Hydrodynamics of Water Strider Locomotion
Danielle Wain
19 September 2003
Water striders are capable of moving across the water surface without penetration.
Unlike the basilisk lizard, which is too large to rest on the surface of the water and thus
can only remain on the surface while running, the water strider is capable of remaining
still on the surface in addition to locomotion on the surface. This is due to a balance
between the force exerted downward on the water surface by the weight of the water
strider and upward on the water strider by a buoyancy force and a curvature force. Hu et
al. (2003) show that for a water strider, the buoyancy force is negligible compared to the
curvature force. The curvature force is due to the surface tension of water. Water
molecules in the interior of a body of water feel an equal attraction in all directions
because they are surrounding by similar molecules in all directions. At the surface
however, the water molecules are unequally attracted inwards towards the rest of the
water mass. This inward attraction is what creates surface tension (McMurray and Fay
1995). The small hairs on the legs of the water striders assist in their ability to rest on the
water surface without penetrating the surface because their legs do not get wet. Where
the legs of the water strider sit on the water, they create depressions in the surface
without penetrating it and the surface tension acting on the length of the legs in contact
with the water surface to support the weight of the water strider. Hu et al. (2003) show
that for 342 species of water strider, the length of the leg in contact with the water surface
increases with the body weight of the water strider, thus the force due to surface tension
increases with the force due to the body weight. MC is the ratio between these two
forces, and as can be seen in the figure 2 from Hu et al. (2003), none of the water striders
evaluated fall below the line MC = 1, where the water strider would be too heavy to be
supported by the surface tension. This figure also shows that water striders do not scale
isometrically. The heavier water striders measured would not be able to remain the water
surface given this scaling. The best fit to the data does show an allometric scaling
though: Fs = 48Fg0.58, where Fs is the force due to the surface tension and Fg is the force
due to the weight of the water strider. Fs is a function of the length of the leg in contact
with the water, so this shows that the length of the legs increases with the body mass,
although more slowly. All of the water striders measured though are smaller than the
critical point where the length of the legs would not be sufficient to support the body
mass.
Figure 2 from Hu et al. (2003): The relation between the maximum curvature force and
body weight for 342 species of water striders.
The hydrodynamics of locomotion for the water strider is very different from that of the
basilisk lizard, but they both must follow Newton’s third law of motion. This states that
when a force is exerted on one object by another, the second object will exert and an
equal and opposite force on the first object. In the case of the basilisk lizard, the slap and
stroke impulses act to create an upward force that supports the weight of the lizard. The
water strider is shown to exert a force on the water when it strokes that is less than the
upward force due to surface tension, which it allows it to remain on the surface. But in
order for the water strider to propel itself forward, Newton’s third law says that it must
transfer its momentum to the water. Denny proposed that the mechanism by which the
momentum was transferred was through capillary waves. Capillary waves are waves due
to surface tension, as opposed to the more commonly known gravity waves, which are
what we see in the ocean for instance (Kundu 2002). The primary difference is the force
generating the waves. In the ocean, gravity drives the waves, whereas in this case the
surface tension drives the waves. The problem with this theory of momentum transfer is
that infant water striders do not stroke their legs fast enough to produce capillary waves.
This inconsistency is called Denny’s paradox. Hu et al. (2003) use experiments to
attempt to resolve this paradox. They conclude that when a water strider strokes, his legs
shed hemispherical vortices, that are approximately the depth of the depression formed
by the leg in the water surface. The structure of these vortices can be seen in figure 3c of
Hu et al. (2003). The momentum in these vortices is comparable to the momentum of the
strider, whereas the momentum in the capillary waves produced was an order of
magnitude smaller. Consequently, it can be concluded that capillary waves are not an
important mode of momentum transfer and that all water striders, both infant and adult,
move by means of these vortices. A lower limit is placed on when these vortices can be
shed though. The Reynolds number is the ratio between inertial and viscous forces. A
representation of the inertial forces for the movement of the water strider is the product of
the stroke velocity and the length of the leg in contact with the water. The viscosity of
the water represents the resistance that the stroke will encounter. The inertial forces have
to be sufficient enough to overcome the viscosity of the water. In this case the Reynolds
number has to be greater than 100 for vortices to be shed. Hu et al. (2003) suggest that
this represents the minimum size that a water strider can be and be able to propel itself
across the water.
Figure 3c from Hu et al. (2003): An illustration of the structures generated by the stroke:
capillary waves and vortices.
The water strider, like all animals, must transfer momentum in order to move. They have
the added physical challenge of transferring this momentum while remaining on the
surface of the water. The discovery of these hemispherical vortices provides a
mechanism for the movement of infant and adult water striders.
REFERENCES
Hu, D.L., Chan, B., and Bush, J.W.M., 2003. The hydrodynamics of water strider
locomotion. Nature 424, 663 – 666.
Kundu, P.K., 2002. Fluid Mechanics. San Diego, CA: Academic Press.
McMurray, J., and Fay, R.C., 1995. Chemistry. Englewood Cliffs, NJ: Prentice Hall.