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 重庆大学学报  2018, Vol. 41 Issue (12): 1-9  DOI: 10.11835/j.issn.1000-582X.2018.12.001 RIS（文献管理工具） 0

### 引用本文

ZHOU Ping, ZENG Zhong, QIAO Long. Simulation of shear-thinning droplets impact on solid surfaces by using Lattice Boltzmann method[J]. Journal of Chongqing University, 2018, 41(12): 1-9. DOI: 10.11835/j.issn.1000-582X.2018.12.001.

### 文章历史

Simulation of shear-thinning droplets impact on solid surfaces by using Lattice Boltzmann method
ZHOU Ping , ZENG Zhong , QIAO Long
College of Aerospace Engineering, Chongqing University, Chongqing 400044, P. R. China
Supported by the National Natural Science Foundation of China (11572062) and Program for Changjiang Scholars and Innovative Research Team in University (IRT_17R112)
Abstract: The droplet spreading and deposition appear widely in industrial applications, and it is of practical significance to study the effect of non-Newtonian rheology on droplets impact on solid surfaces. In this research, we developed a two-phase lattice Boltzmann model based on the phase field method for power-law fluid flows. By introducing a contact angle condition, power-law droplets impact on solid surfaces was investigated, and the effects of power exponent n (0.5 ≤ n ≤ 1.0) and Weber number We (5 ≤ We ≤ 45) on shear-thinning droplets impact were evaluated. The results indicate that power-law liquid inhibits the droplet spreading and splashing, and it becomes easier for deposition with the decrease of n. In addition, droplets are easier to reach stationary state as weber number increases.
Keywords: Lattice Boltzmann method    power law liquid    pseudoplastic fluid    phase interfaces    phase field method    droplet deposition    droplet spreading

1 理论 1.1 幂律(Power-law)模型

 $\eta (\dot \gamma ) = {\eta _0}|\dot \gamma |n - 1, n > 0,$ (1)

1.2 Call-Hilliard相场模型

Call-Hilliard对流扩散方程[25-27]

 $\left\{ \begin{array}{l} \frac{{\partial C}}{{\partial t}} + {\mathit{\boldsymbol{u}}} \cdot \nabla C = M{\nabla ^2}{\mu _{\rm{c}}}, \\ {\mu _{\rm{c}}} = {C^3} - 1.5{C^2} + 0.5C - {\varepsilon ^2}{\nabla ^2}C, \end{array} \right.$ (2)

 $\nabla \cdot {\mathit{\boldsymbol{u}}} = 0,$ (3)
 $\rho (\frac{{\partial {\mathit{\boldsymbol{u}}}}}{{\partial t}} + {\mathit{\boldsymbol{u}}} \cdot \nabla {\mathit{\boldsymbol{u}}}) = - \nabla p + \nabla \cdot [\eta (\nabla {\mathit{\boldsymbol{u}}} + \nabla {{\mathit{\boldsymbol{u}}}^{\rm{T}}})] + {{\mathit{\boldsymbol{F}}}_{\rm{s}}},$ (4)

1.3 两相流格子Boltzmann方法

 $f_\alpha ^{{\rm{eq}}} = \rho {\Gamma _\alpha }({\mathit{\boldsymbol{u}}}) = \rho {\omega _\alpha }[1 + \frac{{{{\mathit{\boldsymbol{e}}}_\alpha } \cdot {\mathit{\boldsymbol{u}}}}}{{c_{\rm{s}}^{\rm{2}}}} + \frac{{{{({{\mathit{\boldsymbol{e}}}_\alpha } \cdot {\mathit{\boldsymbol{u}}})}^2}}}{{2c_{\rm{s}}^4}} - \frac{{{\mathit{\boldsymbol{u}}} \cdot {\mathit{\boldsymbol{u}}}}}{{2c_{\rm{s}}^{\rm{2}}}}],$ (5)

 $\frac{{\partial {g_\alpha }}}{{\partial t}} + {e_\alpha } \cdot \nabla {g_\alpha } = - {{\mathit{\pmb{\Lambda}}} _{\alpha \beta }}({g_\beta } - g_\beta ^{{\rm{eq}}}) + ({{\mathit{\boldsymbol{e}}}_\alpha } - {\mathit{\boldsymbol{u}}}) \cdot [\nabla \rho c_{\rm{s}}^{\rm{2}}({\Gamma _\alpha } - {\Gamma _\alpha }(0)) + {\mathit{\boldsymbol{F}}}{\Gamma _\alpha }],$ (6)

 ${{\bar g}_\alpha }({\mathit{\boldsymbol{x}}} + {{\mathit{\boldsymbol{e}}}_\alpha }\delta t, t + \delta t) - {{\bar g}_\alpha }({\mathit{\boldsymbol{x}}}, t) = - ({S_{\alpha \beta }} + 2{I_{\alpha \beta }})({{\bar g}_\beta } - \bar g_\beta ^{{\rm{eq}}}) + \\\;\delta t({{\mathit{\boldsymbol{e}}}_\alpha } - {\mathit{\boldsymbol{u}}}) \cdot [\nabla \rho c_{\rm{s}}^{\rm{2}}({\Gamma _\alpha } - {\Gamma _\alpha }(0)) + {{\mathit{\boldsymbol{F}}}_{\rm{s}}}{\Gamma _\alpha }],$ (7)

 $\begin{array}{l} p = \sum\limits_{\alpha = 0}^8 {{{\bar g}_\alpha }} + \frac{{\delta t}}{2}{\mathit{\boldsymbol{u}}} \cdot \nabla \rho c_{\rm{s}}^{\rm{2}}, \\ {\mathit{\boldsymbol{u}}} = \frac{1}{{\rho c_{\rm{s}}^{\rm{2}}}}(\sum\limits_{\alpha = 0}^8 {{{\bar g}_\alpha }} {{\mathit{\boldsymbol{e}}}_\alpha } + \frac{{\delta t}}{2}{{\mathit{\boldsymbol{F}}}_{\rm{s}}})。\end{array}$ (8)

2 模型和程序的验证 2.1 方腔流

 图 1 幂律流体方腔流在竖直中心线上u和水平中心线上v的分布和文献解的对比 Figure 1 Comparision between our results and Neofytou's results for power-law flow: u-velocity profiles along vertical centerline and v-velocity profiles along horizontal centerline
2.2 Laplace定律

 图 2 幂律液滴达到稳态时Δp和1/R的关系与理论解的对比 Figure 2 Comparision between the numerical and theoretical solution of the relationship of Δp and 1/R when power law drops reach steady state
2.3 液滴在壁面上的动态接触

 图 3 幂律液滴在不同静态接触角的稳态 Figure 3 Stable configurations of power law droplet at different equilibrium contact angles

 图 4 量纲一的铺展长度r/R和量纲一的时间t*=t/ $\sqrt {\rho {R^3}/\sigma }$的关系 Figure 4 Relationship between dimensionless spreading length r/R and the square root of dimensionless time t*=t/ $\sqrt {\rho {R^3}/\sigma }$
3 液滴撞击固壁上的铺展

 图 5 液滴撞击壁面的示意图 Figure 5 The sketch of droplet impact on a solid surface

 图 6 不同幂律指数n下铺展长度D随时间变化的关系 Figure 6 Spreading length D as a function of time for different power exponents n

 图 7 幂律液滴在不同时间的形态 Figure 7 Droplets configurations at different time

 图 8 不同韦伯数We下幂律指数n=0.5以及1.0液滴铺展长度随时间的变化关系 Figure 8 Spreading length as a function of time at different We when the power exponent n=0.5 and n=1.0
4 结论