重力沉降效應對水平奈米流體層熱與表面張力驅動不穩定性的影響

Abstract

本研究之目的是探討表面張力效應對水平奈米流體層對流穩定性的影響，模型為一水平奈米流體層，上表面存在表面張力，溫度固定為 Tc，為不可變形之自由液面且與外界環境存在熱通量，下板溫度固定為 Th，考慮熱泳動、布朗運動及奈米粒子重力沉降效應，依照流體穩定學分析流程自統御方程式開始，經由無因次化、基態流場分析、微小擾動分析以及正規模態展開，其中採用連續方程式、動量守恆式、奈米粒子質量守恆式及能量守恆式作為統御方程式，搭配適當的邊界條件，最後使用 Chebyshev polynomials 進行數值分析。 本研究選擇水作為流體介質，氧化鋁作為奈米粒子製成奈米流體進行分析，其中奈米粒子直徑選擇 20、40 和 60 奈米做為小、中和大尺寸奈米粒子代表，考慮不同奈米粒子濃度或 Biot 數對系統穩定性的影響，並繪製出中性曲線、震盪頻率及流動模式圖。得出的結果為奈米粒子直徑增加或濃度增加會使系統更穩定且臨界波數越大，而 Biot 數增加也會得到相同的結果。在系統中無奈米粒子存在時屬於固定模態，加入奈米粒子後則呈現震盪模態，在流動模式中亦呈現相同的結果。本研究探討表面張力隨奈米粒子體積分率增加而增加及表面張力與奈米粒子體積分率無關案例得出了相似的結果。 本研究與前人對奈米流體對流穩定性研究不同之處在於考慮奈米粒子重力沉降效應，可得出更加符合實際情況的方程式。

The objective of this study is to investigate the effect of surface tension on the convective stability of a horizontal nanofluid layer. The model considers a horizontal nanofluid layer with an undeformable free surface subject to surface tension, maintained at a constant temperature Tc, and exchanging heat flux with the external environment. The bottom surface is held at a higher constant temperature Th. Thermophoresis, Brownian motion, and gravity-induced nanoparticle settling are taken into account. Following the standard procedure in fluid stability analysis, the governing equations—comprising the continuity equation, momentum conservation, nanoparticle mass conservation, and energy conservation—are nondimensionalized, and steady-state and linear stability analyses are conducted. The perturbation equations are expanded using normal modes and solved numerically via Chebyshev polynomials. Water is selected as the base fluid and alumina (Al₂O₃) as the dispersed nanoparticles. Particle diameters of 20, 40, and 60 nm are used to represent small, medium, and large nanoparticles, respectively. The effects of nanoparticle volume fraction and Biot number on system stability are examined. Neutral curves, oscillatory frequencies, and flow patterns are presented. The results indicate that increasing nanoparticle diameter or volume fraction enhances system stability and leads to larger critical wavenumbers. A similar stabilizing effect is observed with increasing Biot number. The system exhibits a stationary mode in the absence of nanoparticles and transitions to an oscillatory mode upon nanoparticle inclusion, which is also reflected in the flow patterns. This study demonstrates that similar results are obtained for cases where surface tension increases with nanoparticle volume fraction and where surface tension is independent of nanoparticle volume fraction. This study differs from previous work on nanofluid convective stability by incorporating the effect of nanoparticle gravity settling, resulting in a more realistic and physically consistent model.