cien

ot

bS.forn6; a

e 2

id n

d; LRecently, Brady et al. [1] computed the bulk viscosity (alsoknown as the second or volume viscosity) of a dilute colloidaldispersion to O(2) in the volume fraction of the (rigid spher-ical) suspended particles. This calculation requires determina-tion of the dispersion microstructure, which reflects a balancebetween an imposed uniform expansion flow and Brownian dif-fusion of the particles. The O(2) contribution to the bulk vis-cosity arises from two-particle interactions, whereas the O()contribution is due to the disturbance flow generated by a sin-gle particle in the expansion flow. We call the O() contributionthe Einstein correction, in homage to Einstein [2] who com-puted the O() contribution to the shear viscosity of a dilutecolloidal dispersion of rigid spheres. In this Note, we computethe Einstein correction to the bulk viscosity for a dispersion ofn-dimensional hyperspheres. The correction for n = 2 repre-sents the main outcome of this work; it gives the bulk viscosityfor a dilute dispersion of two-dimensional rigid cylindersa result of practical significance. Furthermore, the general n-dimensional problem provides a useful exercise for students oftransport processes and fluid dynamics, by introducing basicconcepts in suspension mechanics, while, at the same time, of-fering a glimpse into an active research area.

E-mail address: aditya@caltech.edu.

We begin with the constitutive equation for the stress in an-dimensional compressible Newtonian fluid (see, e.g., Batch-elor [3]), viz.

(1) = pthI + 2e +( 2

n

)( u)I ,

where pth is the thermodynamic pressure (as defined by thefluids equation of state), e = 12 [u + (u)] is the rate ofstrain tensor with u the fluid velocity, is the shear viscosity, is the bulk viscosity, and I is the identity tensor. In zero-Reynolds-number flow the velocity field can be decomposedinto an uniform expansion everywhere in the fluid and a distur-bance flow created by any immersed particles, which satisfiesthe usual incompressible Stokes equations; that is,

u = 1ner + us ,

where r is the position vector, e is the expansion rate, u = e,and us = 0. In turn, the fluid stress can be split into a con-tribution due to the uniform expansion flow, e = (e pth)I ,and a disturbance stress s = psI + 2es , with s = 0.Here, ps is the dynamical pressure field of the incompressibleStokes flow.

Consider a force- and torque-free n-dimensional hyper-sphere of radius a immersed in the uniform expansion flow.Exploiting the linearity of the Stokes equations, and noting theJournal of Colloid and Interface S

N

The Einstein correction to the

AdityaDivision of Chemistry and Chemical Engineering, Cali

Received 16 June 200

Available onlin

Abstract

We calculate the effective bulk viscosity of a dilute dispersion of rigReynolds number. 2006 Elsevier Inc. All rights reserved.

Keywords: Colloidal dispersion; Bulk viscosity; Hypersphere; Compressible flui

1. Introduction0021-9797/$ see front matter 2006 Elsevier Inc. All rights reserved.doi:10.1016/j.jcis.2006.07.076ce 302 (2006) 702703www.elsevier.com/locate/jcis

e

ulk viscosity in n dimensionsKhairia Institute of Technology, Pasadena, CA 91125, USA

ccepted 30 July 2006

August 2006

-dimensional hyperspheres in a compressible Newtonian fluid at zero

ow-Reynolds-number flow; Effective properties; Two-phase material

2. Analysis

InteA.S. Khair / Journal of Colloid and

fact that the only vector present is r , one immediately concludesthat the (harmonic) disturbance pressure ps is identically zero.Thus, the disturbance flow us is also a (vector) harmonic func-tion and is given by

us = 1ne

(a

r

)nr,

where r = |r|. Note, in one dimension (n = 1) the disturbanceflow is spatially constant, as required by the incompressibilitycondition, us = 0.

To calculate the effective bulk viscosity of a dilute suspen-sion of hyperspheres we form the volume average of the Cauchystress tensor (Brady et al. [1]) to obtain

(2) = pthf I + 2e +( 2

n

) uI + NS,

where . . . denotes an average over the entire dispersion (par-ticles plus fluid), . . .f is an average over the fluid phase only,N is the particle number density, and the average extra particlestress is a number average defined by S = (1/N)N=1 S ,where the contribution from particle is given by

(3)S = 12

S

(r n + nr)dS,

with n the unit normal pointing out of the particle. (Note,Eq. (3) is applicable to rigid particles only; a more generalexpression for the extra particle stressvalid for drops andbubbles in addition to rigid particlesis given by Eq. (3.1) ofBrady et al. [1].)

The effective bulk viscosity, eff, relates the deviation of thetrace of the average stress from its equilibrium (e = 0) value tothe average rate of expansion e, namely,

(4)effe = 1n

(ii eqii ),where the summation convention is applied to repeated indices.The trace of the average stress is

(5)ii = n(e pthf

)+ NSii.The trace of average extra particle stress is calculated as

Sii =

riij nj dS

=[(e pth)a + 2(n 1)

nea

]dSrface Science 302 (2006) 702703 703

(6)=[(e pth)a + 2(n 1)

nea

]an1 2

n/2

(n/2),

where (z) is the Gamma function. Thus, from (5) we have(7)ii = n(e pth) + 2(n 1)e,

where we have used e = e(1 ) and pthf = pth(1 ),and the volume fraction of hyperspheres is defined by

(8) = an

n

2n/2

(n/2)N .

From (4), the effective bulk viscosity is found to be

eff =( + 2(n 1)

n

)1

1 (9) + 2(n 1)

n as 0.

The case n = 3 gives a correction of 43, as reported by Bradyet al. [1]. And for n = 2two-dimensional rigid cylindersthecorrection is . In one dimension, n = 1, the spatially uni-form disturbance flow does not generate any viscous stresses inthe fluid; consequently, the correction is zero. Interestingly, asn the correction approaches a limiting value of 2; incontrast to the Einstein correction for the shear viscosity, whichgrows as n for large n (Brady [4]). A limiting value existsfor the bulk viscosity correction for the following reason. The(constant) expansion rate e is (equally) distributed over an in-creasing number of spatial dimensions; hence, the expansionrate per dimension is O(1/n). On the other hand, the extra par-ticle stress scales as O(n), for large n (recall, the particle stressis proportional to the disturbance velocity gradient, which in-creases with increasing n owing to the more rapid decay of thedisturbance velocity with distance). Thus, from (4), their prod-uct gives a eff that is independent of n as n .Acknowledgment

The author thanks Dr. John F. Brady for fruitful discussionsand critical reading of the manuscript.

References

[1] J.F. Brady, A.S. Khair, M. Swaroop, J. Fluid Mech. 554 (2006) 109.[2] A. Einstein, Ann. Phys. 19 (1906) 289.[3] G.K. Batchelor, An Introduction to Fluid Dynamics, Cambridge Univ.

Press, Cambridge, 1973.[4] J.F. Brady, Int. J. Multiphase Flow 10 (1984) 113.

The `Einstein correction' to the bulk viscosity in n dimensionsIntroductionAnalysisAcknowledgmentReferences