Analysis of the viscosity of dilute suspensions beyond Einstein's formula

Abstract

We provide a mathematical analysis of the effective viscosity of suspensions of spherical particles in a Stokes flow, at low solid volume fraction $\varphi$. Our objective is to go beyond the Einstein's approximation ${\mu }_{eff}=(1+\frac{5}{2}\varphi )\mu$. Assuming a lower bound on the minimal distance between the $N$ particles, we are able to identify the $O({\varphi }^2)$ correction to the effective viscosity, which involves pairwise particle interactions. Applying the methodology developped over the last years on Coulomb gases, we are able to tackle the limit $N\to +\infty$ of the $O({\varphi }^2)$-correction, and provide explicit formula for this limit when the particles centers can be described by either periodic or stationary ergodic point processes.

arXiv comment51 pages. V3: to appear in Arch. Rat. Mech. Anal

References in corpus (4)

• The Mean-Field Limit for Solid Particles in a Navier-Stokes Flow
• The Method of Reflections, Homogenization and Screening for Poisson and Stokes Equations in Perforated Domains
• Analyticity of homogenized coefficients under Bernoulli perturbations and the Clausius-Mossotti formulas
• Renormalized energy concentration in random matrices

Cited by in corpus (1)

• A simple justification of effective models for conducting or fluid media with dilute spherical inclusions