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2 Experiments in Pebble Flows
zone, which can be called the form resistance. For the convenience of the discussion, the numerous actual trajectories are averaged to obtain the average streamline.
Accordingly, the form resistance and friction resistance should be transformed to the
equivalent normal force and the equivalent shear force. The former is perpendicular
to the average streamline and the latter is parallel to the average streamline. However, it is the equivalent shear force to directly determine the relative motion between
pebbles but not the equivalent normal force, so the equivalent shear force should be
analyzed.
The equivalent shear force is equal to the sum of the tangential components
of the form resistance and friction resistance along the average streamline. Along
different streamlines, their contribution is different. Near the stagnant zone, because
the average streamline is flat, the contribution of the form resistance will be more
important; however, close to the central region, because the average streamline is
steeper, the contribution of the friction resistance will be more critical.
Because of the existence of the stagnant zone with a rough boundary, it can offer
a much higher resistance force than that in the case of using an arched smooth plank
to replace the rough edge of the stagnant zone. So the stagnant zone considerably
blocks the flowing of the neighbor pebbles in the outer zone, and spread toward the
middle layer by layer, e.g., up to the place where the contribution of the friction
resistance to the equivalent shear force is more important. In this way, the stagnant
zone slows down a large area of pebbles’ flowing, and result in the non-uniformity of
the overall flow field. This is the physical mechanism of the stagnant zone’s influence
on the overall flow field. However, how to determine such a place is related to the
friction coefficient and the distribution of the voids.
When the blocking spreads to the place where the contribution of the friction
resistance to the equivalent shear force is more significant, and the friction coefficient
of particle to particle is small, it is possible that the equivalent shear force is not high
enough to endure the weight of the pebbles in the central region. This produces the
collapse in the place of closing to the central region, just like the landslide. This is
called the internal collapse of the dense pebble packing.
On the other hand, the equivalent shear force is the resistance force for the part
of quicker flowing; meanwhile, it is the driving force for the part of slower flowing.
In other words, if the equivalent shear force is more significant, the resistance to the
quick flow area is more significant. Then the driving force to the slow flow area is
more significant, to lead to a more uniform pebble flow. However, this does not mean
that if the friction coefficient is larger, the pebble flow is more uniform because the
equivalent shear force is related to not only the friction coefficient but also the normal
contact force and distribution of the voids. Under a different distribution of the voids,
both the form resistance (the normal contact force) and the friction resistance are
different; conversely, under a different friction coefficient, the distribution of voids
is also different. Thus, it can be seen that the friction coefficient’s influence on the
overall flow field in the pebble packing is very complicated and presents significant
nonlinearity and strongly coupled effects.
The physical mechanism of mixing is complicated, which is related to the nature
of the quasi-static pebble flow. It can be seen that, in the quasi-static pebble flow,
2 Experiments in Pebble Flows
zone, which can be called the form resistance. For the convenience of the discussion, the numerous actual trajectories are averaged to obtain the average streamline.
Accordingly, the form resistance and friction resistance should be transformed to the
equivalent normal force and the equivalent shear force. The former is perpendicular
to the average streamline and the latter is parallel to the average streamline. However, it is the equivalent shear force to directly determine the relative motion between
pebbles but not the equivalent normal force, so the equivalent shear force should be
analyzed.
The equivalent shear force is equal to the sum of the tangential components
of the form resistance and friction resistance along the average streamline. Along
different streamlines, their contribution is different. Near the stagnant zone, because
the average streamline is flat, the contribution of the form resistance will be more
important; however, close to the central region, because the average streamline is
steeper, the contribution of the friction resistance will be more critical.
Because of the existence of the stagnant zone with a rough boundary, it can offer
a much higher resistance force than that in the case of using an arched smooth plank
to replace the rough edge of the stagnant zone. So the stagnant zone considerably
blocks the flowing of the neighbor pebbles in the outer zone, and spread toward the
middle layer by layer, e.g., up to the place where the contribution of the friction
resistance to the equivalent shear force is more important. In this way, the stagnant
zone slows down a large area of pebbles’ flowing, and result in the non-uniformity of
the overall flow field. This is the physical mechanism of the stagnant zone’s influence
on the overall flow field. However, how to determine such a place is related to the
friction coefficient and the distribution of the voids.
When the blocking spreads to the place where the contribution of the friction
resistance to the equivalent shear force is more significant, and the friction coefficient
of particle to particle is small, it is possible that the equivalent shear force is not high
enough to endure the weight of the pebbles in the central region. This produces the
collapse in the place of closing to the central region, just like the landslide. This is
called the internal collapse of the dense pebble packing.
On the other hand, the equivalent shear force is the resistance force for the part
of quicker flowing; meanwhile, it is the driving force for the part of slower flowing.
In other words, if the equivalent shear force is more significant, the resistance to the
quick flow area is more significant. Then the driving force to the slow flow area is
more significant, to lead to a more uniform pebble flow. However, this does not mean
that if the friction coefficient is larger, the pebble flow is more uniform because the
equivalent shear force is related to not only the friction coefficient but also the normal
contact force and distribution of the voids. Under a different distribution of the voids,
both the form resistance (the normal contact force) and the friction resistance are
different; conversely, under a different friction coefficient, the distribution of voids
is also different. Thus, it can be seen that the friction coefficient’s influence on the
overall flow field in the pebble packing is very complicated and presents significant
nonlinearity and strongly coupled effects.
The physical mechanism of mixing is complicated, which is related to the nature
of the quasi-static pebble flow. It can be seen that, in the quasi-static pebble flow,