Experimental and numerical investigations for vane flow

The aim was to find out if, and if yes subject to what restrictions, the vane could be
used as a viscometer, even for viscoelastic fluids. The rationale of such an attempt is
based on the fact that in all viscometers, which require fluid inertia to be neglected,
a viscometric flow does not exist, no matter how small Re is. Prominent examples
are cone–plate flow and torsional (i. e. plate–plate) flow. Secondary motions always
affect the ideal local flow kinematics. Yet, only at sufficiently large Re–numbers
do these changes in the local velocity field lead to measurable global relationships,
which are used in determining ?. For purely rotational devices this is theM–O relationship.
The fluids used were taken from the class of aqueous polymer–solutions. Three
different types of polymers were used, namely an industrial one (polyacrylamide),
a biopolymer (hydroxypropyl guar) produced industrially by adding polypropylene
to guar gum and the biopolymer xanthan gum with its helical backbone. For
either type of polymer solution three or four different concentrations were used.
Using a Couette viscometer (concentric cylinder, CC) the flow curves were obtained.
It turned out that, depending upon concentration, the flow curves differed quantitatively
but showed qualitatively similar behavior. While a Cross–like model sufficed
for the two biopolymers, a Carreau–Yasuda–like model was required for the polyacrylamide
solutions. Irrespective of these details a master curve allows the flow
curve to be determined for any concentration without actually measuring ?(?? ).
To use the vane as a viscometer requires its characteristic curve (ChC) Ne = Ne(Re)
to be established. To this end various Newtonian fluids (NFs) were used, in our case
various silicon oils of lowmolecular weight. The influence of ? on the ChC is largest
in the creepingmotion regime (CMR), when Ne = c/Re is bound to hold. In our case
up to Re ˜ 10 one is in the CMR, where c = 13.51 was established experimentally.
Since this constant differs rather drastically from the one used in our commercial
CC–viscometer (of similar dimensions as the vane device) it is clear that substantially
different flow fields have to prevail in these two devices.
For non–Newtonian fluids (purely viscous or viscoelastic ones) ? is not a constant
but rather depends upon ??. Thus, ? ref was utilized (in our case the solvent water
was used as the reference fluid) to define a reference Reynolds number Reref.
If Ne Re = c characterizes the CMR then a change of Ne Reref vs Reref to Ne Re vs
Re requires, in a log–log plot a shift along a 45? line. The magnitude of this shift furnishes
?. For each O the function ? = ?(O) can thus be determined. Equating then
?(O) with the viscometrically established flow curve ? = ?(?? ) furnishes a relation
between O and ??. For O small enough (small Re) the relation is linear, i. e. ?? = c? O
prevails. Although c? differs slightly from fluid to fluid, the fluid–independent approximation
c? ˜ 4 (in our case) produces satisfactory results in all cases. It is this
fact which allows the vane to be used as a viscometer.
Having succeeded in showing that the vane can be used as a viscometer there are
limitations. For viscoelastic fluids deviations from NeRe = c show up at higher Re
numbers, even when NeRe = c holds in case of Newtonian fluids. This can be most
clearly seen in a direct comparison between ?(?? ) from viscometric measurements
with ?(O) from vane flow. Elastic effects seem to be responsible for such behavior.
Being non–linear they start to influence the global M–O relation at higher shear–
rates, the more so the more elastic the fluid is. As soon as elasticity affects the global
M–O relation the vane ceases to furnish flow curves. For slightly elastic fluids it
is fluid inertia which limits the vane’s use as a viscometer (as it does in cone–plate
and, respectively in torsional flow).
This gets strengthened from PIV results. Even for NFs these results show clearly
that the streamlines between the blades are not circular. Thus, a rigid body motion
between the blades and a viscometric flow outside the blades does not exist. This,
however, would be the requirement for the fluid’s elasticity to have no effect on the
flow field.
The numerical results were obtained in the true CMR regime, i. e. Re = 0. In this
limit fluid inertia plays no role and the flow can be approximated by a steady one.
This fact can be understood if one uses a rotating coordinate system in which the
vane is at rest. In this system the flow is indeed steady. But Coriolis and centrifugal
forces have to be reckoned with the fact that either one scales with Re implies that
for Re = 0 they play no role. The name inertial forces is quite appropriate. While
the local streamlines and contour lines (lines of constant speed) quite clearly deviate
from the ideal ones (in agreement with the PIV measurements) theM–O relation
shows good agreement with experimental results for small O, with noticeable deviations
at larger O. This does not come unexpected, given the fact that generalized
NFs cannot account for any elasticity of the fluid. All polymer solutions used were
viscoelastic. Thus the conclusion is that the vane can be used as a viscometer, subject
to the limitations that the fluid’s elasticity and/or fluid inertia will sooner or later
limit its use for viscoelastic fluids.