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Influence of rotating magnetic field strength on three-dimensional thermocapillary flow in a floating half-zone model

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Abstract

The effects of rotating magnetic field (RMF) on the three-dimensional thermocapillary flow of semiconductor melt (Pr = 0.01) in a floating half-zone model under microgravity are investigated numerically by the finite volume method. The results indicate that the thermocapillary flow without magnetic field is a steady three-dimensional convection for Ma = 40 in a floating half-zone model with As = 1, and the convection evolves to an oscillatory three-dimensional flow by applying 1–6 mT RMF with 50 Hz rotating frequency. Based on the fast Fourier transform spectrum, the convection is confirmed to be a periodically oscillating flow, the oscillatory main frequency, 1.59 × 10−3 Hz for 1 mT RMF and 5.84 × 10−2 Hz for 6 mT RMF, increases with the magnetic strength. However, with increasing the magnetic field strength up to 7 mT, the three-dimensional thermocapillary flow is effectively controlled and the convection turns into a steady axisymmetrical one.

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Abbreviations

As:

Aspect ratio (=H/R)

B 0 :

Amplitude of RMF

\( {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {f} }}_{{\hbox {rot}}}^{*} \) :

Dimensionless Lorentz force

\( f_{x}^{*} \) :

Dimensionless component of the Lorentz force in the x direction

\( f_{y}^{*} \) :

Dimensionless component of the Lorentz force in the y direction

\( f_{z}^{*} \) :

Dimensionless component of the Lorentz force in the z direction

\(\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {\text {F}_{{\text s}}^{*} }\) :

Dimensionless surface tension

H :

Height of the liquid bridge

k :

Thermal diffusion coefficient

Ma:

Marangoni number \( \left(={\frac{{\sigma_{k} \Updelta TH}}{\rho \upsilon k}} \right) \)

P* :

Dimensionless pressure

Pr:

Prandtl number (=υ/k)

R :

Radius of the liquid bridge

Reω :

Rotation magnetic Reynolds number \( ( ={{\omega H^{2} } \mathord{\left/ {\vphantom {{\omega H^{2} } \upsilon }} \right. \kern-\nulldelimiterspace} \upsilon }) \)

T :

Temperature

T * :

Dimensionless temperature

\( T_{\text{top}} \) :

Temperature of the top disc

T bottom :

Temperature of the bottom disc

t * :

Dimensionless time

Ta:

Taylor number \( (=\frac{{\sigma_{e} B_{0}^{2} \omega H^{4} }}{2\upsilon \mu }) \)

\( {\mathbf{\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}} {U} }}^{*} \) :

Dimensionless velocity vector

\( u^{*} \) :

Dimensionless velocity component in the x direction

v * :

Dimensionless velocity component in the y direction

w * :

Dimensionless velocity component in the z direction

α :

Skin depth

δ :

Kronecker operator

λ:

Rotating frequency of RMF

μ :

Dynamic viscosity

υ :

Kinematic viscosity

ρ:

Density

σ:

Surface tension

σe :

Electrical conductivity

σ0 :

Surface tension in the reference temperature

σ k :

Surface tension coefficient

ϕ :

Electric potential

ϕ * :

Dimensionless electric potential

ω :

Rotating angular frequency

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Acknowledgments

The authors are grateful to the staff of the Center for Computer Materials Science at the Institute for Materials Research, Tohoku University, for their continuous support of the SR11000 supercomputing facilities. This work is supported by the National Natural Science Foundation of China (Grant No. 10872222 and 50921063), Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20110191110037).

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Authors and Affiliations

  1. Department of Engineering Mechanics, College of Resources and Environmental Science, Chongqing University, Chongqing, 400044, People’s Republic of China

    Li** Yao, Zhong Zeng, Yi Zhang, Zhouhua Qiu, Huan Mei, Liangqi Zhang & Yongxiang Zhang

  2. State Key Laboratory of Coal Mine Disaster Dynamics and Control, Chongqing University, Chongqing, 400044, People’s Republic of China

    Li** Yao & Zhong Zeng

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  1. Li** Yao
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Correspondence to Li** Yao.

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Yao, L., Zeng, Z., Zhang, Y. et al. Influence of rotating magnetic field strength on three-dimensional thermocapillary flow in a floating half-zone model. Heat Mass Transfer 48, 2103–2111 (2012). https://doi.org/10.1007/s00231-012-1051-5

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