Fluid mechanics of fibre suspensions related to papermaking - DiVA

3.2. INFLUENCE OF FIBRES AT LOW CONCENTRATION 17
1
0.8
0.6
0.4
1
0.8
0.6
z
0.4
0.2
y 0
0.2
0.4
0.6
0.8
1
C=1
C=0.5
C=0.05
0.4 0.3 0.2 0.1 0 0.1 0.2 0.3 0.4
x
Figure 3.2. The Jeffery orbit for different constants of C, the
pathline of the fibre end.
if an effective aspect ratio re is used.
However, several studies has tried to determine a uniquely defined effective
aspect ratio, but this has not been realized. The values depend on the aspect
ratio regime and shape of the fibre ends. Experimentally Mason & Manley
(1957) estimated that re = 0.7r for aspect ratios r ≈20. Generally, the effective
aspect ratio tend to be less than the actual aspect ratio, which is a results of
that fibres tend to rotate faster than predicted by theory.
3.2. Influence of fibres at low concentration
Theoretical work by Cox (1970) resulted in a model of torque and forces on a
solid, long, slender body in a given undisturbed Stokes flow. In his slender body
theory Cox included interaction (no collisions) between two or more slender
bodies as well the effect of a wall. The force and torque were derived for straight
slender bodies as well as for curved slender bodies. Cox also derived a theory for
the forces acting on flexible fibres and incorporated buckling, which is observed
for flexible bodies in shear flow. In addition, he proposed a modification for
the case of small but finite particle Reynolds number. For this case, the inertia
effects resulted in a modification to the orbit constant C, given by the theory by
Jeffery. Cox applied the model on a shear flow and the result was a rheological
behaviour of the suspension in an efficient viscosity.
Contemporary to Cox’s work Batchelor (1970b) presented a theoretical
model for slender bodies and the contribution to bulk stress. The bulk stress
related to the instantaneous particle orientation, averaged over an ambient fluid
volume and a number of particles (m), is given by
(m) 1
ij =
V
σikxjpkdA (3.5)
A0