The indirect boundary element method for the axisymmetric free surface Stokes flow

Abstract

The authors present the formulation of the indirect boundary element method (IBEM) for an axisymmetric Stokes flow with a free surface in the presence of gravity. The formulae of the fundamental solutions of the Stokes equations are found for velocities and tractions in the axisymmetric case. These expressions are written in the cylindrical coordinate system and contain the elliptic integrals of the first and second kind. For the integral equations discretization the constant elements are used. The necessary integrals are evaluated numerically except for the singular ones. The analytical formulae are obtained for them. Two boundaryvalue problems with mixed conditions are considered. The problem of the Poiseuille flow of a viscous fluid in a round tube with the exact solution was calculated to verify the IBEM algorithm and to demonstrate its approximation convergence. Another problem of the cylindrical tube filling by a viscous fluid with a free surface was calculated to prove the IBEM in the case of a moving boundary. The simulation in a steady-state formulation showed that the stationary advancing front shapes exist in both cases when the gravity acts against the flow (Stokes number St<0) and aids the flow (0<St≤0.94). In the case of the Stokes parameter values greater than 1 the fountain flow was replaced by a jet flow. The stationary advancing front shapes were calculated in the range-400≤St≤0.94 and compared with the famous data.