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The surface tension driven convection with the bidirectional temperature differences plays a very important role in many natural processes. However, most of the previous researches have focused only on the convection induced by a unidirectional temperature difference. In this paper, under the coexistence of bidirectional temperature differences, we conduct a series of numerical simulations to investigate the effect of horizontal temperature difference on the Marangoni-thermocapillary convection in a shallow annular pool. The critical values of bottom heat flux Qcri for transition from an axisymmetric steady flow to a three-dimensional unsteady flow at different values of Ma are determined. The result shows the horizontal temperature difference has a negative effect on the stability of Marangoni-thermocapillary convection. The simulation predicts two new state evolutions which do not appear in the convection with a unidirectional temperature difference. When Q is less than the Qcri value of 2.4×10-3, the Marangoni convection without horizontal temperature difference is steady and axisymmetric. When a small horizontal temperature difference is imposed, the convection called basic flow keeps steady and axisymmetric. When the value of Ma exceeds a certain threshold value Macri, the convection becomes a three-dimensional unsteady flow. After this unsteady flow happens, with the increase of Ma, the surface temperature fluctuation evolves from a punctate wave to a hydrothermal wave, and finally to a chaotic wave. Accordingly, the temperature oscillation with time is a periodically regular oscillation at first, then turns into a chaotic mess. When Q is larger than the corresponding Qcri value of 2.4×10-3, without a horizontal difference, the convection is unsteady and no basic flow exists in the variation process of Ma. With the increase of Ma, the surface temperature fluctuation evolves from a double hydrothermal wave to a single hydrothermal wave, and finally to a chaotic wave. The vertical heat transfer and horizontal temperature difference have different effects on the fluid, and their separate roles in driving fluid are determined. The bottom heat flux causes the surface fluid to flow in two opposite radial directions as the highest surface temperature is located in the middle region, while the horizontal temperature difference induces the surface fluid to flow in a single radial direction as the highest surface temperature appears at the hot wall. The combined action of these two forces generates different flows. The increase of horizontal temperature difference leads to the highest surface temperature, which originally appears in the middle region due to the bottom heat flux, and moves toward the hot wall. In this process, the horizontal temperature difference has a positive effect on the enhancement of flow near inner wall but it has a negative effect on the flow near outer wall.
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Keywords:
- Marangoni-thermocapillary convection /
- bidirectional temperature differences /
- shallow annular pool /
- oscillatory flow
[1] Bénard H 1901 Ann. Chim. Phys. 23 62
[2] Pearson J R A 1958 J. Fluid. Mech. 4 486
[3] Schatz M F, Vanhook S J, Mccormick W D, Swift J B, Swinney H L 1999 Phys. Fluids 11 2577
[4] Lebon G, Dauby P C, Regnier V C 2001 Acta Astronaut. 48 617
[5] Kim J, Choi C K, Kang Y T 2004 Int. J. Heat and Mass Transfer 47 2395
[6] Xu B, Ai X, Li B Q 2007 Int. J. Heat and Mass Transfer 50 3035
[7] Siri Z, Hashim I 2008 Int. Commun. Heat. Mass 35 1130
[8] Siri Z, Mustafa Z, Hashim I 2009 Int. J. Heat and Mass Transfer 52 5770
[9] Guo W D, Narayanan R 2007 J. Colloid. Interface. Sci. 314 727
[10] Zheng L C, Sheng X Y, Zhang X X 2006 Acta Phys. Sin. 55 5298 (in Chinese) [郑连存, 盛晓艳, 张欣欣 2006 55 5298]
[11] Zhang Y, Zheng L C, Zhang X X 2009 Acta Phys. Sin. 58 5506 (in Chinese) [张艳, 郑连存, 张欣欣 2009 58 5506]
[12] Bammou L, Omari K E, Blancher S, Guer Y L, Benhamou B, Mediouni T 2013 Int. J. Heat Fluid Flow 42 265
[13] Rachid E S, Kamal E O, Yves L G, Serge B 2014 Int. J. Therm. Sci. 86 198
[14] Smith M K, Davis S H 1983 J. Fluid. Mech. 132 119
[15] Garnier N, Chiffaudel A 2001 Eur. Phys. J. B 19 87
[16] Li Y R, Imaishi N, Azami T, Hibiya T 2004 J. Crystal Growth 260 28
[17] Shi W Y, Imaishi N 2006 J. Crystal Growth 290 280
[18] Kuhlmann H C, Albensoeder S 2008 Phys. Rev. E 77 036303
[19] Gong Z X, Li Y R, Peng L, Wu S Y, Shi W Y 2013 Acta Phys. Sin. 62 040201 (in Chinese) [龚振兴, 李友荣, 彭岚, 吴双应, 石万元 2013 62 040201]
[20] Takagi Y, Okano Y, Minakuchi H, Dost S 2014 J. Crystal Growth 385 72
[21] Li Y R, Zhang H R, Wu C M 2012 Heat Mass Transfer 48 241
[22] Peng Z, Li D, Qi K 2013 Int. J. Heat and Mass Transfer 57 457
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[1] Bénard H 1901 Ann. Chim. Phys. 23 62
[2] Pearson J R A 1958 J. Fluid. Mech. 4 486
[3] Schatz M F, Vanhook S J, Mccormick W D, Swift J B, Swinney H L 1999 Phys. Fluids 11 2577
[4] Lebon G, Dauby P C, Regnier V C 2001 Acta Astronaut. 48 617
[5] Kim J, Choi C K, Kang Y T 2004 Int. J. Heat and Mass Transfer 47 2395
[6] Xu B, Ai X, Li B Q 2007 Int. J. Heat and Mass Transfer 50 3035
[7] Siri Z, Hashim I 2008 Int. Commun. Heat. Mass 35 1130
[8] Siri Z, Mustafa Z, Hashim I 2009 Int. J. Heat and Mass Transfer 52 5770
[9] Guo W D, Narayanan R 2007 J. Colloid. Interface. Sci. 314 727
[10] Zheng L C, Sheng X Y, Zhang X X 2006 Acta Phys. Sin. 55 5298 (in Chinese) [郑连存, 盛晓艳, 张欣欣 2006 55 5298]
[11] Zhang Y, Zheng L C, Zhang X X 2009 Acta Phys. Sin. 58 5506 (in Chinese) [张艳, 郑连存, 张欣欣 2009 58 5506]
[12] Bammou L, Omari K E, Blancher S, Guer Y L, Benhamou B, Mediouni T 2013 Int. J. Heat Fluid Flow 42 265
[13] Rachid E S, Kamal E O, Yves L G, Serge B 2014 Int. J. Therm. Sci. 86 198
[14] Smith M K, Davis S H 1983 J. Fluid. Mech. 132 119
[15] Garnier N, Chiffaudel A 2001 Eur. Phys. J. B 19 87
[16] Li Y R, Imaishi N, Azami T, Hibiya T 2004 J. Crystal Growth 260 28
[17] Shi W Y, Imaishi N 2006 J. Crystal Growth 290 280
[18] Kuhlmann H C, Albensoeder S 2008 Phys. Rev. E 77 036303
[19] Gong Z X, Li Y R, Peng L, Wu S Y, Shi W Y 2013 Acta Phys. Sin. 62 040201 (in Chinese) [龚振兴, 李友荣, 彭岚, 吴双应, 石万元 2013 62 040201]
[20] Takagi Y, Okano Y, Minakuchi H, Dost S 2014 J. Crystal Growth 385 72
[21] Li Y R, Zhang H R, Wu C M 2012 Heat Mass Transfer 48 241
[22] Peng Z, Li D, Qi K 2013 Int. J. Heat and Mass Transfer 57 457
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