The treatment of lubrication forces in boundary integral equations

Despite their many advantages over other numerical methods, boundary integral formulations still fail to provide accurate predictions of mesoscale motion in dense suspensions of rigid particles because the nearly singular flow between surfaces in close proximity cannot be resolved accurately. A procedure for incorporating analytical solutions for the lubrication flow within a large-scale boundary integral equation method is shown. Although the method is applied to the case of spherical particles, in conjunction with the completed double layer boundary integral equation, it can be developed further to treat more complex geometries and can be adapted to other numerical techniques. In contrast to other apparently similar approaches, the present method does not resort to effective medium approximations, and in principle retains all the advantages typical of boundary integral approaches. The framework also allows for forces other than those due to hydrodynamic lubrication between particles, provided that they are a linear function of the relative velocity or at least can be linearized; for example, forces due to sub-continuum fluid behaviour or forces resulting from surface chemistry. It is shown using several benchmarks that the relative motion between two particles in various flows is captured accurately, both statically and dynamically, in situations where uncorrected simulations fail. Moreover, the computational effort is reduced substantially by the application of the method.