# Drag anisotropy of cylindrical solids in fluid-saturated granular beds

Ankush Pal and Arshad Kudrolli  Department of Physics, Clark University,
Worcester, MA 01610, USA

###### Abstract

We study the direction-dependent drag acting on a cylindrical solid intruder
with length $L$ and diameter $D$ as it moves in water-saturated granular beds
at constant depth. Polysterene and hydrogel grains with diameter $d$ are used
to investigate materials which have high contact friction and those which are
nearly frictionless, respectively. The drag on the intruder is measured while
oriented perpendicular $F_{\perp}$ and parallel $F_{\parallel}$ to its axis as
a function of speed $U$ from the quasi-static to the rate-dependent regime. We
find that the drag anisotropy $\xi=F_{\perp}/F_{\parallel}$ is not constant,
and increases significantly with driving rate and $L/D$ in both mediums. In
particular for $L/D=40$, $\xi$ increases from 2.6 to 4.5, and from 7.0 to 8.2
in the high and low friction beds, respectively, as the nondimensional Froude
number $Fr=U/\sqrt{g(D+d)}$ is varied between $10^{-4}$ and $2\times 10^{-2}$.
On average, $\xi$ is observed to increase logarithmically with $L/D$ for
$L/D\gg 1$. Exploiting the index-matched nature of hydrogel grains in water,
we further show that the sediment flow around the cylinder in the two
orientations is consistent with skin friction dominated drag. The relative
contributions of the cylindrical side and the circular flat-ends on $\xi$ are
estimated with thin disks to understand the observed variation of drag with
aspect ratio and surface friction.

##  I Introduction

The drag experienced by an intruder moving in a granular bed is fundamental to
probing the unsteady rheology of granular materials and important for
engineering mixers, diggers, and ploughs across a wide range of industries
from mining to food processing [1, 2, 3, 4]. The direction-dependent drag
experienced by a cylindrical solid in a granular bed is important to
understanding anchoring and drag-assisted propulsion of organisms and robots
in sandy terrestrial and submarine environments [5, 6, 7, 8].

The drag acting on an intruder depends on a number of factors including its
shape and material properties, and those of the medium. For a circular
cylinder with a length $L$ much greater than its diameter $D$, the drag while
moving with speed $U$ perpendicular and parallel to its axis are given in the
Stokes regime by [9, 10, 11]:

$F_{\perp}=-\frac{4\pi\eta L}{\ln(L/D)+\gamma_{\perp}}U,$ (1)

and

$F_{\parallel}=-\frac{2\pi\eta L}{\ln(L/D)+\gamma_{\parallel}}U,$ (2)

respectively, where, $\eta$ is the viscosity of the medium, and
$\gamma_{\parallel}\approx-0.114$ and $\gamma_{\perp}\approx 0.866$ are
constants which further depend weakly on the aspect ratio $A_{r}=L/D$. Thus,
while the drag is proportional to the relative speed, the drag anisotropy
given by the ratio of the drag while moving perpendicular to its axis
$F_{\perp}$, and the drag while moving parallel to its axis $F_{\parallel}$,
i.e.

$\xi=\frac{F_{\perp}}{F_{\parallel}},$ (3)

is approximately $2$ in the case of long thin rods, independent of speed. The
fact that $\xi\neq 1$ is important for example to swimming, because it enables
a net propulsive force while using cyclic non-reciprocal strokes of the body,
with a greater difference leading to faster calculated locomotion speeds [12,
13].

Granular sediments display a yield stress and shear thinning rheology
different from a Newtonian fluid, and $\xi$ is therefore expected to be
different. The drag experienced by spherical and cylindrical intruders has
been observed to scale linearly with the overburden weight of the sediments
and crosssectional area when the intruder size is much greater than the
sediment size [14, 15, 16]. The dynamic force resulting from granular
collisions becomes important with speed and has been calculated in dry
granular materials [17]. Presence of interstitial liquid influences the drag
in granular beds [18, 19, 20], but remains relatively less studied compared to
the dry granular case [21, 22]. Various $\xi$ have been reported in studies
with granular beds composed of dry glass beads including $\xi\approx 3$ in
experiments with rods with $L/D\approx 2.5$ after subtracting end effects [23,
5, 24]. Whereas, $\xi$ ranging between 5 and 7 were measured for long thin
rods moving slowly through granular hydrogels sedimented in water motivated by
burrowing and biolocomotion [8]. Thus, it is difficult to have an integrated
picture from these reports because of their different focus, and a systematic
study examining the issue of drag anisotropy itself in granular mediums is
still necessary.

In this paper, we report systematic measurements of drag and drag anisotropy
experienced by cylindrical solids moving in two complementary fluid-saturated
sediment beds with high and low grain-grain contact friction. The drag from
the essentially quasistatic regime observed at low speeds, to the rate-
dependent friction regime at higher speeds is analyzed. The drag anisotropy is
measured as a function of aspect ratio of the cylinders and compared across
the two kinds of mediums. We analyze the relative contribution of the drag
acting on the end-faces of the cylinders and the shear drag on the cylindrical
surface of the solid by contrasting with measured drag in the thin disk limit.
We visualize the fluorescence tagged grains around the cylinder in the case of
the transparent hydrogel medium and illustrate the nature of the flow in the
two orientations which gives rise to the observed drag.

##  II Experimental System

Figure 1: (a) A schematic diagram of a cylindrical intruder dragged at constant depth inside a rectangular tank filled with granular sediments immersed in water.  The diagram shows the intruder moving while oriented perpendicular to its axis ($\theta=90^{0}$) to measure $F_{\perp}$. Complementary measurements are also performed while the intruder moves parallel to its axis ($\theta=0^{0}$) to measure the drag $F_{\parallel}$. (b) Measured torque $T$ as a function of time $t$ for a cylinder dragged through water-saturated polystyrene bed ($A_{r}=5$; $z_{p}=4$ cm; $U=1$ mm/s). The axis of the cylinder is perpendicular to the direction of motion. The drag is calculated by averaging the torque measured over the time interval indicated by the red box, once steady state is established.  Length $L$ (cm) | Diameter $D$ (cm) | Aspect Ratio $A_{r}$
---|---|---
8 | 1.6 | 5
8 | 0.8 | 10
16 | 0.8 | 20
16 | 0.4 | 40
Table 1: The length $L$, diameter $D$, and aspect ratio $A_{r}=L/D$ of the
cylindrical intruders used to measure drag. Measurements were also performed
with a set of thin disks ($L=3$ mm) with the same set of diameters to estimate
the relative contributions of the circular ends.

A schematic of the experimental system used to measure drag is shown in Fig.
1(a). Cylindrical solids composed of clear resin were fabricated using a 3D
printer (Formlabs Form 3). Intruders with dimensions listed in Table 1 are
used to vary aspect ratio $A_{r}=L/D$. The intruder is attached rigidly to a
thin steel arm, which is connected to a torque sensor (Mark 10 Model 5i)
placed on a translating stage with a computer controlled stepper motor
(Zaber). The intruder can be thus moved with a prescribed speed $U$ ranging
between 0.05 mm/s and 8 mm/s. Then, the flow inertia relative to the
gravitational field is given by the Froude Number ${\rm
Fr}=\frac{U}{\sqrt{g(D+d)}}$ [25], where $g=9.8$ m/s2, ranges between
$10^{-4}$ and $2\times 10^{-2}$ over the system parameters investigated. The
rod is immersed in the granular bed with its major axis oriented perpendicular
or parallel to the direction of motion as it moves horizontally at constant
depth as shown in Fig. 1(a).

Spherical polystyrene grains with mass density $\rho_{g}=$ 1050 kg/m3, mean
diameter $d=0.45$ mm, and granular hydrogels with density $\rho_{g}$= 1005
kg/m3 and mean diameter $d=1$ mm immersed in distilled water with density
$\rho_{w}=997$ kg/m3, viscosity $\eta_{f}=1$ mPa s are used in the study. The
polystyrene and hydrogel grains have a coefficient of friction $\mu_{s}\approx
0.5$, and $10^{-3}$ [26], respectively, spanning the range typically observed
in granular mediums ranging from sand to tapioca. The medium is placed inside
a rectangular tank with a length $L_{c}=32$ cm, width $W_{c}=22$ cm, and
height $H_{c}=22$ cm, and settles to the tank bottom because
$\rho_{g}>\rho_{w}$. A granular volume fraction $\phi\approx 0.6$ is observed
in both kinds of sedimented beds consistent with previous observations [18,
19]. The depth of the bed is maintained at 17 cm in both cases, with a 4 cm
water layer above to eliminate any capillary effects. The drag of the water
layer is observed to be negligible compared to the sediment drag. For
consistency, the bed was initialized before each trial by stirring and
allowing the grains to settle over approximately 10 minutes for the
polystyrene grains, and 13 minutes for the granular hydrogels. The lab
temperature is controlled and maintained at $25^{\circ}$C $\pm\,2^{\circ}$C.

Figure 1(b) shows a measured torque $T$ over time $t$ once the translating
stage is prescribed to move. The torque increases rapidly before reaching a
maximum, corresponding to the yield stress of the medium. The torque then
falls and becomes nearly constant as the intruder moves with the prescribed
speed. This torque combines both the drag due to the intruder and the thin
arm. Therefore the torque due to the arm is measured separately, and
subtracted from the total torque. The drag is obtained by dividing the
resulting torque due to the intruder by the length of the arm.

Preliminary measurements were performed at various depths $z_{p}$ and drag was
observed to scale linearly with depth provided $z_{p}\gtrsim 2D$. This is
consistent with previous studies which found that drag in fluid-saturated
sediment beds scale with the overburden pressure
$P_{p}=\phi\left(\rho_{g}-\rho_{w}\right)gz_{p}$, where $g$ is gravitational
acceleration, in the quasi-static regime [2], and the rate-dependent regime as
well [27, 19]. Thus, we chose $z_{p}=4$ cm in the case of the polystyrene
grains, and $z_{p}=12$ cm in the case of the hydrogel beads, to optimize for
the sensitivity range of the torque sensor and to reduce the number of
experimental parameters to be explored. At this depth, the drag was measured
while the rods were away from the container walls in the front by $z_{p}$ to
avoid boundary effects which were otherwise observed to lead  to
systematically higher drag in the case of the high-friction grains [19]. We
estimated the inertial number $I=\frac{Ud}{D\sqrt{P_{p}/\rho_{g}}}$ [28, 27]
which measures relative importance of inertia of grains to imposed forces, and
viscous number $J=\frac{\eta_{f}U}{P_{p}D}$ [29, 19] which measures the
viscous effects of the interstitial fluid versus imposed forces. We find
$2\times 10^{-5}<I<2\times 10^{-2}$, and $1\times 10^{-7}<J<2\times 10^{-4}$
in the two mediums. Thus, grain inertia and viscous forces are small compared
with imposed forces in the experiments.

##  III Drag Measurements

Figure 2: (a-d) The measured drag $F_{\perp}$ and $F_{\parallel}$ for a
cylindrical rod with $A_{r}=5$ in water-saturated (a) polystyrene and (b)
hydrogel granular beds as a function of speed $U$. The drag approaches a non-
zero value at low speeds and increases with speeds in each case. The errors
bars correspond to RMS fluctuations observed over 6 to 12 trials each.  Figure
3: (a-d) The effective friction as a function of speed is plotted for
different aspect ratio of the cylinder in two mediums while oriented
perpendicular (a,b), and parallel (c,d) to its major axis.  A second-order
polynomial fit given by Eq. (6) is also shown in (a) and (b). The goodness of
fit is 0.99 in both cases. The linear term is observed to be at least an order
of magnitude smaller compared to the quadratic term in both mediums (see
text). Figure 4: (a-d) The effective friction in granular polystyrene beds and
hydrogel beds as a function of Froude Number ${\rm
Fr}=\frac{U}{\sqrt{g(D+d)}}$ while oriented perpendicular in beds composed of
(a,b), and while oriented parallel (c,d).  The data is described by a linear
fit to $\mu_{\perp}=\mu_{o}+\eta\,{\rm Fr}$, where $\mu_{o}=8.9$ and 1.7,
$\eta=242.5$ and 246.5, in the case of polystyrene, and hydrogel beds,
respectively. The goodness of fit is 0.99 in both cases.

Figure 2 shows $F_{\perp}$ and $F_{\parallel}$ moving in the two mediums
corresponding to a cylinder with aspect ratio $A_{r}=5$ over the wide range of
speeds studied. Each measurement of drag corresponds to an average over at
least 6 to 12 trails, and the error bars noted correspond to the root mean
square (RMS) deviation from the mean. We observe that $F_{\perp}$ is much
greater than $F_{\parallel}$ over the entire range of speeds, and approach
non-zero values at vanishing speeds consistent with yield stress materials in
both low and high friction grains [21]. Further, $F_{\perp}$ and
$F_{\parallel}$ increases with speed showing a rate-dependent regime where
viscous and collisional stresses can be expected to become important. In order
to estimate the relative contribution of skin friction drag and pressure drag,
we estimate the force due to pressure difference as $F_{p}=\frac{1}{2}\rho
U^{2}LD$, and find that the ratios $F_{\perp,\parallel}/F_{p}\gg 10^{2}$ and
$10^{3}$, in the case of polysterene and hydrogel grains, respectively. Thus,
it appears that the friction drag dominates, and that the pressure drag is
unimportant over the range of speeds in our study.  Pressure drag is typically
dominant when there is separation of boundary layer rear of the intruder.
However, the visualization of flow (see Fig. 8) around the cylinder does not
reveal any vortex or wake, supporting the fact that pressure drag is not
significant in our system.

To cast the measured drag in nondimensional form, we obtain the effective
friction as the ratio of the drag, and the weight of the granular bed given by
the overburden pressure $P_{p}$ acting on the projected area of the cylinder
$LD$. Then,

$\mu_{\bot}=\frac{F_{\bot}}{P_{p}DL},$ (4)

and

$\mu_{\|}=\frac{F_{\|}}{P_{p}DL}.$ (5)

With these definitions, we have $\xi=\mu_{\bot}/\mu_{\|}$ consistent with
$\xi=F_{\bot}/F_{\|}$.

Figure 3 shows a plot of the measured effective friction as a function of
speed in both orientations for the two mediums. In the quasi-static limit, we
observe $\mu_{\bot}\sim 9$ in the case of the polystyrene grains which have
high grain surface friction. A significantly lower $\mu_{\bot}\sim 1.5$ is
observed in the hydrogel grains which are nearly frictionless. An effective
friction $\mu_{\bot}\sim 13$ has been reported in the quasi-static limit of a
steel rod moving in the perpendicular orientation in dry glass beads which
have a grain surface friction $\mu_{s}$ in the range 0.5 and 0.7, more
comparable to polystyrene [15]. Further, Fig. 3(a,b) shows that $\mu_{\bot}$
increases roughly together with increasing speed for the intruders with
various $A_{r}$ in both mediums. While Fig. 3(c,d) shows that $\mu_{\|}$ also
increases with speed, it is systematically lower compared with $\mu_{\bot}$.
However, it can be observed that $\mu_{\|}$ does not collapse, but is lower at
higher $A_{r}$. Thus, the effective friction encountered by the intruder can
be rate dependent besides varying with its aspect ratio, and grain-grain
friction properties of the medium.

We fit $\mu_{\perp}$ with a second-order polynomial given by

$\mu_{\perp}=\mu_{o}+\gamma_{1}U+\gamma_{2}U^{2},$ (6)

where $\mu_{o}$ corresponds to the yield-stress, $\gamma_{1}$ and $\gamma_{2}$
are fitting constants corresponding to the linear and quadratic terms in $U$.
From the fits to Eq. (6) shown in Fig. 3(a,b), we find $\gamma_{1}=0.63$ s/mm
and $\gamma_{2}=0.029$ s2/mm2 in the case of polystyrene beds, and
$\gamma_{1}=1.16$ s/mm and $\gamma_{2}=0.04$ s2/mm2 in the case of hydrogel
beds. Since $\gamma_{2}\ll\gamma_{1}$, the linear term dominates considering
that $U<10$ mm/s in our experiments. Thus, the effective friction encountered
by the cylindrical intruder while moving perpendicular to its axis can be
described as increasing linearly starting from a value corresponding to the
yield stress over the range speeds investigated in our experiments in both
mediums.

Because various $D$ are used in the investigation of the effect of aspect
ratio, we use the Froude Number ${\rm Fr}$ to nondimensionalize the speed
variable, and plot $\mu_{\bot}$ and $\mu_{\|}$ versus ${\rm Fr}$ in Fig. 4 in
the two mediums. While the collapse is arguably better in the perpendicular
direction, the systematic variation in the parallel orientation remain even
after accounting for the differences in the diameters.  Because the increase
in $\mu_{\perp}$ could be described by the linear term in Eq. (6), we fit
$\mu_{\perp}$ as a function of Fr in Fig. 4(a,b) to
$\mu_{\perp}=\mu_{o}+\eta\,{\rm Fr}$, where $\mu_{o}=8.9$ and 1.7,
$\eta=242.5$ and 246.5, in the case of polystyrene, and hydrogel beds,
respectively. (We also plotted $\mu_{\bot}$ and $\mu_{\|}$ as a function of
the viscous number $J$ and the inertial number $I$ in the case of the
polysterene and hydrogel grains, respectively, but did not find any
significant improvement in the collapse.) Thus, these plots show that while
$\mu_{\bot}$ is independent of the aspect ratio over $5\leq L/D\leq 40$,
$\mu_{\|}$ decreases over the same range. This is in contrast with a Newtonian
fluid in the viscous regime, where a little over 50% reduction can be
calculated in the drag per unit length in both orientations according to Eqs.
(1) and (2) over the same range of $L/D$.

##  IV Drag Anisotropy

Figure 5: The drag anisotropy $\xi=F_{\perp}/F_{\parallel}$ for cylinder with
various aspect ratio as a function of speed in water-saturated granular (a)
polystyrene and (b) hydrogel beds. The error bars correspond to RMS values.

We plot $\xi=F_{\perp}/F_{\parallel}$ to analyze the degree of drag anisotropy
in the two mediums as a function of speed for the various intruders in Fig. 5,
and also list them in Table 2. We observe that the overall drag anisotropy is
significantly greater in the low-friction hydrogels compared to the high-
friction polystyrene. Further, we observe that $\xi$ appears to be more or
less constant at lower $A_{r}$. Whereas, $\xi$ increases with speed as $A_{r}$
increases in the case of the hydrogel bed, and more so in the case of the
polystyrene bed. Thus, $\xi$ in both kinds of granular sediments are greater
than that for a Newtonian fluid, and further show significant rate and aspect
ratio dependence. We understand from Fig. 3 that this dependence arises from
the fact that $\mu_{\|}$ decreases with $A_{r}$, but $\mu_{\bot}$ remains
relatively unchanged with $A_{r}$. The rate dependence of $\xi$ can be linked
to the fact that $\mu_{\|}$ changes more slowly compared with $\mu_{\bot}$
with increasing speed.

Polystyrene

$A_{r}$ = 5 |  $A_{r}$ = 10 |  $A_{r}$ = 20 |  $A_{r}$ = 40
---|---|---|---
$U$ (mm/s) | $\xi$ | $\overline{\xi}$ |  $U$ (mm/s) | $\xi$ | $\overline{\xi}$ |  $U$ (mm/s) | $\xi$ | $\overline{\xi}$ |  $U$ (mm/s) | $\xi$ | $\overline{\xi}$
0.05 | 2.1 $\pm$ 0.2 | 2.2 | 0.05 | 2.3 $\pm$ 0.5 | 2.6 | 0.05 | 2.4 $\pm$ 0.5 | 2.8 | 0.05 | 2.6 $\pm$ 0.3 | 3.1
0.1 | 2.1 $\pm$ 0.4 | 0.1 | 2.3 $\pm$ 0.3 | 0.1 | 2.4 $\pm$ 0.5 | 0.1 | 2.5 $\pm$ 0.3
1 | 2.1 $\pm$ 0.3 | 1 | 2.4 $\pm$ 0.4 | 1 | 2.6 $\pm$ 0.6 | 1 | 2.7 $\pm$ 0.3
4 | 2.2 $\pm$ 0.3 | 4 | 2.8 $\pm$ 0.6 | 4 | 2.8 $\pm$ 0.8 | 4 | 3.3 $\pm$ 0.5
8 | 2.3 $\pm$ 0.5 | 8 | 3.2 $\pm$ 0.9 | 8 | 3.9 $\pm$ 0.9 | 8 | 4.5 $\pm$ 1

Hydrogel

$A_{r}$ = 5 |  $A_{r}$ = 10 |  $A_{r}$ = 20 |  $A_{r}$ = 40
---|---|---|---
$U$ (mm/s) | $\xi$ | $\overline{\xi}$ |  $U$ (mm/s) | $\xi$ | $\overline{\xi}$ |  $U$ (mm/s) | $\xi$ | $\overline{\xi}$ |  $U$ (mm/s) | $\xi$ | $\overline{\xi}$
0.05 | 3.8 $\pm$ 0.3 | 3.6 | 0.05 | 3.8 $\pm$ 1 | 3.9 | 0.05 | 4.9 $\pm$ 0.4 | 5.5 | 0.05 | 7.0 $\pm$ 1.3 | 7.4
0.1 | 3.0 $\pm$ 0.4 | 0.1 | 4.4 $\pm$ 1 | 0.1 | 4.9 $\pm$ 1.3 | 0.1 | 7.1 $\pm$ 1.7
1 | 3.2 $\pm$ 0.5 | 1 | 3.4 $\pm$ 0.6 | 1 | 4.5 $\pm$ 0.2 | 1 | 6.9 $\pm$ 1.4
4 | 3.8 $\pm$ 0.2 | 4 | 3.8 $\pm$ 0.5 | 4 | 6.1 $\pm$ 0.3 | 4 | 7.8 $\pm$ 0.6
8 | 4.1 $\pm$ 0.1 | 8 | 4.0 $\pm$ 0.3 | 8 | 7.3 $\pm$ 0.5 | 8 | 8.2 $\pm$ 0.5

Table 2: The measured drag anisotropy of cylinders of various aspect ratios
moving through water-saturated sedimented polystyrene and hydrogel mediums.

To show the overall trends with $A_{r}$, we average over $\xi$ observed across
$U$, and plot the resulting average drag anisotropy $\overline{\xi}$ in Fig.
6. The minimum and maximum range of $\xi$ observed over $U$ are shown with
error bars. We clearly observe that $\overline{\xi}$ increases with $A_{r}$ in
both mediums. While $\overline{\xi}$ increases from about 2.2 to $3.1$ in the
high-friction polystyrene, $\overline{\xi}$ increases from 3.6 to 7.4 in the
low-friction medium as $A_{r}$ increases from 5 to 40. Noting the fact that
the drag appears to scale with $L$ in the perpendicular orientation, we fit
the data with the logarithmic form inspired by Eq. (2), i.e.

$\overline{\xi}=\alpha\ln(L/D)+\beta$ (7)

where, $\alpha$ and $\beta$ are material dependent fitting constants. We fit
$\overline{\xi}$ in Fig. 6 for $L/D\geq 5$, and find that it captures the
overall trends well with a stronger dependence on the $L/D$-term in the case
of the low-friction hydrogel grains. The fact that $\alpha$ is much greater in
the high-friction material compared to the low-friction material is notable,
and further work is needed to fully explore the connection between the implied
stronger dependence and material friction.

Figure 6: The average drag anisotropy over speed $\overline{\xi}$ is observed
to increase with $A_{r}$, and is greater in low-friction hydrogels compared
with high-friction polystyrene grains. The data is described by the logarithm
fit (dashed lines) given by Eq. 7 (Polystyrene: $\alpha=0.44,\beta=1.49$;
Hydrogel: $\alpha=1.89,\beta=0.085$).

##  V Cylindrical side versus flat end drag

Figure 7: (a,b) The fractional disk to cylinder drag $\psi_{\perp,\parallel}$
given by the ratio of $F_{\perp,\parallel}$ corresponding to the disk and the
$A_{r}=5$ cylinder with same diameter as a function of speed for (a)
polystyrene, and (b) hydrogels. The fractional contribution is nearly constant
across intruder velocity. (c,d) The fractional disk to cylinder drag averaged
over speeds $\overline{\psi}$ decreases with $A_{r}$ in both polystyrene (c),
and hydrogels (d).

Besides the stresses acting on the cylindrical sides of the intruder, the drag
acting on the flat circular ends is important to determining the total drag in
the two orientations, and thus the drag anisotropy. In order to estimate the
relative contribution of these surfaces, we manufactured 2 to 3 mm thick
circular disks with the same set of diameters as the cylindrical intruders. We
measure $\overline{\xi}\approx 0.35$ and $0.53$ corresponding to
$A_{r}=0.3/1.6\approx 0.2$, in the case of the hydrogel and polystyrene
mediums, respectively. These values of $\overline{\xi}<1$ may be expected when
$L/D\ll 1$ because only the shear stress on the end-faces contribute to drag
in the disk limit while in the perpendicular orientation, whereas the normal
stress on the end-faces remained essentially unchanged while in the parallel
orientation. Further, $\overline{\xi}<1$ can be also expected, extrapolating
from the fact that $\overline{\xi}\sim 1$ when $L/D\sim 1$.

Figure 7(a,c) shows the fraction of drag $\psi_{\perp,\parallel}$
corresponding to the ratio of drag of a thin disk with $A_{r}=0.2$ and a
cylinder with $A_{r}=5$ across the range of $U$ investigated in the two
mediums. We find in both mediums that the $\psi_{\perp}$ is less than 20%
across the range of $U$ studied. Whereas $\psi_{\parallel}$ is about 50% and
60% in the polystyrene and hydrogel mediums, respectively. We further plot the
fractional disk to cylinder drag averaged over speeds $\psi_{\perp}$ and
$\psi_{\parallel}$ as a function of $A_{r}$ in Fig. 7(c,d). These ratios are
observed to decrease rapidly with $A_{r}$. Thus, the drag contributions of the
ends in the parallel orientation are always greater and decay slower, than
when in the perpendicular orientation. As $A_{r}$ increases above 20, the
contribution of the ends decrease to less than 10%. By $A_{r}=40$, the
contributions of the cylinder ends to drag were found to be less than the
sensitivity of our instruments. Since $\mu_{\bot}$ continues to decrease on
average at these large aspect ratios, the increase in $\xi$ appears to be
linked to the continued evolution of the tangential stresses acting on the
cylindrical surface of the intruder while oriented in the parallel
orientation.

Although the number of sediment grains in contact with the cylinder at a given
time instant can be assumed to be approximately similar in both orientations,
the number of grains which actually need to rearrange as the intruder moves
through the same distance is vastly different when $L/D\gg 1$.  The time scale
over which a typical sediment grain is sheared by the intruder can be expected
to scale as $D/U$ in the perpendicular orientation, and as $L/U$ in the
parallel orientation. Thus, a smaller fraction of granular sediments are
sheared over a sustained longer time in the parallel orientation compared to
the perpendicular orientation as the intruder moves by. The resulting
difference in shear induced dilatancy and the lowering of local granular
packing fraction [30] may be the reason for the lower $\mu_{\|}$ compared to
$\mu_{\bot}$, and why it continues to decrease with $A_{r}$.  Skin drag exists
for the cylinder moving in either orientation. The granular medium fluidized
by the advancing intruder stays attached to the intruder over a longer length
while orientated parallel to the direction of motion. Because a lubrication
layer between grains can occur when the grains are in motion, this may result
in a lower skin friction, and hence in a higher $F_{\perp}$ compared to
$F_{\parallel}$.

##  VI Flow profiles

Figure 8: The granular flow in the frame of reference of a cylindrical
intruder moving (a) perpendicular and (b) parallel to its axis ($D=1.6$ cm;
$L=8$ cm; $A_{r}=5$; $U=1$ mm/s). The flow field is superimposed on a time
integrated image obtained by tracing bright pixels over 50 s. A central plane
is illuminated using a laser sheet. The flow can be observed to be nearly
symmetric about the central vertical axis. Systematic asymmetry can be
observed above and below the intruder because of gravitational symmetry
breaking.

We examine the sediment flow around the cylinder while moving in the two
orientations to complement the $\xi$ measurements. The flow around the
cylinder moving through water-saturated hydrogels grains can be visualized
exploiting the fact that their refractive index nearly matches that of water.
This method enables us to measure the flow in a frame of reference in which
the intruder is fixed to make the flow measurements simpler. A smaller 15 cm
cubical tank is used for this purpose and is filled with hydrogel beads to a
depth of 12 cm with a 3 cm water layer on top. The tank is placed on a linear
track which can move forward or backward with the same velocities as those
used in measuring drag. The cylindrical intruder with $L/D=5$ is attached with
a fixed arm and placed inside the water saturated granular particles at a
depth $z_{p}=6$ cm. About 10% of the hydrogel beads were hydrated with water
mixed with a dye (Rhodamine 6G, Eastman) to use as tracers and mixed in with
the rest of the grains. A thin laser sheet (Z Laser, Germany) illuminates the
fluorescent dye soaked hydrogel beads around the intruder. A low-light 16-bit
camera (C 11440, Hamamatsu) fitted with a lens and bandpass filter captures
the motion of the hydrogel beads around the cylinder as the rod is moved
relative to the medium. A sequence of images between 1.25 and 30 frames per
second was captured as $U$ was increased from 0.05 mm/s to 8 mm/s, when the
container with the medium was moved through 5 cm.

Figure 8 shows the granular flow observed past the cylindrical intruder while
oriented perpendicular and parallel to the direction of relative motion with
speed 1 mm/s. The captured images are time averaged over a 50 second time
interval using ImageJ to generate bright steaks corresponding to the
fluorescent grains to illustrate the flow past the cylinder. No vortices are
observed in the wake which would signal the importance of form drag over the
entire range of speeds studied. Similar flow profiles are observed over the
entire range of speeds studied, and appear to be consistent with those where
drag is dominated by skin friction.  This is in contrast to the observation of
vortex structures behind a sphere-shaped intruder moving through similar
granular sediments [31]. However, the diameter of the intruders studied there
were considerably larger. Thus, the differences in the geometry of the flow,
and greater $D$ appear to imply that the greater speeds needed to typically
see vortices in dense granular flows [32, 33], are not reached in our
experiments.

We tracked the tracer grains by using standard software which finds the
centroid associated with the bright pixels and their motion over consecutive
frames [34]. We then obtained the mean velocity over approximately $1{\rm
cm}\times 1{\rm cm}$ area grid points, and further time-averaged while the
container and the medium are moved. These fields are also superimposed on the
images in Fig. 8. The flow appears relatively symmetric about the central
vertical axis through the intruder, and approaches the imposed speed
sufficiently far away from the cylinder. Whereas, differences can be observed
in the relative flow above and below the intruder, with the stagnation of flow
occurring below the central axis of the cylindrical rod below the center while
moving in either orientation. The magnitude of the sediment field velocity was
found to reach $U$ within the 20% accuracy of our measurements at a distance
of $2D$ in front and behind the intruder, and in a distance of $D$ above and
below the intruder in both orientations.

We measure the horizontal line corresponding to the stagnation points to be
$\approx 0.2D$ below the cylinder axis while oriented perpendicular, and to be
also $\approx 0.2D$ below the cylinder axis while oriented parallel. This
symmetry breaking occurs because of gravity which results in a greater $P_{p}$
with increasing depth leading the flow resistance to be systematically lower
above the rod. Some of these features of the flow have been noted previously
in dry granular flow simulations past a cylinder oriented with its axis
perpendicular to the flow in the quasi-static regime [15]. Thus, while we are
unable to visualize the flow in the case of the opaque polystyrene medium, we
expect these features of the flow illustrated in Fig. 8, to be present there
as well.

Significant slip can be observed past the smooth sides of the intruder in Fig.
8, similar to previous observations of granular hydrogels around spherical
solid intruders [31], as well as in the case of the motion of glass beads
moving past smooth glass surfaces which have significantly higher surface
friction [35]. From the nature of these observed flows, one may expect that
the drag due to normal and tangential stress distributions on the intruder
would continue to contribute to the drag as $L/D$ increases in the
perpendicular orientation, whereas only the drag due to tangential stress
distribution may be expected to remain relatively important in the parallel
orientation. We anticipate that the normal and tangential stress distribution
scale differently in the two orientations from the observations that $\xi$
increase systematically with intruder speed and aspect ratio. The tangential
stress given by the effective viscosity and the strain rate can be expected to
change [30, 19] as the moving intruder increasingly fluidizes the athermal
granular medium with speed. However, further modeling work is required to
connect the observed flow profiles with the measured trends in $\mu_{\bot}$,
$\mu_{\|}$, and $\xi$.

##  VII Conclusions

In summary, we measured the direction-dependent drag of cylindrical intruders
moving in sediment beds, and found a significant variation of drag anisotropy
with aspect ratio and system material properties. The drag is examined from
the quasi-static regime to the rate-dependent regime when inertial effects are
unimportant. A higher drag is found in the granular sediments with higher
grain-grain contact friction, but the drag anisotropy is found to be
significantly greater in the lower-friction hydrogels. The drag anisotropy is
observed to be greater in both mediums compared to that in a Newtonian fluid
in the Stokes regime, and systematically increases with speed. On average,
drag anisotropy is observed to increase consistent with a logarithmic form as
the aspect ratio of the cylinder increases in both kinds of mediums. This form
of increase appears to arise due to the relative decrease of the drag per unit
length in the parallel orientation, while the drag per unit length in the
perpendicular direction remains relatively unchanged for sufficiently large
aspect ratios. Thus, drag and drag anisotropy of a cylindrical solid moving in
granular sediments with wide ranging material properties is qualitatively
different than in a viscous Newtonian fluid.

We also evaluated the contribution of the flat circular cylinder ends on the
measured drag in parallel and perpendicular orientations by constructing thin
disks with the same diameters. We observe that the end-faces make a greater
contribution in the parallel orientation compared to the perpendicular
orientation at lower aspect ratios. We visualize the flow of the granular
medium around the intruders in the two orientations and show that a symmetric
flow develops fore and aft of the intruder in both orientations. The flow is
observed to remain attached to the intruder surface with no visible vortices
in either orientations over the range of speeds investigated.

Based on the observations on the contributions of end-faces using the disks,
and the fore-aft symmetry in the flow fields around the cylindrical intruders,
it appears that skin friction dominates the drag in granular sediments in our
experiments. While both normal and shear components of the flow are important
all along the length of the cylinder in the perpendicular orientation, the
shear component of the flow past the length of the cylinder dominates in the
parallel orientation, and the flow region around the ends is relativity small.
While our experiments cannot directly measure the normal and shear components
of the drag in the two mediums, we believe numerical simulations can access
this issue, and further illuminate the reason for the observed drag anisotropy
in the two complementary mediums.

###### Acknowledgements.

We thank Samuel Hoang and Benjamin Allen for help with experiments and
discussions, and Brian Chang for a critical reading of the manuscript and
feedback. This work was supported by the U.S. National Science Foundation
under Grant No. CBET-1805398.

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