(110e) Measurement of Rotational Motion in Granular Couette Flow Using Magnetic Resonance Imaging

Granular rheology is complicated
by the dependence of the viscosity upon factors such as solid concentration,
shear rate, granular temperature, and inter-particle friction. Many different
approaches have been developed within the last three decades, including
frictional kinetic theory [1, 2], or non-local granular fluidity models [3].
Testing the validity of these models requires measurements of many different
quantities, including the solid volume fraction, mean velocity, fluctuating
component of velocity (sometimes called the granular temperature), and the
angular velocity variance. Magnetic resonance imaging (MRI) has been used to
make local measurements of the solid fraction, shear rate and velocity
distribution in opaque systems. Here we extend the technique to characterise
the rotational motion of the particles as well.

Granular materials exhibit
behaviour resembling all of the three main states of matter, such as yielding
in solids, gravity flow in liquids and compressibility akin to gases. These
features are the result of the contacts between granules at the scale of the
particle diameter. In order to understand industrial and natural granular
flows, the dynamics of individual granules must be understood. Discrete element
method (DEM) simulations of granular flows are useful since the motions of
individual particles are tracked, providing detailed information about the flow
characteristics [4, 5]. However, DEM simulations are computationally expensive
and are limited to simulating laboratory-scale systems. Therefore, a continuum
description for granular materials is necessary to assist in the design of
industrial processes and to understand natural phenomena.

In order to develop a continuum
model, a rheological model is required to determine the stress distribution
within the granular material. In liquids and gases, such a model determines the
stress from the local shear rate of the material and its viscosity. In granular
materials, the relationship is less clear. Models derived from the kinetic theory
of granular flow use an effective viscosity, much like in the liquid or gas
phase. These models were developed assuming relatively dilute granular flows. By
contrast, models derived from dense systems suggest the stress arises primarily
from the confining pressure on the flow. Several experimental and theoretical
studies now suggest that rotational motion of the granules should also be
considered. In comparison to other variables, particle rotation has not been
widely studied, in part due to the scarcity of suitable measurement techniques.

In MRI, pulsed field gradient
(PFG) measurements encode for spin position during the acquisition time. From
the PFG signal, the local mean velocity and the variance of the velocity
fluctuations (i.e. the granular temperature) may be retrieved [6, 7]. Changes
in spin position may be caused by translational motion resulting from external forcing of a system and by inelastic collisions between granules. Conventional analysis of the signal assumes that the granules
behave as point sources and that there is no contribution to the signal from their
rotation. However, spins move along the encoding axis as the particle rotates.
Thus, it is expected that the measured velocity fluctuations will be greater
than that arising from the translational motion alone.

In this work, we propose a
framework by which the angular velocity distribution of a granular material may
be measured using MRI. A model is developed to describe the PFG signal in terms
of the translational and rotational motion of the granules. The apparent
granular temperature measured by MRI, T_app is related to the
true granular temperature,  and the angular
velocity variance,  by:

where R is the granule
radius. The model was validated against discrete element method simulations of
PFG experiments in granular couette flow. The PFG signal acquisition is
simulated by modelling each particle as a cluster of points. A simulated MRI
signal was then found by numerically integrating the point positions with
respect to time. This method offers the advantage of being able to include or
exclude the effect of particle rotation upon the signal. Hence, the true
granular temperature may be separated from the apparent granular temperature.
These simulated measurements were then analysed as in a conventional MRI experiment
to determine the mean velocity and fluctuating component of the velocity.

The mean velocity profile was in
excellent agreement with physical PFG measurements for the same system.
Granular rotation was found to have no effect upon the imaginary component of
the signal and hence the mean velocity. Each spin has a counterpart positioned
on the other side of the granule with an equal and opposite phase. Thus the net
phase for a granule is equal to the phase arising from translational motion
only.

Measurements of the fluctuating
component of the particle velocity were found to exceed the value obtained from
the simulations of the fluctuations of the translational component of motion. Increasing
the inter-particle friction coefficient and coefficient of restitution caused
the rotational and translational components of the granular temperature to
increase, respectively. However, for all contact parameters tested, the
simulated granular temperature arising from translational motion was lower than
the apparent granular temperature of the PFG experiments. The deviation between
the simulation and experimental results could only be removed by including the
effects of particle rotation in the calculation of the simulated granular
temperature.

Rotational motion increases
signal attenuation when measured by MRI, and hence the apparent fluctuating
component of the particle velocity is overestimated. A detailed study of the
influence of rotation on the signal revealed that the contribution from
rotational motion increased with increasing particle radius, as given in the
equation above. The simulation results also confirmed the additive relationship
between the translational and rotational contributions to the apparent granular
temperature. This result provides a means to quantitatively measure the
rotational and translational components of the velocity fluctuations.

The decomposition of the measured
apparent granular temperature into separate translational and rotational
components is not possible using PFG experiments alone. Instead, the true
granular temperature must be measured by another experimental method. In this
paper we consider using modulated gradient spin echo and dynamic magnetic resonance
scattering as possible means of separating both translational and rotational
components of the motion. These experiments will provide completely
non-invasive measurements of particle dynamics in opaque, particulate flows.
Thus, we will show that MRI may be used to fully quantify the granular dynamics
and hence probe rheological models of granular flow experimentally.

References:

1.         Jenkins, J.T. and C. Zhang, Kinetic theory for
identical, frictional, nearly elastic spheres. Physics of Fluids, 2002. 14(3):
p. 1228-1235.

2.         Lun, C.K.K., Kinetic theory for granular flow of
dense, slightly inelastic, slightly rough spheres. Journal of Fluid
Mechanics, 1991. 233: p. 539-559.

3.         Kamrin, K. and G. Koval, Nonlocal constitutive
relation for steady granular flow. Physical Review Letters, 2012. 108(17).

4.         Cundall, P.A. and O.D.L. Strack, A discrete numerical
model for granular assemblies. Geotechnique, 1979. 29(1): p. 47-65.

5.         Kloss, C., et al., Models, algorithms and validation
for opensource DEM and CFD-DEM. Progress in Computational Fluid Dynamics,
2012. 12(2-3): p. 140-152.

6.         Callaghan, P.T., Principles of nuclear magnetic resonance
microscopy. 1991.

7.         Holland, D.J., et al., Spatially resolved measurement
of anisotropic granular temperature in gas-fluidized beds. Powder
Technology, 2008. 182(2): p. 171-181.