We present a mathematical proof of Einstein's formula for the effectiveviscosity of a dilute suspension of rigid neutrally--buoyant spheres when thespheres are centered on the vertices of a cubic lattice. We keep the size ofthe container finite in the dilute limit and consider boundary effects.Einstein's formula is recovered as a first-order asymptotic expansion of theeffective viscosity in the volume fraction. To rigorously justify thisexpansion, we obtain an explicit upper and lower bound on the effectiveviscosity. A lower bound is found using energy methods reminiscent of the workof Keller et al. An upper bound follows by obtaining an explicit estimate forthe tractions, the normal component of the stress on the fluid boundary, interms of the velocity on the fluid boundary. This estimate, in turn, isestablished using a boundary integral formulation for the Stokes equation. Ourproof admits a generalization to other particle shapes and the inclusion ofpoint forces to model self-propelled particles.
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