爱因斯坦悬浮液有效粘度公式的局部形式
A local version of Einstein's formula for the effective viscosity of suspensions
摘要：
我们证明了爱因斯坦的公式的一个局部变体的有效粘度的稀悬浮液，即$μ^\prime=μ(1+\frac 5 2ϕ+o(ϕ))$，其中$ϕ$是悬浮颗粒的体积分数。到目前为止，严格的理由只得到了流场的耗散泛函。我们证明了该公式在变粘度的斯托克斯方程的水平上成立。我们考虑一个制度，其中悬浮在流体中的颗粒的数量$N$趋于无穷大，而它们的大小$R$和体积分数$ϕ=NR^3$接近零。我们建立了微观解与齐次方程解之差的$L^\infty$和$L^p$估计。在这里，我们假设粒子包含在一个有界区域中，并且在最小距离与平均距离相当的意义上被很好地分离。证明的主要工具是悬浮液流场的偶极近似，以及所谓的反射方法和体积密度的粗粒化。
英文摘要：
We prove a local variant of Einstein's formula for the effective viscosity of dilute suspensions, that is $μ^\prime=μ(1+\frac 5 2ϕ+o(ϕ))$, where $ϕ$ is the volume fraction of the suspended particles. Up to now rigorous justifications have only been obtained for dissipation functionals of the flow field. We prove that the formula holds on the level of the Stokes equation (with variable viscosity). We consider a regime where the number $N$ of particles suspended in the fluid goes to infinity while their size $R$ and the volume fraction $ϕ=NR^3$ approach zero. We establish $L^\infty$ and $L^p$ estimates for the difference of the microscopic solution to the solution of the homogenized equation. Here we assume that the particles are contained in a bounded region and are well separated in the sense that the minimal distance is comparable to the average one. The main tools for the proof are a dipole approximation of the flow field of the suspension together with the so-called method of reflections and a coarse graining of the volume density.
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