# Difference between revisions of "The hydrodynamics of water strider locomotion"

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The force balance on a stationary water strider can be written as Mg = F<sub>b</sub> + F<sub>c</sub>, where F<sub>b</sub> is the buoyancy force and F<sub>c</sub> is the curvature force. F<sub>b</sub> is obtained by integrating the hydrostatic pressure over the body area in contact with the water, while F<sub>c</sub> 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<sub>b</sub>/F<sub>c</sub> ~ w/L<sub>c</sub> <<1, where the leg radius, w ~ 40 um, the capillary length, L<sub>c</sub> = (σ/ρg)<sup>1/2</sup> ~ 2 mm, and ρ, the density of water. The strider's weight is supported almost by surface tension. | The force balance on a stationary water strider can be written as Mg = F<sub>b</sub> + F<sub>c</sub>, where F<sub>b</sub> is the buoyancy force and F<sub>c</sub> is the curvature force. F<sub>b</sub> is obtained by integrating the hydrostatic pressure over the body area in contact with the water, while F<sub>c</sub> 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<sub>b</sub>/F<sub>c</sub> ~ w/L<sub>c</sub> <<1, where the leg radius, w ~ 40 um, the capillary length, L<sub>c</sub> = (σ/ρg)<sup>1/2</sup> ~ 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<sub>m</sub> =
(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 | ||

+ | UL2=n<103; where U < 100 cm s21 is the peak leg speed and | ||

+ | L2 < 0.3 cm is the length of the rowing leg’s tarsal segment (see | ||

+ | Figs 1b and 2), which prescribes the size of the dynamic meniscus | ||

+ | forced by the leg stroke23. 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 cm21. An | ||

+ | applied force per unit length in excess of 2j < 140 dynes cm21 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 leg stroke may also produce a capillary wave packet, whose | ||

+ | contribution to the momentum transfer may be calculated. We | ||

+ | consider linear monochromatic deep-water capillary waves with | ||

+ | surface deflection z(x,t) aei(kx2qt) propagating in the x-direction | ||

+ | with a group speed c g dq/dk, phase speed c q/k, amplitude a, | ||

+ | wavelength l 2p/k and lateral extent W. The time-averaged | ||

+ | horizontal momentum associated with a single wavelength, | ||

+ | Pw pjka2Wc21, may be computed from the velocity field and | ||

+ | relations between wave kinetic energy and momentum21,25. Our | ||

+ | measurements indicate that the leg stroke typically generates a wave | ||

+ | train consisting of three waves with characteristic wavelength | ||

+ | l < 1 cm, phase speed c < 30 cm s21, amplitude a < 0.01– | ||

+ | 0.05 cm, and width L2 < 0.3 cm (see Fig. 3c). The net momentum | ||

+ | carried by the capillary wave packet thus has a maximum value | ||

+ | Pw < 0.05 g cms21, an order of magnitude less than the momentum | ||

+ | of the strider. | ||

+ | |||

+ | The momentum transported by vortices in the wake of the water | ||

+ | strider is comparable to that of the strider, and greatly in excess of that | ||

+ | transported in the capillary wave field; moreover, the striders are | ||

+ | capable of propelling themselves without generating discernible | ||

+ | capillary waves. We thus conclude that capillary waves do not play | ||

+ | an essential role in the propulsion of Gerridae, and thereby circumvent | ||

+ | Denny’s paradox. The strider generates its thrust by rowing, using its | ||

+ | legs as oars and its menisci as blades. As in the case of rowing boats, | ||

+ | while waves are an inevitable consequence of the rowing action, they | ||

+ | do not play a significant role in themomentumtransfer necessary for | ||

+ | propulsion. We note that their mode of propulsion relies on the | ||

+ | Reynolds number exceeding a critical value of approximately 100, | ||

+ | suggesting a bound on the minimum size of water striders. Our | ||

+ | continuing studies ofwater strider dynamics will followthose of birds, | ||

+ | insects and fish11,15,16 in characterizing the hydrodynamic forces acting | ||

+ | on the body through detailed examination of the flows generated | ||

+ | during the propulsive stroke. | ||

==References== | ==References== |

## Revision as of 00:17, 25 April 2009

By Sung Hoon Kang

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 UL2=n<103; where U < 100 cm s21 is the peak leg speed and L2 < 0.3 cm is the length of the rowing leg’s tarsal segment (see Figs 1b and 2), which prescribes the size of the dynamic meniscus forced by the leg stroke23. 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 cm21. An applied force per unit length in excess of 2j < 140 dynes cm21 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 leg stroke may also produce a capillary wave packet, whose contribution to the momentum transfer may be calculated. We consider linear monochromatic deep-water capillary waves with surface deflection z(x,t) aei(kx2qt) propagating in the x-direction with a group speed c g dq/dk, phase speed c q/k, amplitude a, wavelength l 2p/k and lateral extent W. The time-averaged horizontal momentum associated with a single wavelength, Pw pjka2Wc21, may be computed from the velocity field and relations between wave kinetic energy and momentum21,25. Our measurements indicate that the leg stroke typically generates a wave train consisting of three waves with characteristic wavelength l < 1 cm, phase speed c < 30 cm s21, amplitude a < 0.01– 0.05 cm, and width L2 < 0.3 cm (see Fig. 3c). The net momentum carried by the capillary wave packet thus has a maximum value Pw < 0.05 g cms21, an order of magnitude less than the momentum of the strider.

The momentum transported by vortices in the wake of the water strider is comparable to that of the strider, and greatly in excess of that transported in the capillary wave field; moreover, the striders are capable of propelling themselves without generating discernible capillary waves. We thus conclude that capillary waves do not play an essential role in the propulsion of Gerridae, and thereby circumvent Denny’s paradox. The strider generates its thrust by rowing, using its legs as oars and its menisci as blades. As in the case of rowing boats, while waves are an inevitable consequence of the rowing action, they do not play a significant role in themomentumtransfer necessary for propulsion. We note that their mode of propulsion relies on the Reynolds number exceeding a critical value of approximately 100, suggesting a bound on the minimum size of water striders. Our continuing studies ofwater strider dynamics will followthose of birds, insects and fish11,15,16 in characterizing the hydrodynamic forces acting on the body through detailed examination of the flows generated during the propulsive stroke.

## 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).