Improvement of the Stokesian Dynamics method for systems with finite number of particles


Ichiki, Kengo (2002) Improvement of the Stokesian Dynamics method for systems with finite number of particles. Journal of Fluid Mechanics, 452 . pp. 231-262. ISSN 0022-1120

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Abstract:An improvement of the Stokesian Dynamics method for many-particle systems is presented. A direct calculation of the hydrodynamic interaction is used rather than imposing periodic boundary conditions. The two major diculties concern the accuracy and the speed of calculations. The accuracy discussed in this work is not concerned with the lubrication correction but, rather, focuses on the multipole expansion which until now has only been formulated up to the so-called FTS version or the rst order of force moments. This is improved systematically by a real-space multipole expansion with force moments and velocity moments evaluated at the centre of the particles, where the velocity moments are calculated through the velocity derivatives; the introduction of the velocity derivatives makes the formulation and its extensions straightforward. The reduction of the moments into irreducible form is achieved by the Cartesian irreducible tensor. The reduction is essential to form a well-dened linear set of equations as a generalized mobility problem. The order of truncation is not limited in principle, and explicit calculations of two-body problems are shown with order up to 7. The calculating speed is improved by a conjugate-gradient-type iterative method which consists of a dot-product between the generalized mobility matrix and the force moments as a trial value in each iteration. This provides an O(N2) scheme where N is the number of particles in the system. Further improvement is achieved by the fast multipole method for the calculation of the generalized mobility problem in each iteration, and an O(N) scheme for the non-adaptive version is obtained. Real problems are studied on systems with N = 400 000 particles. For mobility problems the number of iterations is constant and an O(N) performance is achieved; however for resistance problems the number of iterations increases as almost N1=2 with a high accuracy of 10¿6 and the total cost seems to be O(N3=2).
1. Introduction
The microstructure of suspensions is governed by the hydrodynamic interactions
among particles immersed in a viscous fluid, which is modelled using the Stokes
approximation, and have attracted much attention from researchers in physics and
chemical engineering. The hydrodynamic interactions have a long-range nature varying
as 1=r, where r is distance measured from a particle, and further they have a
many-body feature, that is, they must be the solution of a boundary-value problem on
the surface of all objects in the system. Therefore, analytical approaches are dicult.
In fact, even for rigid spherical particles, the exact solution has been obtained only
for two-body problems (Jerey & Onishi 1984); of course, this is partially because the
symmetry of the geometry of surfaces for two-body problems is much simpler than
that on systems with three or more particles. Therefore, numerical approaches have
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Copyright:© 2002 Cambridge University Press
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