Modeling of capillary coaxial gap filling with viscous liquid - Chekhonin, Konstantin Aleksandrovich; Vlasenko, Viktor Dmitriyevich (Computational Continuum Mechanics, 2019)
A variational formulation of the boundary value problem on the motion of a viscous incompressible fluid with a free surface and changing dynamic boundary angles is proposed. The mathematical description of the process is based on the equations of motion, continuity and natural boundary conditions on the free surface. The traditional feature of the mathematical model on the three-phase contact lines (LTPC) is eliminated by the slip condition. The boundary angle on the LTPC is included in the variational formulation of the problem by replacing the curvature of the free boundary by the Laplace - Beltrami operator and using integration in parts. To describe the dynamic conditions on the LTPC, linking the speed of the LTPC and the dynamic edge angles on the solid walls of the cylinders, the empirical Jiang ratio is used. The numerical solution of the problem is based on the mixed finite element method with approximation of the main variables of the problem (velocity and pressure vector) satisfying the compatibility condition (LBB - condition). In addition, we use a singular finite element and a discontinuous approximation for the pressure to reduce the pressure oscillations in the vicinity of the LTPC. The numerical implementation of the kinematic condition of the free surface motion is performed according to the predictor-corrector scheme. The algorithm is tested on the problems with an analytical solution. Numerical studies of the flow kinematics and the free surface behavior when filling the coaxial gap in terms of the determining parameters of the Reynolds numbers (Re) in the range from 0 to 5, Stokes (W) in the range from 0 to 300 and capillary number (Ca) in the range from 0.0001 to 10 are carried out. The influence of the main parameters of the problem and dynamic conditions on LTPC on the evolution and maximum deflection of the free boundary is shown. For slow filling conditions, gravitational and capillary forces have the greatest influence on the kinematic characteristics of the filling. In this case, the dominant influence on the evolution of the free surface of the capillary forces begins with the values of the capillary numbers less than CA < 0,1. Increasing the flow rate (Re > 1) leads to a significant increase in the deflection of the free surface.
Keywords: coaxial capillary; free surface; dynamic edge angle; finite element method; three-phase contact line