Accepted Manuscript Short Communication The Einstein shear viscosity correction for non no-slip hyperspheres Charles G. Slominski, Andrew M. Kraynik, John F. Brady PII: DOI: Reference:

S0021-9797(14)00358-0 http://dx.doi.org/10.1016/j.jcis.2014.05.052 YJCIS 19604

To appear in:

Journal of Colloid and Interface Science

Received Date: Accepted Date:

15 March 2014 24 May 2014

Please cite this article as: C.G. Slominski, A.M. Kraynik, J.F. Brady, The Einstein shear viscosity correction for non no-slip hyperspheres, Journal of Colloid and Interface Science (2014), doi: http://dx.doi.org/10.1016/j.jcis. 2014.05.052

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The Einstein shear viscosity correction for non no-slip hyperspheres Charles G. Slominski∗, Andrew M. Kraynik, John F. Brady Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, CA 91125, USA

Abstract We calculate the effective shear viscosity of a dilute dispersion of n-dimensional non no-slip hyperspheres, first for hyperdrops, second for slippery hyperspheres, and third for porous hyperspheres. Keywords: colloidal dispersions, Low-Reynolds-number flow, effective properties, drops, Navier slip, Brinkman equations 1. Introduction The ‘Einstein correction’ is the O (φ), in the volume fraction φ, correction to the viscosity for a dilute colloidal dispersion [1]. Brenner [2] was the first to work with the n-dimensional Stokes equations, calculating the drag/torque on a translating/rotating hypersphere, respectively. Taylor [3] and Brady [4] calculated the correction to the shear viscosity for a spherical drop and a rigid hypersphere, respectively. Khair [5] calculated the correction to the bulk viscosity for a rigid hypersphere. We extend the work of Taylor [3] and Brady [4] to a hyperdrop, a slippery hypersphere, and a porous hypersphere. 2. Analysis Consider a force-free, couple-free hypersphere of radius a located at the origin surrounded by a fluid of viscosity η flowing with a pure straining ∗

Corresponding author Email address: [email protected] (Charles G. Slominski)

Preprint submitted to Journal of Colloid and Interface Science

4th June 2014

motion with rate of strain tensor Eij∞ , which is symmetric and traceless. The fluid outside each hypersphere is governed by the Stokes equations η∇2 u − ∇p = 0,

∇ · u = 0.

(1)

For the hyperdrop, the fluid inside is also governed by (1) but with η i replacing η, thus the viscosity ratio is β = η/η i . At the surface of the hyperdrop, the normal velocity vanishes and the tangential stress and velocity are continuous. At the surface of the slippery hypersphere, we impose the Navier-slip condition [6], in which the normal velocity vanishes and the tangential or ‘slip’ velocity is proportional to the strain rate,     λ  = σij ni tj  , ui ti  η r=a r=a where ti is a vector tangent to the surface and λ is the slip length. For the porous hypersphere, the fluid inside is governed by the Brinkman equations η η∇2 u i − ∇pi − u i = 0, k

∇ · u i = 0,

where k is the permeability [7]. Stress and velocity are continuous at the surface. For each hypersphere, the velocity and pressure fields outside are governed by the Stokes equations and take the form ∞ , ui = Eij∞ xj + Vijk Ejk ∞ , p = p∞ + Pjk Ejk

where Eij∞ xj and p∞ are the undisturbed velocity and pressure fields at infinity, and the quantities Vijk and Pjk are purely geometric and take the form   a 2   a n+2 n xi xj xk , (2) + a n+2 −B + C Vijk = Aδij xk r r r xj xk Pjk = −2Bηan n+2 , (3) r where the constants A, B, and C are given in Table 1. For the hyperdrop and the porous hypersphere, the velocity and pressure fields inside the hypersphere are ∞ , uii = Wijk Ejk 2

∞ pi = p∞ + Qjk Ejk ,

where Wijk and Qjk are again purely geometric. For the hyperdrop   β 1 nβ (n + 2) β  r 2 d Wijk = δij xk − − xi xj xk , 2 (β + 1) a 2 (β + 1) β + 1 a2 n (n + 4) xj xk η , 2 (β + 1) a2 and the normal stress jump at the surface is Qdjk =



  4 (n + 1) β + n2 + 2n + 4 ∞ σiji − σij ni nj r=a = − ηEij ni nj . 2 (β + 1)

For the porous hypersphere   √  n       n+2 2 ak2 I n+2 √ak + ak ar 2 I n2 √rk − 2 ak2 ar 2 I n+2 √rk 2 p    2 = (n + 2) δij xk Wijk √

k a k √ √a n n+2 I nI + 1 + 2 (n + 2) 2 a a 2 k k 2   √   n+4 (n + 2) ak ar 2 I n+4 √rk 1 2     2 xi xj xk , − √

k nI n2 √ak + 1 + 2 (n + 2) ak2 I n+2 √ak a a 2   (n + 2) I n+2 √ak xx 2 p    η j 2k , Qjk = − √

k a nI n √a + 1 + 2 (n + 2) k I n+2 √a a

2

a2

k

2

k

where Im is the modified Bessel function of the first kind of order m. To compute the effective shear viscosity, we form the volume average of the stress tensor σij to obtain σij  = I.T. + 2η Eij∞ + np Sij  , (4) where   denotes a volume average, np is the number density of hyperspheres,

α and Sij  = 1/N N α=1 Sij is the average extra particle stress; the sum being over all particles in the volume. I.T. stands for the isotropic term, which enters into the bulk viscosity. The extra particle stress Sij is determined from the flow outside each hypersphere by    1 1 Sij = (σik xj nk + σjk xi nk ) − δij σlk xl nk − η (ui nj + uj ni ) dS 2 n ∞ = 2BVEij , (5) 3

where V is the volume of the hypersphere. From (4) and (5), the effective viscosity arises solely from B, ηeff = η (1 + Bφ) , where the volume fraction φ = np V. In the no-slip limit β, λ/a, k/a2 → 0, the results of Brady [4] are recovered, the inside velocity vanishes for both the hyperdrop and the porous hypersphere, and the inside pressure vanishes for the hyperdrop and is continuous with the outside pressure for the porous hypersphere. In the tangentialstress-free or bubble limit β, λ/a → ∞,

 d  Wijk β→∞

 xi xj xk Vijk β, λ →∞ = −an n+2 , a r  xj xk Pjk β, λ →∞ = −2ηan n+2 , a r     2 1 n n+2 r − 2 xi xj xk , = δij xk − 2 a 2 a  Qdjk β→∞ = 0,

and B = 1 for all dimensions. In the infinite-permeability limit k → ∞, the pure straining motion is recovered everywhere and B vanishes. The trend observable in Fig. 1 is that B decreases most quickly by increasing permeability and least quickly by decreasing the inside viscosity of a hyperdrop; an increasing slip length has an intermediate effect. This trend is most apparent in the infinite-dimension limit n → ∞, depicted in Fig. 2, in which B diverges for β < ∞ and vanishes for k > 0, and for the slippery hypersphere  1a . B n→∞ = 1 + 2λ Lastly, it is straightforward to adapt the solutions for the porous hypersphere to situations in which the viscosity of the fluid in the hypersphere differs from that in the bulk [8]. 3. Acknowledgements This material is based upon work supported by the National Science Foundation Graduate Research Fellowship under Grant No. DGE-1144469. 4

Figure 1: The Einstein shear viscosity correction B in dimensions n = 2, 3, 4, and 8 as a function of the viscosity ratio β = η/η i for the hyperdrop, the Navier-slip length λ/a for the slippery hypersphere, and the permeability k/a2 for the porous hypersphere

5

Figure 2: The Einstein shear viscosity correction B in the infinite-dimension limit as a function of the viscosity ratio β = η/η i for the hyperdrop, the Navier-slip length λ/a for the slippery hypersphere, and the permeability k/a2 for the porous hypersphere

6

Drop

Slip

A

1 − β+1

− (n+2)1 λ +1 a

B

2β+n+2 2(β+1)

Porous √

C

n+2 2(β+1)

2 √ k nI n a 2

(n+2)(2 λ +1) a 2[(n+2) λ +1] a n+2 2[(n+2) λ +1] a

k I n+4 a 2 √a + k

√a k

2 1+2(n+2) k2 a

[

(n+2)I n+2 √

2

k nI n a 2

2

√a k

(n+2) I n+2 √

2

k nI n a 2

2

√a k

√a k

−I n+2

]I n+2 2

√a k

k I n+2 √a a2 k 2 √ −2 ak I n+4 √a k 2

+[1+2(n+2) √a k

√a k

+[1+2(n+2)

]

k a2

]I n+2 2

√a k

Table 1: The coefficients in (2) and (3) where B is the Einstein shear viscosity correction

References [1] A. Einstein, Eine neue Bestimmung der Molek¨ uldimensionen, Ann Phys 19 (1906) 289 and 34 (1911) 591. [2] H. Brenner, The translational and rotational motions of an ndimensional hypersphere through a viscous fluid at small Reynolds numbers, J Fluid Mech 111 (1981) 197. [3] G. I. Taylor, The Viscosity of a Fluid Containing Small Drops of Another Fluid, P Roy Soc A-Math Phy 138 (1932) 41. [4] J. F. Brady, The Einstein viscosity correction in n dimensions, Int J Multiphas Flow 10 (1984) 113. [5] A. S. Khair, The ‘Einstein correction’ to the bulk viscosity in n dimensions, J Colloid Interf Sci 302 (2006) 702. [6] C. L. M. H. Navier, M´emoire sur les lois du mouvement des fluides, M´emoires de l’Acad´emie des sciences de l’Institut de France (1823), 414416. [7] H. C. Brinkman, A Calculation of the Viscous Force Exerted By a Flowing Fluid on a Dense Swarm of Particles, Appl Sci Res A1 (1949) 27. [8] R. C. Gilver, S. A. Altobelli, A determination of the effective viscosity for the Brinkman-Forchheimer flow model, J Fluid Mech 258 (1994), 355-370. 7

x x x x x

The Einstein shear viscosity correction is calculated for three geometries. Velocity and pressure fields are found in a linear straining flow in n dimensions. The Stokes equations are solved in n dimensions. The Brinkman equations are solved in n dimensions. Limits of the results are analyzed.

n=’ ’

B

Drop Slip Porous

1 0 0

ǯ,

ʲ k , a a2

’

The Einstein shear viscosity correction for non no-slip hyperspheres.

We calculate the effective shear viscosity of a dilute dispersion of n-dimensional non no-slip hyperspheres, first for hyperdrops, second for slippery...
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