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Liquid droplets impacting on the solid surface is an ubiquitous phenomenon in natural, agricultural, and industrial processes. The maximum spreading diameter of a liquid droplet impacting on a solid surface is a significant parameter in the industrial applications such as inkjet printing, spray coating, and spray cooling. However, former models cannot accurately predict the maximum spreading diameter on a superhydrophilic surface, especially under low Weber number (We). In this work, the spreading characteristics of a water droplet impacting on a superhydrophilic surface are explored by high-speed technique. The spherical cap of the spreading droplet, gravitational potential energy, and auxiliary dissipation are introduced into the modified theoretical model based on the energy balance. The model includes two viscous dissipation terms: the viscous dissipation of the initial kinetic energy and the auxiliary dissipation in spontaneous spreading. The energy component analysis in the spreading process shows that the kinetic energy, surface energy, and gravitational potential energy are all transformed into the viscous dissipation on the superhydrophilic surface. The transformation of surface energy into viscous dissipation is dominant at lower We while the transformation of kinetic energy into viscous dissipation is dominant at higher We. It is found that the gravitational potential energy and auxiliary dissipation play a significant role in spreading performance at low We according to the energy component analysis. Moreover, the energy components predicted by the modified model accord well with the experimental data. As a result, the proposed model can predict the maximum spreading diameter of a droplet impacting on the superhydrophilic surface accurately. Furthermore, the model proposed in this work can predict the maximum spreading diameter of the droplet impacting on the hydrophilic surface and hydrophobic surface. The results of this work are of great significance for controlling droplet spreading diameter in spray cooling and falling film evaporation.
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Keywords:
- superhydrophilic surfaces /
- maximum spreading diameter /
- gravitational potential energy /
- auxiliary dissipation
[1] Liang G, Mudawar I 2016 Int. J. Heat Mass Transfer 101 577
Google Scholar
[2] 马学虎, 兰忠, 王凯, 陈彦松, 程雅琦, 杜宾港, 叶轩 2018 化工学报 69 9
Google Scholar
Ma X H, Lan Z, Wang K, Chen Y S, Cheng Y Q, Du B G, Ye X 2018 J. Chem. Ind. Eng. 69 9
Google Scholar
[3] Snoeijer J H, Andreotti B 2013 Annu. Rev. Fluid Mech. 45 269
Google Scholar
[4] 朱君悦, 段远源, 王晓东, 闵琪 2014 化工学报 65 765
Google Scholar
Zhu J Y, Duan Y Y, Wang X D, Min Q 2014 J. Chem. Ind. Eng. 65 765
Google Scholar
[5] 刘海龙, 沈学峰, 王睿, 曹宇, 王军锋 2018 力学学报 50 1024
Google Scholar
Liu H L, Shen X F, Wang R, Cao Y, Wang J F 2018 Acta Mech. Sin. 50 1024
Google Scholar
[6] 闵琪, 段远源, 王晓东, 吴莘馨 2013 热科学与技术 12 335
Google Scholar
Min Q, Duan Y Y, Wang X D, Wu X X 2013 J. Therm. Sci. Technol. 12 335
Google Scholar
[7] Hu H B, Chen L B, Bao L Y, Huang S H 2014 Chin. Phys. B 23 074702
Google Scholar
[8] Sun Z H, Han R J 2008 Chin. Phys. B 17 3185
Google Scholar
[9] Wang Y B, Wang Y F, Gao S R, Yang Y R, Wang X D, Chen M 2020 Langmuir 36 9306
Google Scholar
[10] Hu H B, Huang S H, Chen L B 2013 Chin. Phys. B 22 084702
Google Scholar
[11] Song M, Liu Z, Ma Y, Dong Z, Wang Y, Jiang L 2017 NPG Asia Mater. 9 1
Google Scholar
[12] 王高远, 胥蕊娜, 陈剑楠, 陈学, 姜培学 2018 工程热物理学报 39 1797
Google Scholar
Wang G Y, Xu R N, Chen J N, Chen X, Jiang P X 2018 J. Eng. Thermphys. 39 1797
Google Scholar
[13] Lin S, Zhao B, Zou S, Guo J, Wei Z, Chen L 2018 J. Colloid Interface Sci. 516 86
Google Scholar
[14] Lee J B, Derome D, Guyer R, Carmeliet J 2016 Langmuir 32 1299
Google Scholar
[15] Tang C, Qin M, Weng X, Zhang X, Zhang P, Li J, Huang Z 2017 Int. J. Multiphase Flow 96 56
Google Scholar
[16] 荣松, 沈世全, 王天友, 车志钊 2019 物理学报 68 154701
Google Scholar
Rong S, Shen S Q, Wang T Y, Che Z Z 2019 Acta Phys. Sin. 68 154701
Google Scholar
[17] Ding B, Wang H, Zhu X, Chen R, Liao Q 2018 Int. J. Heat Mass Transfer 124 1025
Google Scholar
[18] 范瑶, 王宏, 朱恂, 黄格永, 丁玉栋, 廖强 2016 化工学报 67 2709
Google Scholar
Fan Y, Wang H, Zhu X, Huang G Y, Ding Y D, Liao Q 2016 J. Chem. Ind. Eng. 67 2709
Google Scholar
[19] Ding B, Wang H, Zhu X, Chen R, Liao Q 2019 Int. J. Heat Mass Transfer 138 844
Google Scholar
[20] Chen M, Wu D, Chen D, Deng J, Liu H, Jiang J 2020 Chem. Eng. Sci. 226 115864
Google Scholar
[21] 焦云龙, 刘小君, 逄明华, 刘焜 2016 物理学报 65 016801
Google Scholar
Jiao Y L, Liu X J, Pang M H, Liu K 2016 Acta Phys. Sin. 65 016801
Google Scholar
[22] 叶学民, 李永康, 李春曦 2016 物理学报 65 234701
Google Scholar
Ye X M, Li Y K, Li C X 2016 Acta Phys. Sin. 65 234701
Google Scholar
[23] Roisman I V 2009 Phys. Fluids 21 052104
Google Scholar
[24] Liang G, Chen Y, Chen L, Shen S 2019 Ind. Eng. Chem. Res. 58 10053
Google Scholar
[25] Laan N, de Bruin K G, Bartolo D, Josserand C, Bonn D 2014 Phys. Rev. Appl. 2 044018
Google Scholar
[26] Lee J B, Laan N, de Bruin K G, Skantzaris G, Shahidzadeh N, Derome D, Carmeliet J, Bonn D 2016 J. Fluid Mech. 786 R4
Google Scholar
[27] Chandra S, Avedisian C T 1991 Proc. R. Soc. A 432 13
Google Scholar
[28] Pasandideh‐Fard M, Qiao Y M, Chandra S, Mostaghimi J 1996 Phys. Fluids 8 650
Google Scholar
[29] Mao T, Kuhn D C S, Tran H 1997 AlChE J. 43 2169
Google Scholar
[30] Ukiwe C, Kwok D Y 2005 Langmuir 21 666
Google Scholar
[31] Huang H M, Chen X P 2018 Phys. Fluids 30 022106
Google Scholar
[32] Park H, Carr W W, Zhu J, Morris J F 2003 AlChE J. 49 2461
Google Scholar
[33] Li Ri, Ashgriz Nasser, Sanjeev C 2010 J. Fluids Eng. 132 061302
Google Scholar
[34] Gao X, Li R 2014 AlChE J. 60 2683
Google Scholar
[35] Wang F, Yang L, Wang L, Zhu Y, Fang T 2019 Langmuir 35 3204
Google Scholar
[36] Panda A, Pati A R, Saha B, Kumar A, Mohapatra S S 2019 Chem. Eng. Sci. 207 527
Google Scholar
[37] Zhang Z, Jiang P X, Christopher D M, Liang X-G 2015 Int. J. Heat Mass Transfer 80 26
Google Scholar
[38] Zheng Y, Ma X, Li Y, Jiang R, Wang K, Lan Z, Liang Q 2017 Appl. Therm. Eng. 111 1548
Google Scholar
[39] Nam Y, Ju Y S 2013 J. Adhes. Sci. Technol. 27 2163
Google Scholar
[40] Vadillo D C, Soucemarianadin A, Delattre C, Roux D C D 2009 Phys. Fluids 21 122002
Google Scholar
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图 1 液滴铺展实验系统(A: 高速摄像机(Photron APX RS); B: 高速摄像机(Photron Mini UX100); C: 微量注射泵(LSP01-1 BH); D: 液滴生成装置; E: 恒温台; F: 高亮光源; G: 数据采集设备; G: 精确自动控制位移的直线位移滑台)
Figure 1. The droplet spreading experiment platform. (A: high speed camera (Photron APX RS); B: high speed camera (Photron Mini UX100); C: micro syringe pump (LSP01-1 BH); D: droplet generating device; E: heating platform; F: diffuse light source; G: data acquisition computer; H: linear displacement slide.
图 2 (a)光滑铜表面的SEM图; (b)超亲水表面SEM图
Figure 2. SEM images of (a) smooth copper surface and (b) superhydrophilic surface.
图 3 液滴铺展过程 (a)光滑铜亲水表面; (b)超亲水表面We = 1.91; (c)超亲水表面We = 25.59
Figure 3. Droplet spreading process: (a) Smooth copper hydrophilic surface; (b) superhydrophilic surface at We = 1.91; (c) superhydrophilic surface at We = 25.59.
图 4 液滴在不同We下撞击超亲水表面铺展因子随时间的变化过程
Figure 4. The variation of spreading factor β with time
图 5 液滴在超亲水表面最大铺展直径形态
Figure 5. Sketch of droplet shape at its maximum spread on superhydrophilic surface.
图 6 (a)模型计算的ΔEk, ΔEs, W, ΔEp随We的变化; (b)低We下ΔEk, ΔEs, W, ΔEp占总能量的占比(青色区域放大)
Figure 6. (a) Variation of the energy component: ΔEk, ΔEs, W, ΔEp with We; (b) comparison of the energy component: ΔEk, ΔEs, W, ΔEp at low We
图 7 (a)超亲水表面液滴铺展过程各项黏性耗散实验和模型计算值对比; b)低We下总黏性耗散中Wvis和Wad的占比(青色区域放大)
Figure 7. (a) Variation of the viscous dissipation components value with We; (b) comparison of the Wvis, Wad at low We.
图 8 液滴在不同We下撞击超亲水表面最大铺展因子的实验和模型预测结果对比(模型包括去除重力势能或辅助耗散的模型及全部考虑的模型)
Figure 8. Comparison of the current experimental measurements of βm with the theoretical prediction from model (models includes without Ep, without Wad and present model).
表 1 基于能量守恒的最大铺展因子的预测模型
Table 1. Theoretical models for predicting the maximum spreading factor.
文献 最大铺展模型预测表达式 表面润湿
性/(°)We 液滴形态 Lee等[14] $\begin{aligned} \rho {V_0}{D_0} + 12\sigma\qquad\qquad\qquad\qquad\quad\qquad\qquad\qquad\qquad \\= 3\sigma (1 - \cos \theta )\beta _{\rm{m} }^2 + 8\sigma \dfrac{1}{ { {\beta _{\rm{m} } } } } + 3\sqrt { {b / c} } \rho V_{\rm{0} }^2{D_0}\beta _{\rm{m} }^{ {5 / 2} }\dfrac{1}{ {\sqrt {Re} } }\end{aligned}$ 60—115 1—290 圆饼 Chandra等[27] $\dfrac{3}{2}\dfrac{ {We} }{ {Re} }\beta _{\rm{m} }^4 + \left( {1 - \cos \theta } \right)\beta _{\rm{m} }^2 - \left( {\dfrac{1}{3}We + 4} \right) = 0$ ～32 ～43 圆饼 Pasandideh-
Fard等[28]${\beta _{\rm{m} } } = \sqrt {\dfrac{ {We + 12} }{ {3\left( {1 - \cos {\theta _{\rm{a} } } } \right) + 4\left( { { {We} / {\sqrt {Re} } } } \right)} } }$ 27—140 27—447 圆饼 Mao等[29] $\left( {\dfrac{ {1 - \cos \theta } }{4} + 0.35\dfrac{ {We} }{ {\sqrt {Re} } } } \right)\beta _{\rm{m} }^4 - \left( {\dfrac{ {We} }{ {12} } + 1} \right)\beta + \dfrac{2}{3} = 0$ 30—120 5—1000 圆饼 Ukiwe等[30] $\left( {We + 12} \right){\beta _{\rm{m} } } = 8 + \beta _{\rm{m} }^3\left[ {3\left( {1 - \cos \theta } \right) + 4\dfrac{ {We} }{ {\sqrt {Re} } } } \right]$ 57—90 18—370 圆饼 Huang等[31] $\begin{aligned}\frac{3}{4}\left( {\frac{ {We} }{ {\sqrt {Re} } } + \frac{ {We^*} }{ {\sqrt {Re^*} } } } \right)\beta _{\rm{m} }^4 + 3\left( {1 - \cos {\theta _{\rm{a} } } } \right)\beta _{\rm{m} }^3 \qquad\\ - \left( {We + 12} \right){\beta _{\rm{m} } } + 8 = 0, ~~{V_0} < V^* \qquad\qquad\qquad\end{aligned}$
$\begin{aligned} \frac{3}{4}\left( {\frac{ {We} }{ {\sqrt {Re} } } + \frac{ {We^*} }{ {\sqrt {Re^*} } }\frac{ {Re^*} }{ {Re} } } \right)\beta _{\rm{m} }^4 + 3\left( {1 - \cos {\theta _{\rm{a} } } } \right)\beta _{\rm{m} }^3 \\ - \left( {We + 12} \right){\beta _{\rm{m} } } + 8 = 0,~~ {V_0} > V^*\qquad\qquad\qquad \end{aligned}$64—110 2—500 圆饼 Park等[32] $\begin{aligned} \left( {0.33\frac{ {We} }{ {\sqrt {Re} } } - \frac{1}{4}\cos \theta + \frac{1}{2}\left( {\frac{ {1 - \cos {\theta _{\rm{a} } } } }{ { { {\sin }^2}{\theta _{\rm{a} } } } } } \right)} \right)\beta _{\rm{m} }^2 \\ - 1 - \frac{ {We} }{ {12} } + \frac{ {\Delta {E_{\rm{s} } } } }{ { {\text{π} }D_0^2\sigma } } = 0 \qquad\qquad\qquad\qquad\quad\end{aligned}$ 31—113 0.2—180 球冠 Li等[33] $\dfrac{ {We} }{ {12} }\left( {1 - {C_{\rm{k} } } - \dfrac{3}{ {2\sqrt {Re} } }\displaystyle\int_{ {H_{\rm{m} } } }^{ {H_{\rm{s} } } } { {d^2}{\rm{d} }h} } \right) = {C_{\rm{S} } }P\left( { {D_{\rm{e} } } } \right) - P\left( { {D_{ {\rm{max} } } } } \right)$ 30—150 0—10 球冠 Gao等[34] $\begin{aligned} 1 + \frac{ {We} }{ {12} } = \frac{1}{6}\left[ {\frac{1}{ { { {\hat r}_{\rm{c} } } } } + \frac{1}{ { { {\hat R}_{\rm{c} } } } } } \right] + 4{\theta _{\rm{a} } }{ {\hat r}_{\rm{c} } }{ {\hat R}_{\rm{c} } } + {\left( { { {\hat R}_{\rm{c} } } - { {\hat r}_{\rm{c} } }\sin {\theta _{\rm{a} } } } \right)^2} \\ + {\left( { { {\hat R}_{\rm{c} } } + { {\hat r}_{\rm{c} } }\sin {\theta _{\rm{a} } } } \right)^2}\left( {\frac{4}{3}\frac{ {We} }{ {\sqrt {Re} } } - \cos {\theta _{\rm{a} } } } \right) \qquad\quad\end{aligned}$ 74—155 135—210 圆环 Wang等[35] $\begin{aligned} We + 12 =\qquad \qquad\qquad \qquad\qquad \qquad\qquad\qquad\qquad\qquad\qquad \qquad\\ \frac{3}{4}\left( {\frac{ {We} }{ {\sqrt {Re} } } + \alpha \frac{ {W{e_{\rm{c} } } } }{ {\sqrt {R{e_{\rm{c} } } } } } } \right)\beta _{\rm{m} }^3 + 3\left( {1 - \cos {\theta _{\rm{a} } } } \right)\beta _{\rm{m} }^2 + 12\bigg\{ \frac{ {\xi _{\rm{r} }^2} }{ { { {\left( {1 - \cos {\theta _{\rm{m} } } } \right)}^2} } } \\ \times \bigg[ {\sin ^2}{\theta _{\rm{m} } } - \frac{ { {\beta _{\rm{m} } } } }{ { {\xi _{\rm{r} } } } }\sin {\theta _{\rm{m} } }(1 - \cos {\theta _{\rm{m} } }) + 2(1 - \cos {\theta _{\rm{m} } }) \bigg] \qquad \qquad \qquad \\ \left. + 2{\xi _{\rm{r} } }\left( {\frac{ { {\beta _{\rm{m} } } } }{2} - {\xi _{\rm{r} } }\frac{ {\sin {\theta _{\rm{m} } } } }{ {1 - \cos {\theta _{\rm{m} } } } } } \right)\left( {\left| {1 - \kappa } \right| + \frac{ { {\theta _{\rm{m} } } } }{ {1 - \cos {\theta _{\rm{m} } } } } } \right) \right\}\qquad\qquad \quad \end{aligned}$ 34—100 0.1—427 环状-薄片 
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表 2 超亲水表面最大铺展因子βm实验结果与以往典型模型预测值[27-32]之间的比较
Table 2. Comparison of previous model[27-32] prediction value of βm with experimental data
V0/(m·s–1) We βm-exp Chandra 等[27] Pasandideh-Fard 等[28] Mao等[29] Park等[32] Ukiwe 等[30] Huang等[31] 0.25 1.91 3.41 2.4 6.37 3.05 11.31 5.84 0.58 0.44 5.90 3.46 2.1 4.82 2.55 7.93 4.42 0.44 0.60 10.77 3.60 1.97 4.37 2.41 6.67 4.02 0.35 0.71 15.26 3.82 1.93 4.20 2.37 6.09 3.90 0.29 0.93 25.59 3.93 1.89 4.07 2.35 5.41 3.82 0.20 130 51.17 4.08 1.90 4.08 2.38 4.9 3.93 0.12 1.50 68.59 4.26 1.91 4.13 2.41 4.78 3.93 0.10 1.89 109.3 4.43 1.93 4.26 2.47 4.67 4.07 0.06 2.35 168.98 4.70 1.97 4.42 2.54 4.66 4.24 0.04 2.8 239.89 4.90 2.00 4.57 2.60 4.72 4.39 0.03 3.08 290.08 5.00 2.02 4.67 2.64 4.76 4.48 0.02
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表 3 本文模型预测值与文献[28,29]中不同润湿性表面的最大铺展因子的实验值对比
Table 3. Comparison of current theoretical model of βm with experimental data in literature[28,29].
固体/液体 D0, mm V0/(m·s–1) We θ/(°) βm-exp βm-model (βm-exp–βm-model)/ βm-model 玻璃/水 2.7 0.55 11.21 37 1.77 2.41 0.26 玻璃/水 2.7 0.82 24.91 37 2.20 2.74 0.19 玻璃/水 2.7 1.00 37.05 37 2.53 2.94 0.14 玻璃/水 2.7 1.58 92.48 37 3.11 3.51 0.11 玻璃/水 2.7 1.86 128.17 37 3.70 3.81 0.03 玻璃/水 2.7 2.77 284.26 37 4.50 4.48 0.00 玻璃/水 2.7 3.72 512.67 37 4.94 4.89 0.01 不锈钢/水 2.7 0.55 11.21 67 1.67 1.95 0.14 不锈钢/水 2.7 0.82 24.91 67 2.16 2.28 0.05 不锈钢/水 2.7 1.00 37.05 67 2.34 2.51 0.06 不锈钢/水 2.7 1.58 92.48 67 3.09 3.13 0.01 不锈钢/水 2.7 1.86 128.17 67 3.67 3.38 0.08 不锈钢/水 2.7 2.77 284.26 67 4.42 4.15 0.06 不锈钢/水 2.7 3.72 512.67 67 4.88 4.65 0.05 石蜡/水 2.7 0.55 11.21 97 1.65 1.58 0.04 石蜡/水 2.7 0.82 24.91 97 2.10 1.91 0.10 石蜡/水 2.7 1.00 37.05 97 2.26 2.13 0.06 石蜡/水 2.7 1.58 92.48 97 3.01 2.79 0.07 石蜡/水 2.7 1.86 128.17 97 3.60 3.09 0.16 石蜡/水 2.7 2.77 284.26 97 4.32 3.89 0.11 石蜡/水 2.7 3.72 512.67 97 4.78 4.44 0.08 蜂蜡/水 0.62 2.61 59 111 2.65 2.19 0.21 蜂蜡/水 0.78 3.29 118 111 3.18 2.76 0.15 蜂蜡/水 0.89 3.71 171 111 3.45 3.09 0.11 蜂蜡/水 0.98 4.00 219 111 3.79 3.33 0.14 蜂蜡/水 1.05 4.28 271 111 3.91 3.53 0.10
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[1] Liang G, Mudawar I 2016 Int. J. Heat Mass Transfer 101 577
Google Scholar
[2] 马学虎, 兰忠, 王凯, 陈彦松, 程雅琦, 杜宾港, 叶轩 2018 化工学报 69 9
Google Scholar
Ma X H, Lan Z, Wang K, Chen Y S, Cheng Y Q, Du B G, Ye X 2018 J. Chem. Ind. Eng. 69 9
Google Scholar
[3] Snoeijer J H, Andreotti B 2013 Annu. Rev. Fluid Mech. 45 269
Google Scholar
[4] 朱君悦, 段远源, 王晓东, 闵琪 2014 化工学报 65 765
Google Scholar
Zhu J Y, Duan Y Y, Wang X D, Min Q 2014 J. Chem. Ind. Eng. 65 765
Google Scholar
[5] 刘海龙, 沈学峰, 王睿, 曹宇, 王军锋 2018 力学学报 50 1024
Google Scholar
Liu H L, Shen X F, Wang R, Cao Y, Wang J F 2018 Acta Mech. Sin. 50 1024
Google Scholar
[6] 闵琪, 段远源, 王晓东, 吴莘馨 2013 热科学与技术 12 335
Google Scholar
Min Q, Duan Y Y, Wang X D, Wu X X 2013 J. Therm. Sci. Technol. 12 335
Google Scholar
[7] Hu H B, Chen L B, Bao L Y, Huang S H 2014 Chin. Phys. B 23 074702
Google Scholar
[8] Sun Z H, Han R J 2008 Chin. Phys. B 17 3185
Google Scholar
[9] Wang Y B, Wang Y F, Gao S R, Yang Y R, Wang X D, Chen M 2020 Langmuir 36 9306
Google Scholar
[10] Hu H B, Huang S H, Chen L B 2013 Chin. Phys. B 22 084702
Google Scholar
[11] Song M, Liu Z, Ma Y, Dong Z, Wang Y, Jiang L 2017 NPG Asia Mater. 9 1
Google Scholar
[12] 王高远, 胥蕊娜, 陈剑楠, 陈学, 姜培学 2018 工程热物理学报 39 1797
Google Scholar
Wang G Y, Xu R N, Chen J N, Chen X, Jiang P X 2018 J. Eng. Thermphys. 39 1797
Google Scholar
[13] Lin S, Zhao B, Zou S, Guo J, Wei Z, Chen L 2018 J. Colloid Interface Sci. 516 86
Google Scholar
[14] Lee J B, Derome D, Guyer R, Carmeliet J 2016 Langmuir 32 1299
Google Scholar
[15] Tang C, Qin M, Weng X, Zhang X, Zhang P, Li J, Huang Z 2017 Int. J. Multiphase Flow 96 56
Google Scholar
[16] 荣松, 沈世全, 王天友, 车志钊 2019 物理学报 68 154701
Google Scholar
Rong S, Shen S Q, Wang T Y, Che Z Z 2019 Acta Phys. Sin. 68 154701
Google Scholar
[17] Ding B, Wang H, Zhu X, Chen R, Liao Q 2018 Int. J. Heat Mass Transfer 124 1025
Google Scholar
[18] 范瑶, 王宏, 朱恂, 黄格永, 丁玉栋, 廖强 2016 化工学报 67 2709
Google Scholar
Fan Y, Wang H, Zhu X, Huang G Y, Ding Y D, Liao Q 2016 J. Chem. Ind. Eng. 67 2709
Google Scholar
[19] Ding B, Wang H, Zhu X, Chen R, Liao Q 2019 Int. J. Heat Mass Transfer 138 844
Google Scholar
[20] Chen M, Wu D, Chen D, Deng J, Liu H, Jiang J 2020 Chem. Eng. Sci. 226 115864
Google Scholar
[21] 焦云龙, 刘小君, 逄明华, 刘焜 2016 物理学报 65 016801
Google Scholar
Jiao Y L, Liu X J, Pang M H, Liu K 2016 Acta Phys. Sin. 65 016801
Google Scholar
[22] 叶学民, 李永康, 李春曦 2016 物理学报 65 234701
Google Scholar
Ye X M, Li Y K, Li C X 2016 Acta Phys. Sin. 65 234701
Google Scholar
[23] Roisman I V 2009 Phys. Fluids 21 052104
Google Scholar
[24] Liang G, Chen Y, Chen L, Shen S 2019 Ind. Eng. Chem. Res. 58 10053
Google Scholar
[25] Laan N, de Bruin K G, Bartolo D, Josserand C, Bonn D 2014 Phys. Rev. Appl. 2 044018
Google Scholar
[26] Lee J B, Laan N, de Bruin K G, Skantzaris G, Shahidzadeh N, Derome D, Carmeliet J, Bonn D 2016 J. Fluid Mech. 786 R4
Google Scholar
[27] Chandra S, Avedisian C T 1991 Proc. R. Soc. A 432 13
Google Scholar
[28] Pasandideh‐Fard M, Qiao Y M, Chandra S, Mostaghimi J 1996 Phys. Fluids 8 650
Google Scholar
[29] Mao T, Kuhn D C S, Tran H 1997 AlChE J. 43 2169
Google Scholar
[30] Ukiwe C, Kwok D Y 2005 Langmuir 21 666
Google Scholar
[31] Huang H M, Chen X P 2018 Phys. Fluids 30 022106
Google Scholar
[32] Park H, Carr W W, Zhu J, Morris J F 2003 AlChE J. 49 2461
Google Scholar
[33] Li Ri, Ashgriz Nasser, Sanjeev C 2010 J. Fluids Eng. 132 061302
Google Scholar
[34] Gao X, Li R 2014 AlChE J. 60 2683
Google Scholar
[35] Wang F, Yang L, Wang L, Zhu Y, Fang T 2019 Langmuir 35 3204
Google Scholar
[36] Panda A, Pati A R, Saha B, Kumar A, Mohapatra S S 2019 Chem. Eng. Sci. 207 527
Google Scholar
[37] Zhang Z, Jiang P X, Christopher D M, Liang X-G 2015 Int. J. Heat Mass Transfer 80 26
Google Scholar
[38] Zheng Y, Ma X, Li Y, Jiang R, Wang K, Lan Z, Liang Q 2017 Appl. Therm. Eng. 111 1548
Google Scholar
[39] Nam Y, Ju Y S 2013 J. Adhes. Sci. Technol. 27 2163
Google Scholar
[40] Vadillo D C, Soucemarianadin A, Delattre C, Roux D C D 2009 Phys. Fluids 21 122002
Google Scholar
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