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

From Soft-Matter
Jump to: navigation, search
(References)
(Soft matter example)
Line 32: Line 32:
 
physical picture, infant water striders cannot swim, an inference referred to as Denny’s paradox [4].
 
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 ˆ
+
The Reynolds number characterizing the adult leg stroke is Re = UL<sub>2</sub>/υ ~ 10<sup>3</sup>; where U ~ 100 cm/s is the peak leg speed and L<sub>2</sub> < 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 < 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.
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
+
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
Figs 1b and 2), which prescribes the size of the dynamic meniscus
+
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
forced by the leg stroke23. For the 0.01-s duration of the stroke, the
+
so has a momentum P = MV ~ 1 g cm/s. The total momentum in the pair of dipolar vortices of mass M<sub>v</sub> ~ 2πR<sup>3</sup>/3 is P<sub>v</sub> = 2M<sub>v</sub>V<sub>v</sub> ~ 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
driving legs apply a total force F < 50 dynes, the magnitude of
+
P<sub>w</sub> ~ 0.05 g cm/s, an order of magnitude less than the momentum of the strider.
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) ˆ aei(kx2qt) propagating in the x-direction
 
with a group speed c g ˆ dq/dk, phase speed c ˆ q/k, amplitude a,
 
wavelength l ˆ 2p/k and lateral extent W. The time-averaged
 
horizontal momentum associated with a single wavelength,
 
Pw ˆ 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
 
The momentum transported by vortices in the wake of the water

Revision as of 00:33, 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).

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 Mc = 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 Mc = 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].

Fig. 1. Natural and mechanical water striders. a, An adult water strider Gerris remigis. b, The static strider on the free surface, distortion of which generates the curvature force per unit leg length 2σ sin θ that supports the strider’s weight. c, An adult water strider facing its mechanical counterpart. Robostrider is 9 cm long, weighs 0.35 g, and has proportions consistent with those of its natural counterpart. Its legs, composed of 0.2-mm. gauge stainless steel wire, are hydrophobic and its body was fashioned from lightweight aluminium. Robostrider is powered by an elastic thread (spring constant 310 dynes cm-1) running the length of its body and coupled to its driving legs through a pulley. The resulting force per unit length along the driving legs is 55 dynes cm-1. Scale bars, 1 cm.

The force balance on a stationary water strider can be written as Mg = Fb + Fc, where Fb is the buoyancy force and Fc is the curvature force. Fb is obtained by integrating the hydrostatic pressure over the body area in contact with the water, while Fc 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 Fb/Fc ~ w/Lc <<1, where the leg radius, w ~ 40 um, the capillary length, Lc = (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, cm =…(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/υ ~ 103; where U ~ 100 cm/s is the peak leg speed and L2 < 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 Mv ~ 2πR3/3 is Pv = 2MvVv ~ 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 Pw ~ 0.05 g cm/s, 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).

[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).