Nonsingular boundary integral method for deformable drops in viscous flows

Abstract

A three-dimensional boundary integral method for deformable drops in viscous flows at low Reynolds numbers is presented. The method is based on a new nonsingular contour-integral representation of the single and double layers of the free-space Green's function. The contour integration overcomes the main difficulty with boundary-integral calculations: the singularities of the kernels. It also improves the accuracy of the calculations as well as the numerical stability. A new element of the presented method is also a higher-order interface approximation, which improves the accuracy of the interface-to-interface distance calculations and in this way makes simulations of polydispersed foam dynamics possible. Moreover, a multiple time-step integration scheme, which improves the numerical stability and thus the performance of the method, is introduced. To demonstrate the advantages of the method presented here, a number of challenging flow problems is considered: drop deformation and breakup at high viscosity ratios for zero and finite surface tension; drop-to-drop interaction in close approach, including film formation and its drainage; and formation of a foam drop and its deformation in simple shear flow, including all structural and dynamic elements of polydispersed foams.

Boundary-integral method for drop deformation between parallel plates

“…26 For large deformations ͑simulations with CaϾ 0.35͒, this alone is not sufficient to keep a stable mesh, as the resulting slender drop shapes give an unfavorable distribution of the nodes over the drop surface. In these cases, nodes are added and redistributed using a remesh algorithm [30][31][32] that includes node subtraction and addition, topology optimization, and mesh relaxation. Remeshing is coupled with the increase in integration points.…”

“…The boundary integral equation ͑1͒ is solved via the procedure documented in Bazhlekov et al 32 Features of this method include the nonsingular contour integration to overcome the singularities of the kernels at x = x 0 ͓see Eq. ͑4͔͒, and a multi-time-step scheme, where the kernels are calculated every M time steps and all other parameters every time step.…”

“…23 In this study, we use a three-dimensional boundaryintegral method 24,25 to study the evolution of the shape of a single drop placed between two parallel plates. Boundaryintegral methods have been successfully used in the past to study drop deformation in bulk flow, [26][27][28][29][30][31][32] or recently for drops squeezing through interparticle constrictions, 33 and have proved to provide highly accurate deformation dynamics. The inclusion of the walls, or other solid objects, is a nontrivial task in a boundary-integral method.…”

Microconfined equiviscous droplet deformation: Comparison of experimental and numerical results

“…37 Details concerning the implementation of the wall contribution to G can be found elsewhere. 23 A standard contour integration is used for the curvature calculation.…”

“…In addition, nodes are advected with an additional tangential velocity that moves nodes to places with high curvature. 38 To limit computation time, a multitime step algorithm is used, 37 in which the kernels are only calculated every 50 time steps. A time step of 5 ϫ 10 −4 is used in all cases.…”