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A Local Version of Einstein's Formula for the Effective Viscosity of Suspensions
SIAM Journal on Mathematical Analysis ( IF 2 ) Pub Date : 2020-05-26 , DOI: 10.1137/19m1251229
Barbara Niethammer , Richard Schubert

SIAM Journal on Mathematical Analysis, Volume 52, Issue 3, Page 2561-2591, January 2020.
We prove a local variant of Einstein's formula for the effective viscosity of dilute suspensions, that is, $\mu^\prime=\mu ({{1+\frac 5 2\phi+o(\phi)}})$, where $\phi$ 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 $\phi=NR^3$ 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.


中文翻译:

爱因斯坦悬架有效粘度公式的本地版本

2020年1月,SIAM数学分析期刊,第52卷,第3期,第2561-2591页。
我们证明了稀悬浮液有效粘度的爱因斯坦公式的局部变体,即$ \ mu ^ \ prime = \ mu({{1+ \ frac 5 2 \ phi + o(\ phi)}})$,其中$ \ phi $是悬浮粒子的体积分数。到目前为止,仅针对流场的耗散功能获得了严格的依据。我们证明该公式在斯托克斯方程(具有可变粘度)的水平上成立。我们考虑一种状态,其中悬浮在流体中的粒子数量$ N $变为无穷大,而它们的大小$ R $和体积分数$ \ phi = NR ^ 3 $接近零。我们为微观解与均化方程解的差建立了$ L ^ infty $和$ L ^ p $估计。在这里,我们假设粒子包含在一个有限区域中,并且在最小距离与平均距离可比的意义上很好地分开了。检验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度。
更新日期:2020-06-30
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