颗粒尺寸对纳米流体自然对流模式影响的格子Boltzmann方法模拟

隋鹏翔

doi:10.7498/aps.73.20241332

摘要

采用无量纲格子玻尔兹曼(non-dimensional lattice Boltzmann method, NDLBM)对方腔内纳米流体的自然对流进行数值模拟, 讨论克努森数(

$10^{-6} \leqslant Kn_{{\rm{f}},{\rm{s}}} \leqslant 10^4$ )、瑞利数(

$10^3 \leqslant Ra_{{\rm{f}},{\rm{L}}} \leqslant 10^6$ )、颗粒体积分数(

$10^{-2} \leqslant$

$\phi_{\rm{s}} \leqslant 10^{-1}$ )等参数对纳米流体流动和传热的影响. 结果表明: 在不同

$Ra_{{\rm{f}},{\rm{L}}}$ 下, 颗粒粒径对传热效率的影响是不同的. 在低

$Ra_{{\rm{f}},{\rm{L}}}$ 的热传导区间, 颗粒粒径对传热影响较小; 在高

$Ra_{{\rm{f}},{\rm{L}}}$ 的热对流区间, 较大的颗粒粒径显著提升了流动强度和传热效率. 若保持

$Ra_{{\rm{f}},{\rm{L}}}$ 和

$\phi_{\rm{s}}$ 不变, 随着颗粒粒径的减小, 纳米流体的传热方式由热传导转变为热对流. 此外, 针对高

$Ra_{{\rm{f}},{\rm{L}}}$ 的热对流区间, 在兼顾了导热和流动性的情况下, 最大传热效率所对应的颗粒体积分数为

$\phi_{\rm{s}} = 8 {\text{%}}$ . 最后, 通过分析平均努塞尔数

$\overline {Nu}_{{\rm{f}},{\rm{L}}}$ 和纳米流体相较于基液增加传热率

$Re_{{\rm{n}},{\rm{f}}}$ 随不同无量纲参数变化的三维等值面图, 发现

$\overline {Nu}_{{\rm{f}},{\rm{L}}}$ 和

$Re_{{\rm{n}},{\rm{f}}}$ 的极值均出现在颗粒粒径为

$Kn_{{\rm{f}},{\rm{s}}} = 10^{-1}$ . 基于数值结果, 构建

$\overline {Nu}_{{\rm{f}},{\rm{L}}}$ 与

$Kn_{\rm{f},\rm{s}}$ ,

$Ra_{\rm{f},\rm{L}}$ ,

$\phi_{\rm{s}}$ 之间的函数关系式, 揭示了这些无量纲参数对传热性能的影响.

关键词:

In this work, numerical simulation of natural convection of nanofluids within a square enclosure are conducted by using the non-dimensional lattice Boltzmann method (NDLBM). The effects of key governing parameters Knudsen number (

$10^{-6} \leqslant Kn_{{\rm{f}},{\rm{s}}} \leqslant 10^4$ ), Rayleigh number (

$10^3 \leqslant Ra_{{\rm{f}},{\rm{L}}} \leqslant 10^6$ ), and nanoparticle volume fraction (

$10^{-2} \leqslant \phi_{\rm{s}} \leqslant 10^{-1}$ ) on the heat and mass transfer of nanofluids are discussed. The results show that in the low

$Ra_{{\rm{f}},{\rm{L}}}$ conduction dominated regime, the nanoparticle size has little effect on heat transfer, whereas in the high

$Ra_{{\rm{f}},{\rm{L}}}$ convection dominated regime, larger nanoparticle size significantly enhances flow intensity and heat transfer efficiency. For fixed

$Ra_{{\rm{f}},{\rm{L}}}$ and

$\phi_{\rm{s}}$ , the heat transfer patterns change from conduction to convection dominated regime with

$Kn_{{\rm{f}},{\rm{s}}}$ increasing. The influence of nanoparticle volume fraction is also investigated, and in the convection-dominated regime, the maximum heat transfer efficiency is achieved when

$\phi_{\rm{s}} = 8 {\text{%}}$ , balancing thermal conduction and drag fore of nanofluid. Additionally, by analyzing the full maps of mean Nusselt number (

$\overline {Nu}_{{\rm{f}},{\rm{L}}}$ ) and the enhancement ratio related to the base fluid (

$Re_{{\rm{n}},{\rm{f}}}$ ), the maximum value of

$\overline {Nu}_{{\rm{f}},{\rm{L}}}$ and

$Re_{{\rm{n}},{\rm{f}}}$ occur when the nanoparticle size is

$Kn_{{\rm{f}},{\rm{s}}} = 10^{-1}$ for both conductive and convection dominated regime. To ascertain the effects of all key governing parameters on

$\overline {Nu}_{{\rm{f}},{\rm{L}}}$ , a new empirical correlation is derived from the numerical results, providing a more in-depth insight into how these parameters influence on heat transfer performance.

Keywords:

PACS:

作者及机构信息

隋鹏翔

首都师范大学初等教育学院, 北京　100048

通信作者: 隋鹏翔, pxsui@cnu.edu.cn

基金项目: 北京市教育委员会科技计划一般项目(批准号: KM202410028009)资助的课题.

Authors and contacts

Sui Peng-Xiang

College of Elementary Education, Capital Normal University, Beijing 100048, China

Corresponding author: Sui Peng-Xiang, pxsui@cnu.edu.cn

Funds: Project supported by the Scientific Research Project of Beijing Education Committee, China (Grant No. KM202410028009).

文章全文

参考文献

[1]	Wang X, Song Y, Li C, Zhang Y, Ali H M, Sharma S, Li R, Yang M, Gao T, Liu M, Cui X, Said Z, Zhou Z 2024 Int. J. Adv. Manuf. Technol. 131 3113 Google Scholar
[2]	Sandhya M, Ramasamy D, Sudhakar K, Kadirgama K, Samykano M, Harun W S W, Najafi G, Mofijur M, Mazlan M 2021 Sustainable Energy Technol. Assess. 44 101058 Google Scholar
[3]	Said Z, Sundar L S, Tiwari A K, Ali H M, Sheikholeslami M, Bellos E, Babar H 2022 Phys. Rep. 946 1 Google Scholar
[4]	Smaisim G F, Mohammed D B, Abdulhadi A M, Uktamov K F, Alsultany F H, Izzat S E, Ansari M J, Kzar H H, Al-Gazally M E, Kianfar E 2022 J. Sol-Gel Sci. Technol. 104 1 Google Scholar
[5]	肖波齐, 范金土, 蒋国平, 陈玲霞 2012 61 317 Google Scholar Xiao B Q, Fan J T, Jiang G P, Chen L X 2012 Acta Phys. Sin. 61 317 Google Scholar
[6]	Azmi W H, Sharma K V, Mamat R, Najafi G, Mohamad M S 2016 Renewable Sustainable Energy Rev. 53 1046 Google Scholar
[7]	Wang X, Xu X, Choi S U S 1999 J. Thermophys. Heat Transfer 13 474 Google Scholar
[8]	Das S K, Putra N, Thiesen P, Roetzel W 2003 J. Heat Transfer 125 567 Google Scholar
[9]	Nguyen C T, Desgranges F, Galanis N, Roy G, Mare T, Boucher S, Minsta H A 2008 Int. J. Therm. Sci. 47 103 Google Scholar
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[19]	Chambre P A, Schaaf S A 1961 Flow of Rarefied Gases (Princeton: Princeton University Press) pp78–146
[20]	Sui P, Su Y, Sin V, Davidson J H 2022 Int. J. Heat Mass Transfer 187 122541 Google Scholar
[21]	Zarki A, Ghalambaz M, Chamkha A J, Ghalambaz M, Rossi D D 2015 Adv. Powder Technol. 26 935 Google Scholar
[22]	Sabour M, Ghalambaz M, Chamkha A 2017 Int. J. Numer. Methods Heat Fluid Flow 27 1504 Google Scholar
[23]	Paul T C, Morshed A, Fox E B, Khan J A 2017 Int. J. Heat Mass Transfer 28 753
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[26]	Trodi A, Benhamza M E H 2017 Chem. Eng. Commun. 204 158 Google Scholar
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[39]	Hua Y C, Cao B Y 2016 Int. J. Heat Mass Transfer 92 995 Google Scholar
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图 1 物理模型和边界条件示意图

Fig. 1. Sketch of the problem and boundary conditions.

下载: 全尺寸图片幻灯片

不同网格数对平均努塞尔数 $\overline{Nu}_{{{\rm{f}}, {\rm{L}}}}$ 的影响

The influence of different mesh grid on the mean Nusselt number $\overline{Nu}_{{{\rm{f}}, {\rm{L}}}}$ .

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平均努塞尔数 $\overline{Nu}_{{{\rm{f}}, {\rm{L}}}}$ 与其他文献数值结果^[31,41]对比

Comparison of mean Nusset number $\overline{Nu}_{{{\rm{f}}, {\rm{L}}}}$ obtained by present study and other numerical results^[31,41].

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在固定颗粒体积分数 $\phi_{\rm{s}} = 8 {\text{%}}$ , 不同颗粒粒径 $1 \times 10^{-4} \leqslant Kn_{{\rm{f}}, {\rm{s}}} \leqslant 1 \times 10^2$ 和瑞利数 $1 \times 10^{3} \leqslant Ra_{{{\rm{f}}, {\rm{L}}}} \leqslant 1 \times 10^6$ 下的无量纲速度场流线分布图像　(a) $Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{3}$ ; (b) $Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{4}$ ; (c) $Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{5}$ ; (d) $Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{6}$

Dimensionless streamlines for different nanoparticle size $1 \times 10^{-4} \leqslant Kn_{{\rm{f}}, {\rm{s}}} \leqslant 1 \times 10^2$ and Rayleigh number $1 \times 10^{3}\leqslant$ $Ra_{{{\rm{f}}, {\rm{L}}}} \leqslant 1 \times 10^6$ with fixed $\phi_{\mathrm{s}} = 8 {\text{%}}$ : (a) $Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{3}$ ; (b) $Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{4}$ ; (c) $Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{5}$ ; (d) $Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{6}$

下载: 全尺寸图片幻灯片

在固定颗粒体积分数 $\phi_{\rm{s}} = 8 {\text{%}}$ , 不同颗粒粒径 $1 \times 10^{-4} \leqslant Kn_{{\rm{f}}, {\rm{s}}} \leqslant 1 \times 10^2$ 和瑞利数 $1 \times 10^{3} \leqslant Ra_{{{\rm{f}}, {\rm{L}}}} \leqslant 1 \times 10^6$ 下的无量纲温度场等温线分布图像　(a) $Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{3}$ ; (b) $Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{4}$ ; (c) $Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{5}$ ; (d) $Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{6}$

Dimensionless isotherms for different nanoparticle size $1 \times 10^{-4} \leqslant Kn_{{\rm{f}}, {\rm{s}}} \leqslant 1 \times 10^2$ and Rayleigh number $1 \times 10^{3} \leqslant$ $Ra_{{{\rm{f}}, {\rm{L}}}} \leqslant 1 \times 10^6$ with fixed $\phi_{\mathrm{s}} = 8 {\text{%}}$ : (a) $Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{3}$ ; (b) $Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{4}$ ; (c) $Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{5}$ ; (d) $Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^{6}$ .

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不同颗粒体积分数和瑞利数下, 平均努塞尔数 $\overline {Nu}_{{{\rm{f}}, {\rm{L}}}}$ 与克努森数 $1 \times 10^{-6} \leqslant Kn_{{\rm{f}}, {\rm{s}}} \leqslant 1 \times 10^4$ 关系　(a) $Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^3$ ; (b) $Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^4$ ; (c) $Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^5$ ; (d) $Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^6$

The mean Nusselt number $\overline {Nu}_{{{\rm{f}}, {\rm{L}}}}$ versus Knudsen number $1 \times 10^{-6} \leqslant Kn_{{\rm{f}}, {\rm{s}}} \leqslant 1 \times 10^4$ with different volume fraction and Rayleigh number: (a) $Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^3$ ; (b) $Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^4$ ; (c) $Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^5$ ; (d) $Ra_{{{\rm{f}}, {\rm{L}}}} = 1 \times 10^6$ .

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不同瑞利数 $1 \times 10^{3} \leqslant Ra_{{{\rm{f}}, {\rm{L}}}} \leqslant 1 \times 10^6$ 和固定颗粒体积分数 $\phi_{\rm{s}} = 8{\text{%}}$ 下, 平均努塞尔数 $\overline {Nu}_{{{\rm{f}}, {\rm{L}}}}$ 与克努森数关系

The mean Nusselt number $\overline {Nu}_{{{\rm{f}}, {\rm{L}}}}$ versus Knudsen number with different Rayleigh number $1 \times 10^{3} \leqslant Ra_{{{\rm{f}}, {\rm{L}}}} \leqslant$ $1 \times 10^6$ and fixed nanoparticle volume fraction $\phi_{\rm{s}} = 8{\text{%}}$ .

下载: 全尺寸图片幻灯片

不同瑞利数 $1 \times 10^{3} \leqslant Ra_{{{\rm{f}}, {\rm{L}}}} \leqslant 1 \times 10^6$ 和固定颗粒体积分数 $\phi_{\rm{s}} = 8{\text{%}}$ 下, 纳米流体相较于基液传热增加率 $Re_{{\rm{n}}, {\rm{f}}}$ 与克努森数关系

The enhancement ratio $Re_{{\rm{n}}, {\rm{f}}}$ versus Knudsen number with different Rayleigh number $1 \times 10^{3} \leqslant Ra_{{{\rm{f}}, {\rm{L}}}} \leqslant$ $1 \times 10^6$ and fixed nanoparticle volume fraction $\phi_{\rm{s}} = 8{\text{%}}$ .

下载: 全尺寸图片幻灯片

不同瑞利数 $1 \times 10^{3} \leqslant Ra_{{{\rm{f}}, {\rm{L}}}} \leqslant 1 \times 10^6$ 和固定颗粒粒径 $Kn_{{\rm{f}}, {\rm{s}}} = 10^{-1}$ 下, 平均努塞尔数 $\overline {Nu}_{{{\rm{f}}, {\rm{L}}}}$ 与颗粒体积分数关系

The mean Nusselt number $\overline {Nu}_{{{\rm{f}}, {\rm{L}}}}$ versus nanoparticle volume fraction with different Rayleigh number $1 \times 10^{3} \leqslant Ra_{{{\rm{f}}, {\rm{L}}}} \leqslant 1 \times 10^6$ and fixed nanoparticle size $Kn_{{\rm{f}}, {\rm{s}}} = 10^{-1}$ .

下载: 全尺寸图片幻灯片

不同瑞利数 $1 \times 10^{3} \leqslant Ra_{{{\rm{f}}, {\rm{L}}}} \leqslant 1 \times 10^6$ 和固定颗粒粒径 $Kn_{{\rm{f}}, {\rm{s}}} = 10^{-1}$ 下, 纳米流体相较于基液传热增加率 $Re_{{\rm{n}}, {\rm{f}}}$ 与颗粒体积分数关系

The enhancement ratio $Re_{{\rm{n}}, {\rm{f}}}$ versus nanoparticle volume fraction with different Rayleigh number $1 \times 10^{3} \leqslant$ $Ra_{{{\rm{f}}, {\rm{L}}}} \leqslant 1 \times 10^6$ and fixed nanoparticle size $Kn_{{\rm{f}}, {\rm{s}}} = 10^{-1}$ .

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不同克努森数 $1 \times 10^{-3} \leqslant Kn_{{\rm{f}}, {\rm{s}}} \leqslant 1 \times 10^2$ 和固定颗粒体积分数 $\phi_{\rm{s}} = 8 {\text{%}}$ 下, 平均努塞尔数 $\overline {Nu}_{{{\rm{f}}, {\rm{L}}}}$ 与瑞利数关系

The mean Nusselt number $\overline {Nu}_{{{\rm{f}}, {\rm{L}}}}$ versus Rayleigh number with different Knudsen number $1 \times 10^{-3} \leqslant Kn_{{\rm{f}}, {\rm{s}}} \leqslant$ $1 \times 10^2$ and fixed nanoparticle volume fraction $\phi_{\rm{s}} = 8 {\text{%}}$ .

下载: 全尺寸图片幻灯片

不同克努森数 $1 \times 10^{-3} \leqslant Kn_{{\rm{f}}, {\rm{s}}} \leqslant 1 \times 10^2$ 和固定颗粒体积分数 $\phi_{\rm{s}} = 8 {\text{%}}$ 下, 纳米流体相较于基液传热增加率 $Re_{{\rm{n}}, {\rm{f}}}$ 与瑞利数关系

The enhancement ratio $Re_{{\rm{n}}, {\rm{f}}}$ versus Rayleigh number with different Knudsen number $1 \times 10^{-3} \leqslant Kn_{{\rm{f}}, {\rm{s}}} \leqslant$ $1 \times 10^2$ and fixed nanoparticle volume fraction $\phi_{\rm{s}} = 8 {\text{%}}$ .

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全参数范围下(a)平均努塞尔数的对数函数 $\lg (\overline {Nu}_{{{\rm{f}}, {\rm{L}}}}) =$ $l \lg \Big(\dfrac{{{k_{\rm{n}}}}}{{{k_{\rm{f}}}}}\overline {Nu}_{{{\rm{n}}, {\rm{L}}}}\Big)$ 和(b)纳米流体相较基液传热增加率 $Re_{{\rm{n}}, {\rm{f}}}$ 随不同克努森数 $Kn_{{\rm{f}}, {\rm{s}}}$ 、瑞利数 $Ra_{{{\rm{f}}, {\rm{L}}}}$ 、颗粒体积分数 $\phi_{\rm{s}}$ 变化的三维等值面图

The three dimensional isosurfaces of (a) logarithmic function of mean Nusselt number $\lg (\overline {Nu}_{{{\rm{f}}, {\rm{L}}}}) =$ $\lg \Big(\dfrac{{{k_{\rm{n}}}}}{{{k_{\rm{f}}}}}\overline {Nu}_{{{\rm{n}}, {\rm{L}}}}\Big)$ and (b) enhancement ratio $Re_{{\rm{n}}, {\rm{f}}}$ over the full parameter range as a function of Knudsen number $Kn_{{\rm{f}}, {\rm{s}}}$ , Rayleigh number $Ra_{{{\rm{f}}, {\rm{L}}}}$ , and nanoparticle volume fraction $\phi_{\rm{s}}$ .

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表 1 水和氧化铝纳米颗粒的物性参数^[37,38]

Physical properties of the water and ${{\rm{Al}}_2{\rm{O}}_3}$ nanoparticle^[37,38].

	ρ/ $(\mathrm{kg {\cdot} m^{-3}})$	$c_{p}/$ $\rm {(J {\cdot} kg^{-1}{\cdot} K^{-1}})$	$k/$ $\rm{(W {\cdot} m^{-1} {\cdot} K^{-1})}$	λ/ nm
水	997.1	4179	0.613	0.3
Al₂O₃纳米颗粒	3970	765	40	35

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map

[1]	Wang X, Song Y, Li C, Zhang Y, Ali H M, Sharma S, Li R, Yang M, Gao T, Liu M, Cui X, Said Z, Zhou Z 2024 Int. J. Adv. Manuf. Technol. 131 3113 Google Scholar
[2]	Sandhya M, Ramasamy D, Sudhakar K, Kadirgama K, Samykano M, Harun W S W, Najafi G, Mofijur M, Mazlan M 2021 Sustainable Energy Technol. Assess. 44 101058 Google Scholar
[3]	Said Z, Sundar L S, Tiwari A K, Ali H M, Sheikholeslami M, Bellos E, Babar H 2022 Phys. Rep. 946 1 Google Scholar
[4]	Smaisim G F, Mohammed D B, Abdulhadi A M, Uktamov K F, Alsultany F H, Izzat S E, Ansari M J, Kzar H H, Al-Gazally M E, Kianfar E 2022 J. Sol-Gel Sci. Technol. 104 1 Google Scholar
[5]	肖波齐, 范金土, 蒋国平, 陈玲霞 2012 61 317 Google Scholar Xiao B Q, Fan J T, Jiang G P, Chen L X 2012 Acta Phys. Sin. 61 317 Google Scholar
[6]	Azmi W H, Sharma K V, Mamat R, Najafi G, Mohamad M S 2016 Renewable Sustainable Energy Rev. 53 1046 Google Scholar
[7]	Wang X, Xu X, Choi S U S 1999 J. Thermophys. Heat Transfer 13 474 Google Scholar
[8]	Das S K, Putra N, Thiesen P, Roetzel W 2003 J. Heat Transfer 125 567 Google Scholar
[9]	Nguyen C T, Desgranges F, Galanis N, Roy G, Mare T, Boucher S, Minsta H A 2008 Int. J. Therm. Sci. 47 103 Google Scholar
[10]	Maxwell J C 1982 A Treatise on Electricity and Magnetism (Vol. 2) (London: Oxford University Press) pp173–215
[11]	Nan C W, Shi Z, Lin Y 2003 Chem. Phys. Lett. 375 666 Google Scholar
[12]	Mintsa H A, Roy G, Nguyen C T, Doucet D 2009 Int. J. Therm. Sci. 48 363 Google Scholar
[13]	Brinkman H C 1952 J. Chem. Phys. 20 571 Google Scholar
[14]	Batchelor G K 1977 J. Fluid Mech. 83 97 Google Scholar
[15]	Nguyen C T, Desgranges F, Roy G, Galanis N, Mare T, Boucher S, Minsta H A 2007 Int. J. Heat Fluid Flow 28 1492 Google Scholar
[16]	Majumdar A 1993 J. Heat Transfer 115 7 Google Scholar
[17]	Mazumder S, Majumdar A 2001 J. Heat Transfer 123 749 Google Scholar
[18]	Su Y, Davidson J H 2018 Int. J. Heat Mass Transfer 127 303 Google Scholar
[19]	Chambre P A, Schaaf S A 1961 Flow of Rarefied Gases (Princeton: Princeton University Press) pp78–146
[20]	Sui P, Su Y, Sin V, Davidson J H 2022 Int. J. Heat Mass Transfer 187 122541 Google Scholar
[21]	Zarki A, Ghalambaz M, Chamkha A J, Ghalambaz M, Rossi D D 2015 Adv. Powder Technol. 26 935 Google Scholar
[22]	Sabour M, Ghalambaz M, Chamkha A 2017 Int. J. Numer. Methods Heat Fluid Flow 27 1504 Google Scholar
[23]	Paul T C, Morshed A, Fox E B, Khan J A 2017 Int. J. Heat Mass Transfer 28 753
[24]	Liu F, Wang L 2009 Int. J. Heat Mass Transfer 52 5849 Google Scholar
[25]	Zahmatkesh I, Sheremet M, Yang L, Heris S Z, Sharifpur M, Meyer J P, Ghalambaz M, Wongwises S, Jing D, Mahian O 2021 J. Mol. Liq. 321 114430 Google Scholar
[26]	Trodi A, Benhamza M E H 2017 Chem. Eng. Commun. 204 158 Google Scholar
[27]	Sheikhzadeh G A, Aghaei A, Soleimani S 2018 Challenges Nano Micro Scale Sci. Technol. 6 27
[28]	Dogonchi A S, Hashemi-Tilehnoee M, Waqas M, Seyyedi S M, Animasaun I L, Ganji D D 2020 Phys. A 540 123034 Google Scholar
[29]	张贝豪, 郑林 2020 69 164401 Google Scholar Zhang B H, Zheng L 2020 Acta Phys. Sin. 69 164401 Google Scholar
[30]	Su Y, Sui P, Davidson J H 2022 Renew. Energy 184 712 Google Scholar
[31]	Lai F, Yang Y 2011 Int. J. Therm. Sci. 50 1930 Google Scholar
[32]	Sheikholeslami M, Gorji-Bandpy M, Domairry G 2013 Appl. Math. Mech. 34 833 Google Scholar
[33]	齐聪, 何光艳, 李意民, 何玉荣 2015 64 328 Google Scholar Qi C, He G Y, Li Y M, He Y R 2015 Acta Phys. Sin. 64 328 Google Scholar
[34]	Taher M A, Kim H D, Lee Y W 2017 Heat Transfer Res. 48 1025 Google Scholar
[35]	袁俊杰, 叶欣, 单彦广 2021 计算物理 38 57 Yuan J J, Ye X, Shan Y G 2021 Chinese Journal of Computational Physics 38 57
[36]	Ganji D D, Malvandi A 2014 Powder Technol. 263 50 Google Scholar
[37]	Hwang K S, Lee J H, Jang S P 2009 Int. J. Heat Mass Transfer 50 4003
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