**
Daniel De Kee
**
, Tulane University

Three dimensional steady state flows in a DCCR/SR and a vane rheometer have been numerically simulated. We analyzed and compared the systematic errors in rheological measurements for different test fluids.

Figure 1: Geometries of (a) the vane rheometer and (b) the DCCR/SR.

Figure (1) illustrates the geometries. There are three factors that are responsible for the deviation of the measured apparent viscosity from its actual value: 1) wall slip effects, 2) end effects, and 3) secondary flow effects. We analyze these separately.

We
define the coefficient *¦' _{slip}*
due to wall slip effects as the ratio of the fluid torque in a rheometer with
slip surfaces of the rotor to the torque in a rheometer in the absence of slip:

| (1) |

Inaccurate estimation of the correction length leads to end
effects-related errors. We define
the coefficient *¦' _{end}*
due to end effects by:

| (2) |

Here, *H* is the
rotor length and superscript '*' refers to the correction length used. We define the coefficient *¦' _{2nd}* due to secondary flow
effects as the ratio between the measured and true values of the fluid apparent
viscosity (no slip) and

| (3) |

Here superscript '*' refers to the measured value of the apparent viscosity. The total error due to these effects is:

| (4) |

The constitutive model used in the numerical study of yield stress fluids is a modified Bingham model:

| (5) |

where ¦" is the extra-stress tensor,
*¦"*_{0} is the yield stress, *m,* *t*_{1}
and *¦Ç*_{1} are constants. Commercial CFD software ANSYS Fluent
12.0 (ANSYS,
Inc., Canonsburg, PA) was used.
We compare the two rheometers under the extreme conditions of free slip
and no wall slip.

Figure 2: Accuracy coefficients due to (a) wall slip effects (b) end effects and (c) secondary flow effects of a vane rheometer (solid bars) and DCCR/SR (open bars) for power law fluids with different n indices.

Figure (2) compares the three
accuracy coefficients between the two designs for power law fluids. In Figure (2a), with free slip *¦'*_{slip}
for both designs less than unity, indicates an underprediction of the rotor
torque. Even for *n* = 0.01, the torque measured with a
vane rheometer is only 87.2 % of the corresponding value under no slip
conditions. The *¦'*_{slip}
further decreases for a vane rheometer with an increase in n, reaching a
minimum of 68.8 % for a Newtonian fluid (*n*
= 1). With 60 % slot area ratio,
the *¦'*_{slip}
of the DCCR/SR is smaller than that of a vane rheometer but less dependent on *n* (~ 60%). In Figure (2b), a vane rheometer has a
much larger error due to end effects (*¦'*_{end} ¨C 1) than the DCCR/SR. This result can be explained by the
small thickness of the slotted rotor, leading to a small area of the end
surfaces in the DCCR/SR. For a vane
rheometer, the accuracy coefficient due to end effects varies significantly
with *n*. For Newtonian fluids, the error due to
end effects is -10% for the measurements with a vane rheometer. This error can be eliminated only with
the calibration of the correction length.
For the DCCR/SR design, *¦' _{end}* is ~ 1 and is independent of

Figure 3: Comparison of the total systematic error in apparent viscosity measurement between a vane rheometer (squares) and a DCCR/SR (circles and triangles) for (a) power-law fluids and (b) a yield stress fluid. Negative values indicate the underprediction of the apparent viscosity.

The total error with the two designs
is plotted in Figure (3) for no slip (closed symbols) and free slip conditions. For power law fluids (Figure 3a), the
error generated by a vane rheometer is more dependent on the power law index *n*.
This variation is much smaller for a DCCR/SR. When there are no wall slip effects, the
DCCR/SR will have higher accuracy than a vane rheometer for any power law
fluid. Under free slip conditions,
the DCCR/SR still has higher accuracy than a vane rheometer when the power law index
n > 0.5. For a yield stress
fluid, (no slip conditions), a vane rheometer will have ~ 46% underprediction
error in the low shear stress region; the DCCR/SR with 60 % slot area ratio has
only 10 % underprediction. In the
case of free slip, the DCCR/SR design with 90 % slot area ratio will be more
accurate than a vane rheometer for the whole range of shear rates.

Our results indicate that: (1) a DCCR/SR is able to accurately measure rheological properties of a wider spectrum of test fluids than a vane rheometer due to significant reductions of end and secondary flow effects; (2) the rheometer design can be optimized by analyzing the accuracy coefficients separately which allows us to determine the dominant source of the measurement error and then to provide a solution for reduction/elimination of this source.

**Impact statement:**
in this research we explore a variety of aspects of determining the yield
stress of complex fluids, via numerical analysis. The project involves a Ph.D. student who
is also being trained in CFD (Fluent) and a postdoctoral fellow who recently
obtained an industrial position with Schlumberger.