On the volume conservation of emulsion drops in boundary integral simulations

Abstract

Boundary Integral simulations are very common to study the microhydrodynamics of viscous drops and predict the rheology of emulsions. The standard boundary integral formulation for the velocity field at the drop surface is given by surface integrals of singular Green’s functions in space and depends on geometric quantities of the mesh, as the local mean curvature and the unitary exterior normal vector. Typically, the drops deform and rotate under the flow action, which leads to a great distortion of the computational mesh and, therefore, might produce several numerical errors in the surface velocity calculation. In this work, we investigate the numerical errors in the calculation of the surface velocity in boundary integral simulations of a viscous drop in simple shear flows. Assuming that the flow is incompressible, such that the total volume of the drops must remain constant, we use the net flow rate across the drop surface as a measure of the numerical errors in the simulations. For both small and moderate regimes of drop surface deformation, we show that the net mass flow rate across the drop surface is positive, and strongly depends on the drop-to-basis fluid viscosity ratio, capillary number of the flow, and mesh refinement. These results indicate that the volume of the drops increases continuously during the simulations. To remove the spurious mass flow rate from the simulations, we present a mesh convergence study based on an extrapolation procedure to an almost continuous mesh, so that the results become independent of mesh refinement and recover the theoretical prediction of a null net flow rate and constant drop volume in incompressible flows. The extrapolation method proposed here can be straightforwardly implemented to improve the accuracy of BI simulations used to study drop microhydrodynamics and emulsion rheology. As a matter of example, we applied the method to predict the surface deformation of a high-viscosity emulsion drop, and the result is compared in very good agreement with asymptotic theories available in the related literature.