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Swimming mechanics and propulsive efficiency in the chambered nautilus

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The chambered nautilus (Nautilus pompilius) encounters severe environmental hypoxia during diurnal vertical movements in the ocean. The metabolic cost of locomotion (Cmet) and swimming performance depend on how efficiently momentum is imparted to the water and how long on-board oxygen stores last. While propulsive efficiency is generally thought to be relatively low in jet propelled animals, the low Cmet in Nautilus indicates that this is not the case. We measured the wake structure in Nautilus during jet propulsion swimming, to determine their propulsive efficiency. Animals swam with either an anterior-first or posterior-first orientation. With increasing swimming speed, whole cycle propulsive efficiency increased during posterior-first swimming but decreased during anterior-first swimming, reaching a maximum of 0.76. The highest propulsive efficiencies were achieved by using an asymmetrical contractile cycle in which the fluid ejection phase was relatively longer than the refilling phase, reducing the volume flow rate of the ejected fluid. Our results demonstrate that a relatively high whole cycle propulsive efficiency underlies the low Cmet in Nautilus, representing a strategy to reduce the metabolic demands in an animal that spends a significant part of its daily life in a hypoxic environment.
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Research
Cite this article: Neil TR, Askew GN. 2018
Swimming mechanics and propulsive
eciency in the chambered nautilus. R. Soc.
open sci. 5: 170467.
http://dx.doi.org/10.1098/rsos.170467
Received: 17 May 2017
Accepted: 19 January 2018
Subject Category:
Biology (whole organism)
Subject Areas:
biomechanics/uid mechanics
Keywords:
uid dynamics, wake structure, vorticity,
cephalopod, mollusc
Author for correspondence:
Graham N. Askew
e-mail: g.n.askew@leeds.ac.uk
Present address: School of Biological Sciences,
University of Bristol, Bristol, BS8 1TQ, UK.
Electronic supplementary material is available
online at https://dx.doi.org/10.6084/m9.
gshare.c.3995766.
Swimming mechanics and
propulsive eciency in
the chambered nautilus
Thomas R. Neiland Graham N. Askew
School of Biomedical Sciences, Faculty of Biological Sciences, University of Leeds,
Leeds LS2 9JT, UK
GNA, 0000-0003-1010-4439
The chambered nautilus (Nautilus pompilius) encounters severe
environmental hypoxia during diurnal vertical movements
in the ocean. The metabolic cost of locomotion (Cmet)and
swimming performance depend on how efficiently momentum
is imparted to the water and how long on-board oxygen
stores last. While propulsive efficiency is generally thought
to be relatively low in jet propelled animals, the low Cmet in
Nautilus indicates that this is not the case. We measured the
wake structure in Nautilus during jet propulsion swimming,
to determine their propulsive efficiency. Animals swam with
either an anterior-first or posterior-first orientation. With
increasing swimming speed, whole cycle propulsive efficiency
increased during posterior-first swimming but decreased
during anterior-first swimming, reaching a maximum of
0.76. The highest propulsive efficiencies were achieved by
using an asymmetrical contractile cycle in which the fluid
ejection phase was relatively longer than the refilling phase,
reducing the volume flow rate of the ejected fluid. Our results
demonstrate that a relatively high whole cycle propulsive
efficiency underlies the low Cmet in Nautilus, representing
a strategy to reduce the metabolic demands in an animal
that spends a significant part of its daily life in a hypoxic
environment.
1. Background
Chambered nautilus (Nautilus pompilius) perform diurnal vertical
movements involving depth changes of 500–600 m. During the
day they either rest at depths of around 200m or forage at depths
up to 700 m, and during the night move almost continuously
between depths of 130 and 700 m [1]. While foraging at depth
animals encounter low concentrations of oxygen (oxygen partial
pressure, PO2, approximately 50mmHg or 30% of air-equilibrated
surface water; [2].) The high capacity of the haemolymph for
oxygen storage [3], the high affinity of haemocyanin for oxygen
2018 The Authors. Published by the Royal Society under the terms of the Creative Commons
Attribution License http://creativecommons.org/licenses/by/4.0/, which permits unrestricted
use, provided the original author and source are credited.
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head retractor muscle
attachment
posterior
wings of funnel mantle cavity
anterior
flexible funnel orifice
Figure 1. Mechanisms of producing jets during swimming in Nautilus: (1) by the contraction of the large head retractor muscle;
(2) through the rhythmic contraction of the funnel wings. Inset demonstrates how the exible jet orice—the funnel—can move to
direct water in multiple directions. Blue lines represent the ow of uid through the animal which facilitates both oxygen exchange and
locomotion. Figure adapted from [9,10].
[4,5], the ability to extract ambient oxygen via the superficial capillaries in the absence of gill perfusion,
and the ability to use oxygen stored in the shell chambers [6] are physiological adaptations that enable
Nautilus to not only survive hypoxia but to also maintain sufficient metabolic scope to perform their
extensive vertical migrations [7]. It is only at PO2below 50 mmHg, encountered in oxygen deficient
water or during retraction of the animal into its shell, that metabolic suppression is required to protect
against hypoxia [7].
A further adaptation that supports hypoxia tolerance relates to their economical locomotion [8].
Nautilus swims by jet propulsion. Powerful jetting is produced by the compression of the mantle cavity
produced by synchronous contraction of the retractor and funnel muscles (figure 1;[9,10]). Compression
of the mantle cavity results in a pressure difference between the mantle cavity and the ambient water,
expelling water from the mantle cavity via the funnel or siphon (as well as along the top edges of the
shell aperture during very powerful contractions) [9,11]. Slower swimming movements and ventilation
are powered by rhythmic contractions of the funnel flaps that result in a wave of movement that moves
anteriorly along the funnel wings, producing unidirectional flow of water across the gills, through the
mantle cavity and exiting through the funnel [11,12]. The fluid jet is formed by the funnel wings that
extend along either side of the head and overlap along the ventral side of the animal terminating in
the funnel. The manoeuvrable funnel allows the water to be ejected at a range of angles giving Nautilus
the ability to swim in all directions.
The efficiency of jet propulsion swimming is generally considered to be lower than undulatory
swimming [13]. This difference in efficiency originates from the fact that, for a given thrust, jet propulsion
swimming involves accelerating a small mass of fluid to a high velocity to achieve propulsion, whereas
greater efficiencies are achievable by accelerating a large mass of water at slower velocities, as can occur
in an undulatory swimmer [14]. However, despite the presence of an external shell and the use of jet
propulsion to power locomotion, Nautilus has a lower minimum metabolic cost of locomotion (Cmet)
compared with squid and, at low speeds, salmon [8]. The low Cmet is advantageous in conserving
the limited oxygen stores when swimming at depth in hypoxic conditions. The efficiency with which
muscular work is transferred to useful hydrodynamic work is one step in the transduction of chemical
energy into useful work in the environment. Therefore, knowledge of the wake structure may give
some insights into the low cost of jet propulsion swimming in Nautilus. For example, Nautilus may be
able to manipulate the structure of the jet, enhancing the propulsive efficiency as has been observed
in jets produced using mechanical pistons [15]. The aim of this study was to measure the wake
structure of Nautilus during jet propulsion swimming, and to determine their propulsive efficiency. It
was hypothesized that Nautilus would have a high whole cycle propulsive efficiency compared to other
jet propelled organisms consistent with the requirement for an economical lifestyle of an animal living
in a hypoxic environment.
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2. Material and methods
2.1. Animals
Chambered nautilus (Nautilus pompilius Linnaeus, 1758, n=5) were obtained from a marine livestock
supplier (Tropical Marine Centre, Manchester, UK) and housed in a 250 l aquarium in artificial seawater
(Instant Ocean, Aquarium Systems, Inc.). The aquarium was maintained at a temperature of 17°C and
a salinity of 34 ppt. Nautilus were fed twice weekly with whole shrimp.
2.2. Wake visualization and analysis
Visualization of the wake structure took place in a 126 l (610×460 ×450 mm, length ×width ×height)
glass aquarium containing artificial seawater at a temperature and salinity matching that of the holding
tank. Nautilus were transferred to the experimental tank and allowed at least 15 min to adjust to their
surroundings. As Nautilus are olfactory foragers [16], a shrimp was added to the water to stimulate
swimming, eliciting a variety of swimming behaviours, e.g. anterior or posterior swimming.
Quantitative analysis of the jet structure of Nautilus was obtained using two-dimensional particle
imaging velocimetry (PIV). The experimental tank was seeded with aluminium oxide (H7881 5 µm,
Sigma-Aldrich, Germany; following [17]) at a density of 30 mg l1. Particles were illuminated with a
1 W continuous 532 nm, green laser (Shanghai Dream Lasers Technology Co., Ltd, Shanghai, People’s
Republic of China) directed through a Powell lens (Thorlabs, Inc., Newton, NJ, USA) creating a 1mm
thick, vertically orientated light sheet. The aim was to visualize the wake of swimming Nautilus in the
sagittal plane; only those sequences in which the laser bisected the jet orifice and thereby the middle of
the vortex structures were used for analysis. The Nautilus and particle movements were recorded using
a high-speed camera (FASTCAM SA3, Photron USA, San Diego, CA, USA) recording at 500 frame s1,
shuttered at 1/500 s1and recording at a 1024 ×1024 pixel resolution.
The positional data of the illuminated particles were analysed using an open source software (PIVlab
v. 1.41 [18]). The image sequences were pre-processed with a contrast-limited adaptive histogram
equalization tool to enhance contrast. The body of the Nautilus was masked on the images to eliminate
edge effects. A cross correlation technique was used with adaptive multi-pass processing to analyse
image pairs and to track particle movement between frames. A total of three passes were used to analyse
images, with an initial interrogation window of 128 ×128 pixels and a final size of 32 ×32 pixels with
a 50% overlap between each pass. A standard deviation filter was used to remove vectors that were
more than 7 deviations away from the mean jet flow. An average of 0.43 ±0.02% of the vectors was
found to be erroneous across all swimming sequences. Missing velocity vectors were interpolated using
a boundary value solver, giving a smooth interpolation that tended towards the average boundary
velocities. The range of pixel displacements across interrogation windows was 2–7 pixels with higher
pixel displacement occurring at the start of the jetting sequence.
Jet thrust, T, is the force propelling the animal and equals the rate of change of momentum in the
surrounding fluid. Thrust was calculated as (following [19])
T=ρ¯
u2
jAj, (2.1)
where ρis seawater density (1025 kg m3), ¯
ujis the average jet velocity (the time average of the average
jet core velocity during the jet period) and Ajis the cross-sectional area of the jet orifice, measured from
still images of the jet orifice (ImageJ v. 1.50i, Bethesda, Maryland, USA).
Whole cycle propulsive efficiency (ηwc) is the ratio of useful power to total power (i.e. the sum of
useful and wasted power), and was calculated using a method developed for jet propulsion swimming,
which accounts for the acceleration of the water during both the refilling (intake) and contraction
(expulsion) phases of the swimming cycle [20]. The useful power is the product of the force propelling
the animal (the rate of change of momentum, mj¯
uj) and the velocity of the animal ( ¯
U), i.e. mj¯
U¯
uj. The rate
of loss of energy in the wake is (1/2)mj¯
u2
jand the kinetic energy is given to the water entering the mantle
cavity at a rate (1/2)mj¯
u2
r, giving a total power of mj¯
U¯
uj+(1/2)mj¯
u2
r+(1/2)mj¯
u2
j. Therefore, whole cycle
efficiency is given by
ηWC =2¯
U¯
uj
2¯
U¯
uj+¯
u2
r+¯
u2
j
, (2.2)
where ¯
Uis the time averaged velocity of the animal, mjis the mass of water passing through the animal
in unit time, ¯
uris the refill velocity, i.e. the velocity of the fluid at the intake orifice during the refilling
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of the mantle. Difficulty in visualizing the flow near the refill orifices of the Nautilus meant that refill
velocities had to be estimated. It was assumed that the total volume of water ejected during jetting was
equal to the volume of water taken in to the mantle during refilling. Therefore, the refill velocity was
estimated as
¯
ur=
¯
ujAjtj
Artr
, (2.3)
where Aris the area of the refill orifice and tjand trare the durations of the jetting and refill periods,
respectively. Jet duration (tj) was taken to be the time period between the beginning of contraction of
the head into the mantle cavity and the beginning of relaxation of the head to its initial position. The
refill duration (tr) was defined as the period between the onset of relaxation of the head and the start of
the next contraction cycle. The sum of these two periods is the total cycle duration (tcd). Duty cycle was
defined as the ratio of tjto tcd. Slip, an indicator of the inverse of propulsive efficiency, was calculated as
¯
uj/¯
U[19].
The ratio of jet length to jet diameter was calculated as Lj/Dj,whereLjis the jet length measured as
the extent of the vorticity field along the jet centreline that exceeded the background flow vorticity by
20%, and Djis the diameter of the vortex ring measured from the two peaks of vorticity that make up the
vortex ring (figure 3a;[21]). Approximately, 8–12 vectors were measured across the jet diameter. Mean
vorticity was calculated as the mean vorticity of the measured jet length during the contraction phase
of the swim cycle.
2.3. Statistical analyses
Statistical analysis was carried out in SPSS for Mac (IBM SPSS Statistics for Mac v. 21.0, Armonk, NY,
USA). Data were checked for normality using a Shapiro–Wilks test. Linear regressions were fit to the
data to test for speed-related changes in swimming mechanics and wake structure. One-way ANOVA
was used to test for differences between swimming orientation. Where differences were detected,
Tukey’s post hoc tests were used to identify where these differences occurred. All data are reported
as mean ±s.e.m.
3. Results
3.1. Swimming behaviour
Morphological data and basic swimming kinematics are reported in electronic supplementary material,
table S1. Two distinct swimming orientations were observed, either ‘anterior-first’ or ‘posterior-first’.
Posterior-first swimming was the most frequently observed with 77% of swims recorded being in
this orientation. Average speed during posterior-first swimming was 0.90 ±0.12 BL s1(range =0.35–
1.60 BL s1) and 0.73 ±0.05 BL s1(range =0.48–1.19 BL s1) during anterior-first swimming. The
average Reynolds number across all swims was 6.9 ×103(average swimming speed 8.31 cm s1and
shell diameter of 9.24 cm).
Cycle duration decreased with increasing speed during both posterior-first (F1,47 =15.28, p<0.001;
figure 2a) and anterior-first (F1,13 =7.01, p<0.05; figure 2b) swimming. Duty cycle was 49.73 ±0.86%
(range =41.72–65.79%) during posterior-first swimming and 51.76 ±1.12 (range =41.88–61.72%) during
anterior-first swimming. Duty cycle was independent of speed during posterior-first swimming (p=0.91;
figure 2c) but decreased with increasing speed during anterior-first swimming (F1,13 =6.50, p<0.05;
figure 2d).
3.2. Jet wake properties
The two-dimensional divergence was non-zero, indicating the jet structures were not perfectly
axisymmetric. The range of variation of two-dimensional divergence was 0.38 to 0.45 s1for posterior-
first swims and 0.17 to 0.26 s1for anterior-first swims. Two categories of jet structures were observed:
jets in which the ejected fluid rolled up into an isolated vortex ring (termed ‘jet mode 1’ jets; figure 3;
electronic supplementary material, S2); and jets that consisted of an elongated jet of ejected fluid (termed
‘jet mode 2’ jets; figure 4; electronic supplementary material, S2). Both types of jet were observed during
both posterior (figure 3a,c;figure 4a,c) and anterior-first (figure 3b,d;figure 4b,d) swimming behaviours.
Both jet modes were observed across the range of speeds during posterior-first swimming. However,
during anterior-first swimming, jet mode 1 jets were never seen at speeds exceeding 0.8BL s–1, while jet
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1.0
0.8
0.6
0.4
0.2
0
0
duty cycle, DC
2.01.51.00.5
swimming speed, U
(BL s–1) swimming speed, U
(BL s–1)
1.2
1.0
0.8
0.6
0.4
0.2
0
cycle duration, tcd (s)
tcd = –0.13 U
+ 0.80
r2= 0.25
1.0
0.8
0.6
0.4
0.2
0
0
duty cycle, DC
2.01.51.00.5
DC = –0.15 U
+ 0.63
r2 = 0.34
1.2
1.0
0.8
0.6
0.4
0.2
0
cycle duration, tcd (s)
tcd = –0.27 U
+ 1.01
r2= 0.33
(a)(b)
(c)(d)
Figure 2. Swimming mechanics as a function of swimming speed in Nautilus. Relationship between cycle duration and swimming speed
during (a) posterior-rst and (b) anterior-rst swimming. The eect of swimming speed on duty cycle during (c) posterior-rst and
(d) anterior-rst swimming.
mode 2 jets were identified at speeds ranging from 0.47 to 1.19 BL s1. The mean area of the jet orifice
during refilling was approximately four times the mean jet orifice area during posterior-first swimming
and seven times the area during anterior-first swimming (electronic supplementary material, table S1).
Refill velocity was estimated to be 0.16–0.36 times the jet velocity.
During posterior-first swimming, Lj/Djranged from 0.79 to 2.16 in jet mode 1 jets and 3.16 to 6.29 in jet
mode 2 jets. During anterior-first swimming Lj/Djranged from 1.08 to 1.52 during jet mode 1 swimming
and 3.29 to 5.51 during jet mode 2 swimming.
Mean jet angle relative to swimming trajectory was 16.15 ±1.58° (range =1.13–33.69°) in posterior-
first swimming and 16.79 ±2.50° (range =5.09–32.47°) in anterior-first swimming. Swimming speed
was independent of jet angle for both posterior-first (p=0.219) and anterior-first swimming (p=0.138).
Swimming speed increased with increasing thrust during both posterior-first (F1,47 =5.82, p<0.05;
figure 5a) and anterior-first swimming (F1,13 =23.99, p<0.001 figure 5b).
During posterior-first swimming whole cycle propulsive efficiency increased with increasing
swimming speed (F1,47 =11.46, p<0.05; figure 5c). By contrast, whole cycle propulsive efficiency
decreased with increasing swimming speed during anterior-first swimming (F1,13 =114.53, p<0.05;
figure 5d). Anterior-first jet mode 1 swimming was more efficient than posterior-first jet mode 2
swimming (p<0.05; electronic supplementary material, figure S1). Thrust varied with swimming
orientation and jet mode (F3,9 =7.01, p<0.05), with anterior-first jet mode 1 swimming producing less
thrust than posterior-first jet mode 2 swimming (p<0.05; electronic supplementary material, figure S1).
4. Discussion
4.1. Jet modes
Two different categories of wake structure were identified during jet propulsion swimming: jet mode 1
jets, in which all of the ejected fluid rolls up into an isolated vortex ring, and jet mode 2 jets, where fluid is
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(a) (b)
(c) (d)
Figure 3. Comparison of instantaneous ow and vorticity between anterior and posterior swimming in Nautilus using ‘jet mode 1’. (a,b)
Vorticity and (c,d) velocity vector elds during (a,c) posterior and (b,d) anterior swimming. Note that the uid is rolledup into an isolated
vortex ring formed, representing ‘jet mode I’. On vorticity plots, red and blue regions denote clockwise and counter-clockwise rotation,
respectively. The jet length and jet diameter as represented by the variables Ljand Dj, are indicated in (a) and are dened in the text.
ejected as an elongated jet. These two jet modes are comparable to the categories of jets described, based
on two-dimensional recordings similar to those used here, in free-swimming brief squid [21,22]andmore
recently quantified using three-dimensional particle image velocimetry in the same species [23]. Thrust
tended to be higher but whole cycle propulsive efficiency was lower during jet mode 2, compared with
jet mode 1, as observed in juvenile and adult squid [21,23]. In Nautilus the transition from jet mode 1 to
jet mode 2 jets occurred at jet length to diameter ratio (Lj/Dj) of approximately 3 (jet mode 1 Lj/Dj<2.16;
jet mode 2 Lj/Dj>3.16), similar to those observed in other jet propulsion swimmers (e.g. squid, [21]).
4.2. Propulsive eciency
The lack of a difference in whole cycle propulsive efficiency and thrust generation between the two
jet modes may explain why the jet mode was not exclusively related to swimming speed in the two
swimming orientations. The average whole cycle propulsive efficiency ranged from 0.30 to 0.75 during
posterior-first swimming and 0.48 to 0.76 during anterior-first swimming in Nautilus. While overlapping,
the whole cycle propulsive efficiency in Nautilus is higher than the range reported in other jet propelled
animals: adult squid (0.42–0.49; [14]), salps (0.47–0.55; [19]), jellyfish (0.09–0.53; [24]). Note that efficiency
has not been calculated in the same way in this and previous studies. For example, in jellyfish, the Froude
efficiency was calculated [24], which does not include momentum losses during fluid intake and refilling
[25]. The calculated jet propulsive efficiency is expected to be higher if the losses during intake and
refilling are not included in the calculations [21,26]. In salps, momentum losses during intake and refilling
periods are included in the calculation of whole cycle efficiency, though a different equation to that used
in this study (assumes refilling occurs passively) was applied [19]. Furthermore, in the study on squid
[14], only the horizontal component of the jet was considered to yield ‘useful’ work; in our equation
2, the vertical component of the jet is also considered ‘useful’, because it contributes to the propulsion
of the animal. However, the difference in the calculated whole cycle efficiency is a reduction of only
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(a) (b)
(c) (d)
Figure 4. Comparison of instantaneous ow and vorticity between anterior and posterior swimming in Nautilus using ‘jet mode 2’. (a,b)
Vorticity and (c,d) velocity vector elds during (a,c) posterior and (b,d) anterior swimming. Note the presence of a trailingjet that results
from vortex pinch-o, representing‘jet mode 2’. On vorticity plots, red and blue regions denote clockwise and counter-clockwise rotation,
respectively.
3.5%, assuming a jet angle of 15°, when considering only the horizontal component of the jet, which
alone is insufficient to explain the higher whole cycle propulsive efficiency in Nautilus, indicating that
the difference is real.
Intriguingly, whole cycle propulsive efficiency increased as a function of speed in posterior-first
swimming, but decreased as a function of speed during anterior-first swimming. The relationship
between whole cycle propulsive efficiency and swimming orientation may be related to the energy losses
associated with re-orienting the funnel during anterior-first swimming. During anterior-first swimming
the funnel is turned back on itself, creating a bend through which the fluid must pass before being
ejected (figure 1). This may result in energy losses due to turbulence that are proportional to fluid velocity
[27], thereby reducing whole cycle propulsive efficiency as swimming speed increases. However, in jet
propelling squids, slip tends to decrease with increasing swimming speed, resulting in an increase in
efficiency with swimming speed that is independent of swimming orientation [23,26].
An additional or alternative explanation for the inverse relationship between swimming speed
and whole cycle propulsive efficiency across swimming orientations could be related to differences in
swimming mechanics between the two orientations. In the anterior-first swims, both duty cycle and
jet period decreased with increasing swimming speed. Therefore, at their slowest swimming speeds
Nautilus have asymmetrical contractile cycles, spending more time ejecting fluid than refilling. This
results in a relatively low speed jet of fluid being ejected, reducing slip and increasing whole cycle
propulsive efficiency [14]. If the area of the orifice through which water is drawn and ejected were
the same, the required increase in flow rate during refilling would negate the benefits gained during
propulsion. However, during both swimming orientations the size of the refill orifice area is larger (4×
during posterior-first swimming and 8×during anterior-first swimming) than during jetting, reducing
the refill velocities and avoiding detrimental effects on whole cycle propulsive efficiency. However, as
swimming speed increases, the shorter jetting period requires a higher velocity jet, leading to a reduction
in whole cycle propulsive efficiency.
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1.0
0.8
0.6
0.4
0.2
0
2.01.51.00.50
hwc = –0.24 U
+ 0.83
r2= 0.40
1.0
0.8
0.6
0.4
0.2
0
2.01.51.00.50
hwc= 0.13 U
+ 0.42
r2= 0.22
2.0
1.5
1.0
0.5
0
543210
thrust, T (mN)
jet mode 1
jet mode 2
2.0
1.5
1.0
0.5
0
543210
thrust, T (mN)
U
= 1.22 T+ 0.84 U
= 1.63 T– 0.84
r2= 0.11 r2= 0.62
swimmin
g
speed, U
(BL s–1) swimmin
g
speed, U
(BL s–1)
swimming speed, U
(BL s–1)
hwc
(a)(b)
(c)(d)
Figure 5. The relationships between jet characteristics and swimming speed in Nautilus. Swimming speed plotted as a function of thrust
for (a) posterior-rst and (b) anterior-rst swimming. Hydrodynamic whole cycle propulsive eciency ηwc as a function of swimming
speed is illustrated for (c) posterior-rst and (d) anterior-rst swimming. The regression lines for all swims combined, are shown.
In squid, jet angle increases with decreasing swimming speed [26] to provide a vertical force to
counteract the negative buoyancy of squid. In squid, this vertical component of the jet reduces the
proportion of the jet’s momentum that propels the animal forward [28]. The neutrally buoyant Nautilus
does not need to generate a vertical force, resulting in the absence of a relationship between swimming
speed and jet angle (jet angle simply relates to swimming direction). Consequently, the large range of jet
angles observed during swimming simply reflects the control of swimming directions. While Nautilus has
an advantage over squid in not needing to produce a vertical force at slow speeds, as speed increases the
relatively high frontal area of Nautilus is expected to have a detrimental effect on swimming performance
compared to the streamlined bodies and hydrodynamic lift producing fins of squid [29].
The swimming performance of Nautilus is ultimately determined by the transfer of mechanical work
by the locomotory muscles into useful hydrodynamic work (i.e. work done against drag) in the jet.
The efficiency of this process is the whole cycle propulsive efficiency. The locomotory muscles convert
chemical energy, ultimately derived from food, into mechanical work; the efficiency of this energy
transfer is the net muscle efficiency (ηmus). Together, the net muscle efficiency and the whole cycle
propulsive efficiency determine the overall locomotor efficiency with which chemical energy is converted
into useful work in the environment. Therefore, the relatively high whole cycle efficiency in Nautilus will
increase the overall locomotor efficiency and will reduce Cmet: the magnitude of these effects depends
on the muscle efficiency, which was not determined here.
As a component of the overall locomotor efficiency and determinant of Cmet, it is not surprising that
whole cycle propulsive efficiency is related to ecological niche in jet propelled swimmers. In cephalopod
molluscs, the slow swimming Nautilus has a higher efficiency than the faster swimming squid [14]; in
salps it is the slow swimming species that are the most efficient [19]; and cruising predatory jellyfish
have a higher efficiency than those that are ambush predators [24]. These observations indicate that
the efficiency with which energy is transferred to the environment is under major selective pressure,
but increased whole cycle propulsive efficiency appears to come at the expense of swimming speed.
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5. Conclusion
Nautilus are unique amongst cephalopod molluscs in their ability to tolerate hypoxic conditions, which
they encounter when undertaking vertical migrations in the water column [6]. These animals employ
a number of different physiological strategies to enable them to survive and remain active in such
conditions [6,7,30]. This study shows that Nautilus are also able to use biomechanical strategies. When
swimming at low speeds with an anterior-first orientation, their high whole cycle propulsive efficiency
corresponds to a low Cmet [8]. Reducing the metabolic cost of swimming through a high whole cycle
propulsive efficiency conserves on-board oxygen supplies and helps avoid anaerobiosis [30].
Data accessibility. Supporting data are available from Research Data Leeds Repository—https://doi.org/10.5518/192.
Authors’ contributions. G.N.A. conceived and designed the project; T.R.N. designed the project, collected the data and
analysed the data; T.R.N. and G.N.A. interpreted the data and wrote the manuscript.
Competing interests. The authors declare that they have no competing interests.
Funding. This work was funded by an Engineering and Physical Sciences Research Council (EPSRC, EP/K503526/1)
UK Institutional sponsorship grant.
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Supplementary resource (1)

... In addition to overall conch geometry, several second-order hydrodynamic factors complicate relationships between shape and hydrodynamics. Propulsive thrust relates to jet duration, frequency, and iteration 16,34 . Propulsive efficiency is tightly linked to the hydrostatics of posture and jet orientation 7,10,11,[34][35][36] . ...
... Propulsive thrust relates to jet duration, frequency, and iteration 16,34 . Propulsive efficiency is tightly linked to the hydrostatics of posture and jet orientation 7,10,11,[34][35][36] . Nuanced external shape features-keels; umbilical exposure; ornament of ribbing or spines 37 -produce substantial but nonlinear impacts on overall drag force 23 . ...
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