sphere collisions and is given by
κP(α)=−3f(2, α)ηφ2b. (3.14)
The single particle contribution to the bulk viscosity is still present. TheO(φ2b) hydrodynamic (κH) and Brownian (κB) contributions are identically zero because they originate from hydrodynamic in- teractions due to the imposed external forcing and Brownian motion, respectively. TheO(φ2b) steady expansion bulk viscosity corresponding toα=0 is given byκP(0)=12ηφ2b. At the other extreme is the high-frequency limit (α→ ∞) where f →0 and so there is no interparticle contribution to the bulk viscosity.
The frequency dependent solution for the microstructure can be broken into its real and imagi- nary parts, from which we obtain the reduced bulk viscosity functions as
κ0r(α)= κ0P(α) κ0P(0) =−1
4<f(2, α)= 1+β
1+2β+2β2, (3.15)
and
κ00r(α)= κ00P(α) κ0P(0) = 1
4=f(2, α)= β
1+2β+2β2, (3.16)
whereβ= √
2α. Plots of the real and imaginary reduced viscosity functions are shown in Figure 3.1 and Figure 3.2 along with the reducedshear viscosity functions derived by Brady [1993b]. In the high frequency limit (α → ∞) both bulk and shear reduced viscosities asymptote to zero with the same dependence onα; only the coefficient is different. In the steady expansion/shear limit (α→0) the real part of both viscosities asymptote to 1 and the imaginary part becomes zero as discussed earlier, but the dependence onαis slower for the bulk viscosity. It is useful to examine the low and high frequency asymptotic limits to understand the nature of the microstructural perturbation in expansion flow and its effect on the bulk viscosity.
A Taylor series expansion of (3.13) for smallαproduces the following asymptotic form of the microstructural deformation at contact:
f(2;α)∼ −4n
1−(2α)1/2+2(2α)3/2−i
(2α)1/2−4α+2(2α)3/2o
+O(α2). (3.17)
The first departure from steady state is O(α1/2) and it is present in both the real and imaginary
parts. Equation (3.18) represents a direct balance between Brownian diffusion and the imposed forcing. The deformation is dominated by Brownian motion except at r ∼ O(α−1/2) and larger where diffusion is balanced by the external forcing. In the case of expansion flow the forcing is radial resulting in concentration gradients that are in the radial direction only. Therefore diffusion due to Brownian motion also takes place only radially. The forcing in expansion/compression flow is therefore monopolar and decays as 1/r, as is evident from the value of f(r, α) = −8/r from equation (3.13) in the limit α → 0. In contrast the microstructural perturbation in shear flow is quadrupolar, decaying as 1/r3, as there is accumulation of particles in the compression axis and depletion of particles in the expansion axis, resulting in Brownian diffusion in the radial as well as tangential directions. Hence the nature of the disturbance in shear flow is fundamentally different from that in expansion flow. This is evident in the Smoluchowski equation for shear flow of hard spheres with no hydrodynamic interactions, given by [Brady 1993b; Cichocki and Felderhof 1991]:
1 r2
∂
∂r r2∂f
∂r
!
−6 f
r2 −iαf =0, (3.18)
∂f
∂r =−2 at r=2, and
f ∼0 as r→ ∞.
The extra term −6f/r2 stems from interparticle interactions perpendicular to the line joining the centers between two particles in the suspension. Equation (3.18) is a modified spherical Bessel differential equation of order 2 after the coordinate transformationz= √
iα, and the exact solution for f(r, α) is
f = 32 3
1 r3
1+z+ 13z2 1+z0+ 49z20+ 19z30
ez0(1−r/2), (3.19)
wherez0 = 2√
iα. Only the hard-sphere stress contributes to the shear viscosity in the absence of hydrodynamics and so the reduced shear viscosity functions depend only on f(2, α), just like for the bulk viscosity. The shear viscosity is given by [Brady 1993b]
ηP = 9
5f(2, α)φ2bη. (3.20)
Expanding f(2, α) in a Taylor series for small values ofαgives the asymptotic form
f(2, α)∼ 4 3
( 1− 16
81(2α)2+ 4
27(2α)5/2−i 2
9(2α)− 4
27(2α)5/2
!)
+O(α3). (3.21)
The first departure from equilibrium is linear inαand is purely imaginary and therefore elastic in nature. The next correction isO(α2) and is purely real. The more dominant response being out-of- phase with the imposed flow is a consequence of the quadrupolar nature of the disturbance in shear flow.
In the infinite frequency limit f = 0 but this solution does not satisfy the no-flux boundary condition at contact. The perturbation about α → ∞ is singular and there is a boundary layer of O(α1/2) around the particle in which diffusion balances the imposed forcing. Rescalingr to get a stretched coordinate for the boundary layery = (r−2)√
iα, and neglecting terms ofO(α−1/2) the Smoluchowski equation becomes simply
d2f/dy2= f, (3.22)
which has the solution
f(r, α)=− 2
√iαe−(r−2)
√iα,
giving
f(2, α)=−(1−i)
√2α. (3.23)
Equation (3.22) is obtained for both expansion flow and shear flow after neglecting theO(α−1/2) terms. On the scale of the boundary layer the surface of the particle appears flat and consequently there is a one-dimensional balance between the oscillatory forcing and Brownian diffusion in both cases. The boundary condition for shear flow is slightly different, giving f(2, α)=(1−i)√
2α. The real and imaginary parts of f(2, α) and hence the shear and bulk viscosities vanish likeα−1/2 as α → ∞. This slow decay withαimplies that the elastic bulk modulus of the suspension given by K0(ω) = K00 +ωκ00(ω) diverges asα1/2 at high frequency; clearly an unphysical result. The hard- sphere potential results in a delta function repulsion force at contact, which is responsible for the divergence because the collision between particles is instantaneous and therefore the stress during
an infinitesimal time-step containing the collision is infinite. The same behavior has been observed for the elastic shear modulusG0(ω) and Brady [1993b] showed that the inclusion of hydrodynamic interactions removes the divergence. In §3.5 we will show that accounting for hydrodynamics has the same effect on the bulk modulus and it reaches a high-frequency plateau asα→ ∞.