TY - JOUR

T1 - Strong and weak blow-up of the viscous dissipation rates for concentrated suspensions

AU - Berlyand, Leonid

AU - Panchenko, Alexander

N1 - Funding Information:
The authors wish to thank Professor John F. Brady for bringing the work of Sierou & Brady (2001) to their attention. L. B. was supported in part by NSF grant DMS-0204637. A. P. was supported in part by ONR grant N00014-001-0853 and by DOE grant DE-FG02-05ER25709.

PY - 2007/5/10

Y1 - 2007/5/10

N2 - 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).

AB - 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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U2 - 10.1017/S0022112007004922

DO - 10.1017/S0022112007004922

M3 - Article

AN - SCOPUS:34548192241

SN - 0022-1120

VL - 578

SP - 1

EP - 34

JO - Journal of Fluid Mechanics

JF - Journal of Fluid Mechanics

ER -