Chiang Mai Journal of Science

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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

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