Heat transfer of power-law fluids with variable thermal conductivities on a horizontal rough surface
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摘要: 根據泰勒展開式和邊界層理論, 推導了變熱導率的Oswad-de Waele冪律流體沿水平波面上的邊界層方程.假設熱傳導系數是依賴于溫度梯度的冪律函數, 構建了變熱導率的能量方程模型. 引入一系列變換，把變量量綱為一化和坐標變換，將原始波面轉換為偏微分方程組, 并用Keller-box方法進行數值求解. 討論了某些參數如波幅與波長的比值、冪律指數以及廣義普朗特數對壁面摩擦和流體傳熱的影響. 計算結果顯示：表面速度和壓力梯度沿波面呈周期性變化，而且它們的變化周期與波面的變化周期完全一致. 而對于壁面的摩擦系數和局部Nusselt數, 在靠近零點的地方會有劇烈震蕩, 沿軸向會呈現波形分布狀態, 隨著波長比率的增大而減小, 且會隨著振幅的增大, 壁面摩擦系數也會震蕩加劇. 隨著冪律指數的增加, 局部Nusselt數呈現遞減的分布狀態. 對于問題的特殊情況，當壁面是光滑平板時，盡管壁面的摩擦系數和局部Nusselt數沿軸向在初始位置會有波動，但會在很短的距離達到穩定的狀態. 從不同參數對周期的影響來看, 周期性波動的壁面摩擦系數和局部Nusselt數與波面曲線的峰頂和波谷并不保持一致.Abstract: Power-law fluids have recently received increasing attention because of their applications in different industrial fields. In previous works, the energy and momentum equations for power-law fluids were considered the same as those for Newtonian fluids. However, as the heat transfer of fluids results from thermomolecular motions, the heat-transfer behavior of non-Newtonian power-law fluids should be different from that of Newtonian fluids. The flow of fluids on a smooth plate is a classical problem. In most situations, the plates are rough. In particular, in industrial fields, many plates are deliberately designed to be rough to enhance heat transfer. Herein, according to the Taylor expansion and boundary-layer theory, the boundary-layer equations for the Ostwald–de Waele power-law fluids with a variable thermal conductivity along a horizontal wavy surface are reduced to partial differential equations. An energy equation with a variable thermal conductivity is constructed, where the heat-conduction coefficient is assumed to be a power-law function dependent on the temperature gradient. Through the introduction of a series of transformations, including nondimensional and coordinate transformations, the original wavy-surface problem is transformed into a system of partial differential equations describing the flow problem with boundary conditions on a flat plate, which is solved numerically using the Keller-box method. The effects of some parameters, such as the amplitude–wavelength ratio
$ \alpha $ , power-law index$ n $ , and generalized Prandtl number$ {N_{zh}} $ , on the local friction coefficient and heat-transfer coefficient are discussed. Numerical results show that the velocity of power-law fluids on the surface and pressure gradient varies periodically along the wavy plate. Furthermore, the cycles of the velocity and pressure gradients are the same as the one of the wavy-shape plate. The results show that the local Nusselt number and the friction coefficient vary periodically in a wavelike manner and increase gradually with the amplitude–wavelength ratio, although a sudden change exists near the zero point. With the increasing amplitude, the friction coefficient oscillates more considerably. With the increasing power-law index, the local Nusselt number decreases. For a special case in which the plate is flat, the local Nusselt number and friction coefficient are in a stable state for a short distance along the plate, although initial oscillations appear near the zero point. Owing to the effects of different parameters on the periodicity, the peak and trough of the local Nusselt number and friction coefficient are not consistent, despite occurring in the same cycle. -
圖 2 速度
$ {U_{\text{w}}}(x) $ 和壓力梯度$ {\text{d}}p/{\text{d}}x $ 關于$ x $ 的變化圖像Figure 2. The variation of inviscid surface velocity
$ {U_{\text{w}}}(x) $ and pressure gradient$ {\text{d}}p/{\text{d}}x $ against$ x $ 
圖 3
$ n = 0.8 $ ，$ {N_{{\text{zh}}}} = 1 $ 時壁面摩擦系數$ ( {{{C_{\text{f}}}} \mathord{\left/ {\vphantom {{{C_{\text{f}}}} 2}} \right. } 2} ) {{Re} _x}^{\frac{1}{{n + 1}}} $ 的軸向分布Figure 3. Axial distribution of
$ ( {{{C_{\text{f}}}} \mathord{\left/ {\vphantom {{{C_{\text{f}}}} 2}} \right. } 2} ) {{Re} _x}^{\frac{1}{{n + 1}}} $ for$ n = 0.8 $ and$ {N_{{\text{zh}}}} = 1 $ 
圖 6
$ \alpha {\text{ = 0}}{\text{.01}} $ ，$ {N_{{\text{zh}}}} = 1 $ 時壁面摩擦系數$ ( {{{C_{\text{f}}}} \mathord{\left/ {\vphantom {{{C_{\text{f}}}} 2}} \right. } 2} ) {{Re} _x}^{\frac{1}{{n + 1}}} $ 的軸向分布Figure 6. Axial distribution of
$ ( {{{C_{\text{f}}}} \mathord{\left/ {\vphantom {{{C_{\text{f}}}} 2}} \right. } 2} ) {{Re} _x}^{\frac{1}{{n + 1}}} $ for$ \alpha {\text{ = 0}}{\text{.01}} $ and$ {N_{{\text{zh}}}} = 1 $ 
圖 4
$ n = 1.0 $ ，$ {N_{{\text{zh}}}} = 1 $ 時壁面摩擦系數$ ( {{{C_{\text{f}}}} \mathord{\left/ {\vphantom {{{C_{\text{f}}}} 2}} \right. } 2} ) {{Re} _x}^{\frac{1}{{n + 1}}} $ 的軸向分布Figure 4. Axial distribution of
$ ( {{{C_{\text{f}}}} \mathord{\left/ {\vphantom {{{C_{\text{f}}}} 2}} \right. } 2} ) {{Re} _x}^{\frac{1}{{n + 1}}} $ for$ n = 1.0 $ and$ {N_{{\text{zh}}}} = 1 $ 
圖 5
$ n = 1.2 $ ，$ {N_{{\text{zh}}}} = 1 $ 時壁面摩擦系數$ ( {{{C_{\text{f}}}} \mathord{\left/ {\vphantom {{{C_{\text{f}}}} 2}} \right. } 2} ) {{Re} _x}^{\frac{1}{{n + 1}}} $ 的軸向分布Figure 5. Axial distribution of
$ {{({C_{\text{f}}}} \mathord{\left/ {\vphantom {{({C_{\text{f}}}} 2}} \right. } 2}){{Re} _x}^{\frac{1}{{n + 1}}} $ for$ n = 1.2 $ and$ {N_{{\rm{zh}}}} = 1 $ 
圖 7
$ n = 0.8 $ ，$ {N_{{\text{zh}}}} = 1 $ 時局部Nusselt數的軸向分布Figure 7. Axial distributions of the local Nusselt number for
$ n = 0.8 $ and$ {N_{{\text{zh}}}} = 1 $ 
圖 10
$ \alpha {\text{ = 0}}{\text{.01}} $ ，$ {N_{{\text{zh}}}} = 1 $ 時局部Nusselt數的軸向分布Figure 10. Axial distributions of the local Nusselt number for
$ \alpha {\text{ = 0}}{\text{.01}} $ and$ {N_{{\text{zh}}}} = 1 $ 
圖 8
$ n = 1.0 $ ，$ {N_{{\text{zh}}}} = 1 $ 時局部Nusselt數的軸向分布Figure 8. Axial distributions of the local Nusselt number for
$ n = 1.0 $ and$ {N_{{\text{zh}}}} = 1 $ 
圖 9
$ n = 1.2 $ ，$ {N_{{\text{zh}}}} = 1 $ 時局部Nusselt數的軸向分布Figure 9. Axial distributions of the local Nusselt number for
$ n = 1.2 $ and$ {N_{{\text{zh}}}} = 1 $ 259luxu-164<th id="5nh9l"></th> <strike id="5nh9l"></strike> <th id="5nh9l"><noframes id="5nh9l"><th id="5nh9l"></th> <strike id="5nh9l"></strike> <th id="5nh9l"><noframes id="5nh9l"> <th id="5nh9l"></th> <strike id="5nh9l"><noframes id="5nh9l"><span id="5nh9l"></span> <span id="5nh9l"><noframes id="5nh9l"><span id="5nh9l"></span> <strike id="5nh9l"><noframes id="5nh9l"><strike id="5nh9l"></strike> <span id="5nh9l"><noframes id="5nh9l"> <span id="5nh9l"><noframes id="5nh9l"> <span id="5nh9l"></span> <span id="5nh9l"><video id="5nh9l"></video></span> <th id="5nh9l"><noframes id="5nh9l"><th id="5nh9l"></th> -
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