The influence of Einstein's effective viscosity on sedimentation at very small particle volume fraction

Höfer, Richard M.; Schubert, Richard

Annales de l'I.H.P. Analyse non linéaire, novembre – décembre 2021, Tome 38 (2021) no. 6, pp. 1897-1927.

We investigate the sedimentation of identical inertialess spherical particles in a Stokes fluid in the limit of many small particles. It is known that the presence of the particles leads to an increase of the effective viscosity of the suspension. By Einstein's formula this effect is of the order of the particle volume fraction ϕ. The disturbance of the fluid flow responsible for this increase of viscosity is very singular (like |x|−2). Nevertheless, for well-prepared initial configurations and ϕ→0, we show that the microscopic dynamics is approximated to order ϕ2|log⁡ϕ| by a macroscopic coupled transport-Stokes system with an effective viscosity according to Einstein's formula. We provide quantitative estimates both for convergence of the densities in the p-Wasserstein distance for all p and for the fluid velocity in Lebesgue spaces in terms of the p-Wasserstein distance of the initial data. Our proof is based on approximations through the method of reflections and on a generalization of a classical result on convergence to mean-field limits in the infinite Wasserstein metric by Hauray.

[1] Carrillo, J.A.; Choi, Y.-P.; Hauray, M. The derivation of swarming models: mean-field limit and Wasserstein distances, Collective Dynamics from Bacteria to Crowds, CISM Courses and Lect., vol. 553, Springer, Vienna, 2014, pp. 1-46 | DOI | MR

[2] Duerinckx, M.; Gloria, A. Corrector equations in fluid mechanics: effective viscosity of colloidal suspensions, Arch. Ration. Mech. Anal. (2020), pp. 1-36 | MR

[3] Duerinckx, M.; Gloria, A. On Einstein's effective viscosity formula, 2020 (arXiv preprint) | arXiv | MR

[4] DiPerna, R.J.; Lions, P.L. Ordinary differential equations, transport theory and Sobolev spaces, Invent. Math., Volume 98 (1989), pp. 511-547 | DOI | MR | Zbl

[5] Einstein, A. Eine neue Bestimmung der Moleküldimensionen, Ann. Phys., Volume 19 (1906), pp. 289-306 | DOI | JFM

[6] Figari, R.; Orlandi, E.; Teta, S. The Laplacian in regions with many small obstacles: fluctuations around the limit operator, J. Stat. Phys., Volume 41 (1985) no. 3–4, pp. 465-487 | DOI | MR | Zbl

[7] Gerard-Varet, D. A simple justification of effective models for conducting or fluid media with dilute spherical inclusions, 2019 (arXiv preprint) | arXiv | MR

[8] Gérard-Varet, D.; Hillairet, M. Analysis of the viscosity of dilute suspensions beyond Einstein's formula, Arch. Ration. Mech. Anal., Volume 238 (2020) no. 3, pp. 1349-1411 | DOI | MR

[9] Gérard-Varet, D.; Höfer, R.M. Mild assumptions for the derivation of Einstein's effective viscosity formula, 2020 (arXiv preprint) | arXiv | MR

[10] Gérard-Varet, D.; Mecherbet, A. On the correction to Einstein's formula for the effective viscosity, 2020 (arXiv preprint) | arXiv | MR

[11] Hauray, M. Wasserstein distances for vortices approximation of Euler-type equations, Math. Models Methods Appl. Sci., Volume 19 (2009) no. 8, pp. 1357-1384 | DOI | MR | Zbl

[12] Haines, B.M.; Mazzucato, A.L. A proof of Einstein's effective viscosity for a dilute suspension of spheres, SIAM J. Math. Anal., Volume 44 (2012) no. 3, pp. 2120-2145 | DOI | MR | Zbl

[13] Höfer, R.M. Sedimentation of inertialess particles in Stokes flows, Commun. Math. Phys., Volume 360 (2018) no. 1, pp. 55-101 | DOI | MR

[14] Höfer, R.M. Convergence of the method of reflections for particle suspensions in Stokes flows, 2019 (arXiv preprint) | arXiv | MR

[15] Höfer, R.M.; Velázquez, J.J.L. The method of reflections, homogenization and screening for Poisson and Stokes equations in perforated domains, Arch. Ration. Mech. Anal., Volume 227 (2018) no. 3, pp. 1165-1221 | DOI | MR

[16] Hillairet, M.; Wu, D. Effective viscosity of a polydispersed suspension, J. Math. Pures Appl. (2020) | DOI | MR

[17] Jabin, P.-E.; Otto, F. Identification of the dilute regime in particle sedimentation, Commun. Math. Phys., Volume 250 (2004) no. 2, pp. 415-432 | DOI | MR | Zbl

[18] Loeper, G. Uniqueness of the solution to the Vlasov–Poisson system with bounded density, J. Math. Pures Appl., Volume 86 (2006) no. 1, pp. 68-79 | DOI | MR | Zbl

[19] Mecherbet, A. Sedimentation of particles in Stokes flow, Kinet. Relat. Models, Volume 12 (2019) no. 5, pp. 995-1044 | DOI | MR

[20] Mecherbet, A. On the sedimentation of a droplet in Stokes flow, 2020 (arXiv preprint) | arXiv | MR

[21] Maremonti, P.; Russo, R.; Starita, G. On the Stokes equations: the boundary value problem, Advances in Fluid Dynamics, vol. 4, Quad. Mat. Dept. Math., Seconda Univ. Napoli, Caserta, 1999, pp. 69-140 | MR | Zbl

[22] Niethammer, B.; Schubert, R. A local version of Einstein's formula for the effective viscosity of suspensions, SIAM J. Math. Anal., Volume 52 (2020) no. 3, pp. 2561-2591 | DOI | MR

[23] Rubinstein, J. On the macroscopic description of slow viscous flow past a random array of spheres, J. Stat. Phys., Volume 44 (1986) no. 5–6, pp. 849-863 | DOI | MR | Zbl

[24] Santambrogio, F. Optimal Transport for Applied Mathematicians: Calculus of Variations, PDEs, and Modeling, Progress in Nonlinear Differential Equations and Their Applications, Springer International Publishing, 2015 | DOI

[25] Serfaty, S. Mean field limit for Coulomb-type flows, Duke Math. J., Volume 169 (2020) no. 15, pp. 2887-2935 | DOI | MR