A 2D boundary element method for simulating the deformation of axisymmetric compound non-Newtonian drops

The boundary integral formulation of the solution to the Stokes equations is used to describe the deformation of small compound non-Newtonian axisymmetric drops suspended in a Newtonian fluid that is subjected to an axisymmetric flow field. The non-Newtonian stress is treated as a source term in the Stokes equations, which yields an extra integral over the domains containing non-Newtonian material. By transforming the integral representation for the velocity to cylindrical co-ordinates and performing the integration over the azimuthal direction analytically, the dimension of the problem can be reduced from three to two. A boundary element method for the remaining two-dimensional problem aimed at the simulation of the deformation of such axisymmetric compound non-Newtonian drops is developed. Apart from a numerical validation of the method, simulation results for a drop consisting of an Oldroyd-B fluid and a viscoelastic material are presented. Moreover, the method is extended to compound drops that are composed of a viscous inner core encapsulated by a viscoelastic material. The simulation results for these drops are verified against theoretical results from literature. Moreover, it is shown that the method can be used to identify the dominant break-up mechanism of compound drops in relation to the specific non-Newtonian character of the membrane.