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Strong and weak blow-up of the viscous dissipation rates for concentrated suspensions

  • LEONID BERLYAND (a1) and ALEXANDER PANCHENKO (a2)
Abstract

We study the overall dissipation rate of highly concentrated non-colloidal suspensions of rigid neutrally buoyant particles in a Newtonian fluid. This suspension is confined to a finite size container, subject to shear or extensional boundary conditions at the walls of the container. The corresponding dissipation rates determine the effective shear viscosity μ* and the extensional effective viscosity λ*. We use recently developed discrete network approximation techniques to obtain discrete forms for the overall dissipation rates, and analyse their asymptotics in the limit when the characteristic interparticle distance goes to zero. The focus is on the finite size and particle wall effects in spatially disordered arrays. Use of the network approximation allows us to study the dependence of μ* and λ* on variable distances between neighbouring particles in such arrays.

Our analysis, carried out for a two-dimensional model, can be characterized as global because it goes beyond the local analysis of flow in a single gap between two particles and takes into account hydrodynamic interactions in the entire particle array. The principal conclusion in the paper is that, in general, asymptotic formulae for μ* and λ* obtained by global analysis are different from the formulae obtained from local analysis. In particular, we show that the leading term in the asymptotics of μ* is of lower order than suggested by the local analysis (weak blow-up), while the order of the leading term in the asymptotics of λ* depends on the geometry of the particle array (either weak or strong blow-up). We obtain geometric conditions on a random particle array under which the asymptotic order of λ* coincides with the order of the local dissipation in a gap between two neighbouring particles, and show that these conditions are generic. We also provide an example of a uniformly closely packed particle array for which the leading term in the asymptotics of λ* degenerates (weak blow-up).

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Bakhvalov, N. & Panasenko, G. 1989 Homogenization: Averaging Processes in Periodic Media. Kluwer.
Ball, R. C. & Melrose, J. R. 1997 a A simulation technique for many spheres in quasi-static motion under frame-invariant pair drag and Brownian forces. Physica A 247, 444472.
Ball, R. C. & Melrose, J. R. 1997 b Colloidal microdynamics: pair-drag simulations of model-concentrated aggregated systems. Phys. Rev. E 56, 70677077.
Batchelor, G. K. & Green, J. T. 1972 The determination of the bulk stress in a suspension of spherical particles to order c 2. J. Fluid Mech. 56, 401427.
Bensoussan, A., Lions, J. L., & Papanicolaou, G. 1978 Asymptotic Analysis in Periodic structures. North-Holland.
Berlyand, L. & Kolpakov, A. 2001 Network approximation in the limit of small inter-particle distance of the effective properties of a high-contrast random dispersed composite. Arch. Rat. Mech. Anal. 159, 179227.
Berlyand, L., Borcea, L. & Panchenko, A. 2005 a Network approximation for effective viscosity of concentrated suspensions with complex geometries. SIAM J. Math. Anal. 36, 15801628.
Berlyand, L., Gorb, Y. & Novikov, A. 2005 b Anomalous blow-up in the effective viscous dissipation rate in 2D model of concentrated suspensions. Preprint.
Brady, J. F. & Morris, J. F. 1997 Microstructure of strongly sheared suspensions and its impact on rheology and diffusion. J. Fluid Mech. 348, 103139.
Carreau, P. J. & Cotton, F. 2002 Rheological properties of concentrated suspensions. In Transport Processes in Bubbles, Drops and Particles (ed. Kee, D. De & Chhabra, R. P.). Taylor & Francis.
Coussot, P. 2002 Flows of concentrated granular mixtures. In Transport processes in Bubbles, Drops and Particles (ed. Kee, D. De & Chhabra, R. P.). Taylor & Francis.
Edelsbrunner, H. 2000 Triangulations and meshes in computational geometry. Acta Numerica, 1–81.
Einstein, A. 1906 a Eine neue Bestimmung der Moleküldimensionen. Annln. Phys. 19, 289.
Einstein, A. 1906 b Eine neue Bestimmung der Moleküldimensionen. Annln. Phys. 34, 591.
Frankel, N. A. & Acrivos, A. 1967 On the viscosity of a concentrated suspension of solid spheres. Chem. Engng Sci. 22, 847853.
Jikov, V., Kozlov, S. & Oleinik, O. 1994 Homogenization of Differential Operators and Integral Functionals. Springer.
Kim, S. & Karilla, S. J. 1991 Microhydrodynamics: Principles and Selected Applications. Butterworth–Heinemann.
Nunan, K. C. & Keller, J. B. 1984 Effective viscosity of periodic suspensions J. Fluid Mech. 142, 269287.
Sanchez-Palencia, E. 1980 Non-homogeneous Media and Vibration Theory. Springer.
Schowalter, W. R. 1978 Mechanics of Non-Newtonian Fluids. Pergamon.
Shikata, T. & Pearson, D. S. 1994 Viscoelastic behavior of concentrated spherical suspensions. J. Rheol. 38, 601616.
Shook, C. A. & Roco, M. C. 1991 Slurry Flow. Principles and Practice Butterworth-Heinemann.
Sierou, A. & Brady, J. F. 2001 Accelerated Stokesian dynamic simulations. J. Fluid Mech. 448, 115146.
Sierou, A. & Brady, J. F. 2002 Rheology and microstructure in concentrated noncolloidal suspensions. J. Rheol. 46, 10311056.
Stickel, J. J. & Powell, R. L. 2005 Fluid mechanics and rheology of dense suspensions. Annu. Rev. Fluid Mech. 37, 129149.
Vander Werff, J. C. der Werff, J. C. deKruif, C. G. Kruif, C. G. Blom, C. & Mellema, J. 1989 Linear viscoelastic behavior of dense hard-sphere dispersions. Phys. Rev. A 39, 795807.
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Journal of Fluid Mechanics
  • ISSN: 0022-1120
  • EISSN: 1469-7645
  • URL: /core/journals/journal-of-fluid-mechanics
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