生物對流雷里數( Rb )對系統流場之影響

【圖十】當 =10、H Rb=1 時，流場分佈圖

【圖十一】當H=10、Rb=10 時，流場分佈圖

【圖十二】當H=10、Rb=100 時，流場分佈圖

【圖十三】當H=10、Rb=500 時，流場分佈圖

【圖十四】當H=10、Rb=1000 時，流場分佈圖

第四章 結論

本文利用ADI數值方法將方程式以有限差分法去模擬趨地性微生 物在牛頓流體飽合多孔性介質內熱生物對流現象，利用 Fortran 帶入 副程式 tridiagonal matrix 演算法去做數值模擬分析。研究重點在於 局部熱不平衡模式來考慮多孔性介質內固體與液體溫度分佈，用兩個 溫度方程式去模擬計算對熱生物對流的影響。再將不同情況下熱雷里 數Ra和生物對流雷里數Rb去做進一步的分析，發現皆會影響趨地性 微生物流場的發展。

由模擬計算的結果，可歸納出下列三點結論:

1.因本文採用局部熱不衡模型，能描述在多孔性介質內熱交換的 影響，當無因次兩相熱傳係數H較小的時候，固體與液體間的熱傳會 降低，此時液體的對流會增加，但無因次兩相熱傳係數H增大之後，

因為固體與液體進行熱交換時，會讓液體的溫度分佈較均勻，有穩定 流場的作用。

2.固定生物對流雷里數Rb和固定無因次兩相熱傳係數 的條件 下，發現熱雷里數

H

Ra逐漸增加時，會影響到流場變化，進而出現分 歧的現象，單獨一個渦流會隨著熱雷里數Ra增大而發展到多個渦流 系統，因為熱雷里數Ra會影響到流體溫度的浮力項，使得趨地性微 生物運動方向會受到溫度分佈的影響，而聚集在上蓋兩個渦流中間或

是上蓋的角落。

3.在相同的兩相熱傳係數 下，發現生物對流雷里數H Rb較小時，

發生首次分岔的熱雷里數Ra值較高，隨著生物對流雷里數Rb增加，

發生首次分岔的熱雷里數Ra會遞減，當生物對流雷里數Rb為1000 時，熱對流已經不足以影響流場結構。

參 考 文 獻

[1] Childress S., Leavndowsky M. and Spiegel E. A., “Pattern formation in a suspension of swimming microorganisms: equations and stability theory,” Journal of Fluid Mechanics, Vol. 63, pp. 591-613, 1975

[2] Fujita S. and Watanabe M., “Transition from periodic to non- periodic oscillation observed in a mathematical model of bioconvection by motile micro-organisms,” Physica D: Nonlinear Phenomena, Vol. 20, pp. 435-443, 1986.

[3] Taheri M., Bilgen E., “Bioconvection of gravitactic micro-organisms in rectangular enclosures,” International Journal of Heat and Mass Transfer, Vol. 50, pp. 4652-4660, 2007.

[4] Wager H., “On the effect of gravity upon the movement and aggregation of Euglena virides Ehrb., and other micro-organism,”

Philosophical Transaction of the Royal Society of London. Series B, Vol. 201, pp. 333-390, 1911.

[5] Ghorai S. and Hill N. A., “Development and stability of gyrotactic plumes in bioconvection,” Journal of Fluid Mechanics, Vol. 400, pp.

1-31, 1999.

[6] Ghorai S. and Hill N. A., “Wavelengths of gyrotactic plumes in bioconvection,” Bulletin of Mathematical Biology, Vol. 62, pp.

429-450, 2000.

[7] Ghorai S. and Hill N. A., “Periodic arrays of gyrotactic plumes in bioconvection,” Physics of Fluids, Vol. 12, pp. 5-22, 2000.

[8] Kuznetsov A. V., “The onset of bioconvection in a suspension of gyrotactic microorganisms in a fluid layer of finite depth heated from below,” International Communications in Heat and Mass Transfer, Vol. 32, pp. 574-582, 2005.

[9] Nield D. A. and Kuznetsov A. V., “The onset of bio-thermal convection in a suspension of gyrotactic microorganisms in a fluid layer: oscillatory convection,” International Journal of Thermal Sciences, Vol. 45, pp. 990-997, 2006.

[10] Alloui Z., Nguyen T. H. and Bilgen E., “Numerical investigation of thermo-bioconvection in a suspension of gravitactic micro- organisms,” International Journal of Heat and Mass Transfer, Vol.

50, pp. 1435-1441, 2007.

[11] Kuznetsov A. V. and Jiang N., “Numerical investigation of bioconvection of gravitactic microorganisms in an isotropic porous medium,” International Communications in Heat and Mass Transfer, Vol. 28, pp. 877-886, 2001.

[12] Kuznetsov A. V. and Avramenko A. A., “A 2D analysis of stability of bioconvection in a fluid saturated porous medium-estimation of the critical permeability value,” International Communications in Heat and Mass Transfer, Vol. 29, pp. 175-184, 2002.

[13] Kuznetsov A. V. and Avramenko A. A., ”Stability analysis of bioconvection of gyrotactic motile microorganisms in a fluid saturated porous medium,” Transport in Porous Media, Vol. 53, pp.

95-104, 2003.

[14] Kuznetsov A. V. and Avramenko A. A., “Analysis of stability of bioconvection of motile oxytactic bacteria in a horizontal fluid saturated porous layer,” International Communications in Heat and Mass Transfer, Vol. 30, pp. 593-602, 2003.

[15] Kuznetsov A. V. and Avramenko A. A., “The effect of deposition and declogging on the critical permeability in bioconvection in a porous medium,” Acta Mechanica, Vol. 160, pp. 113-125, 2003.

[16] Kuznetsov A. V. and Jiang N., “Bioconvection of negatively geotactic microorganisms in a porous medium: the effect of cell deposition and declogging,” International Journal of Numerical Methods in Heat and Fluid Flow, Vol. 13, pp. 341-364, 2003.

[17] Kuznetsov A. V., Avramenko A. A. and Geng P., ”A similarity solution for a falling plume in bioconvection of oxytactic bacteria in a porous medium,” International Communications in Heat and Mass Transfer, Vol. 30, pp. 37-46, 2003.

[18] Nield D. A., Kuznetsov A. V. and Avramenko A. A., “The onset of bioconvection in a horizontal porous-medium layer,” Transport in Porous Media, Vol. 54, pp. 335-344, 2004.

[19] Kuznetsov A. V., “The onset of thermo-bioconvection in a shallow fluid saturated porous layer heated from below in a suspension of oxytactic microorganisms,” European Journal of Mechanics B/Fluids, Vol. 25, pp. 223-233, 2006.

[20] Nguyen-Quang T., Bahloul A. and Nguyen T. H., “Stability of gravitactic micro-organisms in a fluid-saturated porous medium,”

International Communications in Heat and Mass Transfer, Vol. 32,

[21] Fogler H. S. and Stewart T. L. “Biomass piug development and propagation in porous media,”Biotechnology Bioengineering, Vol.

72, pp. 353-363, 2001.

[22] Fogler H. S. and Kim D. S., “Biomass evolution in porous media and its effect on permeability under starvation conditions,” Bio- technology Bioengineering, Vol.69, pp. 47-56, 2000.

[23] Nield D. A. and Bejan A., Convection in Porous Medium, ed., Springer-Verlag, New York, 1999.

2^nd

[24] Pop I., and Ingham D. B., Convection Heat Transfer: mathematical and computation modeling of viscous fluids and porous media, Pergamon, Oxford, UK, 2001.

[25] Rees D.A.S. and Pop I., “Free convective stagnation point flow in a porous medium using thermal non-equilibrium model,” International Communications in Heat and Mass Transfer, Vol. 26, pp. 945-954, 1999.

[26] Rees D.A.S. and Pop I., “Vertical free convective boundary layer flow in a porous medium using a thermal non-equilibrium model,”

Journal of Porous Medium, Vol. 3, pp. 31-44, 2003.

[27] Banu N. and Rees D.A.S., “Onset of Darcy-Benard convection using a thermal non-equilibrium model,” International Journal of Heat and Mass Transfer, Vol. 45, pp. 2221-2228, 2002.

[28] 施人豪, “趨地性微生物在牛頓流體飽合多孔性介質內熱-生物對 流穩定性分析,” 中華大學機械與航太工程研究所, 碩士論文, 2008.