Dynamic simulation of the flow of suspensions of two-dimensional particles with arbitrary shapes

A boundary-element method is implemented for simulating the motion of two-dimensional rigid particles with arbitrary shapes suspended in a viscous fluid in Stokes flow. The numerical implementation results in a system of linear equations for the components of the hydrodynamic traction over boundary elements distributed over the particle surfaces, and for the velocity of translation and angular velocity of rotation of the particles about designated centers. The linear system is solved by the method successive substitutions based on a physically motivated iterative procedure that involves decomposing the influence matrix into diagonal blocks consisting of physical particle clusters, and performing updates by matrix-vector multiplication using the inverses of the diagonal blocks. The iterations are found to converge as long as the particles are separated by a sufficiently large distance that depends on the particle shape and level of discretization. The stiffness of the governing equations due to lubrication forces developing between intercepting particles is removed by preventing the particles from approaching one another by less than a specified distance. Simulations are carried out for doubly-periodic suspensions of circular and elliptical particles in simple shear flow. The simulation algorithm is found to be successful for particles with moderate aspect ratios, and for small and moderate areal fractions up to 0.25. Higher areal fractions require long simulation times due to the large size of spontaneously forming particle clusters. The results illustrate the performance of various aspects of the boundary-element method and provide insights into the effect of the particle areal fraction and aspect ratio on the rheological properties of the suspension and on the geometrical properties of the evolving microstructure.