Axisymmetric non-Newtonian drops treated with a boundary integral method

A boundary integral method for the simulation of the time-dependent deformation of axisymmetric Newtonian or non-Newtonian drops suspended in a Newtonian fluid subjected to an axisymmetric flow field is developed. The boundary integral formulation for Stokes flow is used and the non-Newtonian stress is treated as a source term which yields an extra integral over the domain of the drop. By transforming the integral representation for the velocity to cylindrical coordinates we can reduce the dimension of the computational problem. The integral equation for the velocity remains of the same form as in Cartesian coordinates, and the Green's functions are transformed explicitly to cylindrical coordinates. Besides a numerical validation of the method we present simulation results for a Newtonian drop and a drop consisting of an Oldroyd-B fluid. The results for the Newtonian drop are consistent with results from the literature. The deformation process of the non-Newtonian drop for small capillary numbers appears to be governed by two relaxation times.