Abstract
Consider a colloidal suspension of rigid particles in a steady Stokes flow. In a celebrated work, Einstein argued that in the regime of dilute particles the system behaves at leading order like a Stokes fluid with some explicit effective viscosity. In the present contribution, we rigorously define a notion of effective viscosity, regardless of the dilute regime assumption. More precisely, we establish a homogenization result for when particles are distributed according to a given stationary and ergodic random point process. The main novelty is the introduction and analysis of suitable corrector equations.
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Notes
That is, \(\nabla \psi _E^\omega (x+y)=\nabla \psi _E^{\tau _{-y}\omega }(x)\) and \(\Sigma _E^\omega (x+y)\mathbb {1}_{{\mathbb {R}}^d\setminus {\mathcal {I}}^\omega }(x+y)=\Sigma _E^{\tau _{-y}\omega }(x)\mathbb {1}_{{\mathbb {R}}^d\setminus {\mathcal {I}}^{\tau _{-y}\omega }}(x)\) for all \(x,y,\omega \), cf. Remark 1.2.
References
Allaire, G.: Homogenization of the Stokes flow in a connected porous medium. Asymptotic Anal. 2(3), 203–222, 1989
Allaire, G.: Homogenization of the Navier-Stokes equations in open sets perforated with tiny holes. I. Abstract framework, a volume distribution of holes. Arch. Rational Mech. Anal. 113(3), 209–259, 1990
Almog, Y., Brenner, H.: Global homogenization of a dilute suspension of sphere. Unpublished manuscript (1998)
Ammari, H., Garapon, P., Kang, H., Lee, H.: Effective viscosity properties of dilute suspensions of arbitrarily shaped particles. Asymptot. Anal. 80(3–4), 189–211, 2012
Basson, A., Gérard-Varet, D.: Wall laws for fluid flows at a boundary with random roughness. Commun. Pure Appl. Math. 61(7), 941–987, 2008
Batchelor, G.K.: Sedimentation in a dilute dispersion of spheres. J. Fluids Mech. 52(2), 245–268, 1972
Batchelor, G.L., Green, J.T.: The determination of the bulk stress in suspension of spherical particles to order \(c^2\). J. Fluid Mech. 56, 401–427, 1972
Beliaev, A.Y., Kozlov, S.M.: Darcy equation for random porous media. Commun. Pure Appl. Math. 49(1), 1–34, 1996
Cioranescu, D., Saint Jean Paulin, J.: Homogenization in open sets with holes. J. Math. Anal. Appl. 71(2), 590–607, 1979
Duerinckx, M., Gloria, A.: Analyticity of homogenized coefficients under Bernoulli perturbations and the Clausius-Mossotti formulas. Arch. Ration. Mech. Anal. 220(1), 297–361, 2016
Duerinckx, M., Gloria, A.: On Einstein’s effective viscosity formula. Preprint, arXiv:2008.03837
Duerinckx, M., Gloria, A.: Quantitative homogenization theory for suspensions in steady Stokes flow. In preparation
Duerinckx, M., Gloria, A., Sedimentation of random suspensions and the effect of hyperuniformity. Preprint, arXiv:2004.03240
Einstein, A.: Eine neue Bestimmung der Moleküldimensionen. Ann. Phys. 19(2), 289–306, 1906
Galdi, G.P.: An introduction to the mathematical theory of the Navier-Stokes equations. Steady-state problems, 2nd edn. Springer Monographs in Mathematics Springer, New York 2011
Gérard-Varet, D., Hillairet, M.: Analysis of the viscosity of dilute suspensions beyond Einstein’s formula. Preprint, arXiv:1905.08208
Gérard-Varet, D., Mecherbet, A.: On the correction to Einstein’s formula for the effective viscosity. Preprint, arXiv:2004.05601
Giunti, A., Höfer, R.M.: Homogenization for the Stokes equations in randomly perforated domains under almost minimal assumptions on the size of the holes. Preprint, arXiv:1809.04491
Gloria, A.: A scalar version of the Caflisch-Luke paradox. Preprint, arXiv:1907.08182.
Haines, B.M., Mazzucato, A.L.: A proof of Einstein’s effective viscosity for a dilute suspension of spheres. SIAM J. Math. Anal. 44(3), 2120–2145, 2012
Jikov, V.V.: Averaging of functionals in the calculus of variations and elasticity. Math. USSR, Izvestiya 29, 33–66, 1987
Jikov, V.V.: Some problems of extension of functions arising in connection with the homogenization theory. Differ. Uravnenia 26(1), 39–51, 1990
Jikov, V.V., Kozlov, S.M., Oleĭnik, O.A.: Homogenization of differential operators and integral functionals. Springer, Berlin 1994
Niethammer, B., Schubert, R.: A local version of Einstein’s formula for the effective viscosity of suspensions. Preprint, arXiv:1903.08554
Nunan, K., Keller, J.: Effective viscosity of a periodic suspension. J. Fluid Mech. 142, 269–287, 1984
Papanicolaou, G.C., Varadhan, S.R.S.: Boundary value problems with rapidly oscillating random coefficients. In Random fields, Vol. I, II (Esztergom, 1979), volume 27 of Colloq. Math. Soc. János Bolyai, pages 835–873. North-Holland, Amsterdam, 1981
Sánchez-Palencia, E.: Nonhomogeneous Media and Vibration Theory, vol. 127. Lecture Notes in PhysicsSpringer, Berlin 1980
Tartar, L.: Incompressible fluid flow in a porous medium: convergence of the homogenization process. Nonhomogeneous media and vibration theory (E, pp. 368–377. Sánchez-Palencia), volume 127 of Lecture Notes in Physics. Springer-Verlag, Berlin-New York (1980)
Tartar, L.: The general theory of homogenization, vol. 7. Lecture Notes of the Unione Matematica Italiana. Springer-Verlag, Berlin; UMI, Bologna (2009)
Acknowledgements
MD acknowledges financial support from the CNRS-Momentum program. AG has received funding from the European Union’s Horizon 2020 research and innovation programme under Grant Agreement No. 864066.
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Duerinckx, M., Gloria, A. Corrector Equations in Fluid Mechanics: Effective Viscosity of Colloidal Suspensions. Arch Rational Mech Anal 239, 1025–1060 (2021). https://doi.org/10.1007/s00205-020-01589-1
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DOI: https://doi.org/10.1007/s00205-020-01589-1