A proof of Einstein's effective viscosity for a dilute suspension of spheres (Q2902752)

In this interesting and well written paper the authors give a rigorous proof of the Einstein formula for the effective viscosity of a dilute suspension of rigid spherical particles in a viscous incompressible fluid. The paper is divided in three sections. The first section is devoted to the mathematical model for a suspension of spheres. The second section contains the main result of this paper given in Theorem 2.1. In the last section the authors prove a lemma related to the continuity of solutions to a boundary integral equation corresponding to the Stokes system. This result is mainly used in the proof of Theorem 2.1.NEWLINENEWLINENEWLINELet \(\Omega\) be a bounded smooth domain in \(\mathbb R^3\) and let \(B^l\) be a neutrally buoyant, passive sphere of radius \(a\) centered at \(x^l\), \(l=1,\dots ,N\). These spheres provide a dilute suspension in an ambient fluid of viscosity \(\eta\). The model that describes this suspension is described by the Stokes system in \(\Omega\setminus\cup_{l=1}^NB^l\), subject to balance of forces and torques at the boundary of these spheres. Let \(\hat{\eta}\) be the effective viscosity of the dilute suspension of the spheres and let \(\phi:=\frac{4\pi}{3}a^3\).NEWLINENEWLINENEWLINEIn Theorem 2.1 the authors prove that there exists a constant \(C>0\), which does not depend on \(a\) and \(N\), such that for all \(\phi\in (0,\frac{\pi}{6})\), NEWLINE\[NEWLINE\left|\hat{\eta}-\eta\left(1+\frac{5}{2}\phi\right)\right|\leq C\eta\phi^{\frac{3}{2}}.NEWLINE\]NEWLINE The authors point out that their results in the last section can be extended to the case of other particle shapes.