Hostname: page-component-cd9895bd7-dk4vv Total loading time: 0 Render date: 2024-12-19T12:09:34.734Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  26 April 2007

LEONID BERLYAND  and
ALEXANDER PANCHENKO
LEONID BERLYAND
Affiliation:
Department of Mathematics and Materials Research Institute, Penn State University, University Park, PA 16802, [email protected]
ALEXANDER PANCHENKO
Affiliation:
Department of Mathematics, Washington State University, Pullman, WA 99164, [email protected]
Get access Rights & Permissions [Opens in a new window]

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).

Type
Papers
Information
Journal of Fluid Mechanics , Volume 578 , 10 May 2007 , pp. 1 - 34
Copyright
Copyright © Cambridge University Press 2007

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFRENCES

Bakhvalov, N. & Panasenko, G. 1989 Homogenization: Averaging Processes in Periodic Media. Kluwer.CrossRefGoogle Scholar
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, 444–472.CrossRefGoogle Scholar
Ball, R. C. & Melrose, J. R. 1997 b Colloidal microdynamics: pair-drag simulations of model-concentrated aggregated systems. Phys. Rev. E 56, 7067–7077.Google Scholar
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, 401–427.Google Scholar
Bensoussan, A., Lions, J. L., & Papanicolaou, G. 1978 Asymptotic Analysis in Periodic structures. North-Holland.Google Scholar
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, 179–227.CrossRefGoogle Scholar
Berlyand, L., Borcea, L. & Panchenko, A. 2005 a Network approximation for effective viscosity of concentrated suspensions with complex geometries. SIAM J. Math. Anal. 36, 1580–1628.CrossRefGoogle Scholar
Berlyand, L., Gorb, Y. & Novikov, A. 2005 b Anomalous blow-up in the effective viscous dissipation rate in 2D model of concentrated suspensions. Preprint.Google Scholar
Brady, J. F. & Morris, J. F. 1997 Microstructure of strongly sheared suspensions and its impact on rheology and diffusion. J. Fluid Mech. 348, 103–139.CrossRefGoogle Scholar
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.Google Scholar
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.Google Scholar
Edelsbrunner, H. 2000 Triangulations and meshes in computational geometry. Acta Numerica, 1–81.Google Scholar
Einstein, A. 1906 a Eine neue Bestimmung der Moleküldimensionen. Annln. Phys. 19, 289.CrossRefGoogle Scholar
Einstein, A. 1906 b Eine neue Bestimmung der Moleküldimensionen. Annln. Phys. 34, 591.Google Scholar
Frankel, N. A. & Acrivos, A. 1967 On the viscosity of a concentrated suspension of solid spheres. Chem. Engng Sci. 22, 847–853.CrossRefGoogle Scholar
Jikov, V., Kozlov, S. & Oleinik, O. 1994 Homogenization of Differential Operators and Integral Functionals. Springer.CrossRefGoogle Scholar
Kim, S. & Karilla, S. J. 1991 Microhydrodynamics: Principles and Selected Applications. Butterworth–Heinemann.Google Scholar
Nunan, K. C. & Keller, J. B. 1984 Effective viscosity of periodic suspensions J. Fluid Mech. 142, 269–287.CrossRefGoogle Scholar
Sanchez-Palencia, E. 1980 Non-homogeneous Media and Vibration Theory. Springer.Google Scholar
Schowalter, W. R. 1978 Mechanics of Non-Newtonian Fluids. Pergamon.Google Scholar
Shikata, T. & Pearson, D. S. 1994 Viscoelastic behavior of concentrated spherical suspensions. J. Rheol. 38, 601–616.Google Scholar
Shook, C. A. & Roco, M. C. 1991 Slurry Flow. Principles and Practice Butterworth-Heinemann.Google Scholar
Sierou, A. & Brady, J. F. 2001 Accelerated Stokesian dynamic simulations. J. Fluid Mech. 448, 115–146.Google Scholar
Sierou, A. & Brady, J. F. 2002 Rheology and microstructure in concentrated noncolloidal suspensions. J. Rheol. 46, 1031–1056.CrossRefGoogle Scholar
Stickel, J. J. & Powell, R. L. 2005 Fluid mechanics and rheology of dense suspensions. Annu. Rev. Fluid Mech. 37, 129–149.Google Scholar
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, 795–807.Google Scholar