Chapter 9 Viscous flow

14 CHAPTER 9. VISCOUS FLOW
where µ eff is the effective viscosity of the suspension, and 〈〉 is a volume
average over the entire volume of the suspension.
The volume averaged symmetric traceless part of the stress tensor can be
expressed as
〈τ ij 〉 = 1 ∫
dV τ ij
V V suspension
⎛
⎞
= 1 ∫
∫
⎝ dV (2µE ij ) + dV (τ ij − 2µE ij ) ⎠
V V suspension
V suspension
= 2µ〈E ij 〉 + 1 ∫
dV (τ ij − 2µE ij ) (9.47)
V V suspension
The second term on the right side of 9.47 is identically equal to zero for the
fluid, from the constitutive equation. Therefore, the second integral reduces
to an integral over the particles,
〈τ ij 〉 = 2µ〈E ij 〉 + 1 ∫
dV (τ ij − 2µE ij ) (9.48)
V V particles
In order to determine the effective viscosity, it is sufficient to consider the
symmetric traceless part of the above equation. Further, if the particles are
solid, the rate of deformation E ij within the particles is identically equal to
zero. Therefore, the symmetric traceless part of the above equation reduces
to
〈τ ij 〉 = 2µ〈E ij 〉 + N ∫
dV (τ ij − 2µE ij ) (9.49)
V V 1 particles
= 2µ〈E ij 〉 + N ∫
dV τ ij (9.50)
V V 1 particle
∫
φ
= 2µ〈E ij 〉 +
dV τ
(4πR 3 ij (9.51)
/3) V 1 particle
In deriving equation 9.51 from 9.50, we have used the simplification that the
number of particles per unit volume is the ratio of the volume fraction and
the volume of a particle. The second term on the right side of equation 9.51
can be simplified as follows. Consider the divergence of (τ il x j ),
∂τ il x j
∂x l
= τ il δ lj + x j
∂τ il
∂x l
(9.52)