Potential flow solutions with viscous interface stresses for axisymmetric configurations

The boundary element or integral method (BIM) is a major tool for the simulation of inviscid multiphase flows. However, when simulating flow dynamics of micrometric gas bubbles—that are currently receiving attention e.g. in connection with targeted drug delivery—viscous effects become relevant. Existing viscous extensions of potential flow theories rely on the evaluation of second-order derivatives of the flow potential. While these derivatives are simply calculable in three-dimensional Cartesian grids through the repeated use of the boundary integral equation, a non-linearity complicates a corresponding mathematical formulation in cylindrical coordinates for axisymmetric geometries. Simulating axisymmetric geometries with a Cartesian 3d discretization entails either unnecessarily high computational costs or numerical errors. The latter are especially problematic as numerical grid-resolution errors have a spatially non-local effect in the boundary integral method. As an alternative, we derive in this work an exact modified boundary integral equation for second-order potential derivatives in cylindrical coordinates. The resulting viscous BIM requires only the interface discretization of a cross-section line with few degrees of freedom respectively grid points, while the circumferential integration does not need discretization. The new method is tested in two verification studies and illustrated with two application examples.