Boundary integral equation analysis for suspension of spheres in Stokes flow
This work offers a computationally efficient framework for simulating many-body hydrodynamic interactions in Stokes flow, relevant to porous media, active matter, and magneto-hydrodynamics.
The authors developed a boundary integral method for Stokes flow around spheres, providing analytical diagonalization of operators on vector spherical harmonics and a hybrid near-field/far-field scheme. Numerical results confirm accuracy and linear scaling with particle count.
We show that the standard boundary integral operators, defined on the unit sphere, for the Stokes equations diagonalize on a specific set of vector spherical harmonics and provide formulas for their spectra. We also derive analytical expressions for evaluating the operators away from the boundary. When two particle are located close to each other, we use a truncated series expansion to compute the hydrodynamic interaction. On the other hand, we use the standard spectrally accurate quadrature scheme to evaluate smooth integrals on the far-field, and accelerate the resulting discrete sums using the fast multipole method (FMM). We employ this discretization scheme to analyze several boundary integral formulations of interest including those arising in porous media flow, active matter and magneto-hydrodynamics of rigid particles. We provide numerical results verifying the accuracy and scaling of their evaluation.