∎

11institutetext: Neil Shah 22institutetext: Department of Physics and
Astronomy, Tufts University, 574 Boston Ave., Medford MA
22email<EMAIL_ADDRESS>33institutetext: Brenda Carballo-Ramirez
44institutetext: Nalini Easwar 55institutetext: Department of Physics, Smith
College, Northampton MA 01063
55email<EMAIL_ADDRESS>66institutetext: Sumit Kumar Birwa 77institutetext:
Department of Applied Mathematics and Theoretical Physics, University of
Cambridge, Cambridge CB3 0WA, United Kingdom 88institutetext: International
Centre for Theoretical Sciences, Tata Institute of Fundamental Research,
Shivakote, Bengaluru 560089, India 99institutetext: Shubha Tewari
1010institutetext: Department of Physics, University of Massachusetts,
Amherst, MA 01003
1010email<EMAIL_ADDRESS>
# Effect of wall friction on 2D hopper flow

Neil Shah  Brenda Carballo-Ramirez  Sumit Kumar Birwa  Nalini Easwar  Shubha
Tewari

(Received: date / Accepted: date)

###### Abstract

We report here on experiments and simulations examining the effect of changing
wall friction on the gravity-driven flow of spherical particles in a vertical
hopper. In 2D experiments and simulations, we observe that the exponent of the
expected power-law scaling of mass flow rate with opening size (known as
Beverloo’s law) decreases as the coefficient of friction between particles and
wall increases, whereas Beverloo scaling works as expected in 3D. In our 2D
experiments, we find that wall friction plays the biggest role in a region
near the outlet comparable in height to the largest opening size. However,
wall friction is not the only factor determining a constant rate of flow, as
we observe a near-constant mass outflow rate in the 2D simulations even when
wall friction is set to zero. We show in our simulations that an increase in
wall friction leaves packing fractions relatively unchanged, while average
particle velocities become independent of opening size as the coefficient of
friction increases. We track the spatial pattern of time-averaged particle
velocities and accelerations inside the hopper. We observe that the
hemisphere-like region above the opening where particles begin to accelerate
is largely independent of opening size at finite wall friction. However, the
magnitude of particle accelerations decreases significantly as wall friction
increases, which in turn results in mean sphere velocities that no longer
scale with opening size, consistent with our observations of mass flow rate
scaling. The case of zero wall friction is anomalous, in that most of the
acceleration takes place near the outlet.

###### Keywords:

Granular flow Quasi-2D Friction

##  1 Introduction

The flow of granular materials out of a container under gravity has long been
a topic of both scientific and practical interest. In this paper, we focus on
an age-old observation: the rate of flow of granular materials through a
vertical hopper is constant, independent of column height, until the column is
almost fully drained Nedderman1982Flow . Repeated measurements have shown that
this constant mass flow rate scales as the opening size to a characteristic
power that depends on the dimensionality of the system, a phenomenon labeled
Beverloo’s law Beverloo1961Flow . We perform experiments and simulations in a
quasi-2D hopper that examine the effect of wall friction on gravity-driven
mass flow rates. We find that changing wall friction alters the effective
scaling exponent of the power-law dependence of mass flow rate on opening
size.

At first glance, the idea that wall friction alters how the mass efflux scales
with opening size does not make sense, since the scaling power law is usually
cast in terms of dimensional arguments: $W\propto a^{5/2}(a^{3/2})$ in three
(two) dimensions, where $a$ is the diameter of the opening. A power of 2 in 3D
(1 in 2D) comes from the area (length) of the opening, and the additional
$1/2$ power from the idea that there is a ‘free-fall’ region of height
equivalent to the opening diameter $a$, allowing particles to gain speed
proportional to $a^{1/2}$ as they exit, an idea first proposed by Hagen in
1852 Tighe2007Pressure . Given that the flow rate vanishes for a finite
opening size, Beverloo Beverloo1961Flow  introduced a correction to the
scaling law that accounts for the finite size of the particles as they crowd
the opening, i.e.

$W=W_{0}(a-kd)^{3\over 2}$

in 2D where $W_{0}$ and $k$ are experimentally determined fitting parameters
and $d$ is the particle diameter Beverloo1961Flow . The free-fall region idea
was later refined Hilton2011Granular  to an assumption of a hemispherical arch
that marks the boundary between the region of dense flow and a more dilute
region in which particles lose contact and begin to free-fall. This assumption
allowed a derivation of the Beverloo expression for the flow rate, with a
proportionality constant that is purely geometry-dependent.

There is no clear experimental evidence for the formation of a physical free-
fall arch. Experiments that track particles and contact forces using
photoelastic disks do observe that these arch-like regions occur
intermittently, but only for small openings, and find no direct connection
between arch formation and outflow rate Vivanco2012Dynamical . Rather, it
appears from experiments and simulations that there is a change in kinetic
pressure over a region that scales with the opening size
Rubio2015Disentangling , and the effective acceleration of the exiting
particles increases over this same region in a fashion that also scales with
the opening size. The proximity of the walls also doesn’t appear important -
experiments on granular hoppers have shown that that the mass flow rate is
purely a function of the opening size, independent of the width of the hopper,
as long as the width is at least 2.5 times the size of the outlet
NeddermanBook1992 .

We report here that our 2D experiments and simulations indicate, consistent
with these previous observations, that there is an acceleration region near
the opening. The width of this region (in the horizontal direction) grows with
the size of the opening, but there is little accompanying change in the height
above the opening at which particles first begin to accelerate. While both our
experiments and simulations indicate that friction at the side walls plays
little to no role, we find that increasing friction on the front and back
walls of the quasi-2D hopper strongly impedes the extent to which particles
are able to accelerate. As a result, for high wall friction, sphere velocities
at the outlet no longer scale with opening size, in apparent violation of
Beverloo’s law. We emphasize, though, that this appears to come from the loss
of kinetic energy due to particle collisions with the front and back walls.
Significantly, our 2D experiments find no difference in the flow rates for
hoppers fully lined with a particular frictional backing and hoppers in which
only a region near the opening has that frictional backing. This means the
flow rate is determined almost entirely by the friction in a region near the
outlet. In 3D experiments, however, we observe no significant effect of wall
friction on how the flow rate scales with opening size Smith , and Beverloo’s
law holds.

While the presence of frictional forces is crucial for achieving a constant
rate of flow, the precise form of the frictional interactions needed, as well
as the role of the walls, is less clear. Also not clear is whether these
frictional forces are required to create a condition of constant pressure at
the outlet while grains are flowing out of the hopper. The well-known
observation that grains flow at a constant rate out of a hopper is in stark
contrast with a column of liquid, in which energy conservation dictates that
the outflow rate at the base decreases with the column height because of the
lower potential energy available to the system. The usual explanation for
constant granular flow relies on the assumption of an unchanging net potential
energy at the base, arising from an effective column weight that becomes
independent of column height. It has long been observed that the pressure at
the base of a static column of increasing height saturates over a
characteristic length comparable to the column width, a phenomenon called the
Janssen effect Janssen1895 . The idea is that when this happens, the excess
weight of added grains is distributed laterally rather than downwards,
balanced by the upward frictional force at the walls. The Janssen effect is
also found when grains are in motion relative to the walls Bertho2003Dynamical
, though the apparent weight saturates at values lower than in the static
case. However, measurements of pressure near the outlet in a system of disks
moving at constant speed on a horizontal conveyor belt Aguirre2010  found that
pressure at the outlet and the outflow speed were decoupled – one could be
made to vary while the other was held constant. We report here that,
consistent with this last observation, and in contrast to what we had expected
for vertical gravity-driven flow, our simulations indicate a near-constant
flow rate even when the particle-wall frictional coefficient is set to zero.
We do retain inter-particle friction in these cases, and our results appear to
indicate that the mechanism for slowing down particles is dynamic in nature.
However, what happens to the dynamic pressure in the vicinity of the outlet as
wall friction changes remains to be investigated.

In what follows, we first describe our experiment and simulation method, and
then present results for how flow rates change with wall friction and opening
size for both experiment and simulation in 2D. We then use our simulations to
explore the dynamics further by reporting on the spatial pattern of time-
averaged particle velocities and accelerations inside the hopper as wall
friction and the opening size at the base are changed.

##  2 Methods

###  2.1 Experiment

The experimental setup is a quasi-2D rectangular prism 48 particle diameters
($d$) wide, 400$d$ tall and 1.2$d$ in depth. (All length measurements are
expressed in particle diameters $d$). The front and back walls are made of
plexiglass and are lined with different frictional backings. The three
materials used are Teflon, poster-board (paper), and 60-grit sandpaper. The
right-hand side of Fig. 1 shows the experimental setup with the sandpaper
backing. To study where wall friction plays the biggest role in the hopper, we
created hybrid linings, see left hand-side of Fig. 1, in which one material
was used for the bulk of the lining (white region), and the other material
(shaded region) was used to line a region of height 22$d$ above the opening.
There are two realizations of these hybrid linings: sandpaper at the opening
and paper in the bulk (referred to as OS/BP) and the reverse, OP/BS. The
grains are steel spheres, 2.5 mm in diameter, thus large enough to neglect the
effects of air resistance. The 1.2$d$ depth of the hopper ensures that the
particles form a single layer and undergo significant collisions with the
front and back walls. The grains are filled to the same height for all the
wall frictions and openings, then allowed to flow through a rectangular
opening at the base into a container. The opening size $a$ is varied from 6$d$
$\rightarrow$ 22$d$ in increments of 2$d$. A force sensor placed beneath the
outlet container allows a measurement of the accumulated weight as a function
of time. This measured weight vs time data, converted to a fill percentage of
the draining hopper for better comparison with simulation, is plotted as a
function of time for two different opening sizes and two different hopper
linings in Fig. 2. The data for each opening and wall backing are averaged
over five experimental runs, with the corresponding error bars shown on the
figure. The straight line fits (dashed lines) show that the flow rates remain
linear till the hopper is almost fully drained, and this flow rate is smaller
for a given opening when the wall backing is more frictional, as might be
expected. Doing linear fits to the weight vs time data for all five hopper
linings and all nine openings, we are able to look at how flow rate scales
with opening size as wall friction changes. The results for all five hopper
linings, plotted using open symbols in Fig. 6, indicate that the power law
exponent decreases as friction increases. We thus use simulations to try and
understand this phenomenon better.

###  2.2 Simulations

We set up our simulations using the granular package of LAMMPS LAMMPS . The
simulation geometry is chosen to closely match the specifications of the
experimental setup in Fig. 1: a quasi-2D rectangular box with a rectangular
opening at the base, filled with monodisperse spheres. Length and mass units
are expressed in units of sphere diameter and mass, and the acceleration of
gravity $g=1$. In these units, the box has width 48, height 460 and depth 1.2.
The unit of simulation time is equivalent to the time it takes for a sphere to
fall through its own radius, and the simulation timestep in these units is
$10^{-4}$.

Though the vertical walls of the rectangular prism are set using the set
wall/gran command in LAMMPS, the base of the hopper is constructed using
highly overlapping particles that are frozen in place to obtain a smooth
surface. The Hertzian interaction governs interparticle and particle-wall
collisions on all surfaces, and is given by the following equation
Silbert2001Granular :

$\vec{F_{\text{Hz}}}=\sqrt{\delta}\sqrt{\frac{R_{i}R_{j}}{R_{i}+R_{j}}}[(k_{n}\delta\vec{n_{ij}}-m_{\text{eff}}\gamma_{n}\vec{v_{n}})\\\
-(k_{t}\Delta\vec{s_{t}}-m_{\text{eff}}\gamma_{t}\vec{v_{t}})]=\vec{F}_{n}+\vec{F}_{t}$
(1)

The first and second terms give the normal and tangential components of the
force $\vec{F}_{n}$ and $\vec{F}_{t}$, which are normal and tangential
relative to the intersecting plane of the particles. Here $\delta$ is the
overlap between contacting spheres, with $\vec{n_{ij}}$ the vector connecting
their centers; $R_{i}$ and $R_{j}$ are the radii of the two particles (here
equal), $\Delta\vec{s_{t}}$ is the tangential displacement vector between the
particles, and $v_{n}$ and $v_{t}$ are the normal and tangential components of
their relative velocity. For our simulation, the values for normal and
tangential elasticity constants are: $k_{n}=200000,k_{t}=\frac{2}{7}k_{n}$;
the normal damping constant is $\gamma_{n}=50.0$, while the tangential
$\gamma_{t}=0.0$. Note that the spring constants are smaller than the
corresponding material parameters for steel, but choosing larger values would
make the simulation impossibly slow Silbert2001Granular . The tangential force
$F_{T}$ is truncated according to the Coulomb frictional yield criterion:
$F_{T}/F_{N}\leq\mu$, where we set the inter-particle friction to $\mu=0.86$.
The particle-wall friction assumes the following values in the simulation:
$\mu_{W}=0.0,0.3,0.6,$ and $0.9$.

To initialize each simulation run, we fill the entire container with a random
arrangement of 45,000 spheres. Many of these overlap, and after deleting all
overlapping pairs, we allow the spheres to fall under gravity and settle onto
a plane lining the hopper bottom that prevents them from flowing through the
outlet. This process leaves us with approximately $10^{4}$ particles up to a
height of approximately 205. We wait long enough to ensure that the average
kinetic energy of the particles falls below a threshold value of $10^{-12}$.
Next, we remove the plane and allow particles to flow out under gravity. The
flow rate is found by measuring the rate of change in the number of particles
in the box. As in the experiment, we use 9 opening sizes (from $6d$ to $22d$
in steps of $2d$) at each $\mu_{W}$. The simulation is repeated 10 times with
different initializations for each wall friction and opening size.

As in Fig. 2, we calculate the fill percentage as the ratio of number of
particles remaining in the hopper to the number of particles before flow
started. This is shown as a function of time in Fig. 3 for a single opening,
$14d$, at the four different wall frictions. All four curves begin linear, and
flatten as the hopper empties. The dotted lines show straight line fits to the
linear region. Note that while the curves for non-zero friction are linear,
there is a small deviation from linearity for $\mu_{W}=0$.

This zero wall friction case is shown for all openings in Fig. 4. Here, the
deviations from linearity become more evident as the opening size increases.
The difference between zero and non-zero $\mu_{W}$ cases is more clearly seen
in the next figure, Fig. 5, which shows the variation with time of the
discrete time derivative of the number of particles in the hopper for all
$\mu_{W}$ and openings. Here the $\mu_{W}=0$ case, Fig. 5(a), clearly
distinguishes itself from the others, but in a non-obvious way. For all non-
zero frictional coefficients, Fig. 5(b) - (d), there is a flat region at early
times of constant flow rate; the duration of that constant flow is naturally
smaller for large openings, since the hopper drains faster. For zero friction,
however, though the flow rate at large opening sizes decreases in magnitude
over time till it vanishes, it is close to constant for smaller openings (6$d$
to 10$d$), which is a somewhat surprising result, since it is usually assumed
that wall friction is essential to the constant flow rate exhibited by
draining grains. This result indicates that factors such as geometric
constraints become significant as the opening narrows.

##  3 Results

We now look at how the flow rate for both experiment and simulation varies
with opening size, and how this dependence is affected by wall friction. The
flow rates are extracted from linear fits of the data for the variation in
time of the number of particles in the hopper, some of which have been shown
in the previous figures. These mass flow rates, converted to units of g/s, are
plotted vs. reduced opening size $(a-d)$ in Fig. 6: experimental data (open
symbols), each averaged over 5 runs for each distinct wall lining; and
simulation data (solid symbols), each point averaged over the 10 runs at each
of the four friction values. Note that the experimental OS/BP data,
corresponding to the experimental hopper lining where the bulk is lined with
paper and the opening with sandpaper (see the left panel of Fig. 1), lie on
top of the data for the fully sandpaper-lined hopper, seen in the right panel
of Fig. 1. Likewise the OP/BS data, which has paper at the opening and
sandpaper in the bulk, is indistinguishable from the purely paper-lined hopper
data. This tells us the surprising result that the mass flow rate is primarily
affected by the value of the wall friction in the vicinity of the opening, and
not in the bulk.

The values of the particle-wall static friction coefficient used in simulation
were chosen to resemble those of the experimental materials being used, and
indeed, we observe in Fig. 6 that the simulation data for each wall friction
follows the corresponding data in the experiment. The error bars for each data
point are small enough that most of them are contained within the symbol size.
The dashed lines represent power law fits to the data on this log-log graph,
and these are clearly not parallel to one another. Though the dynamical range
in opening sizes is necessarily small (the hopper clogs intermittently at
smaller openings, and at larger openings, empties too rapidly to establish a
steady rate), the fits indicate that the exponent of the power-law dependence
of flow rate on reduced opening size decreases as wall friction increases. The
power law exponents of all the experimental and simulation data, shown in
Table 1, lie between 1.5 (black solid line in Fig. 6) and 1.0 (green dot-
dashed line), and the difference from one exponent to the next is larger than
the standard deviation in each. The exception is the zero friction $\mu_{W}=0$
case, which has an exponent that appears slightly higher than 1.5, and we
think this is because the hopper drains at a faster rate than linear at early
times for large openings.

The mass flow rate at the hopper exit is a function of two quantities: the
packing fraction and the velocity of the flowing particles at the opening. We
examine both separately in the simulations to glean where wall friction plays
the bigger role. In Fig. 7, we plot the mean volume packing fraction, measured
just above the outlet, as a function of opening size, for all 4 wall friction
values. To do so, we construct a box centered at the opening of height 1$d$,
depth $1.2d$, and width equal to the opening size, and find the time-average
and standard deviation of the packing fraction measured from snapshots of the
simulation taken every $10^{4}$ timesteps. While the mean packing fraction is
higher when wall friction is zero, we see no significant difference in the
curves for all non-zero $\mu_{W}$ values. For a given $\mu_{W}$, the packing
fraction variation with opening size is relatively slight as compared to the
accompanying variation of particle velocities. The largest change in packing
fraction, around 10% from $a=6$ to $a=22$, occurs for $\mu_{W}=0$, with the
corresponding exit velocity changing by 300%, see Fig. 9. The variation of
packing fraction with opening size is even less significant when we do a
similar analysis within the bulk at a height greater than $10d$ (not shown
here). We have also checked that dividing the mass flow rate by the
corresponding packing fraction cannot account for the apparent change in the
Beverloo exponent shown in Fig. 6 and Table 1.

We find this variation of packing fraction with opening size near the outlet
is well fit by an empirical model from Janda et al. Janda2012Flow . This fit,
shown by the dashed lines in Fig. 7, has the form $B(1-Ce^{-a/L})$, where we
find that the parameter $L$ is the most sensitive to friction. This particular
form Janda2012Flow  was motivated by experimental results from Mankoc et al.
Mankoc2007Flow , who showed that Beverloo’s law tends to fail for small
openings, and $L\approx 6$ represents an opening size below which the
likelihood of clogging is high and the flow rate is suppressed due to changes
in the density at the outlet. Our fits show that this parameter $L$ decreases
as wall friction increases, subtly altering the shape of the packing fraction
dependence on opening size. This also suggests, somewhat counter-intuitively,
that the length scale over which the clogging affects the packing fraction at
the exit is smaller at higher friction.

Next, we focus on the effect of changing friction and opening size on particle
velocities at the exit. We analyze these in a few different ways. First, we
look into the putative notion of a free-fall arch. For a particular wall
friction and opening size, we find instantaneous velocities for spheres at the
exit from snapshots of the flow taken at every simulation time. These are then
converted to a height $h$ by setting $h=v^{2}/2g$, which can be interpreted as
the starting height for a particle had it been in free-fall. The dots in Fig.
8 show the resulting heights for each particle (one dot per particle for each
time) as a function of its x-position, where the color represents the
magnitude of the velocity ranging from low (dark blue) to high (yellow). We
then divide the x-axis into bins of 0.25$d$ width and average the $v^{2}/2g$
values in each bin, giving the red solid line running through the data, where
the vertical bars represent the standard deviation in each bin.

The averaged height profiles with error bars for all the openings studied are
shown in Fig. 9 for each opening size, with each panel representing a
different $\mu_{W}$. At zero particle-wall friction, panel (a), $\langle
v^{2}/2g\rangle$ at the exit scales with opening size, consistent with the
expectation of a free fall arch. This is made clearer in Fig. 10(a), which
shows the same figure with both x and y axes scaled by the opening size. The
curves for all openings collapse within error bars, and this shape is well fit
by a parabolic function, shown by the solid black line in the figure, in good
agreement with the shape seen for the velocity profiles in
experimentsJanda2012Flow . This suggests that the description of a free fall
arch does not seem unreasonable in the $\mu_{W}=0$ case.

However, when we examine this further by looking at spatial distributions of
the actual particle velocities, we encounter an apparent contradiction. To
visualize this, we coarse-grain the hopper into square cells of side $1d$ and
find the time-averaged sphere velocity vector in each cell. The top row of
Fig. 11 shows, for $\mu_{W}=0$, the time-averaged velocity at spatial points
in the lower half of the hopper. The figures on the left and the right
represent two opening sizes. Note that the coarse-grained velocity vectors do
not begin to increase until the very bottom of the hopper, implying that the
acceleration region for $\mu_{W}=0$ is far smaller than the opening size, and
there is no real free-fall arch. We conclude that it is misleading to extract
anything definitive about the nature of the acceleration experienced by
particles simply from observing their velocities at the exit.

The picture changes for non-zero wall friction. The remaining panels in Fig. 9
show that the $\langle v^{2}/2g\rangle$ distributions do not scale as expected
with the opening size for all finite values of friction. The vertical axis
scale for Fig. 9(b)-(d) have a maximum of 12, in contrast to the vertical axis
of Fig. 9(a) for $\mu_{W}=0$, which has a maximum of 36. This already shows
that as $\mu_{W}$ increases, the mean square velocity at the exit, or free-
fall arch height, scales more weakly with opening size. At $\mu_{W}=0.9$, Fig.
9(d), the average velocity at the opening becomes fully independent of opening
size. The shape of the arch also changes, and is no longer parabolic.

The change in shape with friction is more clearly seen in Fig. 10(b), where
all the curves collapse when only the x-axis is scaled by the opening $a$.
This shape is no longer parabolic, but closer to the form
$A(1-B\exp[-C(|x/a|-1)])$, where $A$ is set by the maximum value of $\langle
v^{2}/2g\rangle$ for each opening, and $B,C$ are fitting parameters. This
suggests that even though the naïve notion of a free-fall arch is no longer
valid for large values of wall friction, the opening size remains the only
relevant length scale in the problem.

To better understand the origin of the change in exit velocity distributions
with wall friction, we return to the analysis of the spatial distribution of
coarse-grained velocity vector fields in the hopper. Deviations from mean flow
velocity in the lower half of the hopper are shown in Fig. 11, where left and
right represent opening sizes $10d$ and $22d$ respectively, and each row
represents a different wall friction $\mu_{W}$. The color bar indicates the
magnitude of the coarse-grained velocity vector. We observe from the change in
vector size that there is a transition from bulk flow to rapid flow near the
opening where the velocity is changing. To further accentuate this transition,
the curves in blue (dashed)/black (solid) represent the height at which the
velocity increases 10%/50% over the mean particle velocity in the bulk. Thus
these curves also indicate a spatial boundary from a region of constant mean
particle velocity to a region of accelerating flow.

Whether you look at the velocity vector magnitudes using the color bar, or the
dashed blue/solid black curves, there is a clear difference in shape between
the zero and non-zero $\mu_{W}$ values. For the $\mu_{W}=0$ case, top row,
particles flow on average at the same rate till they get to a region less than
$20d$ in height, whereupon they accelerate rapidly out of the opening. This is
also seen in movies of the simulation, where the particles appear to be
unaware of the exit almost until they reach the bottom of the hopper. After
reaching the bottom, particles flow laterally towards the exit. This lateral
flow is seen more clearly for the larger opening: top row, right. The size of
the acceleration region does not change significantly with opening size ($14d$
on the left, $22d$ on the right). However, particle velocities as they exit do
increase with the size of the opening.

In the non-zero friction cases, a conical region of the hopper where particles
are moving is visible, within which the transition from steady to accelerated
flow is demarcated by a change in color for all $\mu_{W}\neq 0$, rows 2 - 4 of
Fig. 11. (Note that the color bar for rows 2 - 4 is the same, i.e. for all
non-zero friction, but half the size of the color bar for the zero friction
case, top row). The width of this conical region changes with opening size,
but for a given opening, there is not much change from one wall friction to
the next. At all $\mu_{W}$, comparing the height at which the acceleration
starts, we see not much difference from opening $14d$ on the left to opening
$22d$ on the right. What does change as friction increases is how much
acceleration occurs, judging by the change in size of the vectors from the
start of the acceleration region to the exit. While the exit velocities
increase significantly with opening size for $\mu_{W}=0.3$, this is negligible
at $\mu_{W}=0.9$ \- this is consistent with the lack of change in the height
of the exit velocity profiles with opening size seen for the same wall
friction, Fig. 9(d).

The decrease in the magnitude of vertical acceleration with increasing
friction $\mu_{W}$ is highlighted in Fig. 12. Here, we calculate the vertical
velocity component $v_{y}(t)$ in a series of square cells of size $1d$ running
down the center of the hopper, and use it to calculate
$a_{y}=dv_{y}/dt=v_{y}dv_{y}/dy$ and plot it as a function of vertical
position $y$. This gives us the rather surprising result that the average
acceleration at the hopper center for $\mu_{W}=0$ becomes larger than $g$,
close to $6g$ at the exit. This occurs even when the size of the averaging box
is increased. We believe this is due to the fact that the flow remains
collisional up until the hopper exit - the instantaneous accelerations come
from collisional impulses from other particles and from the front and back
walls - and these can be very large in the absence of friction. As $\mu_{W}$
increases, the magnitude of the acceleration drops, but the accelerations at
all $\mu_{W}\neq 0$ begin to deviate from zero at about the same height for a
given opening, consistent with what was seen in Fig. 11 – the deviation occurs
at a smaller value of $y$ for $\mu_{W}=0$. Notably, far from taking a constant
free-fall value, accelerations continue to increase as the particles approach
the opening, a pattern that persists for all openings and all values of
$\mu_{W}$.

##  4 Discussion

We have examined the effect of changing particle-wall friction on the flow
patterns of spheres falling under gravity in a quasi-2D hopper of depth
1.2$d$. What appears to matter are particle collisions with the front and back
walls, since changing the friction at the side walls has no effect. We find
that the flow rate is constant at all non-zero wall friction $\mu_{W}$ and
opening sizes, Fig. 3. While the rate of flow in the absence of wall friction,
$\mu_{W}=0$, is not constant, as seen in Figs. 4 and 5(a), the flow behavior
is distinctly different from Newtonian fluid flow, in which the flow rate
would decrease linearly in time. The mass flow rate also approaches a constant
value for small openings, Fig. 5(a), which leads us to conclude that wall
friction is not the only requirement to have a constant rate of flow -
geometric confinement appears to also play an important role.

While the dependence of the flow rate magnitude on the particle-wall friction
coefficient $\mu_{W}$ comes as no surprise, Fig. 3, it is less obvious that
the flow rate scaling with opening size changes as a function of friction, and
no longer follows the expected Beverloo scaling exponent of $3/2$, Fig. 6.
Since the dependence of packing fraction on opening size in the regime we have
studied is weak, Fig. 7, this decrease in the exponent comes primarily from
how the particle velocity field transitions from bulk flow to accelerated
flow. As seen in Fig. 11, the region where the acceleration begins stays
roughly the same for all values of wall friction $\mu_{W}$, but the primary
effect of friction is to impede the acceleration of the particles. Fig. 12
shows that the magnitude of this acceleration is strongly affected by
friction. Indeed, the high values of these accelerations at low friction,
which come from taking finite differences of particle positions separated by
0.1 simulation times, lead us to believe the particles are not, in fact, in
free fall, but are undergoing many collisions right until they exit.

Thus, in conclusion, our experiments and simulations indicate that the idea of
a free-fall arch should not be interpreted literally, but as a region that
marks the boundary between steady flow and accelerated flow, a region where
the dynamics of particle-particle and particle-wall collisions begin to play a
role in how rapidly grains exit the hopper. The height of this region is of
order the opening size, consistent with the fact that the ratio of opening
size to particle diameter is the only relevant dimensionless scale in the
problem. This is borne out by the experiments we reported on here, which
showed no statistical difference between the flow rate of a hopper where the
entire wall was covered with a particular frictional backing, and one in which
merely the region of height 22$d$ near the exit of the hopper had that
frictional backing. Note that even when we set particle-wall friction to zero
in our simulations, inter-particle friction remains non-zero. The fact that
the dependence on particle-wall friction of the flow rate scaling with opening
size vanishes in our 3D experiments and simulations is consistent with
previous experiments that have shown that while flow in 2D experiments is
highly collisional, 3D flow tends to be less collisional and have more
enduring frictional contacts 3Dflowexpt . It remains to be investigated at
what hopper thickness this transition from 2D to 3D flow occurs. Finally, we
plan in the future to measure a dynamical stress tensor during flow to see if
the Janssen effect continues to hold during flow, and whether any change is
seen as a function of changing particle-wall friction.

##  5 Acknowledgements

We thank Narayanan Menon for helpful discussions, and Mollie Pleau for sharing
her 3D data. SKB acknowledges funding from ICTS Bangalore, the APS-IUSSTF
travel grant, and NSF DMR 1506750.

## References

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$\mu_{W}$ | $\gamma$
---|---
$0.0$ | $1.65\pm 0.01$
$0.3$ | $1.41\pm 0.01$
$0.6$ | $1.20\pm 0.02$
$0.9$ | $1.11\pm 0.01$
Teflon | $1.55\pm 0.01$
OP/BS | $1.42\pm 0.03$
Paper | $1.46\pm 0.01$
OS/BP | $1.20\pm 0.02$
Sandpaper | $1.20\pm 0.02$
Table 1: Exponents for power law fits of flow rate vs. opening size shown in
Fig. 6. The first four rows represent the power law exponents $\gamma$ for
simulation data vs wall friction coefficient, and the remaining for the
experimental data vs the material of the wall backing. The uncertainty is
obtained from the fit.  Figure 1: (Left) Schematic of experimental hopper with
the OS/BP (Opening Sandpaper, Bulk Paper) lining: the bulk is lined with
paper, with the rough region of height 22d near the opening lined with
sandpaper. (Right) Photograph of experimental hopper of dimensions 400 x 48 x
1.2 in units of particle diameter, fully lined with sandpaper. The hopper is
filled with steel spheres of diameter $\approx 2.5$ mm, which flow out of an
opening at the base onto a box sitting on a force transducer that records the
weight as a function of time. The slope of this force trace is a measure of
the mass flow rate. We control the friction on the walls by changing the
material backing on the front and back walls.

Figure 2: Experimental fill percentage as a function of time for a draining
hopper at two representative opening sizes (8$d$ and 14$d$) and for two
distinct wall backings: paper and sandpaper. Each curve is averaged over five
experimental runs, with error bars indicating the mean deviation over these
runs. The fill percentage is the ratio of the height of the granular pile at a
given time to the initial fill height and is extracted from the force trace
data. Error bars are shown every 10 data points for better visibility. The
dashed lines represent straight line fits to the data.

Figure 3: Fill percentage vs. time for simulations at fixed opening size $a$ =
14$d$ and varying wall frictions. The markers are not data points, but are
instead used to identify the curve they are on. The lines shown here are
actually connected dots, and there are approximately 7000 data points for each
opening size, with the markers placed every 500 data points. The dashed lines
represent linear fits that give us the flow rate for each case. There is
deviation from linearity when the hopper empties for all wall frictions, but
the $\mu_{W}=0.0$ case shows deviations from linearity during flow as well.

Figure 4: Fill percentage vs. time at different opening sizes for fixed wall
friction 0.0. Flow rates are obtained from linear fits. The flattening of the
curves represents the container running out of particles.

Figure 5: Discrete time derivative of the number of particles $N(t)$ in the
hopper shown as a function of time for all openings, where panels (a) - (d)
represent increasing wall friction $\mu_{W}$ values. Each time point $i$ here
is given by the following:
$\Big{(}\frac{dN}{dt}\Big{)}_{i}=\frac{N_{i+1}-N_{i-1}}{2}$. The arrow in (a)
represents the direction of increasing opening size from $6d$ to $22d$ (as
ordered at t = 0), for all sub-figures. The $\mu_{W}=0.0$ case shows deviation
from constant flow, and these deviations increase with the opening size.

.

Figure 6: Mass Flow rates in g/s as a function of reduced opening size for
both the experiment and simulation data on a log-log scale. The simulation
data is given by the solid symbols, and the experimental data by the open
symbols, and both have error bars obtained by averaging over distinct runs.
The simulation errors are smaller than the marker sizes. OS/BP refers to
opening sandpaper and bulk paper (see left panel of Fig. 1) and OP/BS to
opening paper and bulk sandpaper. The OP/BS and paper data are congruent to
one another, as are the OS/BP and sandpaper data. The dotted lines are power
law fits to the data, where the exponents of power law fit along with their
standard deviation are given in Table 1. Other than the $\mu_{W}=0$ data from
simulation, the slopes fall between the solid black line of power law exponent
1.5 and the green dash-dot line of exponent 1.0.

Figure 7: Packing fractions within a box of width equal to the opening size,
height $1d$ and depth $1.2d$ at the hopper opening as a function of the
opening size. The values plotted represent the average volume fraction in the
box over the duration of the flow with vertical bars showing one standard
deviation. The dashed lines represent an empirical fit of the form:
$B(1-Ce^{-a/L})$ Janda2012Flow , where $B$, $C$ and $L$ are fitting
parameters. In our fits, $B$ and $C$ do not vary much, while $L$ changes from
about 13 for $\mu_{W}=0$ to 4.5 for $\mu_{W}=0.9$.

Figure 8: Scatter plot of extrapolated free-fall heights, or $v^{2}/2g$
values, for each particle with velocity v at the opening, sampled over the
entire flow period and plotted against its x-position for $\mu_{W}=0.3$ and
opening size $14d$. The color is representative of the ”height” or velocity of
the particles, with darker representing lower velocity. The corresponding
distribution, obtained by averaging these points in x-bins of size $0.25d$, is
shown as a solid red line, with vertical error bars representing one standard
deviation.

Figure 9: The four panels show the time-averaged profile of extrapolated ‘free
fall heights’, or $\langle v^{2}/2g\rangle$, along with errors bars, plotted
as a function of x position for openings ranging from $6d$ to $22d$. Each
panel represents a different friction $\mu_{W}$. Note that as wall friction
increases, the ‘arch’ heights stop increasing with the opening size.

Figure 10: $(a)$

Figure 11: $(b)$ Figure 10: (a) The ‘free-fall height’ profiles $\langle
v^{2}/2g\rangle$ for all openings at wall friction $\mu_{W}=0.0$ collapse when
scaled along both axes by the size $a$ of the opening. The dashed black line
represents a fit to a parabolic profile, and the solid green line to the form
$A(1-B\exp[-C(|x/a|-1)])$. (b) The ‘free-fall height’ profiles $\langle
v^{2}/2g\rangle$ for all openings at wall friction $\mu_{W}=0.9$ collapse when
scaled along only the x-axis by the size $a$ of the opening - the height is
independent of the opening size as seen in Fig. 9 (d)

Figure 11: Velocity vector field for two openings (left column $14d$, right
column $22d$), where each row represents wall friction values
$\mu_{W}=0,0.3,0.6,0.9$. The hopper is binned into square cells of side
$0.25d$ to obtain this plot, where the velocity vector in each cell is the
time average of particle velocities in that cell over the course of the run.
The blue dashed (black solid) curve shows the points at which the average
velocity of the particles becomes 10% (50%) larger than the constant bulk
velocity. This is done to highlight the transition from bulk flow (zero
acceleration) to near-opening flow (increasing acceleration). The color bar is
scaled differently for $\mu_{W}=0.0$ in order to capture the full range of
velocities.

Figure 12: This figure shows the acceleration of the particles along the
central line of the hopper for the different wall $\mu$ and varying opening
sizes. The arrow indicates the direction of increasing opening size, and is
representative for all wall $\mu$. The accelerations are found by binning the
central vertical line of the hopper with $1\times 1$ boxes, in which we
average $v_{y}\frac{dv_{y}}{dy}$ over time and initializations.


LHCP 2013

11institutetext: CERN

# Quarkonium Production at LHCb

Monica Pepe Altarelli
On behalf of the LHCb collaboration 11<EMAIL_ADDRESS>
###### Abstract

I will review a selection of LHCb results on the production of heavy
quarkonium states in pp collisions, including recent results on
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and
$\varUpsilon(nS)$ ($n=1,2,3$) production at $\sqrt{s}=8\,\mathrm{TeV}$, as
well as preliminary results on
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ production in
proton-lead collisions at $\sqrt{s_{\rm NN}}=5\,\mathrm{TeV}$.

##  1 Introduction

Heavy quarkonium states are particularly interesting systems to test our
understanding of strong interactions, both at perturbative and non-
perturbative level, and are therefore the subject of intense theoretical and
experimental studies. An effective field theory, non-relativistic QCD (NRQCD)
CaswellLepage1986PL ; PhysRevD.51.1125 , provides the foundation for much of
the current theoretical work. According to NRQCD, the production of heavy
quarkonium factorises into two steps: a heavy quark-antiquark pair is first
created at short distances and subsequently evolves non-perturbatively into
quarkonium at long distances. The NRQCD calculations include colour-singlet
(CS) and colour-octet (CO) amplitudes (see Brambilla:2010cs  and references
therein), which account for the probability of a heavy quark-antiquark pair in
a particular colour state to evolve into a heavy quarkonium state. The CS
model (CSM) was initially used to describe experimental data. However, it
underestimates the observed cross-section for single
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ production at high
transverse momentum ($p_{\rm T}$). To resolve this discrepancy, the CO
mechanism was introduced. More recent higher-order calculations to the CS
predictions raise substantially the cross-sections at large $p_{\rm T}$
bringing them closer to the experimental data. However, none of these
approaches can reproduce in a consistent way the available experimental
results on both cross-section and polarisation (see Brambilla:2010cs  and
references therein, Lourenco ).

Heavy charmonium is also produced from $b$-hadron decays. This production
mechanism can be distinguished from prompt quarkonium production by exploiting
the $b$-hadron finite decay time. In this case QCD predictions are based on
the Fixed-Order-Next-to-Leading-Log (FONLL) formalism FONLL ; Cacciari , which
combines a Fixed-Order calculation with an all-order resummation of Leading
and Next-to-Leading logarithms.

In the following, results from heavy quarkonium production at LHCb will be
compared with various theoretical models to provide direct tests of the
underlying production mechanism. Furthermore, preliminary results on
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ production in
proton-lead collisions at $\sqrt{s_{\rm NN}}=5\,\mathrm{TeV}$ will be
reported.

##  2 Production of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$
and $\varUpsilon(nS)$ mesons

The differential production cross-sections of prompt
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and
$\varUpsilon(nS)$ mesons produced at the $pp$ collision point either directly
or via feed-down from higher mass charmonium or bottomonium states, are
measured at $\sqrt{s}=8\,\mathrm{TeV}$ Aaij:2013yaa  with the LHCb detector
Alves:2008zz . The measurements are performed in the range of rapidity
$2.0<y<4.5$ and $p_{\rm T}<14\,\,\mathrm{GeV}\mskip-2.0mu/\mskip-1.0muc$ for
the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ meson and
$p_{\rm T}<15\,\,\mathrm{GeV}\mskip-2.0mu/\mskip-1.0muc$ for the
$\varUpsilon(nS)$, in bins of $p_{\rm T}$ and $y$ with bin sizes $\Delta
p_{\rm T}=1\,\mathrm{GeV}\mskip-2.0mu/\mskip-1.0muc$ and $\Delta y=0.5$. The
fraction of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\leavevmode\nobreak\ \mathrm{from}\leavevmode\nobreak\ b$ (abbreviated
as “from $b$” in the following) is also measured in the same fiducial region.
The $\varUpsilon(nS)$ meson analysis is based on a data sample, corresponding
to an integrated luminosity of about $51\leavevmode\nobreak\
\,\mathrm{pb}^{-1}$ of $pp$ interactions, while the analysis for the more
abundant ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ mesons is
based on data, corresponding to an integrated luminosity of about
$18\leavevmode\nobreak\ \,\mathrm{pb}^{-1}$. The
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and
$\varUpsilon(nS)$ mesons are reconstructed through their decay into a pair of
muons following the strategy described in Refs. Aaij:2013yaa ; LHCb-
PAPER-2011-003 ; LHCB-PAPER-2011-036 . The resulting dimuon invariant mass
distributions are displayed in Figs.1 and 2 for the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and
$\varUpsilon(nS)$ mesons. The invariant mass resolution is on average
$13.5\,\mathrm{MeV}\mskip-2.0mu/\mskip-1.0muc^{2}$ for the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and
$43\,\mathrm{MeV}\mskip-2.0mu/\mskip-1.0muc^{2}$ for the $\varUpsilon(1S)$
meson.

Figure 1: Dimuon invariant mass of the selected
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\to\mu^{+}\mu^{-}$
candidates in a selected bin in $y$ and $p_{\rm T}$ ($2.5<y<3.0$ and $3<p_{\rm
T}<4\,\,\mathrm{GeV}\mskip-2.0mu/\mskip-1.0muc$).  Figure 2: Invariant mass
distribution of the selected $\varUpsilon(nS)\to\mu^{+}\mu^{-}$ candidates in
the range $p_{\rm T}<15\,\,\mathrm{GeV}\mskip-2.0mu/\mskip-1.0muc$ and
$2.0<y<4.5$. The three peaks correspond to the $\varUpsilon(1S)$,
$\varUpsilon(2S)$ and $\varUpsilon(3S)$ meson signals (from left to right).

The production is studied under the assumption of zero polarisation, and no
corresponding systematic uncertainty is assigned on the cross-section for this
effect. This assumption is justified by the small polarisation measured for
the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ meson at
$\sqrt{s}=7\leavevmode\nobreak\ \,\mathrm{TeV}$ by LHCb LHCb-PAPER-2013-008
and ALICE AliceJpsiPola , in a kinematic range similar to that used in this
analysis, and for the $\varUpsilon(nS)$ by CMS CMSUpsilonPola  at large
$p_{\rm T}$ and central rapidity.

The measured differential cross-sections for the production of prompt
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ mesons as a function
of $p_{\rm T}$ are compared in Fig.3 to three theoretical models: an NRQCD
model at next-to-leading order (NLO) Butenschoen:2011yh ; Butenschoen:2010rq ,
an NNLO* CSM Artoisenet:2008 ; lansberg:2009  where the notation NNLO*
indicates that the calculation at next-to-next leading order is not complete
and neglects part of the logarithmic terms and an NLO CSM Campbell:2007ws . In
these comparisons the predictions are for direct
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ meson production,
i.e., they do not include feed-down from higher charmonium states, whereas the
experimental measurements do include feed-down. In particular, the
contribution from ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$
mesons produced in radiative $\rm\chi_{c}$ decays in the considered fiducial
range was measured to be at the level of $20\%$ at
$\sqrt{s}=7\,\,\mathrm{TeV}$ LHCb:2012af . Allowing for this contribution,
both the NNLO* CSM and the NLO NRQCD models provide reasonable descriptions of
the experimental data. In contrast, the CSM at NLO underestimates the cross-
section by an order of magnitude.

Figure 3: Comparison of the differential cross-section for the production of
prompt ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ meson (under
the assumption of zero polarisation) as a function of $p_{\rm T}$ with direct
production in an NLO NRQCD model Butenschoen:2011yh ; Butenschoen:2010rq
(orange diagonal shading), an NNLO* CSM lansberg:2009  (solid yellow) and an
NLO CSM Campbell:2007ws  (blue vertical shading).

The fraction of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\leavevmode\nobreak\ \mathrm{from}\leavevmode\nobreak\ b$ is shown is
Fig.4 as a function of the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}$ transverse momentum in bins of rapidity. This fraction is measured to
be smaller at low $p_{\rm T}$ and for forward data.

Figure 4: Fraction of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\leavevmode\nobreak\ \mathrm{from}\leavevmode\nobreak\ b$ as a function
of the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ transverse
momentum, in bins of rapidity.

In Fig.5 the data for the differential production cross-section for
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\leavevmode\nobreak\
\mathrm{from}\leavevmode\nobreak\ b$ as a function of $p_{\rm T}$ at
$\sqrt{s}=8\,\,\mathrm{TeV}$ are compared to the FONLL predictions. Very good
agreement is observed.

Figure 5: Differential production cross-section for
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\leavevmode\nobreak\
\mathrm{from}\leavevmode\nobreak\ b$ as a function of $p_{\rm T}$ in the
fiducial range $2.0<y<4.5$. The FONLL prediction FONLL ; Cacciari  is shown in
yellow.

The integrated cross-section for
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\leavevmode\nobreak\
\mathrm{from}\leavevmode\nobreak\ b$ production in the defined fiducial region
is determined to be
$\sigma\left({\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\leavevmode\nobreak\ \mathrm{from}\leavevmode\nobreak\ b,\,p_{\rm
T}<14\,\,\mathrm{GeV}\mskip-2.0mu/\mskip-1.0muc,\,2.0<y<4.5\right)=1.28\pm
0.01\pm 0.11\,\upmu\mathrm{b}$ Aaij:2013yaa , where the first uncertainty is
statistical and the second is systematic, in very good agreement with the
FONLL prediction of $1.34^{+0.63}_{-0.49}\,\upmu{\rm b}$ Cacciari .

The results on ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$
production at $\sqrt{s}=8\,\mathrm{TeV}$ provide an extra measurement, which
complements those already published by LHCb at $\sqrt{s}=2.76$ LHCb-
PAPER-2012-039  and 7 $\,\mathrm{TeV}$LHCb-PAPER-2011-003 . Figure 6 shows the
results for the cross-section for
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\leavevmode\nobreak\
\mathrm{from}\leavevmode\nobreak\ b$ at $\sqrt{s}=2.76$, 7 and
$8\,\,\mathrm{TeV}$ as well as the FONLL prediction as a function of centre-
of-mass energy. Excellent agreement with the theoretical calculation is
observed.

Figure 6: Predictions based on the FONLL formalism FONLL ; Cacciari  for the
production cross-section for ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\leavevmode\nobreak\ \mathrm{from}\leavevmode\nobreak\ b$ in the
fiducial range $0<p_{\rm T}<14\,\,\mathrm{GeV}\mskip-2.0mu/\mskip-1.0muc$ and
$2.0<y<4.5$ (yellow band). The uncertainty includes contributions from the
renormalisation scale, quark masses and the choice of PDF set. The black
dotted line shows the central value of the prediction. The points show the
LHCb measurements at $\sqrt{s}=2.76$ LHCb-PAPER-2012-039 , 7 LHCb-
PAPER-2011-003 , and 8$\,\,\mathrm{TeV}$ Aaij:2013yaa .

From the measurement of the production cross-section for
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\leavevmode\nobreak\
\mathrm{from}\leavevmode\nobreak\ b$ in the fiducial region, using the LHCb
Monte Carlo simulation based on Pythia to extrapolate to the full polar angle
range (with an extrapolation factor of 5.4 from the measured cross-section to
the full kinematic region) and the inclusive
$b{\to}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}X$ branching
fraction, one can derive the total $b\overline{b}$ production cross-section at
$\sqrt{s}=8\,\mathrm{TeV}$, $\sigma(pp\to b\overline{b}X)=298\pm 2\pm
36\,\upmu{\rm b}$, where the first uncertainty is statistical and the second
is systematic.

In Fig.7 the cross-sections times dimuon branching fractions for the three
$\varUpsilon$ meson states are compared to the NNLO∗ CSM Artoisenet:2008  and
an NLO CSM Campbell:2007ws  predictions as a function of $p_{\rm T}$. The
NNLO* CSM provides a reasonable description of the experimental data,
particularly for the $\varUpsilon(3S)$ meson, which is expected to be less
affected by feed-down. As for the prompt
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ meson production,
the CSM at NLO underestimates the cross-section by an order of magnitude.

Figure 7: Comparison of the differential production cross-sections times
dimuon branching fractions for (a) $\varUpsilon(1S)$, (b) $\varUpsilon(2S)$
and (c) $\varUpsilon(3S)$ mesons as a function of $p_{\rm T}$ with direct
production in an NNLO∗ CSM Artoisenet:2008  (solid yellow) and an NLO CSM
Campbell:2007ws  (blue vertical shading) model.

##  3 Production of $\uppsi{(2S)}$ mesons

The production of $\uppsi{(2S)}$ mesons was studied by LHCb in $pp$ collisions
at 7$\,\mathrm{TeV}$ LHCb-PAPER-2011-045 . The $\uppsi{(2S)}$ mesons were
reconstructed in the decay channel $\uppsi{(2S)}\rightarrow\mu^{+}\mu^{-}$ and
$\uppsi{(2S)}\rightarrow{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}\pi^{+}\pi^{-}$ and the results from the two channels were combined. The
$\uppsi{(2S)}$ production from $b$-hadron decays is distinguished from
promptly produced charmonium by exploiting the $b$-hadron finite decay time.
The interest of measuring $\uppsi{(2S)}$ production arises from the fact that
no significant feed-down from excited charmonium states is expected, which
facilitates the comparison with theoretical model. Figures 8 and 9 show the
differential production cross-section as a function of $p_{\rm T}$ for prompt
$\uppsi{(2S)}$ and $\uppsi{(2S)}$ from $b$-hadron-decays. The data are
compared with various theoretical models: MWC Ma:2010jj  and KB
Butenschoen:2010rq  are NLO NRQCD calculations, while AL Artoisenet:2008 ;
lansberg:2009  is a CSM including the dominant NNLO terms. The NLO NRQCD and
FONLL FONLL ; Cacciari  calculations provide a good description of the
$\uppsi{(2S)}$ prompt cross-section and of the cross-section of the
$\uppsi{(2S)}$ from $b$-hadron decays.

Figure 8: Differential production cross-section as a function of $p_{\rm T}$
for prompt $\uppsi{(2S)}$. The data are compared with various theoretical
models: MWC Ma:2010jj  and KB Butenschoen:2010rq  are NLO NRQCD calculations,
while AL Artoisenet:2008 ; lansberg:2009  is a CSM including the dominant NNLO
terms  Figure 9: Differential production cross-section as a function of
$p_{\rm T}$ for $\uppsi{(2S)}$ from $b$-hadron decays. The data are compared
with FONLL calculations FONLL ; Cacciari .

##  4 Double ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$
production

Given the large charmonium production cross-section at the LHC, the question
of multiple production naturally arises. This was studied by LHCb using an
integrated luminosity of about $37\leavevmode\nobreak\ \,\mathrm{pb}^{-1}$ of
$pp$ interactions at 7$\,\mathrm{TeV}$ Aaij:2011yc . Simultaneous production
of two ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ mesons can be
expected in single parton collisions via higher order $O(\alpha_{s}^{4})$
gluon-gluon diagrams. This production could also be enhanced by double parton
scattering (DPS) in which two gluons are extracted from a proton and the two
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ mesons are produced
in two independent sub-processes. The cross-section for double
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ production in the
fiducial range $2<y^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}<4.5$ and $p_{\rm T}^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}<10\,\mathrm{GeV}\mskip-2.0mu/\mskip-1.0muc$ is measured to be
$\sigma^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}}=5.1\pm 1.0\pm
1.1$nb Aaij:2011yc , where the first uncertainty is statistical and the second
is systematic, in good agreement with predictions from leading-order QCD
calculations Berezhnoy:2011xy . The suppression factor with respect to single
prompt ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ production is
$\sigma^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}/\sigma^{{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}}=(5.1\pm 1.0\pm 0.6^{+1.2}_{-1.0})\times 10^{-4}$, where the first
uncertainty is statistical, the second systematic and the third accounts for
the unknown ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$
polarisation. Figure 10 shows the differential production cross-section for
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ pairs as a function
of the invariant mass of the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}$ pair system. The data are compared with a theoretical prediction
Berezhnoy:2011xy , which includes both direct production and feed-down from
$\uppsi{(2S)}$ decays and no contribution from DPS. Within the available
statistics good agreement is observed. For double
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ production, the
predictions from single and double parton scattering are expected to be fairly
close in magnitude PhysRevD.86.034017 , but with somewhat different kinematic
distributions for the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}$ pairs. The analysis of the higher statistics already collected by LHCb
will allow an in-depth study of the kinematic properties of double
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ production and of
the different production models.

[htb]

Figure 10: Differential production cross-section for
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ pairs as a function
of the invariant mass of the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}$ pair system. The shaded area corresponds to the prediction by the
model described in Ref. Berezhnoy:2011xy .

##  5 Exclusive production of
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and $\uppsi{(2S)}$
mesons

Exclusive production of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}$ and $\uppsi{(2S)}$ mesons through processes such as the one displayed
in Fig.11 were observed by LHCb Aaij:2013jxj  in the dimuon channel. LHCb is
very well suited for this type of studies as it has access to high rapidities,
with some sensitivity to backward tracks using the Vertex Locator (VELO), it
operates at relatively low pileup and has sensitivity to low momentum and
transverse momentum particles. Exclusively produced
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and $\uppsi{(2S)}$
mesons into dimuons are characterised by a very distinct topology: their
selection requires the presence of two reconstructed muons in the forward
region and no other tracks and photons in the detector. An additional rapidity
gap is obtained by also excluding tracks in the backward region using the VELO
detector. Based on a data sample, corresponding to an integrated luminosity of
about $36\leavevmode\nobreak\ \,\mathrm{pb}^{-1}$ of $pp$ interactions, the
cross-sections times branching fractions to two muons for exclusive
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and $\uppsi{(2S)}$
with pseudorapidities between 2.0 and 4.5 is measured to be
$\sigma_{pp\to{\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip
2.0mu}(\to\mu^{+}\mu^{-})}=307\pm 21\pm 36\,{\rm pb}$ and
$\sigma_{pp\to\uppsi{(2S)}(\to\mu^{+}\mu^{-})}=7.8\pm 1.3\pm 1.0\,{\rm pb}$,
where the first uncertainty is statistical and the second is systematic, in
good agreement with theoretical predictions HarlandLang:2009qe ;
PhysRevLett.92.142003 ; Motyka:2008ac ; Goncalves:2011vf . During the 2012
run, a new trigger was implemented to increase the rate of exclusive events,
which makes use of “upstream” silicon sensors to veto any backward activity
and of soft $p_{\rm T}$ cuts to gain sensitivity for hadronic final states.
Data corresponding to an integrated luminosity of about
$1.4\leavevmode\nobreak\ \mathrm{fb^{-1}}$ were already collected with such a
trigger, which will allow LHCb to study the production of charmonium decaying
into hadronic final states, the production of open charm and to perform a
search for higher mass charmonium states.

Figure 11: Feynman diagram contributing to exclusive
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ photoproduction.

##  6 Production of ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$
mesons in proton-lead collisions at $\sqrt{s_{\rm NN}}=5\,\mathrm{TeV}$

LHCb has recently presented preliminary results on production of
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ mesons in proton-
lead collisions at $\sqrt{s_{\rm NN}}=5\,\mathrm{TeV}$ LHCb-CONF-2013-008 ;
Burkhard . Data were collected with a proton beam energy of 4$\,\mathrm{TeV}$
and a lead beam energy of 1.58 $\,\mathrm{TeV}$ per nucleon, resulting in a
centre-of-mass energy of the proton-nucleon system of 5 $\,\mathrm{TeV}$. The
directions of the proton and lead beams were swapped during data taking to
produce both $pA$ and $Ap$ collisions. Data collected with the inverted beam
directions (i.e., in $Ap$ collision) allow LHCb to measure backward
production. The proton-nucleon centre-of-mass system has a rapidity in the
laboratory frame of +0.47 for $pA$ and -0.47 for $Ap$ collisions, resulting in
a rapidity coverage in the laboratory frame ranging from about 1.5 to 4.0 for
$pA$ collisions and from -5.0 to -2.5 for $Ap$ collisions. The analysis is
based on data samples corresponding to integrated luminosities of 0.75
$\mathrm{nb^{-1}}$ of $pA$ and 0.30 $\mathrm{nb^{-1}}$ of $Ap$ collisions. The
double-differential production is measured for prompt
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ and
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}\leavevmode\nobreak\
\mathrm{from}\leavevmode\nobreak\ b$ in bins of the kinematic variables $y$
and $p_{\rm T}$. The measured cross-sections for prompt
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ production
integrated over $y$ and $p_{\rm T}$, scaled by a factor of $1/A$ and rescaled
to the common rapidity range of the proton-nucleon system $2.5<y<4.0$ for $pp$
and $pA$ collisions and $-4.0<y<-2.5$ for $Ap$ collisions are shown is Fig.12
as a function of $\sqrt{s_{\rm NN}}$. The results indicate that the
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ production cross-
section is suppressed in proton-lead collisions, as expected from previous
experiments; however such suppression is less enhanced in the backward region.
One can also derive the attenuation factor $R_{pA}(y,\sqrt{s_{\rm
NN}})=\frac{1}{A}\frac{\frac{{\rm d}\sigma_{pA}}{{\rm d}y}(y,\sqrt{s_{\rm
NN})}}{\frac{{\rm d}\sigma_{pp}}{{\rm d}y}(y,\sqrt{s_{\rm NN})}}$ by using for
the ${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ production
cross-section in $pp$ collisions at 5 $\,\mathrm{TeV}$ a linear interpolation
of the published measurements at $\sqrt{s}=2.76$ LHCb-PAPER-2012-039 , 7 LHCb-
PAPER-2011-003 , and $8\,\,\mathrm{TeV}$ Aaij:2013yaa . The result is shown in
Fig.13 and compared to theoretical predictions Arleo:2012rs . Within the
available statistics good agreement is observed.

Figure 12: Integrated cross-sections for prompt
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ production scaled by
a factor of $1/A$ and rescaled to the common rapidity range of the proton-
nucleon system $2.5<y<4.0$ for $pp$ and $pA$ collisions and $-4.0<y<-2.5$ for
$Ap$ collisions as a function of $\sqrt{s_{\rm NN}}$

.

Figure 13: Attenuation factor $R_{pA}$ compared to theoretical predictions
Arleo:2012rs . The black triangles are LHCb measurements, the red solid line
is the theoretical prediction based on parton energy loss effects, the blue
dashed line takes additional saturation effects into account.

##  7 Conclusion

LHCb has performed a wealth of measurements on quarkonium production, and many
more are coming! A few selected ones were briefly illustrated in this report.

Measurements of charmonium LHCb-PAPER-2011-003 ; LHCb-PAPER-2012-039 ; LHCb-
PAPER-2011-045  and bottomonium LHCB-PAPER-2011-036 ; Aaij:2013yaa  production
were performed at various centre-of-mass energies. They allow an in depth
comparison with theoretical models. A simple CSM is disfavoured by the data,
while a combination of CS and CO, as implemented in the NRQCD formalism, or CS
improved by QCD corrections, provide a good description of prompt quarkonium
production. Charmonium production from $b$-hadron decays is very well
reproduced by FONLL calculations. Some other studies were illustrated in this
report, such as charmonium exclusive production Aaij:2013jxj , double
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ production
Aaij:2011yc , as well as first preliminary results on
${\mathrm{J}\mskip-3.0mu/\mskip-2.0mu\uppsi\mskip 2.0mu}$ production in
proton-lead collisions at $\sqrt{s_{\rm NN}}=5\,\mathrm{TeV}$ LHCb-
CONF-2013-008 . Many of these results are based on the analysis of a partial
dataset and will gain in precision with the analysis of the full data sample
available to LHCb.

##  8 Acknowledgments

I would like to thank the Organizers of LHCP 2013 for their kind invitation,
my LHCb colleagues for providing the material discussed here and, in
particular, Giulia Manca, Vanya Belyaev and Burkhard Schmidt for their careful
reading of this article.

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