Chiang Mai Journal of Science

When the effective viscosity of suspensions is modeled, the main gradient flow is perturbed by the presence of spherical inclusions. Here the constant velocity gradient at infinity. The flow around a single sphere allows one to find the average contribution to the effective viscosity within the first order with respect to the volume fractions of the particulate phase. In order to obtain the second asymptotic order, one needs to solve the problem of the flow around two non-equal spheres under constant gradient at infinity, which is essentially a 3D problem. In this study, the underlying symmetries of the flow are used, and the full 3D problem is reduced to five conjugated 2D problems. Each of these 2D problems is formulated in terms of stream functions which requite solving equations with bi-Stokesian operators. Bi-spherical coordinates are used for which the boundaries of the spheres are also coordinate surfaces. To solve the bi-Sotkesian equations, a fast spectral method based on Legendre polynomials is proposed with exponential convergence. The method of generating function is used for both Legendre and associated Legendre polynomials and closed algebraic systems are obtained for the systems under considerations. V is the uniform stream, and G is