Einstein球稀悬浮液有效粘度的证明
A proof of Einstein's effective viscosity for a dilute suspension of spheres
摘要：
本文给出了当刚性中性浮力球以立方晶格顶点为中心时，其稀悬浮液有效粘度的爱因斯坦公式的数学证明。我们保持容器的大小有限的稀释限制，并考虑边界效应。爱因斯坦公式恢复为一个一阶渐近展开的有效粘度的体积分数。为了严格证明这种扩展，我们得到了一个明确的上限和下界的有效粘度。一个下界被发现使用能量的方法让人想起Keller等人的工作。一个上界如下通过获得一个明确的估计的牵引力，在流体边界上的速度的流体边界上的应力的法向分量。这种估计，反过来，建立使用的斯托克斯方程的边界积分公式。我们的证明承认其他粒子形状的推广和包括点力模型自推进粒子。
英文摘要：
We present a mathematical proof of Einstein's formula for the effective viscosity of a dilute suspension of rigid neutrally--buoyant spheres when the spheres are centered on the vertices of a cubic lattice. We keep the size of the container finite in the dilute limit and consider boundary effects. Einstein's formula is recovered as a first-order asymptotic expansion of the effective viscosity in the volume fraction. To rigorously justify this expansion, we obtain an explicit upper and lower bound on the effective viscosity. A lower bound is found using energy methods reminiscent of the work of Keller et al. An upper bound follows by obtaining an explicit estimate for the tractions, the normal component of the stress on the fluid boundary, in terms of the velocity on the fluid boundary. This estimate, in turn, is established using a boundary integral formulation for the Stokes equation. Our proof admits a generalization to other particle shapes and the inclusion of point forces to model self-propelled particles.
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