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变截面微管道中高zeta势下幂律流体的旋转电渗滑移流动

张天鸽 任美蓉 崔继峰 陈小刚 王怡丹

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变截面微管道中高zeta势下幂律流体的旋转电渗滑移流动

张天鸽, 任美蓉, 崔继峰, 陈小刚, 王怡丹

Rotational electroosmotic slip flow of power-law fluid at high zeta potential in variable-section microchannel

Zhang Tian-Ge, Ren Mei-Rong, Cui Ji-Feng, Chen Xiao-Gang, Wang Yi-Dan
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  • 本文研究高zeta势下具有Navier滑移边界条件的幂律流体, 在变截面微管道中的垂向磁场作用下的旋转电渗流动. 在不使用Debye–Hückel线性近似条件时, 利用有限差分法数值计算外加磁场的旋转电渗流的电势分布和速度分布. 当行为指数$n = 1$时得到的流体为牛顿流体, 将本文的分析结果与Debye–Hückel 线性近似所得解析近似解作比较, 证明本文数值方法的可行性. 除此之外, 还详细讨论行为指数n、哈特曼数Ha、旋转角速度$\varOmega$、电动宽度K及滑移参数$\beta $对速度分布的影响, 得到当哈特曼数Ha >1时, 速度随着哈特曼数 Ha 的增加而减小; 但当哈特曼数Ha <1时, x方向速度 u 的大小随着 Ha 的增加而增加.
    In this paper we study the rotating electroosmotic flow of a power-law fluid with Navier slip boundary conditions under high zeta potential subjected to the action of a vertical magnetic field in a variable cross-section microchannel. Without using the Debye–Hückel linear approximation, the finite difference method is used to numerically calculate the potential distribution and velocity distribution of the rotating electroosmotic flow subjected to an external magnetic field. When the behavior index $n = 1$, the fluid obtained is a Newtonian fluid. The analysis results in this paper are compared with the analytical approximate solutions obtained in the Debye–Hückel linear approximation to prove the feasibility of the numerical method in this paper. In addition, the influence of behavior index n, Hartmann number Ha, rotation angular velocity $\Omega $, electric width K and slip parameters $\beta $ on the velocity distribution are discussed in detail. It is obtained that when the Hartmann number Ha > 1, the velocity decreases with the increase of the Hartmann number Ha; but when the Hartmann number Ha < 1, the magnitude of the x-direction velocity u increases with the augment of Ha.
      通信作者: 陈小刚, xiaogang_chen@imut.edu.cn
    • 基金项目: 国家自然科学基金(批准号: 12062018, 12172333)、内蒙古自治区高等学校青年科技英才支持计划资助项目(批准号: NJYT22075)和内蒙古自然科学基金(批准号: 2020MS01015)资助的课题.
      Corresponding author: Chen Xiao-Gang, xiaogang_chen@imut.edu.cn
    • Funds: Project supported by the National Natural Science Foundation of China (Grant Nos. 12062018, 12172333), the Young Talents of Science and Technology in Universities of Inner Mongolia Autonomous Region, China (Grant No. NJYT22075), and the Natural Science Foundation of Inner Mongolia, China (Grant No. 2020MS01015).
    [1]

    Stone H A, Stroock A D, Ajdari A 2004 Annu. Rev. Fluid Mech. 36 381Google Scholar

    [2]

    Patel M, Kruthiventi S S H, Kaushik P 2020 Colloids Surf. B 193 111058Google Scholar

    [3]

    Srinivas, Bhadri 2016 Colloids Surf. A 492 144Google Scholar

    [4]

    Nekoubin N 2018 J. Non-Newtonian Fluid Mech. 260 54Google Scholar

    [5]

    Baños R D, Arcos J C, Bautista O, Méndez F, Merchán-Cruz E A 2021 J. Braz. Soc. Mech. Sci. 43 1Google Scholar

    [6]

    Baños R, Arcos J, Bautista O, Méndez F 2020 Defect Diffus. Forum 399 92Google Scholar

    [7]

    姜玉婷, 齐海涛 2015 物理学报 64 174702Google Scholar

    Jiang Y T, Qi H T 2015 Acta Phys. Sin. 64 174702Google Scholar

    [8]

    Ajdari A 2002 Phys. Rev. E 65 16301Google Scholar

    [9]

    Chang C C, Wang C Y 2011 Phys. Rev. E 84 056320Google Scholar

    [10]

    Song J, Wang S W, Zhao M L, Li N 2020 Z. Naturforsch. A: Phys. Sci. 75 649Google Scholar

    [11]

    Shit G C, Mondal A, Sinha A, Kundu P K 2016 Colloids Surf. A 489 249Google Scholar

    [12]

    刘全生, 杨联贵, 苏洁 2013 物理学报 62 144702Google Scholar

    Liu Q S, Yang L G, Su J 2013 Acta Phys. Sin. 62 144702Google Scholar

    [13]

    段娟, 陈耀钦, 朱庆勇 2016 物理学报 65 034702Google Scholar

    Duan J, Chen Y Q, Zhu Q Y 2016 Acta Phys. Sin. 65 034702Google Scholar

    [14]

    Weston M C, Gerner M D, Fritsch I 2010 Anal. Chem. 82 3411Google Scholar

    [15]

    Jian Y J, Chang L 2015 AIP Adv. 5 057121Google Scholar

    [16]

    Xie Z Y, Jian Y J 2017 Colloids Surf. A 529 334Google Scholar

    [17]

    Habib U, Hayat T, Ahmad S, Alhodaly M S 2021 Int. Commun. Heat Mass Transfer 122 105111Google Scholar

    [18]

    Sarkar S, Ganguly S 2017 J. Non-Newtonian Fluid Mech. 250 18Google Scholar

    [19]

    Yang C H, Jian Y J, Xie Z Y, Li F Q 2020 Micromachines 11 418Google Scholar

    [20]

    Xie Z Y, Jian Y J 2017 Energy 139 1080Google Scholar

    [21]

    Wang S W, Li N, Zhao M L, Azese M N 2018 Z. Naturforsch. A: Phys. Sci. 73 825Google Scholar

    [22]

    Xie Z Y, Jian Y J 2014 Colloids Surf. A 461 231Google Scholar

    [23]

    Bird R B, Armstrong R C, Hassager O, Curtiss C F, Middleman S 1978 Phys. Today 31 54Google Scholar

  • 图 1  变截面微通道中流体流动示意图

    Fig. 1.  Schematic view of the flow in a variable cross-section microchannel.

    图 2  目前数值解与Chang和Wang[9]解析解的比较, 其中 $ \beta = 0, $$ K = 30, $$\varOmega = 100{\text{ }}{\rm{rad}}/{\rm{s}}, {\text{ }}{\bar \psi _\omega } = 1{\text{ }}{\rm{V}}, {\text{ }}a = 0, {\text{ }}Ha = 0, {\text{ }}S = 0$

    Fig. 2.  Comparison of the current numerical solution with the analytical solution of Chang and Wang [9], $ \beta = 0, $$ K = 30, $$\varOmega = 100{\text{ }}{\rm{rad}}/{\rm{s}}, {\text{ }}{\bar \psi _\omega } = 1{\text{ }}{\rm{V}}, {\text{ }}a = 0, {\text{ }}Ha = 0, {\text{ }}S = 0$

    图 3  当无滑移边界条件时, 幂律流体行为指数n对外加磁场的旋转电渗流速度的影响, 其中$\beta = 0, {\text{ }}K = 10, {\text{ }}\varOmega = 100{\text{ }}{\rm{rad}}/{\rm{s}}, $ ${\text{ }}{\bar \psi _\omega } = 5 \;{\rm{V}}, {\text{ }}a = 0.05, {\text{ }}Ha = 1, {\text{ }}S = $1

    Fig. 3.  When there is a no-slip boundary condition, the influence of power-law fluid behavior index n on rotating electroosmotic flow velocity with the external magnetic field, $\beta = 0, {\text{ }}K = 10, {\text{ }}\varOmega = 100{\text{ }}{\rm{rad}}/{\rm{s}}, {\text{ }}{\bar \psi _\omega } = 5 \;{\rm{V}}, {\text{ }}a = 0.05, {\text{ }}Ha = 1, {\text{ }}S = 1$

    图 4  当存在滑移边界条件时, 幂律流体行为指数n对外加磁场的旋转电渗流速度的影响, 其中$ \beta = 0.1, {\text{ }}K = 10, {\text{ }}\varOmega = 100{\text{ }}{\rm{rad}}/{\rm{s}}, $ ${\text{ }}{\bar \psi _\omega } = 5{\text{ }}{\rm{V}}, {\text{ }}a = 0.05, {\text{ }}Ha = 1, {\text{ }}S =$1

    Fig. 4.  When there is a slip boundary condition, the influence of the power-law fluid behavior index n on the rotating electroosmotic flow velocity with an external magnetic field, $\beta = 0.1, {\text{ }}K = 10, {\text{ }}\varOmega = 100{\text{ }}{\rm{rad}}/{\rm{s}}, {\text{ }}{\bar \psi _\omega } = 5{\text{ }}{\rm{V}}, {\text{ }}a = 0.05, {\text{ }}Ha = 1, {\text{ }}S = 1$

    图 5  哈特曼数Ha对外加磁场的旋转电渗流速度的影响, 其中$ n = 0.8, {\text{ }}K = 10, $ $ {\text{ }}\beta = 0.1, {\text{ }}\varOmega = 100{\text{ }}{\rm{rad}}/{\rm{s}}, $ ${\text{ }}{\bar \psi _\omega } = 5{\text{ }}{\rm{V}}, {\text{ }}a = $$ 0.05, {\text{ }}S =$1

    Fig. 5.  The influence of Hartmann number Ha on the velocity of rotating electroosmotic flow with external magnetic field, $ n = 0.8, {\text{ }}K = 10, $$ {\text{ }}\beta = 0.1, {\text{ }}\varOmega = 100{\text{ }}{\rm{rad}}/{\rm{s}}, {\text{ }}{\bar \psi _\omega } = 5{\text{ }}{\rm{V}}, {\text{ }}a = 0.05, {\text{ }}S = 1 $

    图 6  哈特曼数Ha对外加磁场的旋转电渗流速度的影响, 其中$ n = 1.2, {\text{ }}K = 10, $ ${\text{ }}\beta = 0.1, {\text{ }}\varOmega = 100{\text{ }}{\rm{rad}}/{\rm{s}}, $ ${\bar \psi _\omega } = 5{\text{ }}{\rm{V}}, {\text{ }}a = $$ 0.05, {\text{ }}S =$1

    Fig. 6.  The influence of Hartmann number Ha on the velocity of rotating electroosmotic flow with external magnetic field, $ n = 1.2, {\text{ }}K = 10, $$ {\text{ }}\beta = 0.1, {\text{ }}\varOmega = 100{\text{ }}{\rm{rad}}/{\rm{s}}, {\text{ }}{\bar \psi _\omega } = 5{\text{ }}{\rm{V}}, {\text{ }}a = 0.05, {\text{ }}S = 1 $

    图 7  旋转角速度$\varOmega $对外加磁场的旋转电渗流速度的影响, 其中$ n = 0.8, $$ {\bar \psi _\omega } = {\text{5}}\;{\rm{V}}, $$ a = 0.05, {\text{ }}Ha = 1, {\text{ }}S = 1 $ (a)$K = 10, {\text{ }}\beta = $$ 0.1;$ (b) $ K = 10, {\text{ }}\beta = 0.1; $ (c) $ K = 10, {\text{ }}\beta = 0; $ (d)$ K = 20, {\text{ }}\beta = 0.1. $

    Fig. 7.  The influence of the rotational angular velocity $\varOmega $ on the rotational electroosmotic flow velocity of the external magnetic field, $ n = 0.8, $$ {\bar \psi _\omega } = {\text{5 }}V, $$ a = 0.05, {\text{ }}Ha = 1, {\text{ }}S = 1 $(a) $ K = 10, {\text{ }}\beta = 0.1; $ (b) $ K = 10, {\text{ }}\beta = 0.1; $ (c) $ K = 10, {\text{ }}\beta = 0; $ (d)$K = 20, $$ {\text{ }}\beta = 0.1$

    图 8  旋转角速度$\varOmega $对外加磁场的旋转电渗流速度的影响, 其中$ n = 1.2, $$ {\bar \psi _\omega } = 5{\text{ }}{\rm{V}}, {\text{ }} $$ a = 0.05, {\text{ }}Ha = 1, {\text{ }}S = 1 $ (a) $\beta = 0.1, {\text{ }}K = $$ 10.$ (b) $ \beta = 0.1, {\text{ }}K = 10. $ (c) $ \beta = 0, {\text{ }}K = 10. $ (d) $\beta = 0.1, {\text{ }}K = 30$

    Fig. 8.  The influence of the rotational angular velocity $\varOmega $ on the rotational electroosmotic flow velocity of the external magnetic field, $ n = 1.2, $$ {\bar \psi _\omega } = 5{\text{ }}V, {\text{ }} $$ a = 0.05, {\text{ }}Ha = 1, {\text{ }}S = 1 $ (a) $ \beta = 0.1, {\text{ }}K = 10. $ (b) $ \beta = 0.1, {\text{ }}K = 10. $ (c) $ \beta = 0, {\text{ }}K = 10. $ (d) $\beta = 0.1, $$ {\text{ }}K = 30$

    图 9  电动宽度 K 对外加磁场的旋转电渗流速度分布的影响, 其中$ n = 0.8, $ $ \beta {\text{ = 0}}{\text{.1, }}\varOmega = 100{\text{ }}{\rm{rad}}/{\rm{s}}, {\text{ }}{\bar \psi _\omega } = 5{\text{ }}{\rm{V}}, $ ${\text{ }}a = 0.05, {\text{ }}Ha = $$ 1, {\text{ }}S = $1

    Fig. 9.  The influence of the electric width K on the velocity distribution of rotating electroosmotic flow with external magnetic field, $n = 0.8, {\text{ }}\beta {\text{ = 0}}{\text{.1, }}\varOmega = 100{\text{ rad/s}}, {\text{ }}{\bar \psi _\omega } = 5{\text{ }}V, {\text{ }}$$ a = 0.05, {\text{ }}Ha = 1, {\text{ }}S = 1 $

    图 10  电动宽度 K 对外加磁场的旋转电渗流速度分布的影响, 其中$ n = 1.2, $$ \beta {\text{ = 0}}{\text{.1, }}\varOmega = 100{\text{ }}{\rm{rad}}/{\rm{s}}, {\text{ }}{\bar \psi _\omega } = 5{\text{ }}{\rm{V}}, {\text{ }}a = 0.05, {\text{ }}Ha = $$ 1, {\text{ }}S = $1

    Fig. 10.  The influence of the electric width K on the velocity distribution of rotating electroosmotic flow with external magnetic field, $ n = 1.2, {\text{ }}\beta {\text{ = 0}}{\text{.1, }}\varOmega = 100{\text{ }}{\rm{rad}}/{\rm{s}}, {\text{ }}{\bar \psi _\omega } = 5{\text{ }}{\rm{V}}, {\text{ }} $$ a = 0.05, {\text{ }}Ha = 1, {\text{ }}S = 1 $

    图 11  电动宽度 K 对外加磁场的旋转电渗流速度分布的影响, 其中$ n = 1.2, $$\beta {\text{ = 0, }}\varOmega = 100{\text{ }}{\rm{rad}}/{\rm{s}}, {\text{ }}{\bar \psi _\omega } = 5{\text{ }}{\rm{V}}, {\text{ }}a = 0.05, {\text{ }}Ha = $$ 1, {\text{ }}S =$1

    Fig. 11.  The influence of the electric width K on the velocity distribution of rotating electroosmotic flow with external magnetic field, $ n = 1.2, {\text{ }}\beta {\text{ = 0, }}\varOmega = 100{\text{ }}{\rm{rad}}/{\rm{s}}, {\text{ }}{\bar \psi _\omega } = 5{\text{ }}{\rm{V}}, {\text{ }} $$ a = 0.05, {\text{ }}Ha = 1, {\text{ }}S = 1 $

    图 12  滑移参数$\beta $对外加磁场的旋转电渗流速度的影响, 其中$ n = 0.8, $$ K = 10, $$\varOmega = 100{\text{ }}{\rm{rad}}/{\rm{s}}, {\text{ }}{\bar \psi _\omega } = 5{\text{ }}{\rm{V}}, {\text{ }}a = 0.05, {\text{ }}Ha = $$ 1, {\text{ }}S =$1

    Fig. 12.  The influence of the slip parameter$\beta $on the rotating electroosmotic flow velocity with an external magnetic field, $ n = 0.8, $$ K = 10, $$ \varOmega = 100{\text{ }}{\rm{rad}}/{\rm{s}}, {\text{ }}{\bar \psi _\omega } = 5{\text{ }}{\rm{V}}, {\text{ }}a = 0.05, {\text{ }}Ha = 1, {\text{ }}S = 1$

    图 13  滑移参数$\beta $对外加磁场的旋转电渗流速度的影响, 其中$ n = 1.2, $$ K = 10, $$ \varOmega = 100{\text{ }}{\rm{rad}}/{\rm{s}}, {\text{ }}{\bar \psi _\omega } = 5{\text{ }}{\rm{V}}, {\text{ }}a = 0.05, {\text{ }}Ha = $$ 1, {\text{ }}S = 1 $

    Fig. 13.  The influence of the slip parameter$\beta $on the rotating electroosmotic flow velocity with an external magnetic field, $ n = 1.2, $$ K = 10, $$\varOmega = 100{\text{ }}{\rm{rad}}/{\rm{s}}, {\text{ }}{\bar \psi _\omega } = 5{\text{ }}{\rm{V}}, {\text{ }}a = 0.05, {\text{ }}Ha = 1, {\text{ }}S = 1$

  • [1]

    Stone H A, Stroock A D, Ajdari A 2004 Annu. Rev. Fluid Mech. 36 381Google Scholar

    [2]

    Patel M, Kruthiventi S S H, Kaushik P 2020 Colloids Surf. B 193 111058Google Scholar

    [3]

    Srinivas, Bhadri 2016 Colloids Surf. A 492 144Google Scholar

    [4]

    Nekoubin N 2018 J. Non-Newtonian Fluid Mech. 260 54Google Scholar

    [5]

    Baños R D, Arcos J C, Bautista O, Méndez F, Merchán-Cruz E A 2021 J. Braz. Soc. Mech. Sci. 43 1Google Scholar

    [6]

    Baños R, Arcos J, Bautista O, Méndez F 2020 Defect Diffus. Forum 399 92Google Scholar

    [7]

    姜玉婷, 齐海涛 2015 物理学报 64 174702Google Scholar

    Jiang Y T, Qi H T 2015 Acta Phys. Sin. 64 174702Google Scholar

    [8]

    Ajdari A 2002 Phys. Rev. E 65 16301Google Scholar

    [9]

    Chang C C, Wang C Y 2011 Phys. Rev. E 84 056320Google Scholar

    [10]

    Song J, Wang S W, Zhao M L, Li N 2020 Z. Naturforsch. A: Phys. Sci. 75 649Google Scholar

    [11]

    Shit G C, Mondal A, Sinha A, Kundu P K 2016 Colloids Surf. A 489 249Google Scholar

    [12]

    刘全生, 杨联贵, 苏洁 2013 物理学报 62 144702Google Scholar

    Liu Q S, Yang L G, Su J 2013 Acta Phys. Sin. 62 144702Google Scholar

    [13]

    段娟, 陈耀钦, 朱庆勇 2016 物理学报 65 034702Google Scholar

    Duan J, Chen Y Q, Zhu Q Y 2016 Acta Phys. Sin. 65 034702Google Scholar

    [14]

    Weston M C, Gerner M D, Fritsch I 2010 Anal. Chem. 82 3411Google Scholar

    [15]

    Jian Y J, Chang L 2015 AIP Adv. 5 057121Google Scholar

    [16]

    Xie Z Y, Jian Y J 2017 Colloids Surf. A 529 334Google Scholar

    [17]

    Habib U, Hayat T, Ahmad S, Alhodaly M S 2021 Int. Commun. Heat Mass Transfer 122 105111Google Scholar

    [18]

    Sarkar S, Ganguly S 2017 J. Non-Newtonian Fluid Mech. 250 18Google Scholar

    [19]

    Yang C H, Jian Y J, Xie Z Y, Li F Q 2020 Micromachines 11 418Google Scholar

    [20]

    Xie Z Y, Jian Y J 2017 Energy 139 1080Google Scholar

    [21]

    Wang S W, Li N, Zhao M L, Azese M N 2018 Z. Naturforsch. A: Phys. Sci. 73 825Google Scholar

    [22]

    Xie Z Y, Jian Y J 2014 Colloids Surf. A 461 231Google Scholar

    [23]

    Bird R B, Armstrong R C, Hassager O, Curtiss C F, Middleman S 1978 Phys. Today 31 54Google Scholar

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出版历程
  • 收稿日期:  2021-12-16
  • 修回日期:  2022-03-03
  • 上网日期:  2022-06-29
  • 刊出日期:  2022-07-05

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