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.

Classification: 65N30, 65M12, 76D45, 76M30

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 - Mod\'elisation Math\'ematique et Analyse Num\'erique}, publisher = {EDP-Sciences}, volume = {48}, number = {4}, year = {2014}, pages = {969-1009}, doi = {10.1051/m2an/2013130}, zbl = {1299.76064}, mrnumber = {3264343}, language = {en}, url = {http://www.numdam.org/item/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 - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 4, pp. 969-1009. doi : 10.1051/m2an/2013130. http://www.numdam.org/item/M2AN_2014__48_4_969_0/

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