11institutetext: Department of Astronomy, University of Maryland, College
Park, MD 20742 ; 22institutetext: X-ray Astrophysics Laboratory NASA/GSFC,
Greenbelt, MD 20771 33institutetext: Center for Research and Exploration in
Space Science and Technology, NASA/GSFC, Greenbelt, MD 20771, USA
44institutetext: AIM, CEA, CNRS, Université Paris-Saclay, Université de Paris,
F-91191 Gif sur Yvette, France

# The X-ray synchrotron rims in Cassiopeia A narrow with energy

A. Picquenot 112233 B. J. Williams 22 F. Acero 44 B. T. Guest 112233

###### Abstract

Context. Some young supernova remnants exhibit thin filaments of X-ray
synchrotron radiation coinciding with the forward shock due to accelerated
electrons interacting with the local magnetic field. The two main models
accounting for the radial brightness evolution of these filaments differ in
their prediction of the narrowing (or not) of the filaments with increasing
photon energy.

Aims. In this paper, we report our observation of such a narrowing of the
synchrotron filaments in Cassiopeia A at X-ray energies, and how this finding
could help in understanding the mechanisms at stake in their formation.

Methods. We used a new blind source separation method on the $1$ Ms Chandra
observation of Cassiopeia A, in order to obtain detailed and unpolluted images
of the synchrotron emission in three energy bands. We then extracted the
profiles of several filaments at the forward shock and the reverse shock to
estimate and compare their widths.

Results. We find that there is indeed a narrowing with energy of the
synchrotron filaments both at the forward and at the reverse shocks in
Cassiopeia A. The energy dependency of this narrowing seems stronger at high
energy, which is indicative of a damping effect, confirmed by radio
observations.

###### Key Words.:

ISM: supernova remnants – ISM: individual objects: Cassiopeia A - ISM:
magnetic field – ISM: structure

##  1 Introduction

Electrons accelerated in the forward shocks of young supernova remnants (SNR)
can emit synchrotron radiation. This emission is mostly seen at radio
wavelengths, but can also be seen in X-rays thanks to the fast ($\geq 3000$ km
s-1, see Aharonian & Atoyan, 1999) shocks of young SNRs. In some cases, the
X-ray brightness fades downstream and produces a bright shell-like structure
of rims and filaments.

A better understanding of these rims could help constrain the downstream
magnetic field structure. Current models use three physical effects to account
for the existence of the synchrotron filaments in X-ray: advection (bulk
motion of the plasma), diffusion (random motion of electrons at small scales),
and magnetic damping (where the X-ray emission reflects the magnetic field
morphology). The relative importance of these effects has an influence on the
evolution of the filaments widths with energy.

###  1.1 The evolution of rim widths with energy

The models describing the formation of X-ray synchrotron rims can mostly be
divided into two main categories, depending on how they take magnetic damping
effects into account. In the “loss-limited” models, the magnetic field is
considered constant over the width of the rim. Electrons only travel a certain
distance until they lose enough energy through diffusion and advection for
their radiation to drop below the X-ray band (Vink & Laming, 2003; Bamba et
al., 2005; Parizot et al., 2006). If the magnetic field is constant
downstream, most energetic electrons will radiate and cool more quickly
through advection, which results in a narrowing of the synchrotron rims with
energy. However, diffusion will dilute this effect: more energetic electrons
may diffuse further than would be expected from pure advection, weakening the
energy dependence of the widths at higher energies (Araya et al., 2010;
Ressler et al., 2014). Hence, ”loss-limited” models predict a narrowing of the
X-ray synchrotron filaments, with a weakening of the energy dependence with
energy.

“Loss-limited” models require a strong magnetic field amplification to
successfully account for the formation of thin rims. Pohl et al. (2005)
proposed that the rim profiles could reflect the magnetic field morphology,
adding a magnetic damping mechanism to the effects of diffusion and advection.
The magnetic field could be damped downstream of the shock and prevent
electrons from radiating efficiently, thus reducing synchrotron flux at all
wavelengths. Thus, a damping mechanism is supposed to produce relatively
energy-independent rim widths below a threshold energy and may decrease or
increase once advection and/or diffusion controls rim widths (Tran et al.,
2015). Hence, the evolution of these widths with energy is a constraint on the
magnetic field amplification and the amount of damping. However, there are
various physically possible damping mechanisms, so the relationship between
the filaments widths and the actual magnetic field variations is not
straightforward.

Ressler et al. (2014) observed synchrotron narrowing in SN$1006$ and concluded
that it was too strongly energy-dependent to be well described by the damping
mechanism only: the damping lengths would need to be larger than the
synchrotron-loss lengths. Tran et al. (2015) conducted a similar study in
Tycho’s SNR. They proposed a range of radio and X-ray rim profiles showing the
influences of both the magnetic field and the damping length. Their work
highlights the similarity between the effects of a strong magnetic field and a
small damping length on lowering the dependence in energy of the rim widths,
showing that it is difficult to probe a model with no further information on
the magnetic field. In the same paper, a moderate narrowing of the rim widths
was found in Tycho’s SNR in X-rays, but this was insufficient to constrain the
model. However, “loss-limited” models alone cannot account for the formation
of thin radio filaments: hydrodynamic models with diffusive shock acceleration
cannot produce radio profiles with narrow rims in a purely advected magnetic
field (Cassam-Chenai et al., 2007; Slane et al., 2014). The observation of
thin radio filaments hence proved that there is a damping effect in Tycho,
which is not necessarily sufficient to account for the narrowing in X-rays.

###  1.2 Thin X-ray rims in Cassiopeia A

Cassiopeia A (hereafter, Cas A) is among the most studied astronomical objects
at X-ray wavelengths. It benefits from extensive observations (about $3$ Ms in
total with Chandra) and it is surrounded by a synchrotron shell showing
filamentary structures, making it an ideal laboratory to investigate the
potential narrowing of synchrotron rims with energy.

(a)  Figure 1: Images of the synchrotron emission obtained with pGMCA in three
energy bands, with square root scaling. On the bottom right, the boxes used to
define the filament profiles. In black, profiles at the forward shock. In
green, profiles at the reverse shock. In blue, unidentified profiles.

Araya et al. (2010) investigated some properties of the synchrotron rims
visible at the forward shock in the $1$ Ms Chandra observation of Cas A,
including a comparison between the linear profiles of some filaments in the
$0.3$-$2.0$ keV, $3.0$-$6.0$ keV and $6.0-10.0$ keV energy bands. They only
found a slight difference in the widths between the $0.3$-$2.0$ keV and the
$3.0$-$6.0$ keV images, but none between the $3.0$-$6.0$ keV and the
$6.0$-$10.0$ keV images, as shown in Table $1$ of their paper. They also found
that the shape of the decline in emission downstream of the shock was similar
to that upstream, contrary to model predictions.

Here, we obtained more detailed images by using a new method to retrieve
accurate maps of the synchrotron emission around different energy bands. This
method is based on the General Morphological Components Analysis (GMCA, see
Bobin et al., 2015), a blind source separation algorithm that was introduced
for X-ray observations by Picquenot et al. (2019). It can disentangle
spectrally and spatially mixed components from an X-ray data cube of the form
$(x,y,E)$. The new images thus obtained suffer less contamination by other
components, such as thermal emission from lines or continuum. An updated
version of this algorithm, the pGMCA (see Bobin et al., 2020), has been
developed to take into account the Poissonian nature of X-ray data. It was
first used on Cas A data to probe the three-dimensional morphological
asymmetries in the ejecta distribution (Picquenot et al., 2021), and proved
perfectly suited for producing clear, detailed and unpolluted images of both
the ejecta and the synchrotron at different energies. Thanks to these new
images, we are able to study the profiles of some filamentary structures
associated with the forward shock, as well as find some associated with the
reverse shock. We labelled “upstream” and “downstream” the sides of the
filament profiles according to their location relative to the shock they are
associated with. However, the widths of the profiles do not necessarily
correspond to the actual width of the upstream and downstream shock as
projection effects might have an effect.

This paper is structured as follows. In Section 2, we will show the images of
the synchrotron we used, and the way the filaments profiles are defined. In
Section 3, we will present a way to quantify the narrowing of the filaments,
and discuss our results.

(a)  Figure 2: At top left, the northeast region of Cas A from our merged
observations. On the top right, the image of the synchrotron retrieved by our
method on the $2.5$ \- $4$ keV band. In both, the blue and green contours show
the regions of extraction, and the white rectangle shows the 4th reverse shock
box from Fig. 1. On the bottom, the extracted spectra from both regions, both
fitted with a simple phabs*powerlaw model in Xspec in red.

##  2 Using pGMCA to probe the synchrotron rims widths in Cas A

(a)  Figure 3: Linear profiles at the forward shock along the black boxes
presented in Fig. 1, and the models fitted to describe their widths, offset
for clarity. The profiles and the models are normalized so that the peaks
coincide, and the radius is given in arcsec. The arrows show the direction of
the forward shock (upstream is right of the plot). (a)  Figure 4: Linear
profiles at the reverse shock along the green boxes presented in Fig. 1, and
the models fitted to describe their widths, offset for clarity. The profiles
and the models are normalized so that the peaks coincide, and the radius is
given in arcsec. The arrows show the direction of the reverse shock (upstream
is left of the plot).

Being one of the brightest sources in the X-ray sky, Cas A is the perfect
extended source to showcase the capabilities of pGMCA. Cas A benefits from
years of extensive observations, and offers high statistics and strongly
overlapping components that the algorithm is well suited to disentangle. Here,
we used it to obtain three images of the synchrotron at different wavelengths,
from which we will derive the linear profiles of some wisely chosen
synchrotron rims.

###  2.1 Images definition

(a)  Figure 5: Unidentified linear profiles along the blue boxes presented in
Fig. 1, and the models fitted to describe their widths, offset for clarity.
The profiles and the models are normalized so that the peaks coincide, and the
radius is given in arcsec.

For our study, we used Chandra observations of the Cas A SNR, which was
observed with the ACIS-S instrument in 2004 for a total of 980 ks (Hwang et
al., 2004, ObsId 4634, 4635, 4636, 4637, 4638, 4639, 5196, 5319 and 5320). We
used only the 2004 data set to avoid the need to correct for proper motion
across epochs. The event lists from all observations were merged in a single
data cube. The spatial bin size is the native Chandra bin size of $0.5$
arcseconds, in order to produce images as detailed as possible. As we are not
interested in the spectral lines, we chose a spectral bin size of $58.4$ eV to
keep a good number of counts in every pixels.

We applied the pGMCA algorithm on three bandwidths: between $0.4$ and $1.7$
keV, between $2.5$ and $4$ keV and between $5$ and $8$ keV. We chose these
bandwidths on different criteria: large enough for pGMCA to work properly, not
too large in order not to affect a possible dependency in energy of the
filament widths, and we tried as much as possible to avoid any line emission
in the same bandwidths. This last criterion could not be fulfilled at lower
energies, and some pollution from other emission can be seen in the first
image displayed in Fig. 1, particularly in the southwest. The two other images
seem clear and unpolluted, even with a square root scaling. They probably
constitute the most detailed and accurate maps of the synchrotron in X-rays in
Cas A to this day, especially at low energy.

These images present filamentary structures throughout the ejecta, including
some following the known layout of the reverse shock. In order to assess the
non-thermal nature of these filaments, we extracted the spectrum of one and
fitted a phabs*powerlaw model in Xspec. The results are shown in Fig. 2,
together with a spectrum extracted from a bright region with low synchrotron
level fitted with the same model. The filament presents a spectrum that can be
well-fit with a simple power-law nonthermal model, while the other region
obviously cannot. This is consistent with the results from Helder & Vink
(2008), where synchrotron emission from filaments inside of the remnant was
attributed to the reverse shock.

Forward shock (Fig. 3)
FWHM (arcsec) Box Image Downstream Upstream Mean $m_{E}$ $0.4$-$1.7$ keV
$4.46\pm 0.42$ $5.02\pm 0.26$ $4.74\pm 0.25$ 1 $2.5$-$4.0$ keV $3.39\pm 0.23$
$5.06\pm 0.26$ $4.22\pm 0.17$ $-0.06\pm 0.04$ $5.0$-$8.0$ keV $2.72\pm 0.14$
$4.42\pm 0.19$ $3.57\pm 0.12$ $-0.24\pm 0.08$ $0.4$-$1.7$ keV $3.52\pm 0.41$
$5.62\pm 0.30$ $4.57\pm 0.25$ 2 $2.5$-$4.0$ keV $3.04\pm 0.29$ $5.55\pm 0.28$
$4.30\pm 0.20$ $-0.03\pm 0.04$ $5.0$-$8.0$ keV $2.76\pm 0.35$ $4.71\pm 0.41$
$3.73\pm 0.27$ $-0.20\pm 0.13$ $0.4$-$1.7$ keV $4.52\pm 0.31$ $6.50\pm 0.29$
$5.51\pm 0.21$ 3 $2.5$-$4.0$ keV $3.33\pm 0.16$ $5.36\pm 0.21$ $4.35\pm 0.13$
$-0.13\pm 0.03$ $5.0$-$8.0$ keV $3.22\pm 0.26$ $3.52\pm 0.29$ $3.37\pm 0.20$
$-0.37\pm 0.09$ $0.4$-$1.7$ keV $3.08\pm 0.25$ $1.61\pm 0.12$ $2.34\pm 0.14$ 4
$2.5$-$4.0$ keV $2.82\pm 0.25$ $1.63\pm 0.17$ $2.23\pm 0.15$ $-0.03\pm 0.05$
$5.0$-$8.0$ keV $2.25\pm 0.15$ $1.04\pm 0.13$ $1.64\pm 0.10$ $-0.44\pm 0.13$
$0.4$-$1.7$ keV $4.83\pm 0.42$ $4.46\pm 0.39$ $4.65\pm 0.29$ 5 $2.5$-$4.0$ keV
$4.20\pm 0.42$ $4.59\pm 0.45$ $4.39\pm 0.31$ $-0.03\pm 0.05$ $5.0$-$8.0$ keV
$2.92\pm 0.22$ $4.28\pm 0.29$ $3.60\pm 0.18$ $-0.29\pm 0.12$ $0.4$-$1.7$ keV
$6.80\pm 0.80$ $7.22\pm 0.51$ $7.01\pm 0.47$ 6 $2.5$-$4.0$ keV $5.87\pm 0.82$
$7.72\pm 0.57$ $6.80\pm 0.50$ $-0.02\pm 0.05$ $5.0$-$8.0$ keV $5.66\pm 0.89$
$7.74\pm 0.68$ $6.70\pm 0.56$ $-0.02\pm 0.16$ $0.4$-$1.7$ keV $4.54\pm 0.59$
$7.42\pm 0.83$ $5.98\pm 0.51$ 7 $2.5$-$4.0$ keV $4.25\pm 0.61$ $6.91\pm 0.34$
$5.58\pm 0.35$ $-0.04\pm 0.06$ $5.0$-$8.0$ keV $4.81\pm 1.06$ $7.18\pm 0.65$
$6.00\pm 0.62$ $0.10\pm 0.18$ $0.4$-$1.7$ keV $4.54\pm 0.18$ $5.41\pm 0.17$
$4.97\pm 0.12$ Mean $2.5$-$4.0$ keV $3.84\pm 0.17$ $5.26\pm 0.13$ $4.55\pm
0.11$ $-0.05\pm 0.02$ $5.0$-$8.0$ keV $3.48\pm 0.21$ $4.70\pm 0.16$ $4.09\pm
0.13$ $-0.21\pm 0.05$

Table 1: FWHMs and $m_{E}$ values derived from the models fitted on the
forward shock filaments profiles shown in Fig. 3. $m_{E}$ values are
calculated between the energy band of the line on which they are written and
the energy band of the line above. A negative $m_{E}$ indicates a filament
width narrowing with increasing energy. The values’ means are arithmetic, and
the errors are mean square.

###  2.2 The filaments linear profiles

Thanks to the highly detailed images of the synchrotron emission we were able
to find using pGMCA, we could investigate the narrowing with energy of the
filaments both at the forward shock and, for the first time, at the reverse
shock.

In order to compare the widths of the rims, we defined boxes surrounding small
regions crossing a filament at the forward shock, and did likewise for the
reverse shock. The boxes are shown in Fig. 1, and the normalized linear
profiles obtained perpendicularly to the rims are presented in Fig. 3 for the
forward shock, and in Fig. 4 for the reverse shock.

We also looked at some filaments whose positions could not allow us to label
clearly. A line-of-sight effect is likely at stake, and these filaments could
either be attached to the forward or to the reverse shock. The boxes are also
shown in Fig. 1, in blue, and the linear profiles obtained perpendicularly to
the rim are presented in Fig. 5.

##  3 Quantifying the narrowing of the synchrotron rims

###  3.1 Modeling the filaments linear profiles

In order to measure the widths of the filament profiles we extracted, we
fitted them with a piecewise two-exponential model, as in Tran et al. (2015) :

$h(r)=\left\\{\begin{array}[]{ll}A_{d}\exp\Big{(}\frac{r-r_{0}}{w_{d}}\Big{)}+C_{d},\quad
r<r_{0}\\\ A_{u}\exp\Big{(}\frac{r-r_{0}}{w_{u}}\Big{)}+C_{u},\quad r\geq
r_{0}\end{array}\right.$ (1)

All parameters are free, except for $A_{d}$ that is fixed to ensure continuity
at $r=r_{0}$ , with $A_{d}=A_{u}+(C_{u}-C_{d})$.

This model is well adapted to describe the sharp peak displayed by most
profiles, but some filaments (such as the $6$th or $7$th of the forward shock)
present plateaus or gaussian-like features that are not accounted for in this
model. However, this model remains overall a good way to estimate the widths
of the profiles we extracted without neglecting the asymmetry around the peak,
between the upstream and downstream media. The resulting fitted models are
displayed in Fig. 3 (forward shock), Fig. 4 (reverse shock) and Fig. 5 (other
profiles). In some cases, we had to remove secondary peaks in the profiles to
focus on the main one when fitting our model.

From the parameters of our model we can derive two Full Widths at Half Maximum
(FWHM) for each profile, FWHM${}_{u}=2\ln({2})w_{u}$ and
FWHM${}_{d}=2\ln({2})w_{d}$, describing the ”sharpness” of the profile on both
sides, a larger FWHM meaning a wider profile. The mean between these FWHMs
gives an estimation of the actual width of the profile. We also define $m_{E}$
:

$m_{E}=\frac{\ln\big{(}{\text{FWHM}_{2}/\text{FWHM}_{1}}\big{)}}{\ln\big{(}{E_{2}/E_{1}}\big{)}}$
(2)

Where FWHM1 and FWHM2 are the mean FWHMs of a same filament in two energy
bands, and $E_{1}$ and $E_{2}$ are the lower energy values for each energy
band. For each filament, we calculate the $m_{E}$ between the $0.4$-$1.7$ keV
and $2.5$-$4.0$ keV energy bands ($E_{1}=0.4$, $E_{2}=2.5$), and between the
$2.5$-$4.0$ keV and $5.0$-$8.0$ keV energy bands ($E_{1}=2.5$, $E_{2}=5.0$).
This parameter aims to evaluate the narrowing of the filaments widths and
quantify its dependence on energy: positive values mean widening, negative
means narrowing, larger values mean higher energy-dependence while weaker
values mean weaker energy-dependence.

The FWHMu, FWHMd, mean FWHM and $m_{E}$ values derived from our fitted models
are shown in Table 1 (forward shock) Table 2 (reverse shock) and Table 3
(other profiles). As the “upstream” and “downstream” labels do not make sense
for the profiles that were not clearly identified, we named both sides “right”
and “left,” the orientations corresponding to the plots of Fig. 5. As there is
no straightforward way to estimate the pGMCA algorithm errors, the errors
shown in our Tables are the model fitting errors and their propagation in the
calculation of the mean FWHM and $m_{E}$ values.

Reverse shock (Fig. 4)
FWHM (arcsec) Box Image Downstream Upstream Mean $m_{E}$ $0.4$-$1.7$ keV
$2.85\pm 0.22$ $1.33\pm 0.42$ $2.09\pm 0.24$ 1 $2.5$-$4.0$ keV $1.67\pm 0.44$
$2.14\pm 0.19$ $1.90\pm 0.24$ $-0.05\pm 0.09$ $5.0$-$8.0$ keV $1.49\pm 0.40$
$1.79\pm 0.25$ $1.64\pm 0.24$ $-0.22\pm 0.28$ $0.4$-$1.7$ keV $7.74\pm 0.43$
$6.55\pm 0.86$ $7.15\pm 0.48$ 2 $2.5$-$4.0$ keV $6.28\pm 0.76$ $6.43\pm 0.40$
$6.36\pm 0.43$ $-0.06\pm 0.05$ $5.0$-$8.0$ keV $5.82\pm 0.65$ $5.26\pm 0.34$
$5.54\pm 0.37$ $-0.20\pm 0.14$ $0.4$-$1.7$ keV $6.46\pm 1.42$ $2.66\pm 0.37$
$4.56\pm 0.73$ 3 $2.5$-$4.0$ keV $3.05\pm 0.43$ $9.06\pm 0.71$ $6.06\pm 0.42$
$0.15\pm 0.10$ $5.0$-$8.0$ keV $3.01\pm 0.41$ $5.85\pm 0.48$ $4.43\pm 0.32$
$-0.45\pm 0.14$ $0.4$-$1.7$ keV $2.41\pm 0.51$ $16.87\pm 1.11$ $9.64\pm 0.61$
4 $2.5$-$4.0$ keV $6.14\pm 0.61$ $1.73\pm 0.27$ $3.94\pm 0.33$ $-0.49\pm 0.06$
$5.0$-$8.0$ keV $4.53\pm 0.33$ $1.72\pm 0.23$ $3.13\pm 0.20$ $-0.33\pm 0.15$
$0.4$-$1.7$ keV $2.97\pm 1.20$ $1.43\pm 0.75$ $2.20\pm 0.71$ 5 $2.5$-$4.0$ keV
$1.37\pm 0.27$ $1.74\pm 0.17$ $1.55\pm 0.16$ $-0.19\pm 0.18$ $5.0$-$8.0$ keV
$1.49\pm 0.19$ $1.52\pm 0.12$ $1.50\pm 0.11$ $-0.05\pm 0.18$ $0.4$-$1.7$ keV
$4.49\pm 0.40$ $5.77\pm 0.34$ $5.13\pm 0.26$ Mean $2.5$-$4.0$ keV $3.70\pm
0.24$ $4.22\pm 0.18$ $3.96\pm 0.15$ $-0.13\pm 0.05$ $5.0$-$8.0$ keV $3.27\pm
0.19$ $3.23\pm 0.14$ $3.25\pm 0.12$ $-0.25\pm 0.08$

Table 2: FWHMs and $m_{E}$ values derived from the models fitted on the
forward shock filaments profiles shown in Fig. 4. $m_{E}$ values are
calculated between the energy band of the line on which they are written and
the energy band of the line above. The values’ means are arithmetic, and the
errors are mean square.

Other profiles (Fig. 5)
FWHM (arcsec) Box Image Left Right Mean $m_{E}$ $0.4$-$1.7$ keV $2.92\pm 0.70$
$9.70\pm 0.59$ $6.31\pm 0.46$ 1 $2.5$-$4.0$ keV $2.67\pm 0.38$ $6.20\pm 0.96$
$4.43\pm 0.51$ $-0.19\pm 0.07$ $5.0$-$8.0$ keV $2.47\pm 0.35$ $4.98\pm 0.69$
$3.73\pm 0.39$ $-0.25\pm 0.22$ $0.4$-$1.7$ keV $1.23\pm 0.16$ $1.18\pm 0.10$
$1.21\pm 0.09$ 2 $2.5$-$4.0$ keV $1.15\pm 0.15$ $1.41\pm 0.11$ $1.28\pm 0.09$
$0.03\pm 0.06$ $5.0$-$8.0$ keV $1.21\pm 0.19$ $1.59\pm 0.17$ $1.40\pm 0.13$
$0.13\pm 0.17$ $0.4$-$1.7$ keV $2.43\pm 0.31$ $1.89\pm 0.28$ $2.16\pm 0.21$ 3
$2.5$-$4.0$ keV $2.51\pm 0.51$ $1.92\pm 0.52$ $2.22\pm 0.36$ $0.01\pm 0.10$
$5.0$-$8.0$ keV $2.15\pm 0.46$ $1.89\pm 0.62$ $2.02\pm 0.38$ $-0.13\pm 0.36$
$0.4$-$1.7$ keV $5.58\pm 3.53$ $4.16\pm 1.59$ $4.87\pm 1.93$ 4 $2.5$-$4.0$ keV
$4.90\pm 1.47$ $2.92\pm 0.82$ $3.91\pm 0.84$ $-0.12\pm 0.25$ $5.0$-$8.0$ keV
$2.61\pm 0.57$ $1.53\pm 0.53$ $2.07\pm 0.39$ $-0.92\pm 0.41$ $0.4$-$1.7$ keV
$3.23\pm 1.03$ $0.88\pm 0.20$ $2.06\pm 0.52$ 5 $2.5$-$4.0$ keV $2.13\pm 0.38$
$0.68\pm 0.24$ $1.40\pm 0.23$ $-0.21\pm 0.16$ $5.0$-$8.0$ keV $2.15\pm 0.27$
$0.56\pm 0.31$ $1.35\pm 0.20$ $-0.06\pm 0.32$ $0.4$-$1.7$ keV $3.08\pm 0.75$
$3.56\pm 0.35$ $3.32\pm 0.41$ Mean $2.5$-$4.0$ keV $2.67\pm 0.33$ $2.63\pm
0.28$ $2.65\pm 0.22$ $-0.09\pm 0.07$ $5.0$-$8.0$ keV $2.12\pm 0.17$ $2.11\pm
0.22$ $2.11\pm 0.14$ $-0.25\pm 0.14$

Table 3: FWHMs and $m_{E}$ values derived from the models fitted on the
forward shock filaments profiles shown in Fig. 5. $m_{E}$ values are
calculated between the energy band of the line on which they are written and
the energy band of the line above. The values’ means are arithmetic, and the
errors are mean square.

###  3.2 Discussion

A quick look at the mean FWHMs or at the $m_{E}$ signs shows that there is
indeed a narrowing of the filaments profiles with energy. In all of the seven
forward shock profiles, five reverse shock profiles, and five unidentified
profiles, only two present a widening between $0.4$-$1.7$ keV and $5.0$-$8.0$
keV: FS $7$ and unidentified profile $2$. In both cases, this widening is so
low that the fitting errors allow for the possibility of a narrowing as well.
Hence, we can reasonably conclude that the narrowing of the filaments with
energy in Cas A is a global effect, that can be observed on several filaments
both at the forward and at the reverse shock. In Sect. 3.3 we will see that
this narrowing is not due to the evolution of Chandra’s point spread function
(PSF) with energy. Future studies could also take into account the possible
effects of dust scattering on the observed widths of the filaments. However,
in first approximation, scattering effects have an energy dependence in
$E^{-2}$ (Costantini & Corrales, 2022; Corrales & Paerels, 2015, for example),
which is not consistent with our observations. Hence, the narrowing we observe
is likely not primarily due to dust scattering.

We can see in the results a common trend regarding the evolution of $m_{E}$
values: the mean $m_{E}$ between $2.5$-$4.0$ keV and $5.0$-$8.0$ keV are
significantly larger, in absolute, than the mean $m_{E}$ between $0.4$-$1.7$
keV and $2.5$-$4.0$ keV. While this result is to handle cautiously, given the
importance of the errors, the low statistics on which the means are calculated
and the non-negligible number of outliers, it would mean that the observed
narrowing of the synchrotron rims has a stronger energy-dependence at higher
energies than at lower energies. Following Tran et al. (2015), this result
would be characteristic of a damping mechanism: the rim widths are relatively
energy-independent below a threshold energy and increase once advection
controls rim widths. This is in apparent contradiction with Araya et al.
(2010), where a moderate narrowing was observed between $0.3$-$2.0$ keV and
$3.0$-$6.0$ keV, but not between $3.0$-$6.0$ keV and $6.0$-$10.0$ keV.
However, the energy bands we compared are not the same, and our “high energy
$m_{E}$” is defined between $2.5$ and $5.0$ keV, while theirs is between $3.0$
and $6.0$ keV. The highly detailed synchrotron images we obtained with pGMCA
may also have an influence, allowing for a more precise profile definition and
width measurement with less thermal emission contamination.

In Araya et al. (2010), the model used to describe the forward shock filament
profiles predicts a sharp decline upstream, after the peak, that they did not
observe. We can see from the FWHMu and FWHMd values from Table 1 that we did
not observe it either. The FWHMu values even tend to be larger than FWHMd
values, meaning that the profiles are sharper downstream than upstream of the
forward shock. On the contrary, we can see in Table 2 that at the reverse
shock, FWHMu values are mainly smaller than FWHMd values, meaning that the
profiles are sharper upstream (even though it is not apparent in the mean
FWHMu and FWHMd values, that are driven by a few outliers). However, we will
see in Sect. 3.3 that this could be linked to the PSF.

Not much can be said to identify our “other” profiles as belonging to the
forward or to the reverse shock. Although it could be tempting to attempt an
identification based on their FWHM on each side, line of sight effects are
likely involved and for a filament facing the observer, “left” and “right” of
the peak are not trivially equivalent to “upstream” or “downstream”.
Nonetheless, Table 1 mostly follows the same trend of narrowing, with stronger
energy-dependence at high energies, that we observed on both forward and
reverse shock rim profiles.

###  3.3 Influence of the PSF

Previous studies by Araya et al. (2010), Ressler et al. (2014) and Tran et al.
(2015) did not take into account the PSF in their filaments widths
measurements. Yet, the PSF evolves with the energy, and could have an impact
on our widths comparison between energy ranges.

(a)  Figure 6: A Dirac delta function is convolved with the PSF profiles along
the third FS box for our three energy ranges and normalized. This box was
chosen because it was the one along which the PSF varied the most, both
spatially and with energy. The FWHM values downstream and upstream are
obtained by fitting our piecewise two-exponential model; values are given in
arcsec.

In a first attempt to take it into account, we generated the 1-sigma PSF maps
of the merged 2004 observations of Cas A using the merge_obs routine from CIAO
around the $0.4$-$1.7$ keV, $2.5$-$4.0$ keV and $5.0$-$8.0$ keV energy bands.
We then intended to convolve our piecewise two-exponential model to the PSF
profiles along each box for each energy range while fitting. However, the
results we obtained were endowed with disproportionate errors due to the
additional uncertainty brought by the spreading. Hence, we renounced taking
the PSF directly into account in our fitting, and decided to present a “worst
case scenario” to give an idea of the possible effects of the PSF on the
results shown in Tables 1, 2 and 3.

To do so, we searched the box along which the PSF evolved the most, both
spatially and with energy, which was the third box of our FS profiles, the
furthest from the optical center. We then convolved the profiles of the three
energy ranges PSF maps along this box with an infinitely thin filament, i.e. a
Dirac function. The results were then fitted with our model, and the
differences between the retrieved FWHMs for each energy range give an idea of
the influence of the PSF in the worst scenario, with infinitely thin filaments
in our worst box as regards to the PSF.

The results are shown in Fig. 6, and it appears that the PSF can potentially
have a significant influence. The FWHMs upstream are consistently larger
upstream than downstream, and the FWHMs on both sides increase with energy. It
is important to note that the PSF profiles all behave in a similar way: the
spreading radius values increase both with energy and with the distance from
the center (downstream to upstream for the forward shock, upstream to
downstream for the reverse shock). Hence, the narrowing of the filaments with
energy we observed might be underestimated because of the PSF, and our
observations regarding the sharper decline downstream than upstream of the
forward shock might be due to the PSF. The $m_{E}$ between the $0.4$-$1.7$ keV
and $2.5$-$4.0$ keV profiles is $0.05\pm 0.03$ and the $m_{E}$ between the
$2.5$-$4.0$ keV and $5.0$-$8.0$ keV profiles is $0.22\pm 0.08$, indicating
that the PSF would tend to widen the filaments more at higher energies.

###  3.4 Interpretation

The calculated $m_{E}$ values tend to indicate a stronger energy-dependence of
the narrowing at high energies, which is likely underestimated by the PSF.
According to Tran et al. (2015), this would be characteristic of a damping
mechanism, where the rims widths are supposed to be energy-independent below a
threshold energy, and decrease or increase once advection and/or diffusion
controls the widths. Following the same paper, the action of a damping
mechanism can be more confidently assessed by the observation of thin
synchrotron filaments in radio. In the radio VLA observations of Cas A, such
filaments can be seen (see for example DeLaney et al., 2014), which also
suggests that damping effects are at stake in this SNR’s synchrotron emission.
A thorough joint study of both radio and X-ray filaments could allow for an
estimation of the damping lengths.

##  4 Conclusions

From our study of the filament profiles in Cas A, we can conclude the
following :

  1. 1.

Our blind source separation method was able to produce clear, detailed and
unpolluted maps of the synchrotron emission around three energy bands:
$0.4$-$1.7$ keV, $2.5$-$4.0$ keV, and $5.0$-$8.0$ keV. These images clearly
show filamentary structures all around the remnant, some associated to the
forward shock, and some to the reverse shock. Some profiles can also be seen
that cannot be clearly attributed to the forward or to the reverse shock,
possibly because of line of sight effects.

  2. 2.

There is a noticeable narrowing of the synchrotron rims from lower to higher
energies. Contrary to a previous study by Araya et al. (2010), we find that
this effect is also visible between $2.5$-$4.0$ keV and $5.0$-$8.0$ keV, not
only at lower energies. It is observed for filaments both at the forward and
at the reverse shock, and for non-identified filaments. The evolution of the
PSF with energy cannot account for this narrowing; on the contrary, the PSF
would tend to make filaments appear wider with energy.

  3. 3.

There seems to be a stronger energy-dependence of this narrowing at higher
energies, which would be a sign that there is a damping mechanism at stake.
The observation of thin synchrotron profiles in the radio 6cm VLA observation
of Cas A is another indicator of this damping mechanism. However, it does not
exclude the possible influence of the loss-limited effect as well.

  4. 4.

The filament profiles at the forward shock tend to be sharper downstream than
upstream. At the reverse shock, the profiles mainly tend to be sharper
upstream. However, this could be due to the radial degradation of the PSF away
from the aim point. It could also be caused by projection effects that do not
have the same impact on the forward and the reverse shock as they propagate in
opposite directions.

###### Acknowledgements.

The material is based upon work supported by NASA under award number
80GSFC21M0002. We thank D. Castro for the useful discussion about the PSF
potential influence, and E. Costantini for the interesting discussion about
potential dust scattering effects.

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# Experimental validation of fluid inertia models for a cylinder settling in a
quiescent flow

F. Cabrera  Univ Lyon, ENS de Lyon, Univ Claude Bernard Lyon 1, CNRS,
Laboratoire de Physique, F-69342 Lyon, France  M.Z. Sheikh  Univ Lyon, ENS de
Lyon, Univ Claude Bernard Lyon 1, CNRS, Laboratoire de Physique, F-69342 Lyon,
France  Department of Mechanical Engineering, University of Engineering and
Technology, 54890, Lahore, Pakistan  B. Mehlig  Department of Physics,
Gothenburg University, 41296 Gothenburg, Sweden  N. Plihon  Univ Lyon, ENS de
Lyon, Univ Claude Bernard Lyon 1, CNRS, Laboratoire de Physique, F-69342 Lyon,
France  M. Bourgoin  Univ Lyon, ENS de Lyon, Univ Claude Bernard Lyon 1, CNRS,
Laboratoire de Physique, F-69342 Lyon, France  A. Pumir  Univ Lyon, ENS de
Lyon, Univ Claude Bernard Lyon 1, CNRS, Laboratoire de Physique, F-69342 Lyon,
France  Max-Planck Institute for Dynamics and Self-Organization, Göttingen,
D-37077, Germany  A. Naso  Univ Lyon, CNRS, Ecole Centrale de Lyon, INSA Lyon,
Univ Claude Bernard Lyon 1, Laboratoire de Mécanique des Fluides et
d’Acoustique, UMR 5509, 69130 Ecully, France

###### Abstract

The precise description of the motion of anisotropic particles in a flow rests
on the understanding of the force and torque acting on them. Here, we study
experimentally small, very elongated particles settling in a fluid at small
Reynolds number. In our experiments, we can, to a very good approximation,
relate the rate of rotation of cylindrical tungsten rods, of aspect ratios
$\beta=8$ and $\beta=16$, settling in pure glycerol to the torque they are
experiencing. This allows us to compare the measured torque with expressions
obtained either in the slender-rod limit, or in the case of spheroids. Both
theories predict a simple angle dependence for the torque, which is found to
capture very well the experimental results. Surprisingly, the slender-rod
approximation predicts much better the results for $\beta=8$, than for
$\beta=16$. In the latter case, the expression obtained for a spheroid
provides a better approximation. The translational dynamics is shown to be in
qualitative agreement with the slender-rod and spheroid models, the former one
being found to represent better the experimental data.

##  I Introduction

The settling of non-spherical particles at low Reynolds number in a fluid at
rest is a subtle problem. It has been known for a long time that describing
the angular degrees of freedom with the Stokes approximation leads to an
indeterminacy in the settling angle: the particle may settle at any angle with
respect to the vertical. This degeneracy, however, is lifted by the action of
the fluid, already in the limit of a small but non-zero particle Reynolds
number ${\rm Re}_{p}$. Whereas the correction to the translational velocity
has been understood for a long time, the torque acting on an non-spherical
particle is not as well understood. Cox Cox (1965) determined the torque
acting on particles. Later, Khayat and Cox Khayat and Cox (1989) determined
the torque for slender particles. More recently, the torque acting on
spheroids was determined for spheroids of arbitrary dimensions Dabade  _et
al._ (2015). These explicit analytic expressions are very useful to describe
the motion of particles at small Reynolds numbers.

Because of the importance of the problem, many contributions have been devoted
to its numerical solution, to determine approximate torque parameterizations
as a function of the particle Reynolds number ${\rm Re}_{p}$ and particle
shape Hölzer and Sommerfeld (2009); Zastawny  _et al._ (2012); Jiang  _et al._
(2014); Ouchene  _et al._ (2015, 2016); Andersson and Jiang (2019); Fröhlich
_et al._ (2020). The effect of small to moderate Reynolds numbers for
spheroids over a large range of aspect ratios $\beta$ was recently analysed by
means of numerical simulations Jiang  _et al._ (2021), allowing a direct
comparison with the small-${\rm Re}_{p}$ theory Dabade  _et al._ (2015) in the
limit of small ${\rm Re}_{p}$. It was found that the theory Dabade  _et al._
(2015) generally provides the correct functional form and the correct
magnitude for the torque. However, it tends to over predict the numerically
determined values when ${\rm Re}_{p}\gtrsim 1$. The hydrodynamic loads on a
fixed finite-length circular cylinder in a uniform flow have also been
obtained numerically, from creeping-flow conditions to strongly inertial
regimes Kharrouba  _et al._ (2021). Semi-empirical models based on theoretical
predictions and incorporating finite-length and inertial corrections extracted
from the numerical data were then derived. There have been, in comparison, far
fewer experimental studies.

In experiments where rods settle through a simple vortical flow, it was shown
that the effect of fluid inertia could not be ignored Lopez and Guazzelli
(2017). More recently, the force and torque measured in settling experiments
of symmetric and asymmetric fibers were accurately predicted by the model of
Khayat and Cox Roy  _et al._ (2019). The expression of torque by Khayat and
Cox for symmetric fibers overpredicts the torque experimentally measured for
$\beta=20$ and ${\rm Re}_{p}=1.6$, while it underpredicts the measurements for
$\beta=100$ and ${\rm Re}_{p}=8.6$ – however the origin was argued to lie in
the defects in fibers which produce slight asymmetries Roy  _et al._ (2019);
Candelier and Mehlig (2016).
Here, we consider again the problem in a different setup, using small tungsten
rods, roughly $12$ times denser than the surrounding fluid, which is taken as
pure glycerol. The resulting particle Reynolds number, $U_{s}l/\nu$, where $l$
is the half-length of the rod, $U_{s}$ the settling velocity and $\nu$ the
fluid viscosity, is always smaller than 0.39. Our experimental setup (based on
3D tracking of the particle rotation and translation) allows us to determine
precisely all the degrees of freedom of the motion, and to deduce from it the
torque acting on the particle. Our experimental results are then
systematically compared to the predictions for slender-rods and spheroids.

One motivation for studying the problem comes from the modeling of processes
involving non-spherical particles settling in turbulent flows. This includes,
in an engineering context, problems involving paper fibers Lundell  _et al._
(2011). The question is also particularly relevant in the environmental
sciences, such as the settling of planktons in the oceans Durham  _et al._
(2013); Pedley and Kessler (1992); Ruiz  _et al._ (2004), or ice crystals
settling in clouds Pruppacher and Klett (1980); Siewert  _et al._ (2014);
Gustavsson  _et al._ (2017); Jucha  _et al._ (2018); Gustavsson  _et al._
(2019); Sheikh  _et al._ (2020).

The article is organized as follows. We first introduce in Sec. II the
translational and rotational equations of motion for spheroids and slender-
rods. The experimental design is then described in Sec. III. The results are
presented in Sec. IV. Finally, our conclusions are summarized in Section V.

##  II Theoretical background

Our experiments were performed with cylindrical rods of density $\rho_{p}$,
half-length $l$ and radius $a$. The particle aspect ratio is $\beta=l/a$. The
motion of the particles is characterized by the center of mass, $\mathbf{x}$,
and the unit vector, $\hat{\mathbf{n}}$, characterizing the orientation, as
illustrated in Fig. 1. The orientation of $\hat{\mathbf{n}}$ is defined by the
angles $\psi$ and $\theta$. The vector $\hat{\mathbf{p}}$ is orthogonal to the
projection of $\hat{\mathbf{n}}$ onto the horizontal
$(\hat{\mathbf{e}}_{x},\hat{\mathbf{e}}_{y})$ plane,
$\hat{\mathbf{p}}=\sin(\psi)\hat{\mathbf{e}}_{x}-\cos(\psi)\hat{\mathbf{e}}_{y}$,
and $\hat{\mathbf{e}}_{z}$ is the vertical. In the experiments, the initial
angular velocity was orthogonal to the plane spanned by $\mathbf{g}$ and
$\hat{\mathbf{n}}$ (See Section IV). In other words, the angle $\psi$ was
found to remain constant, so the angular dynamics can be described by $\theta$
only.

The particle Reynolds number is defined as

${\rm Re}_{p}=\frac{l|\mathbf{w}|}{\nu},$ (1)

where $\mathbf{w}$ is the velocity of the center of mass. The particle
Reynolds number allows us to quantify the respective role of the inertial and
viscous effects in the problem. In our experiments, ${\rm Re}_{p}$ does not
exceed $1$. As a consequence, the force and the torque acting on the particle
originate from the viscous term (the Stokes forces and torques,
$\mathbf{f}_{S}$ and $\mathbf{T}_{S}$), plus contributions due to fluid
inertia, which are derived in a systematic perturbation expansion in the
parameter ${\rm Re}_{p}$, denoted here as $\mathbf{f}_{I}$ and
$\mathbf{T}_{I}$.

The equations of motion for the translational degrees of freedom read as

$\frac{d\mathbf{x}}{dt}=\mathbf{w}~{}~{}~{}~{}{\rm
and}~{}~{}~{}\frac{d\mathbf{w}}{dt}=\mathbf{g}+\frac{\mathbf{f}_{H}}{m_{p}},$
(2)

where $\mathbf{f}_{H}\equiv\mathbf{f}_{S}+\mathbf{f}_{I}$ is the total
hydrodynamic force acting on the particle, and $m_{p}$ is its mass. Note that
the effect of buoyancy can be simply taken into account by the transformation:
$\mathbf{g}\rightarrow\mathbf{g}(1-\rho_{f}/\rho_{p})$. The general expression
for the Stokes force is: $\mathbf{f}_{S}=6\pi
a\mu\mathbb{A}(\hat{\mathbf{n}})(\mathbf{u}-\mathbf{w})$, where $\mu$ is the
dynamic viscosity of the fluid, $\mathbf{u}$ is the undisturbed velocity of
the fluid at the particle position (equal to zero in a quiescent flow), and
$\mathbb{A}(\hat{\mathbf{n}})$ is the resistance tensor Kim and Karrila
(1991). The tensor $\mathbb{A}$ can be expressed in terms of two coefficients,
$A_{\perp}$ and $A_{\|}$:
$A_{ij}=A_{\perp}(\delta_{ij}-n_{i}n_{j})+A_{\|}n_{i}n_{j}$. The expressions
of $A_{\perp}$ and $A_{\|}$ depend on the shape of the particle (rod or
spheroid), and in particular on its aspect ratio $\beta$, and are given in
Appendix A. In the limit $\beta\gg 1$, the expressions reduce, for the two
shapes considered, to $A_{\perp}=2A_{\|}=(4/3)[\beta/\log(\beta)]$. Equation
(2), together with the specific form of $\mathbf{f}_{S}$, imply that if
$|\mathbf{f}_{I}|\ll|\mathbf{f}_{S}|$ the particle velocity $\mathbf{w}$
relaxes with a characteristic (Stokes) time
$\tau_{p}=a^{2}\log(\beta)(\rho_{p}/\rho_{f})/(3\nu)$. In our experiments, we
observe that the characteristic time scales of the particles (of the order of
seconds) are much longer than $\tau_{p}$ (of the order of a milliseconds).
This implies that the dynamics is overdamped, and that the center-of-mass
velocity can be obtained by solving $\mathbf{g}+\mathbf{f}_{H}/m_{p}=0$ at any
time.

The effect of finite fluid-inertia leads to a correction to the Stokes force.
In the case of a spheroid, this correction to the resistance tensor can be
expressed as: $\mathbb{A}^{s}\rightarrow\mathbb{A}^{s}+\mathbb{A}^{s}_{I}$.
Similarly, the expressions for the corrections to the force for a slender rod,
$\mathbf{f}^{r}_{I}$, can be found in Khayat and Cox (1989), and are given in
the Appendix A.

(a)(b)

Figure 1: Representation of the settling cylindrical particle (a) in 3D and
(b) in the vertical plane $(\hat{\mathbf{n}},\mathbf{g})$. See text for
details.

As stated before, in our experiments the change in the orientation of a
particle is due to variations of the angle $\theta$ only. The equation of
motion for this angle is obtained by projecting the equation for the angular
momentum along the $\hat{\mathbf{p}}$-direction:

$I_{p}\ddot{\theta}=(\mathbf{T}_{S}+\mathbf{T}_{I})\cdot\hat{\mathbf{p}},$ (3)

where $I_{p}$ is the moment of inertia of the particle with respect to its
center of mass, perpendicular to its axis ($I_{p}=\frac{1}{12}ml^{2}$). As it
was the case for the force, the torque can be written as the sum of a
contribution due to viscous forces (Stokes), $\mathbf{T}_{S}$, and a
contribution due to inertia, $\mathbf{T}_{I}$. The general expression for the
Stokes torque is Jeffery (1922): $\mathbf{T}_{S}=6\pi
a\mu[\mathbb{C}(\boldsymbol{\Omega}-\boldsymbol{\omega})+\mathbb{H}:\mathbb{S}]$,
where $\boldsymbol{\omega}$ is the angular velocity of the solid,
$\boldsymbol{\Omega}=1/2(\nabla\wedge\mathbf{u})$ the vorticity, and
$\mathbb{C}$ is the resistance tensor. The last term, $\mathbb{H}:\mathbb{S}$,
involves the strain in the fluid, which is $0$ in a fluid at rest, as
considered here. In a quiescent fluid, the torque is therefore proportional to
the particle angular velocity $\omega$ which reads $\dot{\theta}$ since the
$\psi$ angle is constant:

$T_{S}\equiv\mathbf{T}_{S}\cdot\hat{\mathbf{p}}=-C_{S}\dot{\theta},$ (4)

where $C_{S}$ is a resistance coefficient proportional to $\mu a^{3}$. The
dimensionless constant of proportionality depends on the details of the
particle. In the very large $\beta$ limit, the expression for $C_{S}$ reduces
to

$C_{S}=\frac{8\pi}{3}\frac{\mu l^{3}}{\log\beta}.$ (5)

As it was the case for the dynamics of the center of mass, the resistance term
$-C_{S}\dot{\theta}$ provides a characteristic relaxation time scale for the
particle orientation which is, up to a numerical factor of order $1$, equal to
the characteristic time $\tau_{p}$ already introduced Anand  _et al._ (2020).
The time scale over which the angle $\theta$ evolves is also found to be very
long compared to $\tau_{p}$, which ensures that, to a very good approximation,
the left-hand side of Eq. (3) can be set to $0$. This allows us to relate
directly, via Eq. (4), the projection of the torque due to the inertial
forces, $\mathbf{T}_{I}$, perpendicularly to $\hat{\mathbf{n}}$,
$T_{I}\equiv\mathbf{T}_{I}\cdot\hat{\mathbf{p}}$:

$T_{I}=C_{S}\dot{\theta}.$ (6)

Determining the torque $\mathbf{T}_{I}$ is a very challenging problem, even in
the small-${\rm Re}_{p}$ limit.

We compare our experimental results, carried out with rods, to the results of
the slender-rod theory of Khayat and Cox (1989) , valid asymptotically when
$\beta^{-1}{\rm Re}_{p}\ll 1$ and $\beta\gg 1$. The first criterion is very
well satisfied in the experiments. The second one is not, since $\beta=8$ or
$16$ in our experiments, and since the small parameter in the slender-rod
theory is $1/\log\beta$. We therefore compared also with a second perturbative
theory for the torque, valid when the particle has a spheroidal shape Dabade
_et al._ (2015). This theory is valid for any arbitrary aspect ratio, $\beta$,
but only to leading order in ${\rm Re}_{p}$. Furthermore, it was found
numerically to describe the torque acting on spheroidal particles quite well,
with an accuracy of $\sim 20\%$ for $\beta=6$ over the range of ${\rm Re}_{p}$
in the experiment Jiang  _et al._ (2021). We emphasize that the theory was
derived for spheroidal particles, but we expect that it nevertheless works
qualitatively for rod-like particles of the same aspect ratio and the same
particle mass.

At small Reynolds numbers, symmetry considerations Subramanian and Koch
(2005); Jiang  _et al._ (2021) indicate that the torque takes the form

$\mathbf{T}_{I}=F(\beta)\rho_{f}l^{3}(\mathbf{w}\times\hat{\mathbf{n}})(\mathbf{w}\cdot\hat{\mathbf{n}}).$
(7)

The expression (7) shows that $\mathbf{T}_{I}$ is along $\hat{\mathbf{p}}$,
and that its norm reads

$T_{I}=F(\beta)\rho_{f}l^{3}|{\bf
w}|^{2}\sin\phi\cos\phi=C_{I}|\mathbf{w}|^{2}\sin 2\phi.$ (8)

The shape factor $F(\beta)$ depends only on the particle aspect ratio. As a
result, the factor $C_{I}$ is known for spheroids Dabade  _et al._ (2015) and
in the slender-rod limit Khayat and Cox (1989); Lopez and Guazzelli (2017); we
denote its values as $C_{I}^{s}$ and $C_{I}^{r}$. The corresponding formulae
are summarized in Appendix B. We note that the expressions for the two
theories coincide in the limits ${\rm Re}_{p}\rightarrow 0$ and
$\beta\rightarrow\infty$, where:
$C_{I}(\beta)\approx-5\pi/[3(\log\beta)^{2}]\times(\rho_{f}l^{3}/2)$.
Combining Eqs. (6) and (8) leads to the prediction of the model that

$\dot{\theta}=\frac{C_{I}}{C_{S}}|\mathbf{w}|^{2}\sin 2\phi.$ (9)

This overdamped torque model for rods was found to qualitatively reproduce
experimental results of rods settling in a cellular flow Lopez and Guazzelli
(2017), and allowed to investigate the settling of anisotropic particles in
turbulent flows Kramel (2017). The expressions of the torque for spheroids
have been validated numerically Jiang  _et al._ (2021), and used to study
theoretically and numerically the settling of spheroids in turbulence
Gustavsson  _et al._ (2019); Sheikh  _et al._ (2020); Anand  _et al._ (2020);
Gustavsson  _et al._ (2021). We notice that when the fluid is in motion, the
local velocity gradients may contribute to the torque expression Einarsson
_et al._ (2015). In our problem, with $\rho_{p}/\rho_{f}\gg 1$, the
corresponding contributions are negligible.

##  III Experimental setup

Figure 2: (a) Schematic view of the experimental setup. The origin (0,0,0) is
set at the geometric center of the vessel. (b) Typical raw image obtained from
one camera showing the particle of aspect ratio $\beta=16$, the detected
particle center line (in red) and center of mass (white dot). (c) Successive
positions of a rod’s center line and center of mass during one experiment (for
$\beta=16$), color coded by the value of the angle $\theta$. For the sake of
clarity, only one center of mass data-point out of 80, and one center line
each 5 center of mass data-points are shown.

The experiments were performed in a PMMA tank with a square cross-section of
side $150$ mm and height of $710$ mm (see Fig. 2 (a)). The tank was filled
with pure glycerol (Sigma-Aldrich W252506-25KG-K), with a density
$\rho_{f}=1216$ kg m-3 and dynamic viscosity $\mu=1.05$ Pa.s at 22∘C in which
we observed the settling of heavy cylindrical particles. The cylindrical
particles were made of Tungsten-Carbide (WC with $5\%$ Cobalt), manufactured
by Comac Europe, with a density $\rho_{p}=14800$ kg/m3, resulting in a fluid-
to-particle density ratio of $\rho_{p}/\rho_{f}=12.1$. Cylindrical rods were
milled and filed at lengths $2l=(16.0\pm 0.5)$ mm and ($8.0\pm 0.5$) mm, from
long cylinders of diameter $2a=1$ mm, resulting in cylindrical particles with
aspect ratios $\beta$ of 16 and 8. The surface roughness was measured using a
Scanning Electron Microscope, leading to an average arithmetic roughness value
of $15$ $\mu$m. The particle Reynolds number $Re_{p}$ was measured to lie
between 0.255-0.390 for $\beta=16$, and 0.105-0.150 for $\beta=8$. The
particles trajectories were reconstructed from images acquired using two
cameras (fps1000 model from The Slow Motion Company), placed orthogonal, as
shown in Fig. 2(a). The cameras acquire images at a frame rate of $1400$ fps
with a resolution of $720\times 1280$ px2. The volume over which the
trajectories of the particles are reconstructed is $130\times 130\times 200$
mm3 to avoid wall effects Chhabra  _et al._ (2003) – the distance between the
rods and the walls is thus always larger than $10$ mm.

The particles are backlight-illuminated by using two white and homogeneous
light panels (light-red rectangles in Fig. 2 (a)). A typical image captured
from one camera shows the projection of the cylindrical particle in the
$(y,z)$ plane in Fig. 2(b). For each image, and on both cameras, the detection
of the particle is computed using the MATLAB function regionprops, which
detects the particle’s major axis and minor axis, center of mass and
orientation (see Fig. 2(b)). The 3D orientation of the particle is then
obtained from a 3D Particle Velocity Tracking (PTV) algorithm Bourgoin and
Huisman (2020): a set of five points on the major axis are matched in 3D from
the projections in the $(y,z)$ and $(x,z)$ planes (the five points are equally
spaced between the two extrema). The particle position and orientation is thus
reconstructed in 3D, as shown in Fig. 2(c), and all parameters related to the
dynamics of the particles can be subsequently computed. The evolution of a
typical sedimentation experiment is displayed in Fig. 2(c), from top to bottom
as time increases, clearly demonstrating that the angle $\theta$ decreases in
time. The tracking error of the PTV system was shown to be inferior to
$130~{}\mu$m, or equivalently $50\%$ of the pixel size. Additionally, the raw
data was filtered via the convolution with a Gaussian kernel of variance
$\sigma=1.2\times 10^{3}$ and $\sigma=8\times 10^{2}$, for $\beta=8$ and 16
respectively. Note that to avoid perspective distortion of the projected
contour of an anisotropic object Huang  _et al._ (1996); Toupoint  _et al._
(2019), the cameras were installed sufficiently far from the sedimentation
tank. An air-conditioning system keeps a constant room temperature at (22
$\pm$ 1)∘C, which bounds the viscosity variations to 5$\%$. Moreover, to reach
thermal equilibrium, a 48-hours delay was systematically respected between the
filling of the tank with pure glycerol and the experiments. The calibration of
the PTV system then consisted on the displacement of a target with known
dimensions over the visualization volume Bourgoin and Huisman (2020). The
particles were released in the fluid with chemical tweezers: we completely
submerged the particle and released it when the glycerol’s free surface was at
rest (approximately after $15$ s). A minimum time of $90$ s is taken between
two successive realisations to assure that the fluid has no motion left from
the previous drop. For both $\beta=8$ and $\beta=16$, 25 independent
realisations have been acquired.
With the values of the physical parameters in the experiment, the
characteristic time scale of the particles, $\tau_{p}$, introduced in Section
II, does not exceed $10^{-3}$ s, which, as we will document, is very short
compared to the characteristic time of the evolution. Note that, additionally,
there is $150$ mm of fluid above and below the visualization volume. A
particle then travels for approximately 1000$\tau_{p}$ before and after it
enters the detection volume: the fluid above makes the particle loose memory
of any transient produced in the release; whereas the fluid below keeps the
particle away from the bottom-wall at all times.

##  IV Results

Before we proceed to present our results, we recall that the motion of the
rods was found to be, to a very good approximation, planar. Namely, the
variations of the angle $\psi$ between the projection of the ${\bf n}$ vector
on a horizontal plane and an arbitrary fixed horizontal vector, as illustrated
in Fig. 1 (a), are less than $5^{\circ}$ during a given experiment, with a
typical value of 2∘ over all realisations. We therefore used the values of the
torques valid in the case of a planar motion, where the orientation is
parametrized solely by the angle $\theta$.

###  IV.1 Translational motion

$\beta=8$(a)$\beta=16$(b)

Figure 3: Evolution of the vertical component of the velocity wg as a function
of the angle $\theta$ for (a) $\beta=8$ and (b) $\beta=16$. Each line
represents one realization, color-coded by the value of $\mathrm{Re}_{p}$.
Prediction for the spheroid model (solid black line) and for the slender-rod
theory Khayat and Cox (1989) (dashed-dotted black line).

For all experimental realizations, the ratio between the particle inertial
term $m_{p}\textrm{d}\mathbf{w}/\textrm{d}t$ and the Stokes force term ${\bf
f_{S}}$ in Eq. (2) was computed to be of order $10^{-3}$. In other words, the
particle inertia contribution in the translational dynamics can be neglected
and the particle dynamics is overdamped, as anticipated from the ratio between
$\tau_{p}$ and the characteristic time for the evolution of $\theta$.

Figure 3 shows the settling velocity of the rod wg, defined as the projection
of ${\bf w}$ on the vertical vector ${\bf g}/|{\bf g}|$, as a function of its
orientation $\theta$, for the two values of $\beta$ considered. As for all
experimental figures in this article, each trajectory is color-coded with its
instantaneous particle Reynolds number $\mathrm{Re}_{p}$ based on the particle
half-length $l$. The time series of the evolution of $\theta$ are displayed in
Fig. 4, and, as anticipated, we observe that the position of the particle,
hence its velocity, evolve over a time scale which is much longer than
$\tau_{p}$. We also observe in Fig. 3 that the settling velocity is an
increasing function of $\theta$: in other words, the settling velocity
increases when particles become vertical. This can be readily understood,
since the drag decreases when $\theta$ increases from $0$ to $90^{\circ}$.
Therefore, the particle Reynolds number increases linearly with the settling
velocity. The measured settling velocity is compared with the settling
velocity calculated by integrating the equation of motion, Eq. (2), using for
the hydrodynamic force the expressions for the spheroids model or for the
slender-rod model (see Section II), and including the particle inertia,
despite its weak contribution in the overdamped regime. We recall here that,
in the framework of the spheroid model, we use the values of $\beta$ and the
mass $m_{p}$ of the cylindrical particles used in the experiments, and not the
values for a spheroid with exactly the same geometrical dimensions as that of
the particles used in the experiments, for the computation of the resistance
tensors $\mathbb{A}_{S}$ and $\mathbb{A}_{I}$ respectively due to the Stokes
and to the fluid inertia corrections. The velocity predicted by the spheroid
model is shown as a full black line in Fig. 3, and the prediction from the
slender-rod model is displayed as a dash-dotted line.

For the two values of $\beta$ considered, the predicted velocities
overestimate the measured values, or equivalently, the predicted values of the
hydrodynamic forces underestimate the actual force. However, the qualitative
agreement is satisfactory: the relative difference is always smaller than 16%
(for the spheroid model) and 6% (for the slender-rod model) for $\beta=16$,
and respectively 20% and 10% for $\beta=8$. As expected, the agreement with
the predictions is better for the slender-rod model than for the spheroid
model, as the particles used in the experiment have a cylindrical shape.
Nonetheless, as the aspect ratio increases, the agreement improves from
$\beta=8$ to $\beta=16$, even with the spheroid model. These observations can
be rationalized by noticing that the sharp ends at the extrema of the cylinder
play a less important role when $\beta$ increases.

The horizontal component of the measured velocity remains smaller than 20% of
the vertical one, see Figure 7 in Appendix C. The figure also shows the ratio
of the horizontal and the vertical velocities predicted by the two models
discussed in Section II for spheroids and slender-rods. It can be seen from
Figure 7 that the ratio of the experimentally measured horizontal and vertical
velocities is qualitatively consistent with the predictions of the two models.

###  IV.2 Angular motion

Figure 4 shows the time dependence of the angle $\theta$ for the two aspect
ratios considered, for a number of different initial rod orientations. The
experimental data is color-coded with the instantaneous particle Reynolds
number as indicated by the colorbar. The different realizations correspond to
different initial orientations covering the range $\theta(t=0)\in(0,\pi/2)$.
For all realizations, $\theta$ decreases with time, meaning that particles
tend to orient their broadside facing down, resulting in a maximal drag, as
previously illustrated in Fig. 2(c). This trend, which has already been
reported Jayaweera and Mason (1965); Bragg  _et al._ (1974); Lopez and
Guazzelli (2017); Roy  _et al._ (2019); Gustavsson  _et al._ (2019); Sheikh
_et al._ (2020), is a consequence of the action of the inertial torque
$T_{I}$. Equation (8) shows (for spheroids), in the case of a fluid at rest,
that under the effect of this torque, the particle orientation has two fixed
points, a vertical one ($\theta=\pi/2$), and an horizontal one ($\theta=0$),
but only the latter is stable. This is consistent with the observation that in
Fig. 4 the magnitude of the angular velocity reduces as the particle
approaches horizontal orientation, i.e. as $\theta$ tends to $0$.

Finally, it is worth mentioning that the particle Reynolds number,
$\mathrm{Re}_{p}$, is smaller when the particle settles horizontally
($\theta=0^{\circ}$) as compared to vertically ($\theta=90^{\circ}$). This is
a consequence of the fact that the drag exerted on a fiber settling broadside
down is higher than that on a fiber settling with its narrow edge first. The
later results in a velocity increase with $\theta$, as seen in Figure 3.

$\beta=8$(a)$\beta=16$(b)

Figure 4: Time evolution of the orientation angle $\theta$ for a) $\beta=8$
and b) $\beta=16$ for all realizations, color-coded by the value of
$\mathrm{Re}_{p}$.

We now estimate the importance of the particle inertia term in the rotational
dynamics, Eq. (3), from the ratio $I_{p}\ddot{\theta}/C_{S}\dot{\theta}$. This
ratio is very small, well below $10^{-3}$, for the two $\beta$ values. As a
consequence, the left-hand side term of Eq. (3) is negligible with respect to
either of the two terms on the right-hand side. In other words, the particle
inertia contribution in the rotational dynamics can be neglected: the fluid-
inertia torque ($T_{I}$) and the Stokes torque ($T_{S}$) essentially balance
each other, so the particle angular dynamics is overdamped. Neglecting the
second order derivative in Eq. (3) allows us to simply relate the torque
$T_{I}$ to the instantaneous rotation rate, $\dot{\theta}$, which therefore
provides us with an elementary way to determine $T_{I}$.

$\beta=8$(a)$\beta=16$(b)

Figure 5: The measured angular velocity, $\dot{\theta}$, divided by
$|\mathbf{w}|^{2}$, color coded by the local value of ${\rm Re}_{p}$. The
predictions of Eq. (9) are shown for the spheroid model and for the slender-
rod model, as indicated by the legend. Panel (a) corresponds to $\beta=8$, and
(b) to $\beta=16$.

To compare the results of the experiments with the prediction of Eq. (9), Fig.
5 shows, for $\beta=8$ (panel a) and for $\beta=16$ (panel b), the measured
ratio $\dot{\theta}/|\mathbf{w}|^{2}$ as a function of the pitch angle,
$\phi$. The trajectories are color coded by the recorded value of the particle
Reynolds number, ${\rm Re}_{p}$, as indicated by the color bars. Equation (9)
predicts that the points should be along a curve
$\dot{\theta}/|\mathbf{w}|^{2}=(C_{I}/C_{S})\sin 2\phi$. The expected values
for rods (empty circles) and for spheroids (stars) are indicated in the
figure. We also tested the torque expression proposed by Kharrouba  _et al._
(2021): it leads in the regime of our experiments ($Re_{p}<0.4$) to curves
virtually undistinguishable from those plotted by using the slender-rod model
from Khayat and Cox (1989) (empty circles). As shown in Fig. 5, the constants
$C_{I}/C_{S}$ for rods and spheroids differ by a factor $\approx 2$ for
$\beta=8$, and $\approx 1.5$ for $\beta=16$.

The dependence of $\dot{\theta}/|\mathbf{w}|^{2}$ qualitatively follows the
$\sin 2\phi$ prediction, although with a very large dispersion, particularly
for $\beta=8$. Interestingly, for both $\beta=8$ and $\beta=16$, the slender-
rod theory overpredicts the value of $\dot{\theta}/|\mathbf{w}|^{2}$. On the
other hand, the spheroid theory underpredicts the observed dependence,
although the agreement becomes better when $\beta=16$.

In view of the limitations of the theory, it is reassuring to see that the
theoretical predictions provide the right order of magnitude for the
experimental results. To discuss further, we recall that the two theories used
here for comparison have different shortcomings. Namely, the slender-rod
theory (open circles) is valid over a larger range of Reynolds numbers (it
only requires that, with our notation, ${\rm Re}_{p}\ll 1$); but the theory is
valid only when $\log\beta\gg 1$, which is not particularly well satisfied. On
the other hand, the theory for spheroids works for any aspect ratio ($\beta$
does not need to be very large); but the predictions of the torque deviate by
$\sim 30\%$ for values of ${\rm Re}_{p}=0.3$, and deviations grow when ${\rm
Re}_{p}$ increases Jiang  _et al._ (2021). In this context, our observations
fall in the delicate regime where both theories have shortcomings.

The relatively good agreement between the predictions of the slender-rod
theory Khayat and Cox (1989) when $\beta=8$ is surprising, as the value
$\beta=8$ is not particularly large. In fact, the agreement when $\beta$
increases to $\beta=16$ tends to deteriorate, which indicates that the
slender-rod approximations does not provide a satisfactory description of the
results. In a similar spirit, the relatively better agreement between our
measurements for $\beta=16$ and the prediction for spheroids is very likely to
be coincidental: we do not expect it to indicate a definite trend, valid at
larger values of $\beta$.

It is interesting to compare the trends in our experiments with previous
experimental investigations Roy  _et al._ (2019). As already noticed, the
expression for the torque obtained in the slender-rod approximation provides a
very good description of our results at $\beta=8$ (${\rm Re}_{p}\lesssim
0.15$). At the comparable, higher values $\beta=16$ in our case, and
$\beta=20$ in Roy  _et al._ (2019), the prediction of the slender-rod theory
Khayat and Cox (1989) overpredicts the torque. We notice that the Reynolds
number in Roy  _et al._ (2019) is significantly larger: ${\rm Re}_{p}\approx
1.6$, although the values of $\rm{Re}_{p}/\beta$ are both small. At the larger
value of $\beta=100$ in Roy  _et al._ (2019), the theoretical prediction of
Khayat and Cox (1989) underestimates the experimentally measured torque. The
Reynolds number, however, is even larger: ${\rm Re}_{p}\approx 8.6$, which
adds some extra uncertainties when comparing the measurements with the
theoretical expressions, obtained by assuming $\mathrm{Re}_{p}\ll 1$. Note,
however, that the value of ${\rm Re}_{p}/\beta$ remains small: ${\rm
Re}_{p}/\beta\approx 0.09$. For this reason, it would be extremely interesting
to obtain results at the large aspect ratios considered in Roy  _et al._
(2019), but at a lower Reynolds number, as could be obtained by using a more
viscous fluid, as done here.

##  V Conclusions

In this work, we studied experimentally the dynamics of rods settling in a
quiescent flow. Our experiments were in the regime where the angular dynamics
is overdamped, so the torque and the force acting on the object could be
readily determined from the settling and from the angular velocities, which
was made possible via a 3D-PTV. We considered two particles, with aspect
ratios $\beta$ = 8 and $\beta$ = 16. In the two cases, the Reynolds numbers
were small, making comparisons with predictions at low particle Reynolds
numbers ($\rm Re_{p}\ll 1$) meaningful.

We compared the measured torques with theoretical predictions in the slender-
rod limit, and in the case of spheroids. Our results show that both models
qualitatively predict the translational dynamics, in particular the slender-
rod theory was found to represent better the experimental data.

Regarding the rod angular dynamics, the models took a simpler form as the
rotational dynamics was found to be planar and overdamped. The particles were
seen to orient broadside on, i.e. with the maximal drag orientation.

The torque is found to be qualitatively well described by a sin $2\phi$
functional form, and differences in the torque pre-factor that both models
provide were seen. The spheroid theory was found to describe correctly the
dynamics of the rods with $\beta$ = 16, whereas for the case $\beta$ = 8 the
agreement is less satisfactory, the experimental pre-factor being roughly
twice as large as the theory predictions. On the other hand, the slender-rod
theory happens to quantitatively agree with the data in the case $\beta$ = 8
and, for $\beta$ = 16, tends to over-predict the magnitude of fluid-inertia
torque.

###### Acknowledgements.

The authors are grateful to Vincent Dolique for his help with the Scanning
Electron Microscopy. This work was supported by the French research program
IDEX-LYON of the University of Lyon in the framework of the French program
“Programme Investissements d’Avenir” (ANR-16-IDEX-0005). This research was
also supported by Higher Education Commission Pakistan and Campus France.

##  Appendix A Expression for the force acting on anisotropic particles

We begin by recalling the general expression for the Stokes force:

$\mathbf{f}_{S}=6\pi\mu a\mathbb{A}(\hat{\mathbf{n}})(\mathbf{u}-\mathbf{w}),$
(10)

where the components of the tensor $\mathbb{A}$ are given by
$A_{ij}=A_{\|}n_{i}n_{j}+A_{\perp}(\delta_{ij}-n_{i}n_{j})$. The expression
for $A_{\|}$ and $A_{\perp}$ are, in the case of a spheroid:

$A^{s}_{\|}=\frac{8}{3}\frac{\beta}{\chi_{0}+\beta^{2}\gamma_{0}}~{}~{}~{}{\rm
and}~{}~{}~{}A^{s}_{\perp}=\frac{8}{3}\frac{\beta}{\chi_{0}+\alpha_{0}},$ (11)

with

$\alpha_{o}=\frac{\beta^{2}}{\beta^{2}-1}-\beta\frac{\mathrm{cosh^{-1}}\beta}{(\beta^{2}-1)^{3/2}}~{}~{}{\rm,}~{}~{}\gamma_{o}=\frac{-2}{\beta^{2}-1}+2\beta\frac{\mathrm{cosh^{-1}}\beta}{(\beta^{2}-1)^{3/2}}~{}~{}{\rm
and}~{}~{}\chi_{0}=2\beta\frac{\cosh^{-1}\beta}{(\beta^{2}-1)^{1/2}}.$ (12)

In the case of a rod, the expression of the coefficients $A^{r}_{\perp}$ and
$A^{r}_{\|}$ are:

$A^{r}_{\|}=\frac{2}{3}\frac{\beta}{\log\beta}~{}~{}~{}{\rm
and}~{}~{}~{}A^{r}_{\perp}=\frac{4}{3}\frac{\beta}{\log\beta}.$ (13)

The correction to the force due to inertial effects has been derived for
spheroids in Oberbeck (1876). The expression for the coefficients
$A^{s}_{I,\perp}$ and $A^{s}_{I,\|}$ are:

$A^{s}_{I,\|}=[3A_{\|}-(A_{\|}\cos^{2}\phi+A_{\perp}\sin^{2}\phi)]A_{\|}~{}~{}~{}{\rm
and}~{}~{}~{}A^{s}_{I,\perp}=[3A_{\perp}-(A_{\|}\cos^{2}\phi+A_{\perp}\sin^{2}\phi)]A_{\perp},$
(14)

where $\phi=\cos^{-1}({\hat{\bf w}\cdot\hat{\bf n}})$.

In the case of a rod, the fluid-inertia correction to the force,
$\mathbf{f}^{r}_{I}$, is given by:

$\frac{\mathbf{f}^{r}_{I}}{2\pi\mu|{\bf
w}|l}=\bigg{(}\frac{1}{\log\beta}\bigg{)}^{2}\bigg{[}$ (15)
$\quad\frac{2\cos\phi\hat{\bf w}-(2-\cos\phi+\cos^{2}\phi){\bf
n}}{2\mathrm{Re}_{p}(1-\cos\phi)}\bigg{\\{}E_{1}[\mathrm{Re}_{p}(1-\cos\phi)]+\log[\mathrm{Re}_{p}(1-\cos\phi)]+\gamma-\mathrm{Re}_{p}(1-\cos\phi)\bigg{\\}}$
$-\frac{2\cos\phi\hat{\bf w}-(2+\cos\phi+\cos^{2}\phi){\bf
n}}{2\mathrm{Re}_{p}(1+\cos\phi)}\bigg{\\{}E_{1}[\mathrm{Re}_{p}(1+\cos\phi)]+\log[\mathrm{Re}_{p}(1+\cos\phi)]+\gamma-\mathrm{Re}_{p}(1+\cos\phi)\bigg{\\}}$
$-[\cos\phi{\bf n}-2\hat{\bf
w}]\bigg{\\{}E_{1}[\mathrm{Re}_{p}(1-\cos\phi)]+\log(1-\cos\phi)+E_{1}[\mathrm{Re}_{p}(1+\cos\phi)]+\log(1+\cos\phi)$
$+\frac{1-e^{-\mathrm{Re}_{p}(1-\cos\phi)}}{\mathrm{Re}_{p}(1-\cos\phi)}+\frac{1-e^{-\mathrm{Re}_{p}(1+\cos\phi)}}{\mathrm{Re}_{p}(1+\cos\phi)}+2(\gamma+\log(\mathrm{Re}_{p}/4))\bigg{\\}}+3\cos\phi{\bf
n}-2\hat{\bf w}\bigg{]},$

where $\hat{\bf w}$ is the unit vector along fiber velocity ${\bf w}$, and
$\gamma$ is Euler constant.

##  Appendix B Expression for the torque acting on anisotropic particles

The crucial parameter for our work is the ratio $C_{I}/C_{S}$ in Eq. (9),
where $C_{S}$ and $C_{I}$ are defined by Eq. (4) and (8) respectively.

#### Spheroids

In the case of spheroids, the shape factor, $F(\beta)$, has been determined in
Dabade  _et al._ (2015) (see in particular Eq. 4.1). The function $F(\beta)$
is represented e.g. in Fig.1 of Jiang  _et al._ (2021). In the limit of very
large aspect ratio, $\beta\gg 1$, $F(\beta)\approx-5\pi/[3(\log\beta)^{2}]$.

The ratio between the two coefficients $C_{I}$ and $C^{s}_{S}$, appearing in
Eq. (9), is independent of particle size and only depends on $\beta$ and
$\nu$; it is given as

$\frac{C^{s}_{I}}{C^{s}_{S}}=\frac{\rho_{f}l^{3}F(\beta)}{2}\frac{3}{16}\frac{1}{\pi\mu
a^{3}\beta}\bigg{(}\frac{\alpha_{o}+\beta^{2}\gamma_{o}}{1+\beta^{2}}\bigg{)}=\frac{3}{32}\frac{\beta^{2}F(\beta)}{\pi\nu}\bigg{(}\frac{\alpha_{o}+\beta^{2}\gamma_{o}}{1+\beta^{2}}\bigg{)}$
(16)

The ratio in Eq. (16) is plotted in figure 6.

Figure 6: The ratio $-C^{s}_{I}/C^{s}_{S}$ in the case of spheroids, plotted
as a function of $\beta$ at $\nu=1.05/1216\mathrm{m^{2}s^{-1}}$.

#### Slender-Rods

In the case of rods, the expression for the torque $T^{r}_{I}$ is given by
Khayat and Cox (1989):

$T_{I}^{r}=H({\rm Re}_{p},\phi)|\mathbf{w}|\sin 2\phi=C^{r}_{I}|{\bf
w}|^{2}\sin 2\phi,$ (17)

where $C^{r}_{I}=H(\mathrm{Re_{p},\phi})/|{\bf w}|$ with the following
expression for $H$:

$\frac{(\log\beta)^{2}}{2\pi\mu
l^{2}}H(\mathrm{Re_{p},\phi})=\frac{1}{2\mathrm{Re}_{p}(1-\cos\phi)}\bigg{\\{}2+2\frac{e^{-\mathrm{Re}_{p}(1-\cos\phi)}-1}{\mathrm{Re}_{p}(1-\cos\phi)}-E_{1}[\mathrm{Re}_{p}(1-\cos\phi)]-\log[\mathrm{Re}_{p}(1-\cos\phi)]-\gamma\bigg{\\}}$
$+\frac{1}{2\mathrm{Re}_{p}(1+\cos\phi)}\bigg{\\{}2+2\frac{e^{-\mathrm{Re}_{p}(1+\cos\phi)}-1}{\mathrm{Re}_{p}(1+\cos\phi)}-E_{1}[\mathrm{Re}_{p}(1+\cos\phi)]-\log[\mathrm{Re}_{p}(1+\cos\phi)]-\gamma\bigg{\\}}$
$-\frac{1}{\mathrm{Re}_{p}(1-\cos\phi)\cos\phi}\bigg{\\{}1-\frac{1-e^{-\mathrm{Re}_{p}(1-\cos\phi)}}{\mathrm{Re}_{p}(1-\cos\phi)}\bigg{\\}}+\frac{1}{\mathrm{Re}_{p}(1+\cos\phi)\cos\phi}\bigg{\\{}1-\frac{1-e^{-\mathrm{Re}_{p}(1+\cos\phi)}}{\mathrm{Re}_{p}(1+\cos\phi)}\bigg{\\}}.$
(18)

In the limit $\mathrm{Re}_{p}\rightarrow 0$, appropriate in the case of our
study, and for $\beta\gg 1$, we find that:

$T_{I}^{r}\approx-\frac{5\pi}{6(\log\beta)^{2}}\rho_{f}|\mathbf{w}|^{2}l^{3}\sin
2\phi,$ (19)

which coincides with the expression derived in Dabade  _et al._ (2015). The
ratio $C^{r}_{I}/C^{r}_{S}$ for rods reads

$\frac{C^{r}_{I}}{C^{r}_{S}}=\frac{H(\mathrm{Re}_{p},\phi)}{|{\bf
w}|}\frac{3}{8\pi}\frac{\log\beta}{\mu
l^{3}}=\frac{3}{8\pi}\frac{H(\mathrm{Re}_{p},\phi)\log\beta}{|{\bf w}|\mu
l^{3}}.$ (20)

For a particle settling in a viscous fluid, the ratio in Eq. (20) is a
function of particle orientation through the particle Reynolds number,
$\mathrm{Re}_{p}$, the settling velocity, ${\bf w}$, and the pitch angle,
$\phi$. The averaged values (over all orientations) of $C^{r}_{I}/C^{r}_{S}$
for the cylindrical particles under consideration are -170 and -115 for
$\beta=8$ and $\beta=16$, respectively. The corresponding values in case of
spheroids are -82 and -75, see Fig. 6.

##  Appendix C Horizontal translational dynamics

Figure 7 shows the ratio between the horizontal, wh, and the vertical, wg,
components of velocity. As it was the case in Fig. 5, we show the expected
ratios from the spheroid model (full line) and from the slender-rod model
(dashed-dotted lines), for $\beta=8$ (panel a) and for $\beta=16$ (panel b).
The ratios are all found to be small, less than $\sim 20\%$, in agreement with
the predictions from the two models, which differ by no more than $\sim 20\%$
from each other.

$\beta=8$(a)$\beta=16$(b)

Figure 7: Ratio between horizontal (wh) and vertical (wg) velocity components,
as a function of $\phi$.

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