Paradox lost: answers and questions about walking on water
Hopkins Marine Station, Stanford University, Pacific Grove, CA 93950-3094, USA
e-mail: mwdenny{at}leland.stanford.edu
Accepted 2 February 2004
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Summary |
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Key words: Gerris, water strider, capillary wave, Denny's paradox, spider
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Introduction |
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The fascination here is actually twofold. Before one can understand how a
water strider can move about, one has first to explain how they can even stand
on the water's surface. In this case, the explanation is well known. First,
the attraction of one water molecule to another requires that considerable
energy be expended to create new area of airwater interface. Pure water
has a surface energy of approximately 0.07 J m2
(Denny, 1993). Now, surface
energy (J m2) is dimensionally equivalent to a capillary
tension (N m1), and it is in this disguise that it will be
employed here. Second, when a hydrophobic object is pressed into the interface
between air and water, the water attempts to minimize its contact with the
object, often at the expense of creating new surface area. As a result, when a
water strider presses one of its hydrophobic legs down onto the surface of a
pond, a dimple is formed in the water's surface, and the surface is stretched
(Fig. 1). Thevertical component
of the resulting capillary force resists the downward push of the leg
(Fig. 2), and the water strider
is supported (e.g. Vogel,
1988
; Denny,
1993
).
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This explanation leads to a classic example of biological scaling. The
capillary force that supports a water strider is proportional to the perimeter
of the legs in contact with the liquid, and therefore scales roughly in
proportion to some linear dimension of the organism. In contrast, the weight
of the animal (the force pushing the legs downward) is proportional to the
animal's volume, and therefore approximately to the cube of its linear
dimension. In other words, with an increase in size, the tendency to sink into
the water increases much more rapidly than the ability to resist. As a
consequence, standing on water is a knack confined to small organisms. Hu et
al. (2003) show that large
water striders have disproportionately longer legs, allowing these insects to
reach somewhat larger sizes than we might expect. But the allometric change in
leg length is not sufficient to completely offset the drastic increase in
mass, and water striders are, indeed, confined to small body size.
This scaling argument has long been standard fare in introductory biology classes, but in my experience, its presentation is immediately followed by a pertinent question. Granted, small insects and spiders can stand on water, but how do they move about? Early attempts at an answer led to an apparent paradox.
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Denny's paradox |
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It was at this point that the scientific study of water-strider locomotion initially went astray. When faced with a basic question in locomotion, it is often best to start by filming the animal as it moves. Water striders can be brought into the laboratory, where they busily dart about on the surface of water in a shallow tray, and when lit with bright lights, their motion is readily photographed. The most strikingly apparent aspect of these photographs is the pattern of waves that is produced each time an adult strider moves, waves that move in the opposite direction from the insect. Could the momentum associated with these waves be the momentum required for locomotion?
The idea has a certain appeal. The relatively large water waves with which we are most familiar propagate as a result of the inertial interaction between the water's mass and the restoring force of gravity. In contrast, waves with the short wavelengths produced by water striders (capillary waves) move in part as a result of the interaction between mass and surface tension. Wouldn't it be lovely if the same property of water that accounted for the water striders' ability to stand (surface tension), could also account for their ability to move?
There is a problem, however. As the wavelength, , of a pure
capillary wave increases, the speed of the wave, cc, slows
down (see Denny, 1993
):
![]() | (1) |
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![]() | (3) |
The net result of the combined influences of gravity and surface tension is
that there is a minimum speed at which waves can move on the surface of a
liquid (Fig. 3). Taking the
derivative of Eq. 3 with respect
to wavelength and setting it equal to zero, we find that the wavelength at
minimum speed is:
![]() | (4) |
![]() | (5) |
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Therein lies the problem. 23 cm s1 is a relatively high speed for the leg of a small insect. For example, the middle leg of a juvenile water strider may be only 2 mm long. In order for the tip of this leg to move at 23 cm s1, the leg must swing with an angular velocity of 115 rad s1. The entire propulsive stroke (which involves a rotation of about 1.5 rad) must therefore occur in about 13 ms. If the leg can't rotate that fast, it can't produce waves. And if waves are the only means by which it can impart momentum to the fluid, the inability to move the legs at 23 cm s1 means that the animal can't move about.
Indeed, juvenile water striders do not swing the tips of their legs at 23
cm s1, and they do not produce waves. They do, however,
scamper over the water's surface just fine. This disparity between locomotory
theory and organismal reality (noted by briefly in
Denny, 1993) became known as
`Denny's paradox' (Suter et al.,
1997
; Hu et al.,
2003
).
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The role of a paradox |
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Paradox solved |
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The study left several questions unanswered, however. Although Suter et al. measured a drag force under steady flow, they did not quantify the pattern of flow that was responsible. In particular, their experiments did not allow them to describe what happens as the leg's motion stops at the end of a rowing stoke. Exactly how is momentum imparted to the water? Furthermore, all their measurements were conducted at a flow speed below the critical wave speed. What happens when waves are present?
These questions served as the basis for the recent study by Hu et al.
(2003). Through careful use of
high-speed video and the presence of dye and particles in the water, Hu et al.
showed that the locomotory motion of each rowing leg of the water strider
Gerrus remigis imparts momentum to the water through the formation of
surface waves, but, more importantly, also through the formation of a
hemispherical, dipolar vortex. This unusual vortical structure can be
visualized as half of a typical toroidal vortex ring in which the ring has
been sliced parallel to its axis of symmetry. The `cut surface' of the torus
lies at the water's surface, and each vortex travels in the opposite direction
from the water strider at a speed of approximately V=4 cm
s1 (Fig.
4)
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Having visualized the flow imparted to the water by the strider, Hu et al.
easily calculated the associated momentum. Because the vortex is approximately
hemispherical, it volume is
, where
R is the radius of the hemisphere (about 4 mm for an adult strider).
The mass of each vortex is thus
,
its momentum is MvV, and the overall momentum
imparted to the water by the two rowing legs is
2MvV, approximately 105 kg m
s1. The strider itself has a mass of approximately
105 kg and moves at a speed of 1 m s1, so
it, too, has a momentum of 105 kg m s1. In
other words, even when surface waves are produced (as they are by adult
striders) the waves account for at most a negligible fraction of the overall
momentum necessary for locomotion. Here, then, is conclusive proof from freely
moving animals that Denny's paradox can be circumvented.
In fact, the rowing locomotion of water striders appears to be quite
efficient. When an insect of mass Mi moves forward at
speed U, it its body has a kinetic energy equal to
MiU2. In terms of the animal's
locomotion this is `useful' energy. In the process of accelerating its body,
however, the strider does work on the water. To a first approximation, this
`wasted' energy is MvV2 (that is, half
the mass of a vortex times the square of its velocity for each of the two
vortices). This information can be used to construct an index of the
hydrodynamic efficiency of this form of locomotion:
![]() | (6) |
A water strider with a mass of 0.01 g moves forward at 100 cm s1 after producing vortices with a radius of 4 mm that move backwards at a velocity of 4 cm s1. Inserting these values into Eq. 6, we find that the efficiency of this rowing stroke is about 96%! By utilizing vortices to propel a large volume of water backwards at a low speed, water striders create a large amount of momentum with the expenditure of little work.
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Open questions |
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There is also a potential problem associated with surface tension itself.
For example, the dimple of a water strider's leg moving at a steady velocity
is akin to a bubble rising through a liquid (beer, for example). In both
cases, the pattern of flow in the liquid is due to the motion of an
airwater interface. Fluid dynamicists have long realized that this type
of motion is unusual in that, unlike motion relative to a solid object, fluid
motion relative to an airwater interface allows for slippage of water
at the interface itself. For example, the theoretical drag coefficient of a
small air bubble rising in water is only 2/3 that of a buoyant sphere made
from a solid material (Happel and Brenner,
1973), and slippage at the airwater interface may help to
explain why the apparent drag coefficient measured by Suter et al.
(1997
) is lower than might be
expected. Furthermore, there can be discrepancies between the theoretical drag
coefficient for a bubble and that measured in an actual fluid. Small bubbles
rising in beer move slower than simple theory predicts; instead, they act as
if the airwater interface has some `stiffness'
(Happel and Brenner, 1973
).
The apparent solidity of the bubble's surface may be due to surface-active
agents in the interface. As these molecules are swept back by the flow, they
can accumulate at the downstream end of the bubble, and thereby resist
slippage in much the same fashion as the surface of a solid. One supposes that
surface-active molecules might accumulate along the surface upstream of a
water strider's leg, thereby affecting the flow. Alternatively, in 1913
Boussinesq (as cited in Happel and
Brenner, 1973
) pointed out that surface tension is a static
property of a fluid, and therefore it may be inappropriate to use it to
explain dynamic processes such as flow around a bubble or dimple. Building on
this thought, Boussinesq explained the anomalous motion of small bubbles by
hypothesizing that under nonsteady flow (and even in the absence of
surface-active molecules), an airwater interface can exhibit an
intrinsic elasticity. I should note that bubbles rising in beer are smaller
than the leg dimples of water striders and move at a substantially slower
speed [that is, they have a lower Reynolds number (see below)], but the issue
of slippage at the airwater interface and the possibility of surface
elasticity may nonetheless have important consequences for any attempt to
precisely model the drag acting on the leg of a water strider or spider.
There is also much to be learned about the scaling of surface locomotion.
Hu et al. (2003) note that in
order for vortices to be shed from the leg of a water strider, the Reynolds
number of the dimple must be greater than approximately 100. As suggested by
Hu et al., one can calculate Reynolds number using L, the length of
the distal segment of the leg (the tarsus), as an estimate of the flow-wise
dimension of the dimple:
![]() | (7) |
Here u is the speed of the dimple over the water (which we
approximate using the velocity of the rowing leg relative to the insect's
body) and is the kinematic viscosity of water (approximately
106 m2 s1 for pure water). If
Re >100, the product of tarsus length and leg velocity must therefore
exceed approximately 104 m2 s1.
Given that smaller bugs are likely to have both smaller legs and slower
velocities, this relationship potentially places a severe lower limit on the
effective size of water striders. If the animals are too small, they cannot
move their legs fast enough to create either vortices or surface waves, and
they therefore are unlikely to be able to move. Exactly where this limit
occurs depends on the scaling of leg length and angular velocity in surface
insects, as well as on a more precise determination of the critical Reynolds
number that must be exceeded if vortices are to be shed.
We have seen that surface tension sets a maximal size at which animals can
support themselves on water; if they get too big, they sink. Vortex shedding
is likely to set the minimal size, a limit that appears to fall just below the
size of the smallest juvenile water striders. There are other limitations as
well. For example, Suter and Wildman
(1999) have shown that D.
triton, the water-walking spider, changes its gait from a rowing motion
(of the same sort used by water striders) to a galloping motion as its speed
increases. They propose that the change in gait occurs when the rowing legs
exceed the speed at which surface tension can maintain the integrity of the
surface dimple. Above this critical speed, the legs are stabbed vertically
into the water, incurring no appreciable dimple, and the legs subsequently act
as simple oars, relying on the drag of the leg alone.
To fully understand this gait transition, we again need to be able to
account for the complex dynamics of the leg's surface dimple, and precise
answers are therefore unavailable. We can, however, make a rough guess as to
the critical speed. Batchelor
(1967) suggests that bubbles
rising in a liquid begin to deform from their spherical shape if the dynamic
pressure of the flow (
u2) is a substantial
fraction of the pressure increase that surface tension imposes across the
airwater interface. In turn, the magnitude of the pressure increase is
inversely related to the local curvature of the interface, which,
unfortunately, we do not know for the dynamic dimple of a moving water
strider. For the sake of argument, let us assume that the radius of curvature
of the dimple is approximately equal to r, the radius of the tarsus.
The resulting surface-tension-induced pressure is
/r
(Denny, 1993
). For a tarsus 1
mm in radius (such as that of the spiders used by
Suter and Wildman, 1999
), this
implies that the dynamic pressure is equal to the surface-tension-induced
pressure at a velocity of 38 cm s1. We might therefore
expect that the dimple will become unstable at velocities somewhat slower than
this. Indeed, Suter and Wildman
(1999
) showed that the maximum
leg-tip velocity in a rowing spider was about 30 cm s1.
Water striders have tarsi with smaller radii (approximately 40 µm),
implying that their legs must move at 191 cm s1 before the
dynamic pressure is equal to the surface-tension pressure. Hu et al.
(2003
) recorded leg velocities
of 100 cm s1 with no evidence that the dimple had become
unstable.
So, one more locomotory paradox bites the dust, but interesting questions remain before the question of `how do they do that?' is fully resolved. For the time being, my curiosity will continue to itch.
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Acknowledgments |
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