Bulk Density
Content
Journal of Colloid and Interface Science 302 (2006) 702–703
www.elsevier.com/locate/jcis
Note
The ‘Einstein correction’ to the bulk viscosity in n dimensions
Aditya S. Khair
Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, CA 91125, USA
Received 16 June 2006; accepted 30 July 2006
Available online 2 August 2006
Abstract
We calculate the effective bulk viscosity of a dilute dispersion of rigid n-dimensional hyperspheres in a compressible Newtonian fluid at zero
Reynolds number.
© 2006 Elsevier Inc. All rights reserved.
Keywords: Colloidal dispersion; Bulk viscosity; Hypersphere; Compressible fluid; Low-Reynolds-number flow; Effective properties; Two-phase material
1. Introduction
Recently, Brady et al. [1] computed the bulk viscosity (also
known as the second or volume viscosity) of a dilute colloidal
dispersion to O(φ 2 ) in the volume fraction φ of the (rigid spherical) suspended particles. This calculation requires determination of the dispersion microstructure, which reflects a balance
between an imposed uniform expansion flow and Brownian diffusion of the particles. The O(φ 2 ) contribution to the bulk viscosity arises from two-particle interactions, whereas the O(φ)
contribution is due to the disturbance flow generated by a single particle in the expansion flow. We call the O(φ) contribution
the ‘Einstein correction,’ in homage to Einstein [2] who computed the O(φ) contribution to the shear viscosity of a dilute
colloidal dispersion of rigid spheres. In this Note, we compute
the Einstein correction to the bulk viscosity for a dispersion of
n-dimensional hyperspheres. The correction for n = 2 represents the main outcome of this work; it gives the bulk viscosity
for a dilute dispersion of two-dimensional rigid cylinders—
a result of practical significance. Furthermore, the general ndimensional problem provides a useful exercise for students of
transport processes and fluid dynamics, by introducing basic
concepts in suspension mechanics, while, at the same time, offering a glimpse into an active research area.
E-mail address: [email protected].
0021-9797/$ – see front matter © 2006 Elsevier Inc. All rights reserved.
doi:10.1016/j.jcis.2006.07.076
2. Analysis
We begin with the constitutive equation for the stress σ in a
n-dimensional compressible Newtonian fluid (see, e.g., Batchelor [3]), viz.
2
σ = −pth I + 2ηe + κ − η (∇ · u)I ,
(1)
n
where pth is the thermodynamic pressure (as defined by the
fluid’s equation of state), e = 12 [∇u + (∇u)† ] is the rate of
strain tensor with u the fluid velocity, η is the shear viscosity,
κ is the bulk viscosity, and I is the identity tensor. In zeroReynolds-number flow the velocity field can be decomposed
into an uniform expansion everywhere in the fluid and a disturbance flow created by any immersed particles, which satisfies
the usual incompressible Stokes equations; that is,
1
u = er + us ,
n
where r is the position vector, e is the expansion rate, ∇ · u = e,
and ∇ · us = 0. In turn, the fluid stress σ can be split into a contribution due to the uniform expansion flow, σ e = (κe − pth )I ,
and a disturbance stress σ s = −p s I + 2ηes , with ∇ · σ s = 0.
Here, p s is the dynamical pressure field of the incompressible
Stokes flow.
Consider a force- and torque-free n-dimensional hypersphere of radius a immersed in the uniform expansion flow.
Exploiting the linearity of the Stokes equations, and noting the
A.S. Khair / Journal of Colloid and Interface Science 302 (2006) 702–703
fact that the only vector present is r, one immediately concludes
that the (harmonic) disturbance pressure p s is identically zero.
Thus, the disturbance flow us is also a (vector) harmonic function and is given by
1 a n
us = − e
r,
n r
where r = |r|. Note, in one dimension (n = 1) the disturbance
flow is spatially constant, as required by the incompressibility
condition, ∇ · us = 0.
To calculate the effective bulk viscosity of a dilute suspension of hyperspheres we form the volume average of the Cauchy
stress tensor σ (Brady et al. [1]) to obtain
2
¯
σ = −pth f I + 2ηe + κ − η ∇ · uI + NS,
(2)
n
where . . . denotes an average over the entire dispersion (particles plus fluid), . . .f is an average over the fluid phase only,
N¯ is the particle number density, and the average extra
particle
stress is a number average defined by S = (1/N) N
α=1 S α ,
where the contribution from particle α is given by
1
Sα =
(3)
(rσ · n + σ · nr) dSα ,
2
∂Sα
with n the unit normal pointing out of the particle. (Note,
Eq. (3) is applicable to rigid particles only; a more general
expression for the extra particle stress—valid for drops and
bubbles in addition to rigid particles—is given by Eq. (3.1) of
Brady et al. [1].)
The effective bulk viscosity, κeff , relates the deviation of the
trace of the average stress from its equilibrium (e = 0) value to
the average rate of expansion e, namely,
eq
1
σii − σii ,
(4)
n
where the summation convention is applied to repeated indices.
The trace of the average stress is
¯ ii .
σii = n κe − pth f + NS
(5)
κeff e =
The trace of average extra particle stress is calculated as
Sii = ri σij nj dS
2(n − 1)
dS
ηea
= (κe − pth )a +
n
703
2π n/2
2(n − 1)
= (κe − pth )a +
ηea a n−1
,
n
Γ (n/2)
(6)
where Γ (z) is the Gamma function. Thus, from (5) we have
σii = n(κe − pth ) + 2(n − 1)ηeφ,
(7)
where we have used e = e(1 − φ) and pth f = pth (1 − φ),
and the volume fraction of hyperspheres is defined by
φ=
a n 2π n/2 ¯
N.
n Γ (n/2)
From (4), the effective bulk viscosity is found to be
2(n − 1)
1
κeff = κ +
ηφ
n
1−φ
2(n − 1)
∼κ +
ηφ as φ → 0.
n
(8)
(9)
The case n = 3 gives a correction of 43 ηφ, as reported by Brady
et al. [1]. And for n = 2—two-dimensional rigid cylinders—the
correction is ηφ. In one dimension, n = 1, the spatially uniform disturbance flow does not generate any viscous stresses in
the fluid; consequently, the correction is zero. Interestingly, as
n → ∞ the correction approaches a limiting value of 2ηφ; in
contrast to the Einstein correction for the shear viscosity, which
grows as nηφ for large n (Brady [4]). A limiting value exists
for the bulk viscosity correction for the following reason. The
(constant) expansion rate e is (equally) distributed over an increasing number of spatial dimensions; hence, the expansion
rate per dimension is O(1/n). On the other hand, the extra particle stress scales as O(n), for large n (recall, the particle stress
is proportional to the disturbance velocity gradient, which increases with increasing n owing to the more rapid decay of the
disturbance velocity with distance). Thus, from (4), their product gives a κeff that is independent of n as n → ∞.
Acknowledgment
The author thanks Dr. John F. Brady for fruitful discussions
and 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.
www.elsevier.com/locate/jcis
Note
The ‘Einstein correction’ to the bulk viscosity in n dimensions
Aditya S. Khair
Division of Chemistry and Chemical Engineering, California Institute of Technology, Pasadena, CA 91125, USA
Received 16 June 2006; accepted 30 July 2006
Available online 2 August 2006
Abstract
We calculate the effective bulk viscosity of a dilute dispersion of rigid n-dimensional hyperspheres in a compressible Newtonian fluid at zero
Reynolds number.
© 2006 Elsevier Inc. All rights reserved.
Keywords: Colloidal dispersion; Bulk viscosity; Hypersphere; Compressible fluid; Low-Reynolds-number flow; Effective properties; Two-phase material
1. Introduction
Recently, Brady et al. [1] computed the bulk viscosity (also
known as the second or volume viscosity) of a dilute colloidal
dispersion to O(φ 2 ) in the volume fraction φ of the (rigid spherical) suspended particles. This calculation requires determination of the dispersion microstructure, which reflects a balance
between an imposed uniform expansion flow and Brownian diffusion of the particles. The O(φ 2 ) contribution to the bulk viscosity arises from two-particle interactions, whereas the O(φ)
contribution is due to the disturbance flow generated by a single particle in the expansion flow. We call the O(φ) contribution
the ‘Einstein correction,’ in homage to Einstein [2] who computed the O(φ) contribution to the shear viscosity of a dilute
colloidal dispersion of rigid spheres. In this Note, we compute
the Einstein correction to the bulk viscosity for a dispersion of
n-dimensional hyperspheres. The correction for n = 2 represents the main outcome of this work; it gives the bulk viscosity
for a dilute dispersion of two-dimensional rigid cylinders—
a result of practical significance. Furthermore, the general ndimensional problem provides a useful exercise for students of
transport processes and fluid dynamics, by introducing basic
concepts in suspension mechanics, while, at the same time, offering a glimpse into an active research area.
E-mail address: [email protected].
0021-9797/$ – see front matter © 2006 Elsevier Inc. All rights reserved.
doi:10.1016/j.jcis.2006.07.076
2. Analysis
We begin with the constitutive equation for the stress σ in a
n-dimensional compressible Newtonian fluid (see, e.g., Batchelor [3]), viz.
2
σ = −pth I + 2ηe + κ − η (∇ · u)I ,
(1)
n
where pth is the thermodynamic pressure (as defined by the
fluid’s equation of state), e = 12 [∇u + (∇u)† ] is the rate of
strain tensor with u the fluid velocity, η is the shear viscosity,
κ is the bulk viscosity, and I is the identity tensor. In zeroReynolds-number flow the velocity field can be decomposed
into an uniform expansion everywhere in the fluid and a disturbance flow created by any immersed particles, which satisfies
the usual incompressible Stokes equations; that is,
1
u = er + us ,
n
where r is the position vector, e is the expansion rate, ∇ · u = e,
and ∇ · us = 0. In turn, the fluid stress σ can be split into a contribution due to the uniform expansion flow, σ e = (κe − pth )I ,
and a disturbance stress σ s = −p s I + 2ηes , with ∇ · σ s = 0.
Here, p s is the dynamical pressure field of the incompressible
Stokes flow.
Consider a force- and torque-free n-dimensional hypersphere of radius a immersed in the uniform expansion flow.
Exploiting the linearity of the Stokes equations, and noting the
A.S. Khair / Journal of Colloid and Interface Science 302 (2006) 702–703
fact that the only vector present is r, one immediately concludes
that the (harmonic) disturbance pressure p s is identically zero.
Thus, the disturbance flow us is also a (vector) harmonic function and is given by
1 a n
us = − e
r,
n r
where r = |r|. Note, in one dimension (n = 1) the disturbance
flow is spatially constant, as required by the incompressibility
condition, ∇ · us = 0.
To calculate the effective bulk viscosity of a dilute suspension of hyperspheres we form the volume average of the Cauchy
stress tensor σ (Brady et al. [1]) to obtain
2
¯
σ = −pth f I + 2ηe + κ − η ∇ · uI + NS,
(2)
n
where . . . denotes an average over the entire dispersion (particles plus fluid), . . .f is an average over the fluid phase only,
N¯ is the particle number density, and the average extra
particle
stress is a number average defined by S = (1/N) N
α=1 S α ,
where the contribution from particle α is given by
1
Sα =
(3)
(rσ · n + σ · nr) dSα ,
2
∂Sα
with n the unit normal pointing out of the particle. (Note,
Eq. (3) is applicable to rigid particles only; a more general
expression for the extra particle stress—valid for drops and
bubbles in addition to rigid particles—is given by Eq. (3.1) of
Brady et al. [1].)
The effective bulk viscosity, κeff , relates the deviation of the
trace of the average stress from its equilibrium (e = 0) value to
the average rate of expansion e, namely,
eq
1
σii − σii ,
(4)
n
where the summation convention is applied to repeated indices.
The trace of the average stress is
¯ ii .
σii = n κe − pth f + NS
(5)
κeff e =
The trace of average extra particle stress is calculated as
Sii = ri σij nj dS
2(n − 1)
dS
ηea
= (κe − pth )a +
n
703
2π n/2
2(n − 1)
= (κe − pth )a +
ηea a n−1
,
n
Γ (n/2)
(6)
where Γ (z) is the Gamma function. Thus, from (5) we have
σii = n(κe − pth ) + 2(n − 1)ηeφ,
(7)
where we have used e = e(1 − φ) and pth f = pth (1 − φ),
and the volume fraction of hyperspheres is defined by
φ=
a n 2π n/2 ¯
N.
n Γ (n/2)
From (4), the effective bulk viscosity is found to be
2(n − 1)
1
κeff = κ +
ηφ
n
1−φ
2(n − 1)
∼κ +
ηφ as φ → 0.
n
(8)
(9)
The case n = 3 gives a correction of 43 ηφ, as reported by Brady
et al. [1]. And for n = 2—two-dimensional rigid cylinders—the
correction is ηφ. In one dimension, n = 1, the spatially uniform disturbance flow does not generate any viscous stresses in
the fluid; consequently, the correction is zero. Interestingly, as
n → ∞ the correction approaches a limiting value of 2ηφ; in
contrast to the Einstein correction for the shear viscosity, which
grows as nηφ for large n (Brady [4]). A limiting value exists
for the bulk viscosity correction for the following reason. The
(constant) expansion rate e is (equally) distributed over an increasing number of spatial dimensions; hence, the expansion
rate per dimension is O(1/n). On the other hand, the extra particle stress scales as O(n), for large n (recall, the particle stress
is proportional to the disturbance velocity gradient, which increases with increasing n owing to the more rapid decay of the
disturbance velocity with distance). Thus, from (4), their product gives a κeff that is independent of n as n → ∞.
Acknowledgment
The author thanks Dr. John F. Brady for fruitful discussions
and 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.
