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# Investigating neutron transfer in the 9Be + 197Au system

Malika Kaushik1 S.K. Pandit2 V.V. Parkar2 G. Gupta3 Swati Thakur1 V. Nanal3 H.
Krishnamoorthy4,5 A. Shrivastava2,4 C.S. Palshetkar3 K. Mahata2,4 K.
Ramachandran2 S. Pal6 R.G. Pillay1 Pushpendra P. Singh1<EMAIL_ADDRESS>1Department of Physics, Indian Institute of Technology Ropar, Rupnagar - 140
001, Punjab, India  2Nuclear Physics Division, Bhabha Atomic Research Centre,
Mumbai - 400 085, India  3Department of Nuclear and Atomic Physics, Tata
Institute of Fundamental Research, Mumbai - 400 005, India  4Homi Bhabha
National Institute, Anushaktinagar, Mumbai - 400 094, India  5India-based
Neutrino Observatory, Tata Institute of Fundamental Research, Mumbai - 400
005, India  6Pelletron Linac Facility, Tata Institute of Fundamental Research,
Mumbai - 400 005, India

###### Abstract

In this work n-transfer and incomplete fusion cross sections for 9Be + 197Au
system are reported over a wide energy range, Ec.m. $\approx$ 29-45 MeV. The
experiment was carried out using activation technique and off-line gamma
counting. The transfer process is found to be the dominant mode as compared to
all other reaction channels. Detailed coupled reaction channel (CRC)
calculations have been performed for n-transfer stripping and pickup cross
sections. The measured 1n-stripping cross sections are explained with CRC
calculations by including the ground state and the 2+ resonance state (E =
3.03 MeV) of 8Be. The calculations for 1n-pickup, including only the ground
state of 10Be agree reasonably well with the measured cross sections, while it
overpredicts the data at subbarrier energies. For a better insight into the
role of projectile structure in the transfer process, a comprehensive analysis
of 1n-stripping reaction has been carried out for various weakly bound
projectiles on 197Au target nucleus. The transfer cross sections scaled with
the square of total radius of interacting nuclei show the expected Q-value
dependence of 1n-stripping channel for weakly bound stable projectiles.

###### pacs:

Valid PACS appear here

##  I Introduction

The nuclear reactions involving weakly bound projectiles open up many
distinctive facets of reaction dynamics and play a crucial role in
understanding the interplay between quantum tunneling and the breakup process
at energies around the Coulomb barrier vj . The striking characteristics of
weakly bound projectiles, e.g., low breakup threshold (EBU) and underlying
cluster structure, significantly affect nuclear dynamics bb ; lf1 . The weakly
bound projectiles like 6,7Li and 9Be can break up into their constituent
clusters during the interaction process. As a consequence of the weak binding
of the projectile, one of the projectile fragments following breakup can get
absorbed by the target nucleus, known as breakup fusion csni ; brf ; vvpp ;
dhlu ; rr . Single step cluster transfer from bound state of the projectile to
bound/unbound states of the target is also one of the dominant mechanisms, as
shown in recent studies kcook ; kjc ; amm . These two processes are sometimes
also referred to as incomplete fusion (ICF).
Complete fusion (CF) is defined as the amalgamation of the projectile as a
whole with the target nucleus. It should be pointed out that CF and breakup
followed by sequential capture of all the projectile fragments cannot be
distinguished experimentally. In the case of weakly bound projectiles, it was
suggested that the fusion suppression at the above barrier energies occurs due
to the direct breakup of the projectile, while the breakup following transfer
is a dominant mechanism at subbarrier energies md1 . In several existing
studies of 6,7Li, 9Be rr ; dasgupta ; sks , the fusion suppression was
attributed to the breakup of the projectile dasgupta . However, in a recent
exclusive measurement, Cook et al. kcook  have shown that the suppression of
CF is primarily a consequence of strong clustering in weakly bound projectiles
rather than the breakup prior to reaching the fusion barrier. These recent
results emphasize the role of transfer reactions with weakly bound
projectiles.

Investigation of the transfer channel measurements provides insight into low
energy reaction dynamics, particularly to understand the role of valence
nucleons, the spatial correlations of the valence nucleons, and the pairing
properties al1 . Relatively larger (1n and 2n) transfer cross sections have
been observed in reactions involving halo projectiles (6,8He) an ; al1 .
Several investigations with stable weakly bound projectile, 9Be ydf ; rr ; sk
; vvpp , suggest that the neutron transfer channels dominate over CF at
subbarrier energies. The CF excitation functions typically show a steeper fall
off compared to the transfer channels ydf . In 6Li + 96Zr sph  reaction, it is
reported that the neutron transfer channels contribute significantly to the
total reaction cross section, especially at energies around the Coulomb
barrier. In 6He + 64Zn experiment, Pietro et al., adp1  observed that transfer
and breakup channels comprise the largest fraction of the reaction cross
section (about 80$\%$) in near barrier region.

In a recent measurement of 9Be + 197Au system mk , the fusion cross sections
have been found to be suppressed by $\sim 39\%$ w.r.t. theoretical
calculations (CCFULL) at above barrier energies. The 9Be, Borromean nucleus,
may break into $\alpha$ \+ $\alpha$ \+ n (EBU = 1.57 MeV) pp ; tad  or 8Be + n
(EBU = 1.66 MeV) or 5He + $\alpha$ (EBU = 2.31 MeV) nk1 . The 8Be and 5He are
unbound with lifetimes of 10-16s and 10-21s, respectively. In reactions with
9Be, at energies around the Coulomb barrier, the n \+ 8Be cluster
configuration is shown to play a prominent role as compared to the 5He +
$\alpha$ configuration vvpp ; sk . In this work, we have investigated neutron
transfer and incomplete fusion in the 9Be + 197Au system at energies around
the Coulomb barrier. It should be noted that in inclusive measurements,
contributions by different mechanisms breakup fusion and single step cluster
transfer can not be distinguished and are referred as ICF in this paper. The
experimental methodology and analysis are presented in section II. The
detailed theoretical model calculations and their comparison with the
experimental data of transfer channels cross sections in 9Be + 197Au system
are described in section III. The analysis of transfer channels cross sections
in 9Be + 197Au system (present work) and the reactions involving other weakly
bound projectiles, $\rm x$ \+ 197Au systems are delineated in Section IV. The
summary of the present work is outlined in section V.

##  II Experimental details and data analysis

The experiment was performed using 30-47 MeV 9Be beam from the BARC-TIFR
Pelletron Linac Facility, Mumbai, India. A detailed description of the
experimental setup and data analysis is given in Ref. mk . The self-supporting
197Au targets foils (thickness $\sim$ 1.3 - 1.7 mg/cm2) together with the
aluminum catcher foils (thickness $\sim$ 1.5 mg/cm2) to stop recoiling
products were used. Irradiation was done in 1-2 MeV steps with a beam current
of 8-15 pnA. In some cases, stacked target-catcher foil assemblies were
deployed, and beam energy at the center of the target was calculated using
SRIM tr . The activity in the irradiated targets was measured using off-line
$\gamma$-ray counting method with efficiency calibrated HPGe detectors. The
counting geometry, i.e., sample at a distance of 10 cm from the detector face
or at the face, was chosen depending on the activity level in the sample. The
transfer and ICF reaction products were identified by their characteristic
$\gamma$-rays. The identification was independently confirmed by half-life
measurements.

Figure 1: Typical $\gamma$-ray spectra of 9Be + 197Au system measured at Elab
= 36.6 MeV (tcollection = 11.1 hr) and and 46.7 MeV (tcollection = 1.55 hr),
recorded after a cooldown period 4.64 days and 0.71 days, respectively. Some
of intense decay $\gamma$-lines corresponding to different reaction products
are marked.

Typical $\gamma$-ray spectra corresponding to Elab = 46.7 MeV and 36.6 MeV
energies are shown in Fig. 1, where the $\gamma$-rays corresponding to
transfer 198Au (1n-stripping), 196Au (1n-pickup), and ICF (199Tl, 200Tl)
products are marked. The measured reaction products of transfer and ICF in 9Be
+ 197Au system are listed in Table 1 along with the most intense
characteristic $\gamma$-ray and the corresponding intensity. It may be noted
that 199Tl can be populated either via 197Au(4He,2n) or 197Au(5He,3n)
reactions. Similarly, both 197Au(4He,1n) and 197Au(5He,2n) contribute to the
production of 200Tl.
The cross sections for the reaction products populated due to transfer and ICF
channels are extracted from the photopeak yields of characteristic
$\gamma$-rays by taking into account the decay during the irradiation mk . The
measured cross sections at different energies are listed in Table 2 and
plotted in Fig. 2. It should be mentioned that the quoted errors include
statistical and fitting errors.

Table 1: List of reaction products populated via transfer and ICF channels in 9Be + 197Au system with respective half-life (T1/2) together with energy and absolute intensity (Iγ) of the most intense characteristic $\gamma-$ray nndc . Reaction | Nuclide | T1/2 | Eγ (keV) | Iγ($\%$)
---|---|---|---|---
1n stripping |  198Au | 2.69 d | 411.8 | 95.6
1n pickup |  196Au | 6.17 d | 355.7 | 87.0
ICF |  200Tl | 26.1 h | 367.9 | 87.0
ICF |  199Tl | 7.42 h | 455.5 | 12.4

##  III Results and discussion

Figure 2: Measured excitation functions of (a) transfer (198,196Au), and (b)
ICF (200,199Tl) channels populated in 9Be + 197Au system. The barrier value
(VB = 38.4 MeV) is marked on energy axis, and symbols are self explanatory.

Excitation functions for reaction products populated due to 1$n$-stripping
(198Au), 1$n$-pickup (196Au) and ICF followed by neutron evaporation
(199,200Tl) are shown in Fig. 2(a) and Fig. 2(b), respectively. The value of
barrier (VB = 38.4 MeV) mk  is shown by an arrow. As shown in Fig. 2, the
transfer component is measured down to $\approx$ 29 MeV (i.e., $\approx$ 24
$\%$ below the barrier), while the incomplete fusion component is measured
down to $\approx$ 18 $\%$ below the barrier. In order to get a comprehensive
picture of CF mk , ICF and transfer reactions, cross sections of all these
channels are plotted as a function of energy in Fig. 3. As can be seen from
this figure, the transfer process significantly dominates over CF and ICF
processes at subbarrier energies, whereas CF shows higher cross sections at
above barrier energies.

Table 2:  Experimentally measured cross sections of ICF (200,199Tl) evaporation residues (ERs) and transfer (198,196Au) channels populated in 9Be + 197Au system, $V_{\rm B}$ = 38.4 MeV. ELab (MeV) | Ec.m. (MeV) |  200Tl (mb) |  199Tl (mb) |  198Au (mb) |  196Au (mb)
---|---|---|---|---|---
30.4 | 29.1 | - | - | 11.2 $\pm$ 0.1 | -
33.0 | 31.6 | 0.150 $\pm$0.002 | - | 24.3 $\pm$ 0.2 | 0.150 $\pm$ 0.006
34.5 | 33.0 | 0.76 $\pm$ 0.01 | - | 33.2 $\pm$ 0.2 | 0.35 $\pm$ 0.03
35.6 | 34.0 | 3.50 $\pm$ 0.05 | - | 69.3 $\pm$ 0.4 | 0.90 $\pm$ 0.07
36.6 | 35.0 | 10.5 $\pm$ 0.09 | - | 83.7 $\pm$ 0.3 | 1.30 $\pm$ 0.08
37.6 | 36.0 | 22.5 $\pm$ 0.2 | 0.5 $\pm$ 0.2 | 124.2 $\pm$ 0.6 | 2.6 $\pm$ 0.3
38.6 | 36.9 | 36.6 $\pm$ 0.5 | 2.6 $\pm$ 0.6 | 123.8 $\pm$ 0.8 | 3.5 $\pm$ 0.4
39.6 | 37.9 | 55.2 $\pm$ 0.4 | 5.2 $\pm$ 0.9 | 135.4 $\pm$ 0.6 | 5.8 $\pm$ 0.6
39.9 | 38.2 | 68.0 $\pm$ 0.6 | 10.6 $\pm$ 1.5 | 139.7 $\pm$ 1.4 | 5.2 $\pm$ 0.7
40.6 | 38.8 | 92.1 $\pm$ 0.8 | 11.0 $\pm$ 1.2 | 189.6 $\pm$ 1.4 | 6.6 $\pm$ 0.8
42.7 | 40.8 | 95.7 $\pm$ 0.8 | 20.3 $\pm$ 1.9 | 154.1 $\pm$ 1.3 | 7.2 $\pm$ 0.7
44.7 | 42.7 | 144.3 $\pm$ 1.3 | 56.5 $\pm$ 2.8 | 164.7 $\pm$ 1.4 | 8.5 $\pm$ 0.6
46.7 | 44.7 | 119.1 $\pm$ 1.2 | 75.4 $\pm$ 5.1 | 136 $\pm$ 1 | 9.1 $\pm$ 0.4
Figure 3: Comparison of measured transfer, ICF, and CF mk  cross sections as a function of Ec.m. in 9Be + 197Au system. Table 3:  Potential parameters used in CRC calculations for 9Be + 197Au, 6,7Li + 197Au systems. The radius parameter in the potentials are derived from $R_{i}=r_{i}.A^{1/3}$, where i = R, V, S, C and A is the target mass number. System | VR (MeV) | rR (fm) | aR (fm) | WV(MeV) | rV(fm) | aV(fm) | WS(MeV) | rS(fm) | aS(fm) | rC (fm)
---|---|---|---|---|---|---|---|---|---|---
$n$ \+ 197Au dgk  | 50.00 111Depth adjusted to obtain the correct binding energy. | 1.23 | 0.65 | - | - | - | 6.00 | 1.23 | 0.65 | -
$2n$ \+ 197Au dgk  | 50.00 111Depth adjusted to obtain the correct binding energy. | 1.23 | 0.65 | - | - | - | 6.00 | 1.23 | 0.65 | -
$n$ \+ 196Au dgk  | 50.00 111Depth adjusted to obtain the correct binding energy. | 1.23 | 0.65 | - | - | - | 6.00 | 1.23 | 0.65 | -
$n$ \+ 8Be jl  | 50.00 111Depth adjusted to obtain the correct binding energy. | 1.15 | 0.57 | - | - | - | 5.50 | 1.15 | 0.57 | -
$n$ \+ 9Be jl  | 50.00 111Depth adjusted to obtain the correct binding energy. | 1.15 | 0.57 | - | - | - | 5.50 | 1.15 | 0.57 | -
9Be + 197Au yl  | 257.70 | 1.36 | 0.73 | 16.32 | 1.64 | 0.60 | 46.30 | 1.20 | 0.84 | 1.56
$n$ \+ 7Li jc  | 50.00 111Depth adjusted to obtain the correct binding energy. | 1.25 | 0.70 | - | - | - | 6.00 | 1.25 | 0.70 | -
$n$ \+ 6Li jc  | 50.00 111Depth adjusted to obtain the correct binding energy. | 1.25 | 0.70 | - | - | - | 6.00 | 1.25 | 0.70 | -
$2n$ \+ 5Li jc  | 50.00 111Depth adjusted to obtain the correct binding energy. | 1.25 | 0.70 | - | - | - | 6.00 | 1.25 | 0.70 | -
6Li + 197Au jc  | 109.50 | 1.33 | 0.81 | 20.71 | 1.53 | 0.88 | - | - | - | 1.30
7Li + 197Au jc  | 114.20 | 1.29 | 0.85 | 20.71 | 1.74 | 0.81 | - | - | - | 1.30
|  |  |  |  |  |  |  |  |  |

###  III.1  CRC calculations: 9Be + 197Au system

Figure 4: Comparison of measured excitation functions with CRC calculations of
transfer channels for (a) 198Au including the ground state and 2+ resonance (E
= 3.03 MeV) state of 8Be, and only the ground state of 8Be, and (b) 196Au with
ground state of 10Be.

For further analysis of transfer channels presented in Fig. 2, the coupled
reaction channel (CRC) calculations have been performed using a theoretical
model code FRESCO ijt . For CRC calculations optical model potentials for
entrance and exit channels, binding potentials between transferred particle
and core nucleus, and spectroscopic factors for different residual states are
required inputs. The optical model potential is usually taken to be of Woods-
Saxon form for both real and imaginary parts. The potential parameters for $n$
\+ 197Au are taken to be the same as those of $n$ \+ 208Pb from Ref. dgk ,
since 197Au and 208Pb lie in the similar mass region. Since n + 8Be cluster
picture of 9Be has been used in the present analysis, the potential parameters
of the binding of $n$ in 9Be projectile, i.e., the depth of volume potential,
radius, diffuseness along with the spin-orbit component are taken from Ref. jl
. The global optical model potential parameters for 9Be + 197Au system are
taken from Ref. yl . These potential parameters are further cross-checked by
reproducing the experimentally measured elastic scattering cross sections for
this system over a wide energy range, Ec.m. = 32.5$-$46 MeV fg .

The optical model potential parameters for 1n-stripping (198Au) and 1n-pickup
(196Au) channels calculations are given in Table 3. In 1n-stripping, the
spectroscopic factor (C2S) for 9Be/8Be is taken to be 0.42 jl  and 1 for the
ground state and 2+ resonance state of 8Be, respectively. In 1n-pickup, C2S
for 9Be/10Be is taken to be 1.58 mbt . For both 1n-stripping and 1n-pickup,
the spectroscopic factors for target states have been set to 1.0. From the
number of 198Au excited states upto 1.560 MeV given in Ref. nndc , only those
states with J value less than 5 and well determined Jπ have been considered.
Further, the calculations have been performed with a limited number of states
at a time, at three experimental energy points viz. the highest (Ec.m. = 44.7
MeV), lowest (Ec.m. = 33 MeV) and barrier (Ec.m. = 38.2 MeV) energy. This
procedure has been used to select the states which dominantly contribute to
the 1$n$-transfer channel and are listed in Table 4. For all the beam
energies, the final CRC calculations have been performed including these
states. Similarly, the excited states of 196Au upto 0.388 MeV having J value
less than 5 and well determined Jπ have been considered as listed in Table 4.
The couplings of 198Au excited states as given in Table 4 have been included
along with the 0+ ground state and 2+ resonance state of 8Be.

Table 4: Energy levels of transfer products used in the CRC calculations in 6,7Li, 9Be + 197Au systems nndc . 199Au |  198Au |  196Au
---|---|---
E (MeV) | Jπ | E (MeV) | Jπ | E (MeV) | Jπ
0.0 | 1.5+ | 0.0 | 2- | 0.0 | 2-
0.077 | 0.5+ | 0.215 | 4- | 0.006 | 1-
0.317 | 2.5+ | 0.381 | 3+ | 0.213 | 4-
0.323 | 1.5+ | 0.544 | 4- | 0.234 | 3-
0.493 | 3.5+ | 0.764 | 4- | 0.324 | 1-
0.543 | 2.5+ | 0.825 | 3+ | 0.326 | 1-
0.549 | 5.5- | 0.869 | 3- | 0.349 | 2-
0.735 | 3.5- | 0.972 | 3- | 0.375 | 3-
0.791 | 1.5+ | 1.032 | 3- | 0.388 | 3+
0.822 | 0.5+ | 1.061 | 3- |  |
0.907 | 1.5+ | 1.115 | 3- |  |
1.801 | 1.5+ | 1.157 | 3- |  |
2.107 | 0.5+ | 1.160 | 3- |  |
2.205 | 0.5+ | 1.209 | 3- |  |
2.412 | 0.5+ | 1.240 | 3- |  |
2.734 | 0.5+ | 1.272 | 3- |  |
|  | 1.305 | 3- |  |
|  | 1.381 | 3- |  |
|  | 1.396 | 3- |  |
|  | 1.409 | 3- |  |
|  | 1.424 | 3- |  |
|  | 1.444 | 3- |  |
|  | 1.454 | 3- |  |
|  | 1.459 | 3- |  |
|  | 1.472 | 3- |  |
|  | 1.496 | 3- |  |
|  | 1.560 | 3- |  |

The excitation functions of (a) 1n-stripping (198Au), and (b) 1n-pickup
(196Au) channels in 9Be + 197Au system along with the calculations are shown
in Fig. 4. As can be seen from the figure, the calculations including
couplings only to the ground state of 8Be underpredict the $\sigma_{\rm
1n-stripping}$ over the entire measured energy range, while those with
couplings to both the ground state and 2+ resonance state of 8Be well describe
the data. It is evident that the 2+ resonance state plays a crucial role in
the 1n-stripping channel. The Qopt = 0 for $n$-transfer using the semi-
classical expression given in Ref. wh . The inclusion of the 2+ resonance
state of 8Be at 3.03 MeV provides a better matching with the Q-value in the
1$n$-stripping reaction. In the case of 196Au, the couplings to excited states
of 196Au as given in Table 4 and 0+ ground state of 10Be have been included in
the calculations. As shown in Fig. 4(b), the calculations for 196Au agree
reasonably well with the experimental data. However, it overpredicts the
measured cross sections at subbarrier energies. It may be noted that 198Au
cross sections are significantly larger than 196Au cross sections. In
principle, 198Au can also have contribution from the ICF. However, the
observed good agreement with CRC calculations indicates that the dominant
contribution arises from the transfer process.

Table 5: List of dominant breakup channels together with corresponding breakup threshold energy ($E_{\rm BU}$), ground state Q-values for 1n-stripping (Q1n-strip) and 1n-pickup (Q1n-pickup) reactions in MeVs for weakly bound projectiles considered in the present study. Nuclei | Channel | EBU (MeV) | Q1n-strip | Q1n-pickup
---|---|---|---|---
9Be |  $\alpha$ \+ $\alpha$ \+ $n$ | 1.57 | 4.85 | -1.26
|  8Be + $n$ | 1.66 |  |
7Li |  $\alpha$ \+ t | 2.47 | -0.74 | -6.04
6Li |  $\alpha$ \+ d | 1.47 | 0.85 | -0.82
10B |  6Li + $\alpha$ | 4.46 | -1.92 | 3.38
|  9B + $n$ | 8.44 |  |

##  IV Transfer reaction systematics in $\rm x$ \+ $\rm{}^{197}Au$ systems

For a comprehensive understanding of projectile structure effect on transfer
reactions, it is useful to carry out a systematic comparison for transfer
cross sections ($\sigma_{tr}$) of various weakly bound stable projectiles,
i.e., 6,7Li, 9Be, and 10B with 197Au target. For comparison with transfer
reactions of other weakly bound stable projectiles, namely, 6,7Li csa , the
experimental excitation functions have been analyzed in the framework of CRC
calculations following the similar procedure described in the previous
section.

###  IV.1  CRC calculations: 6,7Li + 197Au systems

The excitation functions of transfer reaction products, 196Au, 198Au and
199Au, in 6,7Li + 197Au csa  systems are presented in Fig. 5. The global
optical potential parameters used in the calculation are presented in Table 3.
The potential parameters: VR, WV, WS parameter denote the V0, W0, VS
parameters in the 6,7Li + 197Au systems which are quoted in Ref. jc . In the
case of 6Li + 197Au system, the 6Li/5Li and 6Li/7Li spectroscopic factors are
taken to be 1.12 and 1.85 for 1$n$-stripping and 1$n$-pickup channels,
respectively mbt . Similar to 9Be + 197Au system, spectroscopic factors for
target states (196,198Au) have been fixed at 1.0. In 1$n$-stripping
calculations, the couplings of 198Au excited states up to 1.240 MeV as
tabulated in Table 4 and 3/2- ground state of 5Li have been included.

Figure 5: Comparison of measured cross sections csa  (a) 1n-stripping,
(b)1n-pickup in 6Li + 197Au(VB = 28.9 MeV) system, and (c) 1n-stripping, (d)
2n-stripping in 7Li + 197Au($V_{\rm B}$ = 29.3 MeV) system with CRC
calculations (solid curve). Coulomb barrier(VB) is marked by arrows.

As shown in Fig. 5(a), the CRC calculations for 1$n$-stripping (198Au) agree
well with experimental data at subbarrier energies and underpredict at above
barrier energies. Fig. 5(b) shows the CRC calculations for 1$n$-pickup
channel, performed by including couplings of 196Au excited states up to 0.388
MeV given in Table 4 and 3/2- ground state of 7Li. It can be seen that the
measured excitation function of the 1$n$-pickup channel is well reproduced by
the CRC calculations except for the lowest energy below the barrier.

A similar analysis has been carried out for 7Li + 197Au system and presented
in Fig. 5(c)-(d). For CRC calculations of 1n-stripping channel (198Au), the
7Li/6Li and 197Au/198Au spectroscopic factors have been taken to be 1.85 mbt
and 1.0, respectively. The couplings of 198Au excited states up to 0.972 MeV
are listed in Table 4 and 1+ ground state of 6Li have been included in the CRC
calculations. Similarly, for the 2n-stripping channel (199Au), the couplings
of 199Au excited states up to 2.734 MeV are listed in Table 4 and 3/2- ground
state of 5Li have been included. All the 199Au excited states with well
determined Jπ values have been considered. The 7Li/5Li and 197Au/199Au
spectroscopic factors have been taken as 1.0 in the calculations. As shown in
Fig. 5(c)-(d), the CRC calculations show a reasonable agreement with the
experimental data at subbarrier energies and underpredict the data at above
barrier energies. The increasing trend of underprediction of CRC calculations
at above barrier energies indicate the involvement of more states (high
excitation energy) of target like nuclei, which could not be included in the
present calculations.

As an illustration of the influence of the projectile structure in transfer
reactions, a systematic comparison of $\sigma_{tr}$ with CF cross sections
($\sigma_{CF}$) csa  for sub to above barrier energies has been carried out
for 6,7Li + 197Au systems and is shown in Fig. 6. It can be seen that for both
6,7Li projectiles $\sigma_{tr}$ is higher than $\sigma_{CF}$ at subbarrier
energies as expected. The $\sigma_{tr}$/$\sigma_{CF}$ is larger for 6Li than
that for 7Li, but is considerably smaller than that for 9Be (refer to Fig. 3).
This enhancement of transfer cross section over CF may be attributed to the
structural differences of 6,7Li, 9Be projectiles. In 9Be the last valence
neutron has a large spatial extension csni  and in addition the coupling to
the 2+ resonant state in 8Be plays a crucial role.

Figure 6: Comparison of the measured excitation functions of CF and transfer
reaction in (a) 6Li + 197Au, and (b) 7Li + 197Au systems csa , where VB is the
barrier.

###  IV.2  Comparison of transfer cross sections in $\rm x$ \+ $\rm{}^{197}Au$
systems

The excitation functions of the 1n-stripping channel with different weakly
bound stable projectiles 6,7Li, 9Be (present work) and 10B on 197Au target are
compared in Fig. 7 with suitable scaling. In order to suppress trivial
differences that arise due to the geometrical effects of the interacting
nuclei, a simple semi-classical scaling procedure used in Ref. prsg  for
fusion cross section comparison is adopted for the transfer cross section. The
scaled parameters are defined as,

$\sigma_{sc}~{}=~{}\sigma/R^{2}~{}~{}~{}{\rm
and}~{}~{}~{}E_{sc}~{}=~{}E_{cm}/V_{B},$ (1)

where $R~{}=~{}R_{P}~{}+~{}R_{T},$ RP is the projectile radius and RT = 7.27
fm (1.25.A${}_{T}^{1/3}$) is the target nucleus radius, and $V_{\rm B}$ is the
barrier. The $V_{\rm B}$ for 6,7Li, 9Be + 197Au systems is taken from the
CCFULL calculations, while that for 10B + 197Au is taken from Ref. ma . The
$R_{P}$ values of different projectiles are listed in Table 6.

Table 6:  Radii of different projectiles in $\rm x$ \+ 197Au systems. Projectile | RP (fm) | Ref.
---|---|---
6Li | 2.09 | tk
7Li | 2.23 | tk
9Be | 2.45 | tk
10B | 1.95 | kccp
Figure 7: Comparison of measured transfer excitation functions of
1$n$-stripping (198Au) channel in 9Be + 197Au (present work) and other
reactions, $\rm x$ \+ 197Au, involving 7Li csa , 6Li csa , 10B ma
projectiles.

The projectile breakup thresholds ($E_{\rm BU}$) and 1n-stripping Q-value are
tabulated in Table 5. It can be seen from Fig. 7 and Table 5 that amongst the
weakly bound stable projectiles, $\sigma_{sc}$ is the lowest for 10B, which
has the largest negative Q value and highest for 9Be that corresponds to the
largest positive Q value. Further, it should be mentioned that the choice of
$R_{P}$ is not crucial in the analysis, as $\sigma$ scaled with $R_{B}^{2}$
(where $R_{B}$ is the barrier radius obtained from CCFULL calculations) also
shows a similar trend.

##  V Summary and Conclusions

In summary, the transfer and ICF reaction cross sections have been measured in
9Be + 197Au system in an energy range 0.76 $\leq$ Ec.m./VB $\leq$ 1.16, and
analyzed using coupled reaction channel (CRC) calculations. The experimental
excitation function of 198Au populated via 1n-stripping channel is well
reproduced with the CRC calculations using couplings to the ground state and
2+ resonance state of 8Be. It is shown that the 2+ resonance state of 8Be
plays a significant role in 1$n$-stripping channel. For 1n-pickup channel, the
CRC calculations including only the ground state of 10Be are in reasonable
agreement with the data at above barrier energies, but overpredict the
measured cross sections at subbarrier energies. A systematic analysis of
1$n$-transfer, CF and ICF reaction channels, shows the dominance of
1$n$-transfer over ICF and CF at subbarrier energies in 9Be + 197Au system.

Additionally, the CRC calculations have been performed to analyze the
excitation functions of transfer channels in 6,7Li + 197Au systems csa . In
6Li + 197Au system, it has been found that the CRC calculations for
1n-stripping channel agree well with the experimental data at subbarrier
energies but underpredict at above barrier energies, while the 1n-pickup
channel show excellent agreement over most of the measured energy range. The
calculations of 1n and 2n-stripping transfer channels populated in 7Li + 197Au
system csa  agree well with the experimental data at subbarrier energies.

In the systematics for transfer reactions, the projectile structure effect is
illustrated by comparison of 1n-stripping cross sections of 6,7Li, and 9Be
projectiles on 197Au target. The scaled excitation functions of 1n-stripping
in the $\rm x$ \+ 197Au systems show the highest cross section values for 9Be
projectile, as expected from the ground state Q-value systematics for the
weakly bound stable projectiles.

##  VI Acknowledgments

We thank the PLF staff for providing the steady and smooth beam during the
experiments and the target laboratory personnel for their help in the target
preparation. We acknowledge the support of the Department of Atomic Energy,
Government of India, under Project No. 12-R$\&$DTFR-5.02-0300.

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# Transverse flow under oscillating stimulation in helical square ducts with
cochlea-like geometrical curvature and torsion

N.C. Harte  ARTORG Center for Biomedical Engineering Research, University of
Bern, Bern, Switzerland  Department of Otorhinolaryngology, Head and Neck
Surgery, Inselspital, Bern University Hospital, Bern, Switzerland  D. Obrist
ARTORG Center for Biomedical Engineering Research, University of Bern, Bern,
Switzerland  M. Caversaccio  ARTORG Center for Biomedical Engineering
Research, University of Bern, Bern, Switzerland  Department of
Otorhinolaryngology, Head and Neck Surgery, Inselspital, Bern University
Hospital, Bern, Switzerland  G.P.R. Lajoinie  Physics of Fluids Group,
Department of Science and Technology, MESA+ Institute for Nanotechnology,
Technical Medical (TechMed) Center, University of Twente, Enschede, The
Netherlands  W. Wimmer<EMAIL_ADDRESS>ARTORG Center for Biomedical
Engineering Research, University of Bern, Bern, Switzerland  Department of
Otorhinolaryngology, Head and Neck Surgery, Inselspital, Bern University
Hospital, Bern, Switzerland  Technical University of Munich, Germany; TUM
School of Medicine, Klinikum rechts der Isar, Department of
Otorhinolaryngology

###### Abstract

The cochlea is our fluid-filled organ of hearing with a unique spiral shape.
The physiological role of this shape remains unclear. Previous research has
paid only little attention to the occurrence of transverse flow in the
cochlea, in particular in relation to the cochlea’s shape. To better
understand its influence on fluid dynamics, this study aims to characterize
transverse flow due to harmonically oscillating axial flow in square ducts
with curvature and torsion, similar to the shape of human cochleae. Four
geometries were investigated to study curvature and torsion effects on axial
and transverse fluid flow components. Twelve frequencies from 0.125 Hz to 256
Hz were studied, covering infrasound and low-frequency hearing, with mean
inlet velocity amplitudes representing levels expected for normal
conversations or louder situations. Our simulations show that torsion
contributes significantly to transverse flow in unsteady conditions, and that
its contribution increases with increasing oscillation frequencies. Curvature
has a small effect on transverse flow, which decreases rapidly for increasing
frequencies. Strikingly, the combined effect of curvature and torsion on
transverse flow is greater than expected from a simple superposition of the
two effects, especially when the relative contribution of curvature alone
becomes negligible. These findings could be relevant to understand
physiological processes in the cochlea, including metabolite transport and
wall shear stresses. Further studies are needed to investigate possible
implications on cochlear mechanics.

##  I Introduction

The cochlea, our organ of hearing, is a fluid-filled structure with a peculiar
spiral shape. Despite its importance, the physiological role of its shape in
sound transmission from the cochlear fluids to the sensory epithelium remains
unclear. Performing experimental investigations is challenging because of the
limited access to the cochlea and its small size. Therefore, existing
literature is based on numerical simulations or theoretical studies. For
example, researchers have simulated the interaction between the cochlear
fluids and the basilar membrane (or the organ of Corti) in realistic
geometries [1, 2, 3], while others provide theoretical insight based on
idealized geometries [4, 5, 6, 7]. Manoussaki  _et al._ [8] showed that the
spiral shape redistributes wave energy along the radial direction, altering
the radial vibration profile of the various cochlear structures and affecting
low-frequency hearing. However, geometry-related secondary flow phenomena,
especially the occurrence of transverse flow (i.e., flow in the cross-section
of the cochlea) have received little attention.

Since the physiology of the ear is fundamentally coupled to fluid dynamic
processes, a more detailed understanding of the influence of geometric
properties on the flow field is desirable. In this context, possible
mechanisms could be secondary phenomena introduced by transverse flow. The
combination of oscillating axial flow and transverse flow caused by geometry
leads to steady streaming effects and net transport of particles [34].
Therefore, cross-sectional mixing and longitudinal streaming caused by
transverse flow could be relevant for the transport of metabolites in the
cochlea [35]. In the context of medical treatment, intracochlear fluid-borne
mass transport (drug delivery) is an active area of research [9, 11].
Moreover, transverse flow can generate wall shear stresses and pressures that
could reach magnitudes that may be considered physiologically relevant in the
cochlea.

Curvature and torsion are elementary properties that can be used to describe
duct geometries by the centerline. Relatively simple shapes can generate
complex transverse flow patterns. The geometry of a toroidal duct, only
exhibiting curvature, causes pairs of rotating flow cells oriented in the
transverse plane, known as Dean flow [12]. Inside twisted straight ducts,
which only contain torsion, saddle flow patterns are generated depending on
the cross-section of the duct [13, 14]. More complex phenomena arise in
helical ducts with curvature and torsion combined, as has been extensively
studied by Bolinder [15] and others [16, 17, 18] under laminar steady flow
conditions. Transverse flow occurring under oscillatory stimulation in
toroidal circular ducts was studied by Lyne [19] and Sudo  _et al._ [20].

Motivated by the question of the physiological role of the cochlear shape, the
aim of this study is to fundamentally characterize transverse flow phenomena
under harmonic oscillation in square ducts with curvature and torsion
reflecting the shape of human cochleae. Computational fluid dynamics (CFD) is
used to simulate flow at oscillation frequencies within the infrasonic and
low-frequency hearing range of humans because the apical region of the
cochlea, characterized by a high degree of curvature and torsion, is
particularly sensitive to these frequencies. The results of this study include
axial velocity profiles, transverse flow patterns, relative transverse flow
magnitudes and phase differences between pressure and velocity. Interestingly,
we find that torsion remains the dominant contributor to transverse flow under
oscillating stimulation, similar to its role under steady flow conditions.
Additionally, we observe that the combined contribution of curvature and
torsion to transverse flow is larger than expected based on a simple
superposition of the two effects.

##  II Methods

###  II.1 Duct Geometries

To characterize the contributions of curvature and torsion to the flow field,
we simulated fluid flow in four different geometries (Fig. 1). We defined the
geometries using established methods [15, 21], which are summarized here. The
centerlines of the geometries are described by the position vector
$\mathbf{r}=\mathbf{r}(s)$ and parametrized with the arc length $s$. To
facilitate the decomposition into axial and transverse flow components later
on, we use the orthonormal Frenet-Serret frame consisting of the tangent
($\mathbf{\hat{t}}$), normal ($\mathbf{\hat{n}}$), and binormal
($\mathbf{\hat{b}}$) unit vectors:

$\displaystyle\mathbf{\hat{t}}=\mathbf{r^{\prime}},\quad\mathbf{\hat{n}}=\frac{\mathbf{\hat{t}^{\prime}}}{\lVert\mathbf{\hat{t}^{\prime}}\rVert},\quad\textrm{and}\quad\mathbf{\hat{b}}=\mathbf{\hat{t}}\times\mathbf{\hat{n}},$
(1)

where the prime indicates the derivative with respect to the arc length $s$.
The curvature $\kappa$ and torsion $\tau$ of the centerline are defined as

$\displaystyle\kappa=\left\lVert\mathbf{\hat{t}^{\prime}}\right\rVert\quad\textrm{and}\quad\tau=\mathbf{\hat{n}^{\prime}}\cdot\mathbf{\hat{b}}.$
(2)

For the helical centerline, we obtain

$\displaystyle\kappa=\dfrac{R}{K^{2}+R^{2}}\quad\textrm{and}\quad\tau=\dfrac{K}{K^{2}+R^{2}}\quad,$
(3)

where $R$ is the radius and $2\pi K$ is the pitch of the helix, i.e., the
height of a complete helical turn. The toroidal duct (Fig. 1b) has a
centerline with zero torsion and a curvature of $1/R$. In contrast, the
centerline of the twisted duct (Fig. 1c) has zero curvature and a torsion of
$1/K$. For our geometries, the centerline curvature and torsion were chosen as
$\kappa=1/3\,\text{mm}^{-1}$ and $\tau=1/8\,$mm-1, respectively, to match
average properties observable in human cochleae [22, 23]. To save
computational cost, we reduced the total arc length of the centerline for all
models to $s_{max}=10\,$mm, which is shorter than a human cochlea ($37\,$mm)
[24]. The models have a $2\,$mm$\,\times\,2\,$mm cross-section to capture
typical dimensions of the cochlear cross-section [25].

Figure 1: The geometries with corresponding curvature $\kappa$ and torsion
$\tau$. (a) Straight duct with $\tau=\kappa=0\,\text{mm}^{-1}$, (b) toroidal
duct with $\tau=0\,\text{mm}^{-1}$ and $\kappa=1/3\,\text{mm}^{-1}$, (c)
twisted duct with $\tau=1/8\,\text{mm}^{-1}$ and $\kappa=0\,\text{mm}^{-1}$,
and (d) helical duct with $\kappa=1/3\,\text{mm}^{-1}$ and
$\tau=1/8\,\text{mm}^{-1}$.

###  II.2 Model Implementation

The perilymph flow in the cochlea is modelled as flow of an incompressible
Newtonian fluid [26], by the Navier-Stokes equations

$\displaystyle\begin{split}\nabla\cdot\mathbf{u}&=0\quad\textrm{and}\\\
\rho\frac{\partial\mathbf{u}}{\partial
t}+\rho(\mathbf{u}\cdot\nabla)\mathbf{u}&=-\nabla
p+\mu\Delta\mathbf{u},\end{split}$ (4)

where $\mathbf{u}=\mathbf{u}(x,y,z,t)$ and $p=p(x,y,z,t)$ are the velocity and
pressure fields at time $t$ represented in an Eulerian reference frame. The
dynamic viscosity and the fluid density are denoted by $\mu$ and $\rho$,
respectively (Table 1). At the inlet and outlet surfaces, the pressure
$p_{\rm{out}}$ and $p_{\rm{in}}$ were set to

$\displaystyle\begin{split}p_{\rm{in}}(t)&=P_{0}\cos(2\pi
ft)\quad\textrm{and}\quad p_{\rm{out}}(t)=0,\end{split}$ (5)

with the oscillation frequency $f$ and the pressure amplitude $P_{0}$. No-slip
boundary conditions were imposed on the walls. We performed simulations at 12
frequencies ranging from 0.125 Hz to 256 Hz in powers of two. This frequency
range corresponds to infrasound ($<$$\,16\,$Hz) and the human low-frequency
hearing regime (16 Hz to 256 Hz) and covers quasi-steady (0.125 Hz) to
unsteady inertial flows for which the associated Womersley number is greater
than unity (Table 1). We chose this frequency range for the following reasons.
First, the low stimulation frequencies enable us to verify our results in
quasi-steady-state scenarios with the known solutions of Bolinder [15].
Second, low frequencies are perceived in the apical region of the cochlea,
which exhibits the strongest geometric curvature and torsion. Third, higher
frequencies ($>256\,$Hz) require refined meshes due to steeper velocity
gradients that would make the calculations prohibitively expensive.

Table 1: Model parameters. The viscosity and density are taken from water at
37∘C (body temperature). The inlet pressure amplitude $P_{0}$ was chosen such
that a desired mean velocity amplitude $W_{0}$ (averaged over the cross-
section) was obtained.

Parameter Value Dynamic viscosity $\mu$
$0.69\,$$\mathrm{m}\mathrm{P}\mathrm{a}\,\mathrm{s}$ [26] Density $\rho$
$993\,$$\mathrm{k}\mathrm{g}\mathrm{/}\mathrm{m}^{3}$ [26] Hydraulic diameter
$d_{h}$ $2\,$mm [25] Mean velocity amplitude $W_{0}$
$2-200\,$$\mathrm{\SIUnitSymbolMicro m}\text{/}\mathrm{s}$ [27, 28, 29]
Reynolds number $Re=d_{h}W_{0}\rho/\mu$ $0.0058-0.58$ Oscillation frequency
$f$ $0.125-256\,$Hz Womersley number $\alpha=\frac{d_{h}}{2}\sqrt{2\pi
f\rho/\mu}$ $1-48$

###  II.3 Numerical Model

For the simulations, we used the finite element solver COMSOL Multiphysics®
(COMSOL AB, Stockholm, Sweden). Structured meshes were generated by sweeping a
square cross-section (Fig. 2) along the ducts’ centerlines. The cross-
sectional mesh was aligned along the centerline with $\mathbf{\hat{t}}$,
$\mathbf{\hat{n}}$, and $\mathbf{\hat{b}}$ (see section II.1). Grid stretching
was applied. The number of nodes was chosen based on a convergence analysis,
resulting in $26\times 26$ nodes in the cross-section and a total of $77\,500$
hexahedral elements per geometry. We used Lagrange elements of order two and
one for the velocity and pressure (P2P1), respectively. For the time-dependent
solver, the implicit backward differentiation formula method of variable order
(between 1 and 5) was used. We chose 100 steps per oscillation period of the
pressure boundary condition. The inlet pressure amplitude $P_{0}$ was found
iteratively such that the axial velocity amplitude averaged over the cross-
section $W_{0}$ remained the same across the stimulation frequency range for
the different geometries.

Figure 2: Cross-sectional mesh with $26\times 26$ nodes along the normal
$\mathbf{\hat{n}}$ and binormal $\mathbf{\hat{b}}$ direction (Frenet-Serret
frame).

Based on reported data [27, 28, 29], we chose three mean inlet velocity
amplitudes averaged over the cross-section:
$2\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}\mathrm{s}$,
$20\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}\mathrm{s}$, and
$200\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}\mathrm{s}$. For a constant
velocity amplitude $W_{0}$, decreasing frequencies correspond to increasing
equivalent sound pressure levels at the ear drum. The characteristic middle
ear transfer function (i.e., the stapes velocity versus the sound pressure at
the ear drum) reaches its resonance at about $1\,$kHz, with a mean slope of
approximately 6 dB per octave up to 1 kHz [28, 29]. For example, at 32 Hz, a
stapes velocity of $200\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{/}\mathrm{s}$ can be expected for an external auditory canal pressure
of $\sim 101\,$dB sound pressure level (SPL). At 128 Hz, the same stapes
velocity would correspond to an acoustic stimulus with $\sim 113\,$dB SPL.
Velocity amplitudes of $20\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{/}\mathrm{s}$ and $2\text{\,}\mathrm{\SIUnitSymbolMicro
m}\text{/}\mathrm{s}$ match pressure levels that are 20 dB and 40 dB lower,
respectively. The selected stapes velocities approximately cover the range of
sound pressure levels occurring during normal conversations
($2\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}\mathrm{s}$), shouted
conversations ($20\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}\mathrm{s}$),
and near to the threshold of pain at 256 Hz
($200\text{\,}\mathrm{\SIUnitSymbolMicro m}\text{/}\mathrm{s}$). The selected
parameters result in Reynolds numbers $Re$ well below unity, implying that the
fluid phenomena are in the Stokes regime.

To shorten the duration of the initial transient state, the amplitude of the
inlet pressure $P_{0}$ was ramped up smoothly over the first few cycles. To
ensure that the initial transient is washed out, we retrieved results after a
sufficient number of cycles, e.g., after 21 cycles at 256 Hz. The computations
were performed on a computing cluster with AMD Epyc2 processors. The typical
turnaround times for the computations that were running with 16 tasks and 32
GB RAM per CPU on one node varied from one day for 0.125 Hz, to seven days for
256 Hz.

###  II.4 Axial and Transverse Flow

We apply the Frenet-Serret frame to decompose the velocity field $\mathbf{u}$:

$\mathbf{u}=u\mathbf{\hat{n}}+v\mathbf{\hat{b}}+w\mathbf{\hat{t}},$ (6)

and define the velocity component along the tangent as axial flow
$w=\mathbf{u}\cdot\mathbf{\hat{t}}$, while the components in the normal and
binormal directions, $u=\mathbf{u}\cdot\mathbf{\hat{n}}$ and
$v=\mathbf{u}\cdot\mathbf{\hat{b}}$, constitute the transverse flow.

##  III Results

###  III.1 Axial Flow

Figure 3 shows the axial velocity contours in the geometries for different
oscillation frequencies taken at the peak amplitude. For the straight duct,
the results are in good agreement with the exact solution by Tsangaris and
Vlachakis [30]. The relative velocity error with respect to their solution
(averaged in the cross-section) is 0.09% at 0.125 Hz and increases to 0.25% at
64 Hz and 1.25% at 256 Hz. As expected for Womersley flow, the axial velocity
profiles develop thin boundary layers and steep gradients at the walls with
increasing Womersley numbers [33].

The axial velocity in the twisted duct hardly differs from the one in the
straight duct (relative mean difference of $0.39$%). Also, the flow profile in
the helical ducts differ only little from the toroidal duct profiles
($0.51$%). Figure 4a shows the locations of the maximum axial flow in the
cross-section for different frequencies and geometries. With increasing
Womersley numbers $\alpha$, these locations converge toward the corners in the
straight and the twisted ducts. The maximum axial velocities in the toroidal
and helical geometry are shifted to the inner wall corners for low Reynolds
numbers, as also observed by Pantokratoras [31] and Murata  _et al._ [32].

In the straight and twisted ducts, the axial velocity profile has a central
symmetry with respect to the normal and binormal directions. The axial
velocity profiles in toroidal ducts are symmetrical about the horizontal axis
at $x_{b}=0$, while in the helical duct a small deviation is introduced by the
non-zero pitch.

Figure 3: Axial flow magnitude for different frequencies and geometries shown
at the peak amplitude ($W_{0}=200\,$\mathrm{\SIUnitSymbolMicro
m}\text{/}\mathrm{s}$$). Figure 4: Location of maximum axial (a) and
transverse flow (b) in the toroidal (green), twisted (red), helical (yellow)
and straight (black) ducts for Womersley numbers ranging from 1 to 48 and a
mean velocity amplitude of $W_{0}=200\,$\mathrm{\SIUnitSymbolMicro
m}\text{/}\mathrm{s}$$. Only the first quadrant of the cross-section is shown.

###  III.2 Transverse Flow

Figure 5 shows transverse flow patterns for selected oscillation frequencies
(additional figures are provided in the Supplemental Material at [URL will be
inserted by publisher]). In the quasi-steady situation (i.e., 0.125 Hz), we
found similar patterns as described by Bolinder [15], who studied transverse
flow in steady conditions.

In the toroidal duct, two counter-rotating Dean cells form, caused by inertial
forces that push the fluid along the middle line ($x_{b}=0$) from the inner
towards the outer wall [12]. This effect leads to unidirectional transverse
flow, regardless of the direction of the axial flow. As the oscillation
frequency increases, the Dean cells separate and concentrate on the walls. At
32 Hz ($\alpha\approx 17$) and higher frequencies, an additional pair of
vortices, called Lyne instabilities, can be observed in the center of the
channel, rotating opposite to the Dean cells [19].

The torsion of the twisted duct causes saddle flow in the corners of the
cross-section, alternating with the direction of the axial flow. The observed
pattern stays relatively stable over the simulated frequency range. The
transverse flow pattern in the helical duct shows a similar structure as in
the twisted duct, except that the transverse flow is focussed toward the inner
wall, where the axial flow is stronger as well.

Figure 5: Axial (colored contours) and transverse flow (streamlines) for
different frequencies and geometries shown at the time of maximum transverse
flow ($W_{0}=200\,$\mathrm{\SIUnitSymbolMicro m}\text{/}\mathrm{s}$$).

In general, for all geometries, the location of the maximum transverse flow
starts close to the center of the cross-section, but approaches the corners
(of the inner wall, in helical and toroidal ducts) for increasing Womersley
numbers (Fig. 4b). Figure 6 summarizes the maximum transverse velocity
magnitude versus Womersley number in our geometries. As expected for the
toroidal duct, the transverse flow magnitudes because of curvature are small
compared to the axial velocity (below $1$% of the axial velocity magnitude).
This is because the magnitude of the transverse flow in a toroidal duct is
proportional to $Re^{2}$ [15]. Increasing the oscillation frequency further
reduces the relative magnitude of the transverse flow which scales roughly
with $f^{-1}$ (or $\alpha^{-2}$). As the Womersley number increases, the
contribution of curvature to the transverse flow continues to decrease and
approaches negligible amplitudes in the toroidal duct.

As pointed out by Bolinder [15] for steady flow at low Reynolds numbers,
torsion has a dominating effect on transverse flow, which can be seen by a
5-fold higher maximum transverse flow magnitude in the quasi-steady state
(i.e., 0.125 Hz). Strikingly, in contrast to curvature, the contribution of
torsion is substantial and gets more significant with increasing frequencies
following approximately $\textrm{ln}(\alpha)$, with the relative maximum
exceeding 10% of the axial flow magnitude for Womersley numbers greater than
10. At 256 Hz ($\alpha\approx 48$), the maximum transverse flow velocity
accounts for approximately 15% of the axial flow velocity.

For the twisted duct, the curves for the different Reynolds numbers are
coinciding in Fig. 6. This agrees with Bolinder [15] who found that the
magnitude of transverse flow scales to first order with $Re$ in such
geometries. Also for the helical duct the curves for different $Re$ coincide,
indicating that the maximum transverse flow remains proportional to $Re$, at
least in the frequency range observed in our experiments.

Surprisingly, although the contribution of curvature alone is negligible in
the toroidal duct, the combination of curvature with torsion results in a
2-fold gain of transverse flow magnitude in the helical duct. This gain can be
observed throughout the simulated frequency range, even for cases with
$\alpha>10$, where the isolated contribution of curvature to the transverse
flow is less than $0.1$% of the axial flow magnitude.

Figure 6: Maximum transverse flow magnitude with respect to the mean axial
flow velocity ($W_{0}$) as a function of the Womersley number $\alpha$. The
markers on the left side of the figure show the steady flow solutions for the
toroidal and twisted ducts [15]. Black symbols indicate numerical noise
defined as the maximum transverse flow velocities found in the straight duct
simulations. Note that the maximum transverse flow at $\alpha\approx 48$
($f=256\,$Hz) and $Re=0.0058$ in the toroidal duct is distorted due to its
proximity to the numerical noise floor.

###  III.3 Phase Lag

Figure 7 shows the phase difference between the flow and pressure
oscillations. As the phase depends on the location within the cross-section,
the reported values are shown at the maximum amplitudes for the axial (Fig.
4a) and transverse flow (Fig. 4b). The axial velocity phase lag in the
straight duct is in good agreement with the exact solution by Tsangaris and
Vlachakis [30]. The axial phase lag in the twisted duct resembles the one in
straight ducts (relative difference of $0.15$%), reaching a plateau at a
quarter of a cycle for $\alpha>4$. In the toroidal and helical ducts, the
phase lag is similar but on average higher than in the straight and twisted
ducts.

The phase lag of the transverse flow velocity decreases monotonically in all
geometries and approaches the phase lag of the axial flow velocity with
increasing Womersley numbers, except for the toroidal duct. In the toroidal
duct the phase lag fluctuates around a third of a cycle for $\alpha>4$.

Figure 7: Phase difference between maximum axial (trilateral markers) and
transverse flow velocity (marker style according to the Reynolds number) and
pressure for different Womersley numbers $\alpha$. The black dashed line
indicates the exact solution (phase lag between the axial flow and pressure)
for a straight square duct [30]. Markers for different Reynolds numbers $Re$
coincide. For the toroidal duct, the phase difference is distorted at
$\alpha\approx 48$ ($f=256\,$Hz) and $Re=0.0058$ due to the numerical noise
floor.

##  IV Discussion

###  IV.1 Transverse Flow

In this study, we used simplified geometries to investigate the effects of
curvature and torsion on transverse flow under oscillatory stimulation based
on conditions in the human cochlea. Our models, which have square cross-
sections and do not include fluid interactions with flexible membranes, serve
as abstract representations of the cochlear scalae. Through our simulations,
we discovered two key findings about the influence of geometry on transverse
flow.

First, we found that torsion remains a significant contributor to transverse
flow in unsteady conditions at low Reynolds numbers ($<1$), as pointed out by
Bolinder [15] for steady flow, and its contribution increases with higher
oscillation frequencies. The formation of saddle flow due to torsion could be
interpreted as a kinematic effect, as the twisted geometry forces a change of
direction close to the boundaries.

Second, we observed that the combined effect of curvature and torsion on
transverse flow is greater than what would be expected based on a simple
combination of the two effects, especially when the relative contribution of
curvature alone becomes negligible. The transverse velocities resulting from
curvature are about one order of magnitude smaller than found from torsion,
and decrease rapidly for increasing frequencies. The transverse flow patterns
are similar between the twisted and the helical ducts (Fig. 5), however,
caused by curvature, the maximum axial velocity gets shifted toward the
corners of the inner duct wall in helical ducts (Fig. 4b), leading to stronger
saddle flow there. Therefore, it appears as if the helical shape acts as an
amplifier to transverse flow when stimulated with oscillations within the
observed frequency range.

###  IV.2 Axial Flow

The straight and twisted ducts have nearly identical axial flow profiles in
our simulations. This is in agreement with Kheshgi [13] who found that the
axial flow profile in twisted channels approximates the velocity profile of a
straight channel under steady conditions and low torsion. As soon curvature is
present (i.e., in toroidal and helical ducts), the axial flow profiles are
dominated by curvature over the observed frequencies. At low Reynolds numbers,
the maximum velocity is shifted towards the inner wall of the bend, contrary
to what one would expect for higher Reynolds numbers [31, 32]. With increasing
Womersley numbers, the maximum flow velocity moves closer to the inner wall.
In our geometries, torsion of the helical duct (or the resulting pitch) has
little effect on the axial flow compared to the toroidal duct.

###  IV.3 Phase Lag

We found that the phase difference depends on the duct geometry for transverse
flow and behaves differently than the phase difference between th pressure and
axial flow velocity. In toroidal ducts, the axial and transverse flow
velocities can be considered in-phase when approaching higher frequencies,
with a quarter of a cycle offset to the pressure amplitude. The difference in
the axial phase lag between the geometries with a straight centerline and the
ones with a curved can be explained by the different probing locations, which
were closer to the walls for helical and toroidal ducts (Fig. 4a). Also the
fluctuations in the transverse phase lag in the toroidal duct for $\alpha>4$
likely result from the change in the probing locations with increasing
Womersley numbers (Fig. 4b).

###  IV.4 Study Limitations

The main limitation of our study is the use of abstract geometries to model
the highly complex anatomy of the human cochlea [39]. Further studies on
models featuring more realistic cross-sections and tapering, and in particular
fluid-membrane interactions, are required to investigate possible implications
on cochlear mechanics. Our simulations are limited to the observed frequency
range, which only covers the low-frequency hearing regime of humans.
Additional simulations, with refined mesh geometries, are required to obtain
data covering the whole human hearing range up to 16 kHz and more.

##  V Acknowledgements

We would like to thank Prof. Nathaniel Greene, University of Colorado, for
providing middle ear transfer function data. We acknowledge the support from
the Swiss National Science Foundation (Grant No. 205321_200850). Calculations
were performed on UBELIX (www.id.unibe.ch/hpc), the HPC cluster at the
University of Bern.

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