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  • Journal of Fluid Mechanics, Volume 674
  • May 2011, pp. 307-358

The influence of the inertially dominated outer region on the rheology of a dilute dispersion of low-Reynolds-number drops or rigid particles

  • GANESH SUBRAMANIAN (a1), DONALD L. KOCH (a2), JINGSHENG ZHANG (a3) and CHAO YANG (a3)
  • DOI: http://dx.doi.org/10.1017/jfm.2010.654
  • Published online: 28 April 2011
Abstract

We calculate the rheological properties of a dilute emulsion of neutrally buoyant nearly spherical drops at ORe3/2) in a simple shear flow(u = x211, being the shear rate) as a function of the ratio of the dispersed- and continuous-phase viscosities (λ = /μ). Here, φ is the volume fraction of the dispersed phase and Re is the micro-scale Reynolds number. The latter parameter is a dimensionless measure of inertial effects on the scale of the dispersed-phase constituents and is defined as Re = a2ρ/μ, a being the drop radius and ρ the common density of the two phases. The analysis is restricted to the limit φ, Re ≪ 1, when hydrodynamic interactions between drops may be neglected, and the velocity field in a region around the drop of the order of its own size is governed by the Stokes equations at leading order. The dominant contribution to the rheology at ORe3/2), however, arises from the so-called outer region where the leading-order Stokes approximation ceases to be valid. The relevant length scale in this outer region, the inertial screening length, results from a balance of convection and viscous diffusion, and is O(aRe−1/2) for simple shear flow in the limit Re ≪ 1. The neutrally buoyant drop appears as a point-force dipole on this scale. The rheological calculation at ORe3/2) is therefore based on a solution of the linearized Navier–Stokes equations forced by a point dipole. The principal contributions to the bulk rheological properties at this order arise from inertial corrections to the drop stresslet and Reynolds stress integrals. The theoretical calculations for the stresslet components are validated via finite volume simulations of a spherical drop at finite Re; the latter extend up to Re ≈ 10.

Combining the results of our ORe3/2) analysis with the known rheology of a dilute emulsion to ORe) leads to the following expressions for the relative viscosity (μe), and the non-dimensional first (N1) and second normal stress differences (N2) to ORe3/2): μe = 1 + φ[(5λ+2)/(2(λ+1))+0.024Re3/2(5λ+2)2/(λ+1)2]; N1=φ[−Re4(3λ2+3λ+1)/(9(λ+1)2)+0.066Re3/2(5λ+2)2/(λ+1)2] and N2 = φ[Re2(105λ2+96λ+35)/(315(λ+1)2)−0.085Re3/2(5λ+2)2/(λ+1)2].

Thus, for small but finite Re, inertia endows an emulsion with a non-Newtonian rheology even in the infinitely dilute limit, and in particular, our calculations show that, aside from normal stress differences, such an emulsion also exhibits a shear-thickening behaviour. The results for a suspension of rigid spherical particles are obtained in the limit λ → ∞.

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Email address for correspondence: sganesh@jncasr.ac.in
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E. S. Asmolov & F. Feuillebois 2010 Far-field disturbance flow induced by a small non-neutrally buoyant sphere in a linear shear flow. J. Fluid Mech. 643, 449470.

T. R. Auton 1987 The lift force on a spherical body in a rotational flow. J. Fluid Mech. 183, 199218.

T. R. Auton , J. C. R. Hunt & M. Prud'homme 1988 The force exerted on a body in inviscid unsteady non-uniform rotational flow. J. Fluid Mech. 197, 241257.

S. Bottin , O. Dauchot & F. Daviaud 1997 Intermittency in a locally forced plane Couette flow. Phys. Rev. Lett. 79 (22), 43774380.

S. Bottin , O. Dauchot , F. Daviaud & P. Manneville 1998 Experimental evidence of streamwise vortices as finite amplitude solutions in transitional plane Couette flow. Phys. Fluids 10 (10), 25972607.

J. F. Brady & J. F. Morris 1997 Microstructure of strongly sheared suspensions and its impact on rheology and diffusion. J. Fluid Mech. 348, 103139.

F. P. Bretherton 1962 Slow viscous motion round a cylinder in a simple shear. J. Fluid Mech. 12, 591613.

R. Clever & F. Busse 1997 Tertiary and quaternary solutions for plane Couette flow. J. Fluid Mech. 344, 137153.

A. G. Darbyshire & T. Mullin 1995 Transition to turbulence in constant-mass-flux pipe flow. J. Fluid Mech. 289, 83114.

O. Dauchot & F. Daviaud 1995 Finite amplitude perturbation and spots growth mechanism in plane Couette flow. Phys. Fluids 7 (2), 335343.

E. J. Ding & C. K. Aidun 2000 The dynamics and scaling law for particles suspended in shear flow with inertia. J. Fluid Mech. 423, 317344.

D. E. Elrick 1962 Source functions for diffusion in uniform shear flow. Austral. J. Phys. 15, 283288.

R. T. Foister & T. G. M. Van de Ven 1980 Diffusion of Brownian particles in shear flows. J. Fluid Mech. 96, 105132.

N. A. Frankel & A. Acrivos 1970 The constitutive equation for a dilute emulsion. J. Fluid Mech. 44, 6578.

J. Halcrow , J. F. Gibson , P. Cvitanovic & D. Viswananth 2009 Heteroclinic connections in plane Couette flow. J. Fluid Mech. 621, 365376.

S. Kim & S. J. Karrila 1991 Microhydrodynamics: Principles and Selected Applications. Butterworth-Heinemann.

C. A. Kossack & A. Acrivos 1974 Steady simple shear flow past a circular cylinder at moderate Reynolds number: a numerical solution. J. Fluid Mech. 66, 353376.

M. P. Kulkarni & J. F. Morris 2008 Suspension properties at finite Reynolds number from simulated shear flow. Phys. Fluids 20, 040602.

L. G. Leal 1992 Laminar Flow and Convective Transport Processes. Butterworth-Heinemann.

X. Li & K. Sarkar 2005 Effects of inertia on the rheology of a dilute emulsion of viscous drops in steady shear. J. Rheol. 49, 13771394.

M. J. Lighthill 1956 Drift. J. Fluid Mech. 1, 3153.

C. J. Lin , J. H. Peery & W. R. Schowalter 1970 Simple shear flow around a rigid sphere: inertial effects and suspension rheology. J. Fluid Mech. 44, 117.

J. P. Matas , J. F. Morris & E. Guazzelli 2003 Transition to turbulence in particular pipe flow. Phys. Rev. Lett. 90 (1), 014501–1.

D. R. Mikulencak & J. F. Morris 2004 Stationary shear flow around fixed and free bodies at finite Reynolds number. J. Fluid Mech. 520, 215242.

M. Nagata 1990 Three-dimensional finite-amplitude solutions in plane Couette flow: bifurcation from infinity. J. Fluid Mech. 217, 519527.

I. Proudman & J. R. A. Pearson 1957 Expansions at small Reynolds numbers for the flow past a sphere and a circular cylinder. J. Fluid Mech. 2, 237262.

G. Ryskin 1980 The extensional viscosity of a dilute suspension of spherical particles at intermediate microscale Reynolds numbers. J. Fluid Mech. 99, 513529.

P. G. Saffman 1965 The lift on a small sphere in a slow shear flow. J. Fluid Mech. 22, 385400.

P. J. Schmid & D. S. Henningson 2001 Stability and Transition in Shear Flows. Springer.

W. R. Schowalter , C. E. Chaffey & H. Brenner 1968 Rheological behaviour of a dilute emulsion. J. Colloid Interface Sci. 26, 152160.

G. Subramanian & D. L. Koch 2006 Inertial effects on the transfer of heat or mass from neutrally buoyant spheres in a steady linear velocity field. Phys. Fluids 18, 073302.

R. Vivek Raja , G. Subramanian & D. L. Koch 2010 Inertial effects on the rheology of a dilute emulsion. J. Fluid Mech. 646, 255296.

L. Y. Wang , X. Yin , D. L. Koch & C. Cohen 2009 Hydrodynamic diffusion and mass transfer across a sheared suspension of neutrally buoyant particles. Phys. Fluids 21, 033303.

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