Papers in Physics, vol. 6, art. 060011 (2014)

Received: 16 October 2014, Accepted: 16 October 2014
Edited by: L. A. Pugnaloni
Licence: Creative Commons Attribution 3.0
DOI: http://dx.doi.org/10.4279/PIP.060011

www.papersinphysics.org

ISSN 1852-4249

Reply to the Commentary on “Granular discharge rate for submerged
hoppers”

T. J. Wilson,1, 2 C. R. Pfeifer,1, 3 N. Meysingier,1, 4 D. J. Durian1∗

The commentary of Staron [1] raises interesting
points and helps frame additional lines of research.
Based on her initial report, we have made sev-
eral helpful improvements and clarifications to our
manuscript [2]. Below, we respond to some of the
remaining issues.

First, Staron is absolutely correct in suggesting
that we do not have a clear definitive explanation
for the “surge” effect shown in Fig. 1, where the
discharge rate increases as filling height decreases
toward zero. The interstitial fluid clearly plays a
role, but there are many possibilities. It could be
due to suction into the space between grains as
they move apart near the exit, as Staron suggests.
It could also be due to a reduction in grain–grain
or grain–wall friction, or due to lubrication effects
when grains come together. We hope that on-going
experiments [3] on trends versus system size, etc.,
will help shed light on this issue. A terminal surge
has now been seen for dry grains in air [3], so the
effect is of broader interest. Perhaps in vacuum it
would disappear altogether. In any case, theoreti-
cal and simulation input would be helpful.

∗Email: djdurian@physics.upenn.edu

1 Department of Physics & Astronomy, University of Penn-
sylvania, Philadelphia, PA 19104-6396, USA.

2 Department of Physics, Illinois Wesleyan University,
Bloomington, IL 61702-2900, USA.

3 Department of Physics & Astronomy, Carleton College,
Northfield, MN 55057, USA.

4 Strath Haven High School, Wallingford, PA 19086, USA.

As for the Beverloo scaling, even for dry cohe-
sionless grains, we sympathize with Staron’s point
that intermittent formation and break-up of force
chain arches over the outlet is hard to imagine for
very large apertures —where a continuum model
would be more natural. We all agree that the
Janssen argument for saturation of pressure versus
depth does not apply, or else discharge rates would
grow with hopper diameter. This has been shown
directly by recent experiments with a conveyor belt
[4]. Therefore, some sort of shielding of grain–grain
pressure over the outlet is needed, even if transient
arches do not form. How exactly the grain pres-
sure behaves near the outlet during flow seems like
a fruitful topic for study, perhaps along the lines
suggested by Staron and Fig. 1 of her comment.

Finally, regarding the exit speed for grains in
the submerged case, we did indeed demonstrate a
Beverloo-like scaling simply by replacing the free-
fall speed

√
gD for the dry case with a single-grain

terminal falling speed for the submerged case. As a
minor remark, the terminal speed is not simply the
Stokes speed, which holds only for small grains and
small Reynolds numbers Re. We varied the grain
size over a large enough range that the fluid drag
crossed over from viscous at low Re to inertial at
high Re. The latter is for large grains, in which case
the fluid drag scales as speed-squared and the inter-
stitial fluid flow must be somewhat turbulent. Nev-
ertheless, the modified Beverloo-like scaling holds
across both regimes, with the same small-hole cut-
off (which is significantly larger than in the dry

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Papers in Physics, vol. 6, art. 060011 (2014) / T. J. Wilson et al.

case). Thus, it may not be necessary to grapple
with the full complexities of the interstitial fluid
flow. Further experiments would be helpful, both
to investigate initial transients as Staron suggests
and also to systematically vary an imposed down-
or up-flow of fluid through the packing along the
lines illustrated in Fig. 1d of our manuscript. The
reason for a small-hole cutoff must involve dynam-
ics, and it would be interesting to know how the
size depends on fluid properties.

[1] L Staron, Commentary on “Granular dis-
charge rate for submerged hoppers”, Pap.
Phys. 6, 060010 (2014).

[2] T J Wilson, C R Pfeifer, N Meysingier, D J
Durian, Granular discharge rate for submerged
hoppers, Pap. Phys. 6, 060009 (2014).

[3] J Koivisto, D J Durian, unpublished (2014).

[4] M A Aguirre, J G Grande, A Calvo, L A Pug-
naloni, J-C Géminard, Pressure independence
of granular flow through an aperture, Phys.
Rev. Lett. 104, 238002 (2010).

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