Consider a homogeneous suspension of spherical particles with number densitynin a compressible Newtonian fluid of densityρ, shear viscosityηand bulk viscosityκ. The particles are small enough so that the Reynolds numberRe = ρUa/η(withU being a typical velocity andabeing the radius of the particles) is much less than unity, thus enabling the use of Stokes equations. The fluid is made to expand everywhere in space at a uniform ratee(= ∇ ·u). The imposed flow will cause the suspended particles to move apart but they cannot expand with the fluid because they are rigid.
There will be a disturbance flow as the fluid has to move around the particles to compensate for their rigidity and this disturbance flow will cause the stress on the particles to change, thereby affecting
the bulk stress in the suspension. A uniform compressive flow (negative expansion rate) could be assumed as well without affecting the following derivation.
The bulk viscosity of the suspension is determined by computing the average stress in the mate- rial in a way analogous to that for the shear viscosity [Batchelor and Green 1972b; Brady and Bossis 1988; Bradyet al.2006]. The ensemble or volume average of the Cauchy stressσin the material is given by
hσi=− hpthif I+2ηhei+ κ− 23η
h∇ ·uiI+nhSHi, (2.3) whereeis the rate of strain in the fluid,h. . .idenotes an average over the entire suspension (particles plus fluid), and h. . .if denotes an average over the fluid phase only. The average hydrodynamic stresslet is defined as a number average over all particles by hSHi = (1/N)PN
α=1SHα, where the stresslet of particleαis given by
SHα = 12 Z
Sα
h(rσ·n+σ·nr)−2 κ− 23η
(n·u)I−2η(un+nu)i
dS. (2.4)
The particle stresslet is the symmetric part of the first moment of the surface stress on the particle.
The integral is over the surface of particleαwith normalnpointing into the fluid andris the spatial vector from the center of the particle to a point on its surface. In writing (2.4) it has been assumed that there is no net force on the particle; an assumption that is relaxed below.
The stress resulting from Brownian motion of the particles as well as an interparticle-force contribution−nhxFPimust also be added to the bulk stress. In the present study a simple hard- sphere interparticle potential is assumed that keeps the particles from overlapping. Inertial Reynolds stresses could also be added, but since our interest is in low-Reynolds-number flows only, we ignore the inertial part. The final form of the bulk stress can be written as
hΣi=− hpthif I+2ηhei+ κ− 23η
h∇ ·uiI−nkTI+n[hSBi+hSPi+hSEi], (2.5)
where it is assumed that there are no external couples on the particles [Brady 1993a]. The averaged
particle stresslets can be expressed in terms of hydrodynamic resistance functions as
hSBi = −kTD
∇ ·RS U·R−1FUE , hSPi = −D
RS U·R−1FU+xI
·FPE , hSEi = −D
RS U ·R−1FU·RFE−RS E
E:hei, (2.6)
where the derivative is with respect to the last index of the inverse of the resistance matrixR−1FUand FPis the colloidal interparticle force.
The fluid velocity resulting from a uniform rate of expansion can be decomposed into a uniform expansion and a disturbance (Stokes) velocity
u= 13er+us, (2.7)
such that
∇ ·u=e and ∇ ·us=0.
The disturbance flow created by the particles and the resulting fluid mechanical interactions sat- isfy the usual incompressible equations of motion. The fluid stress associated with the uniform expansion flow is
σe=−(pth−κe)I, (2.8)
while the disturbance stress is given by
σs=−psI+2ηes and satisfies ∇ ·σs=0, (2.9)
where ps is the dynamical pressure distribution associated with the incompressible disturbance Stokes flow. Note that since the disturbance flow (us,σs) satisfies the incompressible Stokes equa- tion, the hydrodynamic interaction functions in (2.6), e.g. RFU, are the usual ones for incompress- ible flow.
The suspension as a whole has a uniform average rate of expansionhei ≡ h∇ ·ui, where the averaging is done over the fluid and the particles; for rigid particles hei = (1−φ)e. The bulk
viscosity is the scalar coefficient that when multiplied with the average rate of expansion gives the difference between the mean suspension stress and the equilibrium stress. The suspension will be in equilibrium whenhei ≡0 and in that case, the bulk stress is given by
hΣieq =− hpthieq
f + Π
I, (2.10)
whereΠis the osmotic pressure:
Π =nkT −13n[hSBieq+hSPieq], (2.11)
and S denotes the trace of the corresponding stresslet, as in hSBieq = 13hSBieqI, and the super- scripteqdenotes an average over the equilibrium distribution of the suspension microstructure. The effective bulk viscosityκe f f is therefore given by
κe f f ≡κ+
−hpthif +hpthieqf
/hei+ 13n[(hSBi − hSBieq)+(hSPi − hSPieq)+hSEi]/hei. (2.12)
Equation (2.12) together with equation (2.6) gives the general expressions which can be used to calculate the effective bulk viscosity of a suspension for all concentrations and expansion or com- pression rates. The Brownian and interparticle force contributions arise from interactions between at least two particles and therefore contributeO(φ2) to the bulk viscosity. The stresslet due to the imposed rate-of-strainhSEi is nonzero for a single particle and therefore contributesO(φ) to the bulk viscosity.
TheO(φ) contribution to bulk viscosity arises from the disturbance flow induced by the presence of a single particle suspended in the uniform expansion flow. Since the particle cannot expand with the fluid, the no-slip condition on its surface causes a disturbance flow:
us=−13ea3 r3r. The particle stresslet from (2.4) is
SE =4πa3
−pth+κe+ 43eη ,
and the effective bulk viscosity from (2.12) is to first order inφ: κe f f =
κ+43ηφ 1 1−φ,
∼ κ+ κ+43η
φ as φ→0, (2.13)
The 43ηφterm corresponds to the ‘Einstein’ correction to the bulk viscosity for a dilute suspension of rigid spheres [Bradyet al.2006]. The factor of 1/(1−φ) represents the difference between the fluid and the bulk’s rate of expansion. The correction to the bulk viscosity is proportional to the shear viscosityηand therefore may not be negligible even for very dilute suspensions, depending on the magnitude of the fluid’s bulk viscosityκin comparison toη.
The same formulation can be used to recover G. I. Taylor’s result for the bulk viscosity of a dilute suspension of bubbles. In this case the suspending fluid is incompressible and the bubbles are compressible. Consider a single bubble of radiusa, bulk viscosityκp, and zero shear viscosity in an incompressible fluid expanding uniformly with rateep. Since only the volume occupied by the bubble is expanding the average rate of expansion in the dispersion is hei = epφ. The expanding bubble surface creates an incompressible disturbance flow in the surrounding fluid
us= 13ep a3/r3
r. (2.14)
In contrast to the rigid particles problem the bubbles cause the pressure in the surrounding fluid to change from its equilibrium value. The fluid pressure is determined through a normal stress balance on the surface of the bubble, (σp−σ)·n=0 (neglecting surface tension effects, which can be added but do not alter the final result anyway), and is given by
pth= pp−κpep− 43ηep, (2.15)
where ppis the equilibrium pressure inside the bubble and equal to the equilibrium pressure in the surrounding fluid neglecting the capillary pressure due to surface tension. The resulting stresslet on the bubble is
SE =4πa3
−pp+κpep+ 43ηep
, (2.16)
and from (4.10) the effective bulk viscosity for the dispersion is found to be κe f f = h
κpφ+ 43η(1−φ)i ep
hei (2.17)
= h
κpφ+ 43η(1−φ)i1
φ (2.18)
∼ κp+43η/φ as φ→0. (2.19)
Thus we get the correction to the bulk viscosity as 43η/φ, a result first derived by Taylor [1954a,b].
The inverse dependence onφis unusual and entails further comment. The smallO(φ) concentra- tion of bubbles produces a small rate of expansion (hei= φep) throughout the dispersion, however the pressure field is perturbed over the entire surrounding fluid, which occupies a large O(1−φ) fraction of the total volume. Thus the dominant contribution comes from the fluid pressure term −hpthif +hpthieqf
/hei in (4.10), which is ofO((1−φ)/φ) ∼ O(1/φ) asφ → 0. In contrast the change in the stresslet exerted by a rigid particle is localized to its surface and therefore contributes O(φ) to the total stress, while the expanding fluid gives an average rate of expansion ofO(1−φ), resulting in a correction to the bulk viscosity ofO(φ) asφ→0. Both problems do have the same co- efficient, namely43ηin the bulk viscosity correction because the disturbance flow in the surrounding fluid is the same in both cases but with opposite sign.