric field are imposed on a dielectric fluid that fills the gap of two concentric infinite-length cylinders. The resulting dielectric force is regarded as thermal buoyancy with a radial effective gravity. This electric gravity varies in space and may change its sign depending on the temperature gradient and the cylinder radius ratio. The linear stability problem is solved by a spectral-collocation method. The critical mode is stationary and non-axisymmetric. The critical Rayleigh number and wavenumbers depend sensitively on the electric gravity and the radius ratio. The mechanism behind the instability is examined from an energetic viewpoint. The instability in wide gap annuli is an exact analogue to the gravity-driven thermal instability.

This work has been partly supported by the CNES (Centre National d'Etudes Spatiales) and the CNRS (Centre National de la Recherche Scientifique). H.N.Y. thanks the Université du Havre and the FEDER (Fonds Européen de Développement Régional) for his postdoctoral grant.

I. INTRODUCTION

II. GOVERNING EQUATIONS AND BASIC STATE

A. Governing equations

B. Basic conductive state

III. LINEAR STABILITY PROBLEM

IV. RESULTS

A. Dispersion relation and marginal curves

B. Critical Rayleigh number

C. Wavenumber selection

D. Eigenmodes

V. ENERGY ANALYSIS

VI. DISCUSSION AND CONCLUSION

### Key Topics

- Thermal convection
- 24.0
- Convection
- 22.0
- Dielectrophoresis
- 17.0
- Electric fields
- 17.0
- Liquid dielectrics
- 7.0

## Figures

Schematic illustration of the geometrical configuration in a plane transversal to the annulus axis.

Schematic illustration of the geometrical configuration in a plane transversal to the annulus axis.

Basic electric gravity . (a) Its direction as function of the dimensionless temperature γ e and the radius ratio η. CP and CF mean centripetal and centrifugal, respectively. (b) Some profiles of calculated for the electric tension V 0 = 1.08 × 104 V. The cylinder radii and liquid properties are taken from the Chandra and Smylie's experiment 7 (R 1 = 1.711 × 10−2 m, R 2 = 1.903 × 10−2 m, ρ = 937.7 kg m−3, α = 1.08 × 10−3 K−1, and e = 3.7 × 10−3 K−1).

Basic electric gravity . (a) Its direction as function of the dimensionless temperature γ e and the radius ratio η. CP and CF mean centripetal and centrifugal, respectively. (b) Some profiles of calculated for the electric tension V 0 = 1.08 × 104 V. The cylinder radii and liquid properties are taken from the Chandra and Smylie's experiment 7 (R 1 = 1.711 × 10−2 m, R 2 = 1.903 × 10−2 m, ρ = 937.7 kg m−3, α = 1.08 × 10−3 K−1, and e = 3.7 × 10−3 K−1).

Dispersion relation for different azimuthal mode number n. (Pr = 100, η = 0.5, γ e = 0.01, L = 1498).

Dispersion relation for different azimuthal mode number n. (Pr = 100, η = 0.5, γ e = 0.01, L = 1498).

Definitions of the wavenumber q and the inclination angle ψ of convection rolls.

Definitions of the wavenumber q and the inclination angle ψ of convection rolls.

Marginal curves for different azimuthal mode number n at small and large radius ratios η: (a) η = 0.5 and (b) η = 0.9. The dimensionless temperature γ e is fixed at 0.01.

Marginal curves for different azimuthal mode number n at small and large radius ratios η: (a) η = 0.5 and (b) η = 0.9. The dimensionless temperature γ e is fixed at 0.01.

Critical electric Rayleigh number L c as function of the radius ratio η for different dimensionless temperatures γ e . The horizontal lines show the critical Rayleigh number of the Rayleigh-Bénard instability (1708) and L c of the DEP convection in plane geometry (2129). 9

Critical electric Rayleigh number L c as function of the radius ratio η for different dimensionless temperatures γ e . The horizontal lines show the critical Rayleigh number of the Rayleigh-Bénard instability (1708) and L c of the DEP convection in plane geometry (2129). 9

Critical wavenumber q c and inclination angle ψ for an outward heating (γ e = 0.01).

Critical wavenumber q c and inclination angle ψ for an outward heating (γ e = 0.01).

Critical wavenumber q c for inward and outward heatings.

Critical wavenumber q c for inward and outward heatings.

Critical eigenmodes for small and large radius ratios η. For η = 0.3, n = 2, k = 1.91, and L = 1177. For η = 0.9, n = 25, k = 1.68, and L = 1732. The heating is outward with the dimensionless temperature γ e = 0.01 for both cases.

Critical eigenmodes for small and large radius ratios η. For η = 0.3, n = 2, k = 1.91, and L = 1177. For η = 0.9, n = 25, k = 1.68, and L = 1732. The heating is outward with the dimensionless temperature γ e = 0.01 for both cases.

Normalized different contributions to the rate of change of the convection flow kinetic energy (Eq. (29) ) at critical conditions.

Normalized different contributions to the rate of change of the convection flow kinetic energy (Eq. (29) ) at critical conditions.

Normalized different contributions to the rate of change of the convection flow kinetic energy (Eq. (29) ) at critical Rayleigh numbers: (a) at a small radius ratio η for different azimuthal mode number n in an outward heating (η = 0.2, γ e = 0.01, and L = 888.4) and (b) at a large η for different heating directions (η = 0.98, L = 2005, and 2355 for γ e = 0.1 and −0.1, respectively). The Prandtl number is fixed at 10 for all.

Normalized different contributions to the rate of change of the convection flow kinetic energy (Eq. (29) ) at critical Rayleigh numbers: (a) at a small radius ratio η for different azimuthal mode number n in an outward heating (η = 0.2, γ e = 0.01, and L = 888.4) and (b) at a large η for different heating directions (η = 0.98, L = 2005, and 2355 for γ e = 0.1 and −0.1, respectively). The Prandtl number is fixed at 10 for all.

Stability boundaries in the voltage–temperature plane for different radius ratio η. Theoretical predictions of the existing theoretical works are also shown (· · · · · · · · · · · ·: Takashima and Hamabata 9 for plane geometry; - - - -: Chandra and Smylie 7 for η = 0.89; ▲ and □: Takashima 8 for η = 0.91 and η = 0.99, respectively).

Stability boundaries in the voltage–temperature plane for different radius ratio η. Theoretical predictions of the existing theoretical works are also shown (· · · · · · · · · · · ·: Takashima and Hamabata 9 for plane geometry; - - - -: Chandra and Smylie 7 for η = 0.89; ▲ and □: Takashima 8 for η = 0.91 and η = 0.99, respectively).

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