Numerical simulation of a concentrated emulsion in shear flow

Abstract

A three-dimensional computer simulation of a concentrated emulsion in shear flow has been developed for low-Reynolds-number finite-capillary-number conditions. Numerical results have been obtained using an efficient boundary integral formulation with periodic boundary conditions and up to twelve drops in each periodically replicated unit cell. Calculations have been performed over a range of capillary numbers where drop deformation is significant up to the value where drop breakup is imminent. Results have been obtained for dispersed-phase volume fractions up to 30% and dispersed- to continuous-phase viscosity ratios in the range of 0 to 5. The results reveal a complex rheology with pronounced shear thinning and large normal stresses that is associated with an anisotropic microstructure that results from the alignment of deformed drops in the flow direction. The viscosity of an emulsion is only a moderately increasing function of the dispersed-phase volume fraction, in contrast to suspensions of rigid particles or undeformed drops. Unlike rigid particles, deformable drops do not form large clusters.

A fast algorithm for simulating vesicle flows in three dimensions

“…For simulation of multiple interacting drops, Loewenberg and Hinch [41] proposed a heuristic formula for artificial tangential velocity reducing mesh distortion. Zinchenko et al [76] observed that the technique of [41] leads to instabilities in simulations with gravity induced motions, and constructed a tangential velocity field by global minimization of the sum of squares of the rates of change of the distances between adjacent vertices (tangency of the field is enforced as a constraint), the first instance of the (passive stabilization) approach. This method was further developed by introducing different objectives for minimization in [70,77], which adapted the sampling density to curvature and controlled triangle quality.…”

Nonsingular boundary integral method for deformable drops in viscous flows

“…The two most commonly used approaches are local mesh refinement and/or higher-order integration rules in the vicinity of the singular point, 1, 16 and near singularity subtraction, where for a closed interface, using volume conservation identities, the integrals over the singular point vicinity are represented via the integral over the residual of the interface. 6,10,17,18 Both approaches, however, cannot give satisfactory results when the singular point is close to the residual of the interface. 16 In our method, the recently proposed 19 nonsingular contour-integral representation of the layer potentials is used for the calculation of the boundary integrals.…”

A mathematical formulation of the boundary integral equations for a compressible stokes flow

“…Trapezoid-rule integration with singularity subtraction and near-singularity subtraction for closely-spaced interfaces of drops (if surface tension gradients are not present) [7] can be used to accurately evaluate the integrals. Equation (36) has eigensolutions that cause unphysical changes in the dispersed-phase volume at small viscosity ratios, corrupt numerical solutions at large viscosity ratios, and slow the iterative convergence.…”

“…Periodic boundary conditions are enforced through the use of periodic Greens functions. These are obtained by Ewald summation [16] using accurate computationally-efficient tabulation of the nonsingular background contribution [7]. The appropriate formulation was derived in §3.…”