# American Institute of Mathematical Sciences

October  2021, 26(10): 5551-5566. doi: 10.3934/dcdsb.2020366

## The effect of surface pattern property on the advancing motion of three-dimensional droplets

 1 School of Mathematics and Statistics, Guangxi Normal University, Guilin 541004, China 2 School of Mathematical Sciences, Guizhou Normal University, Guiyang 550025, China 3 Computational Transport Phenomena Laboratory, King Abdullah University of Science and Technology, Thuwal 23955-6900, Saudi Arabia

* Corresponding author

Received  December 2017 Revised  October 2020 Published  October 2021 Early access  December 2020

We investigate numerically the advancing motion of 3D droplets spreading on physically flat chemically heterogeneous surfaces with periodic structures. We use the Navier-Stokes-Cahn-Hilliard equations with the generalized Navier boundary conditions to model the motion of droplets. Based on a convex splitting scheme, we have done numerical simulations and compared the results between different surface patterns quantitatively. We study the effect of pattern property on the advancing motion of three phase contact lines, the critical volume at the contact line jump and the effective advancing angles. By increasing the volume of droplet slowly on heterogeneous surfaces with different pattern property, we find that the advancing contact line approaches an equiangular octagon for the patterned surface with periodic squares separated by channels and approaches a regular hexagon for the patterned surface with periodic circles in regular hexagonal arrays. The shape of three-phase contact line is much more determined by the macro structure of the pattern than the micro structure of the pattern in each period.

Citation: Hua Zhong, Xiaolin Fan, Shuyu Sun. The effect of surface pattern property on the advancing motion of three-dimensional droplets. Discrete & Continuous Dynamical Systems - B, 2021, 26 (10) : 5551-5566. doi: 10.3934/dcdsb.2020366
##### References:

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##### References:
Schematic diagram of a flat surface with a ring pattern
A spherical cap
Schematic diagram of physically flat surfaces with square-channel like pattern (upper), circular patches in square arrays (middle) and in regular hexagonal arrays (lower)
Effective advancing angle versus Young's angle of channel with Young's angle of square $60^\circ$
Advancing contact line and cubic root of critical volume with $a:b = 12:5, \theta_a:\theta_b = 60^\circ:100^\circ$
Advancing contact line (part) and cubic root of critical volume with $a:b = 12:5, \theta_a:\theta_b = 60^\circ:110^\circ$
Advancing contact line and cubic root of critical volume with $a:b = 12:5, \theta_a:\theta_b = 60^\circ:120^\circ$
Advancing contact line and cubic root of critical volume with $a:b = 12:12, \theta_a:\theta_b = 60^\circ:110^\circ$
Circle pattern in square arrays and in regular hexagon arrays
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