# The hydrodynamics of water strider locomotion

Original entry: Sung Hoon Kang, APPHY 226, Spring 2009

Title: The hydrodynamics of water strider locomotion

Reference: David L. Hu, Brian Chan and John W. M. Bush, *Nature* 424, 663 (2003).

## Contents

## Soft matter keywords

surface tension, hydrophobic, capillary

## Abstract from the original paper

Water striders Gerridae are insects of characteristic length 1 cm and weight 10 dynes that reside on the surface of ponds, rivers, and the open ocean. Their weight is supported by the surface tension force generated by curvature of the free surface, and they propel themselves by driving their central pair of hydrophobic legs in a sculling motion. Previous investigators have assumed that the hydrodynamic propulsion of the water strider relies on momentum transfer by surface waves. This assumption leads to Denny’s paradox: infant water striders, whose legs are too slow to generate waves, should be incapable of propelling themselves along the surface. We here resolve this paradox through reporting the results of high-speed video and particle tracking studies. Experiments reveal that the strider transfers momentum to the underlying fluid not primarily through capillary waves, but rather through hemispherical vortices shed by its driving legs. This insight guided us in constructing a self-contained mechanical water strider whose means of propulsion is analogous to that of its natural counterpart.

## Soft matter example

Hydronamics of the surface locomotion of semiaquatic insects is an interesting subject which is not well understood. In general, there are two ways of walking on water depending on the relative magnitudes of the body weight (Mg) and the maximum curvature force (σP), where M is the body mass, g is the gravitational acceleration,σ is the surface tension and the P is the contract perimeter of the water-walker [1]. Water-walkers with M_{c} = Mg/σP > 1, such as the basilisk lizard, uses the force generated by their feet slapping the surface and propelling water downward, whereas creatures with M_{c} = Mg/σP < 1, such as the wter strider rely on the curvature force by distortion of the free surface as shown in Fig. 1. They have non-wetting body and legs covered by thousands of hairs [2-3].

The force balance on a stationary water strider can be written as Mg = F_{b} + F_{c}, where F_{b} is the buoyancy force and F_{c} is the curvature force. F_{b} is obtained by integrating the hydrostatic pressure over the body area in contact with the water, while F_{c} is deduced by integrating the curvature pressure over this area, or equivalently the vetical component of the surface tension, σ sin θ, along the contact perimeter (Fig. 1b). For a long thin water-strider leg, this ratio is F_{b}/F_{c} ~ w/L_{c} <<1, where the leg radius, w ~ 40 um, the capillary length, L_{c} = (σ/ρg)^{1/2} ~ 2 mm, and ρ, the density of water. The strider's weight is supported almost by surface tension.

For the water strider to move, it should transfer momentum to the underlying fluid. It has been previously assumed that capillary waves are the sole means to accomplish this momentum transfer. Denny suggested that the leg speed of the infant water strider is less than the minimum phase speed of surface waves, c_{m} =
(4gσ/ρ) ~ 23.2 cm/s [4-5]; consequently, the infants are incapable of generating waves and so transferring momentum to the underlying fluid. According to this
physical picture, infant water striders cannot swim, an inference referred to as Denny’s paradox [4].

The Reynolds number characterizing the adult leg stroke is Re = UL_{2} / υ ~ 10^{3}; where U ~ 100 cm/s is the peak leg speed and L_{2} < 0.3 cm is the length of the rowing leg’s tarsal segment, which prescribes the size of the dynamic meniscus forced by the leg stroke. For the 0.01-s duration of the stroke, the driving legs apply a total force F < 50 dynes, the magnitude of which was deduced independently by measuring the strider’s acceleration and leaping height. The applied force per unit length along its driving legs is thus approximately 50/0.6 < 80 dynes/cm. An applied force per unit length in excess of 2σ < 140 dynes/cm will result in the strider penetrating the free surface. The water strider is thus ideally tuned to life at the water surface: it applies as great a force as possible without jeopardizing its status as a water-walker.

The propulsion of a one-day-old first-instar is shown in Fig. 2. Particle tracking revealed that the infant strider transfers momentum to the fluid through dipolar vortices shed by its rowing motion. Video images captured from a side view indicated that the dipolar vortices were roughly hemispherical, with a characteristic radius R < 0.4 cm. The vertical extent of the hemispherical vortices greatly exceeds the static meniscus depth, 120 um, but is comparable to the maximum penetration depth of
the meniscus adjoining the driving leg, 0.1 cm. A strider of mass M ~ 0.01 g achieves a characteristic speed V ~ 100 cm/s and
so has a momentum P = MV ~ 1 g cm/s. The total momentum in the pair of dipolar vortices of mass M_{v} ~ 2πR^{3}/3 is P_{v} = 2M_{v}V_{v} ~ 1 gcm/s, and so comparable to that of the strider. The leg stroke may also produce a capillary wave packet, whose contribution to the momentum transfer may be calculated. According to the author's calculation from measurements, the net momentum carried by the capillary wave packet thus has a maximum value
P_{w} ~ 0.05 g cm/s, an order of magnitude less than the momentum of the strider.

The authors argued that capillary waves did not play an essential role in the propulsion of *Gerridae* because the momentum transported by vortices in the wake of the water strider was comparable to that of the strider, and greatly in excess of that transported in the capillary wave field; moreover, the striders were capable of propelling themselves without generating discernible capillary waves. They also noted that the mode of the propulsion relied on the Reynolds number exceeding a critical value of approximately 100, suggesting a bound on the minimum size of water striders.

This paper was quite interesting to me because they did analysis on one of the unique example of capillarity and wetting phenomena based on scaling analysis and experimental observatioini using high-speed camera. In nature, there are a lot of wonders and it was very interesting to learn underlying principles related to the topic of our course.

## References

[1] Vogel, S. Life in Moving Fluids (Princeton Univ. Press, Princeton, NJ, 1994).

[2] Andersen, N. M. A comparative study of locomotion on the water surface in semiaquatic bugs (Insecta, Hemiptera, Gerromorpha). Vidensk. Meddr. Dansk. Naturh. Foren. 139, 337–396 (1976).

[3] de Gennes, P.-G., Brochard-Wyart, F. & Quere, D. Gouttes, Boules, Perles et Ondes (Belin, Collection Echelles, Paris, 2002).

[4] Denny, M.W. Air andWater: The Biology and Physics of Life’sMedia (Princeton Univ. Press, Princeton, NJ, 1993).

[5] Lamb, H. Hydrodynamics, 6th edn (Cambridge Univ. Press, Cambridge, 1932).