Browsing by Author "Ren, W"
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Item Open Access A Level Set Method for the Simulation of Moving Contact Lines in Three Dimensions(Communications in Computational Physics, 2022-01-01) Zhao, Q; Xu, S; Ren, WWe propose an efficient numerical method for the simulation of the two-phase flows with moving contact lines in three dimensions. The mathematical model consists of the incompressible Navier-Stokes equations for the two immiscible fluids with the standard interface conditions, the Navier slip condition along the solid wall, and a contact angle condition (Ren et al. (2010) [28]). In the numerical method, the governing equations for the fluid dynamics are coupled with an advection equation for a level-set function. The latter models the dynamics of the fluid interface. Following the standard practice, the interface conditions are taken into account by introducing a singular force on the interface in the momentum equation. This results in a single set of governing equations in the whole fluid domain. Similarly, the contact angle condition is imposed by introducing a singular force, which acts in the normal direction of the contact line, into the Navier slip condition. The new boundary condition, which unifies the Navier slip condition and the contact angle condition, is imposed along the solid wall. The model is solved using the finite difference method. Numerical results are presented for the spreading of a droplet on both homogeneous and inhomogeneous solid walls, as well as the dynamics of a droplet on an inclined plate under gravity.Item Open Access Derivation of a continuum model and the energy law for moving contact lines with insoluble surfactants(Physics of Fluids, 2014-06-05) Zhang, Z; Xu, S; Ren, WA continuous model is derived for the dynamics of two immiscible fluids with moving contact lines and insoluble surfactants based on thermodynamic principles. The continuum model consists of the Navier-Stokes equations for the dynamics of the two fluids and a convection-diffusion equation for the evolution of the surfactant on the fluid interface. The interface condition, the boundary condition for the slip velocity, and the condition for the dynamic contact angle are derived from the consideration of energy dissipations. Different types of energy dissipations, including the viscous dissipation, the dissipations on the solid wall and at the contact line, as well as the dissipation due to the diffusion of surfactant, are identified from the analysis. A finite element method is developed for the continuum model. Numerical experiments are performed to demonstrate the influence of surfactant on the contact line dynamics. The different types of energy dissipations are compared numerically. © 2014 AIP Publishing LLC.Item Open Access Reinitialization of the Level-Set Function in 3d Simulation of Moving Contact Lines(Communications in Computational Physics, 2016-11-01) Xu, S; Ren, WThe level set method is one of the most successful methods for the simulation of multi-phase flows. To keep the level set function close the signed distance function, the level set function is constantly reinitialized by solving a Hamilton-Jacobi type of equation during the simulation. When the fluid interface intersects with a solid wall, a moving contact line forms and the reinitialization of the level set function requires a boundary condition in certain regions on the wall. In this work, we propose to use the dynamic contact angle, which is extended from the contact line, as the boundary condition for the reinitialization of the level set function. The reinitialization equation and the equation for the normal extension of the dynamic contact angle form a coupled system and are solved simultaneously. The extension equation is solved on the wall and it provides the boundary condition for the reinitialization equation; the level set function provides the directions along which the contact angle is extended from the contact line. The coupled system is solved using the 3rd order TVD Runge-Kutta method and the Godunov scheme. The Godunov scheme automatically identifies the regions where the angle condition needs to be imposed. The numerical method is illustrated by examples in three dimensions.