Two-phase fluid flows on substrates (i.e. wetting phenomena) are important in many industrial processes, such as micro-fluidics and coating flows. These flows include additional physical effects that occur near moving (three-phase) contact lines. We present a new 2-D variational (saddle-point) formulation of a Stokesian fluid with surface tension that interacts with a rigid substrate. The model is derived by an Onsager type principle using shape differential calculus (at the sharp-interface, front-tracking level) and allows for moving contact lines and contact angle hysteresis and pinning through a variational inequality. Moreover, the formulation can be extended to include non-linear contact line motion models. We prove the well-posedness of the time semi-discrete system and fully discrete method using appropriate choices of finite element spaces. A formal energy law is derived for the semi-discrete and fully discrete formulations and preliminary error estimates are also given. Simulation results are presented for a droplet in multiple configurations to illustrate the method.
Keywords: mixed method, Stokes equations, surface tension, contact line motion, contact line pinning, variational inequality, well-posedness
@article{M2AN_2014__48_4_969_0,
author = {Walker, Shawn W.},
title = {A mixed formulation of a sharp interface model of stokes flow with moving contact lines},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {969--1009},
year = {2014},
publisher = {EDP Sciences},
volume = {48},
number = {4},
doi = {10.1051/m2an/2013130},
mrnumber = {3264343},
zbl = {1299.76064},
language = {en},
url = {https://www.numdam.org/articles/10.1051/m2an/2013130/}
}
TY - JOUR AU - Walker, Shawn W. TI - A mixed formulation of a sharp interface model of stokes flow with moving contact lines JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2014 SP - 969 EP - 1009 VL - 48 IS - 4 PB - EDP Sciences UR - https://www.numdam.org/articles/10.1051/m2an/2013130/ DO - 10.1051/m2an/2013130 LA - en ID - M2AN_2014__48_4_969_0 ER -
%0 Journal Article %A Walker, Shawn W. %T A mixed formulation of a sharp interface model of stokes flow with moving contact lines %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2014 %P 969-1009 %V 48 %N 4 %I EDP Sciences %U https://www.numdam.org/articles/10.1051/m2an/2013130/ %R 10.1051/m2an/2013130 %G en %F M2AN_2014__48_4_969_0
Walker, Shawn W. A mixed formulation of a sharp interface model of stokes flow with moving contact lines. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 48 (2014) no. 4, pp. 969-1009. doi: 10.1051/m2an/2013130
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