A new mesh relaxation approach and automatic time‐step control method for boundary integral simulations of a viscous drop

Summary We present a numerical methodology for the simulation of a viscous drop under simple shear flows by using the Boundary Integral Method. The present work treat only a single drop in an unbounded fluid-flow, but the results can be directly applied to studies on the rheology of dilute emulsions, in which the hydrodynamic interactions between two or more drops can be neglected. Singular and non-singular integral representations of the velocity field are considered. Several aspects of the method are presented, including a new mesh relaxation approach and an automatic time-step control method. The relaxation strategy is used in order to contain the distortion of the mesh and is performed by using relaxation iterations in a virtual temporal march between each physical time step of the simulation and monitoring the standard deviation of the areas of the elements. The automatic time-step control method uses a global quantity related to the drop deformation in order to automatically set the temporal integration time step. It is done in a way to keep the local integration error less than a given tolerance. This strategy reduces the computational cost of the simulation by dramatically reducing the number of time steps in the temporal integration process. Copyright © 2016 John Wiley & Sons, Ltd.