Brian Chan



Development of a Mechanical Water Strider

by

Brian Chan

Submitted to the Department of Mechanical Engineering

in Partial Fulfillment of the Requirements for the Degree of

Bachelor of Science

at the

Massachusetts Institute of Technology

May 2002

© 2002 Brian Chan

All rights reserved

The author hereby grants to MIT permission to reproduce and to distribute publicly paper and electronic copies of this thesis document in whole or in part.

Signature of Author______________________________________________________________

Department of Mechanical Engineering

May 10, 2002

Certified by____________________________________________________________________

John Bush

Assistant Professor of Applied Math

Thesis Supervisor

Accepted by____________________________________________________________________

Ernest Cravalho

Chairman, Undergraduate Thesis Committee

DEVELOPMENT OF A MECHANICAL WATER STRIDER

By

BRIAN CHAN

Submitted to the Department of Mechanical Engineering

on May 10, 2002 in partial fulfillment of the

requirements for the Degree of Bachelor of Science in

Mechanical Engineering

ABSTRACT

We have built a self-contained mechanical water strider capable of being supported by surface tension, and capable of propelling itself without breaking through the water surface. The machine has a mass of 0.35 grams and measures 13 cm long. An elastic chord wrapped around a 2.2 mm pulley provides the energy for five strokes of the middle pair of legs. The maximum distance traveled was 20 cm with the pulley fully wound. We have concluded that the primary propulsive force is generated by the distortion of the water surface caused by the legs, creating a forward-acting component of the surface tension force, or curvature force.

Thesis Supervisor: John Bush

Title: Assistant Professor

TABLE OF CONTENTS

INTRODUCTION…………………………………………………………………………………4

THEORY………………………………………………………..…………………………………6

DESIGN……………………………………………….…………………………………………14

PROCEDURE……………………………………………………………………………………17

RESULTS…………………………………..……………………………………………………17

CONCLUSION……………………..……………………………………………………………23

ACKNOWLEDGEMENTS………...……………………………………………………………23

APPENDIX………………………………………………………………………………………24

REFERENCES…………………………………………………………………………………...25

INTRODUCTION

The purpose of this study was to test the feasibility of a mechanical surface swimmer that mimics the motion of a water strider (Insects, Hemiptera, Gerromorpha). The major challenge was constructing a self-contained device capable of producing the rowing motion of a water strider yet light enough to stand on surface tension. We have developed a functional surface swimmer composed of sheet aluminum, plastic, and wire.

The live water striders that we researched for this project were Gerris remigis. They were captured in freshwater ponds in Massachusetts.

[pic]

Figure 1: Common water strider, Gerris remigis. Body length = 1.3 cm.

The adults weighed on average 2e-4 Newtons and have a typical body length of 1.3 cm. They possess three pairs of segmented legs of length 0.5 cm, 1.5 cm and 1 cm. The outermost segment of each leg is in contact with the water, resulting in a total contact length of about 1 cm. The outer leg segments measure about 50 ìm in diameter. The legs are covered in with a layer of fine hairs of length 20-40 ìm, width 1-2 ìm, and a hair density of 10e5 hairs per square mm (Andersen 1976), that serve to make the legs hydrophobic and non-wetting. Water striders swim by sweeping their middle pair of legs backward upon the water surface, while their first and third pairs of legs act as supports to keep the strider afloat. The adults are capable of top speeds of about 100 cm/s, generating thrust forces on the order of 6e-3 Newtons.

Researchers such as Andersen (1976) have studied the motion of water striders and other surface-dwelling insects in detail. While there has been much work on building land walking insect robots, there are no documented attempts at building a surface swimmer.

Suter et al. (1997) have extensively studied the locomotion of the fisher spider, Dolomedes Triton, another arthropod capable of surface locomotion. They found that hydrodynamic drag on the backstroke of the driving leg was the major contributor to thrust. Compared to the fisher spider, the water strider has a small leg diameter (about one tenth that of the spider), and hence a smaller leg contact area. Hydrodynamic force is proportional to leg area in contact with the water, so the water strider experiences much less hydrodynamic force than the fisher spider. We have found that this force plays a much smaller role in water strider locomotion than it does with the fisher spider. The mechanical strider was built with thin legs (0.2 mm diameter) to decrease the drag on the leg.

The strider was fabricated from easily obtainable materials. A lathe was required for the manufacture of two components, and the only other tools required were metal shears and a scribe for punching holes. The strider was built to be as small as possible, but the minimum size of the strider was limited by the smallest actuation mechanism that could be made in the machine shop. The strider measures 14 cm from the tips of the front legs to the tips of the rear legs.

[pic]

Figure 2. Comparison of the mechanical strider to an adult water strider.

Like a live water strider, the device has one pair of stroking legs. These are the middle pair of legs as seen in Figure 2. They are capable of providing thrust of 3.9 e-3 Newtons without breaking through the water surface. In order for the device to successfully mimic the motion of real water striders, it was necessary for all six legs to remain above or in contact with the surface during operation. The four non-rowing legs have a total contact length of 13 cm, which allows the 350 mg strider to rest stably on the water surface.

THEORY

The most basic requirement was for the mechanical water strider to be able to stand on the surface of the water without breaking through. Water striders have slender non-wetting legs that press down on the surface, creating elongate menisci. It is primarily the motion of the legs and menisci that control the motion of the strider.

Because the legs are long, the flow along the meniscus was modeled in two dimensions. For the purposes of this study, surface tension is a constant value across the surface of the water. Density and viscosity are also assumed to be constant, with values denoted in Table 1.

Table 1: Properties of water

|Surface tension, ( |72.8 e-3 N/m |

|Density, ( |1000 kg/m3 |

|Viscosity, ( |1e-9 m2/kg * s |

Stationary equilibrium

When a water strider is standing still, the legs depress the surface in a curved meniscus as shown in cross-section in figure 1.

[pic]

Figure 3: Meniscus shape during stationary equilibrium

The x dimension is horizontal and perpendicular to the leg, and the y dimension is vertical. Keller (1998) shows that the net force on the leg is equal to the sum of the pressure force on the submerged underside of the leg and the surface tension force where the water surface meets the leg.

F = Mg = FP + FS (Eq. 1)

where M is the mass of the strider and g is the acceleration of gravity. The pressure force FP is equal to the integral of the hydrostatic pressure over the section of leg in contact with water, which turns out to be the weight of the fluid displaced by the volume directly above the leg,

[pic] (Eq. 2)

Where l is the length of the leg and A1 is the area directly above the leg shown in Figure 4.

[pic]

Figure 4: Meniscus showing the different contributions to the

upward force on the leg.

The upward sloping edges of the meniscus pull on either side of the leg with a tangential force along the curve of the water surface.

[pic], (Eq. 3)

where t1 and t2 are the unit tangent vectors shown in figure 3. The force on one side is equal to the surface tension times the length of the leg. Since the meniscus is symmetric when the leg is stationary, the sideways components of this force cancel each other and the surface tension force can be also be expressed as

[pic], (Eq. 4)

where l is the length of the leg and è is the angle from horizontal at which the surface contacts the leg. The sum of the upward forces on the legs is balanced by the weight of the water strider. The surface of the meniscus experiences a buoyancy force equal to the weight of the water displaced by (A2 + A3). Keller shows that the upward component of surface tension must be equal to the weight of the water displaced by the meniscus.

[pic] (Eq. 5)

where ñ is the density of water, 1000 kg/m3, g is the acceleration of gravity, l is the length of the leg and A2 and A3 are the areas above the water surface as shown in figure(4). The net force on the leg is then:

[pic] (Eq. 6)

where A is the entire cross-sectional area shown,

A = A1 +A2 +A3

When the leg diameter is small, as in the case of live water striders and our mechanical strider, A1 is negligible compared to the sum of A2 and A3. When this is so,

Am = A2 + A3 (Eq. 7)

and

F = Mg = FS (Eq. 8)

The angle ( is then:

[pic] (Eq. 9)

We now wish to find a relation between the depth of the meniscus and upward force. In order to determine this relation, the shape of the meniscus must be known. In the static case, hydrostatic pressure increases linearly with depth:

p = -ñgy (Eq. 10)

and the curved surface of the fluid must balance the pressure force. The normal component of the surface tension force on surface element ds equals the pressure force per length p ds.

[pic]

Figure 5. Forces on the surface of a meniscus. In equilibrium,

the surface tension force must equal the pressure force.

Because the pressure is proportional to depth, we find that the radius is inversely proportional to depth:

[pic] (Eq. 11)

[pic] (Eq. 12)

For small values of ( (less than 15 degrees), some simplifications can be made to determine Am. When ( is small, the radius can be approximated by:

[pic] (Eq.13)

The relation between the surface tension force and the pressure force then becomes:

[pic] (Eq.14)

which is satisfied by solutions of the type

[pic] (Eq. 15)

where C1, C2 are constants and

[pic] (Eq. 16)

Since y = 0 at positive and negative infinity, C2 must be zero when x is positive, and C1 must be zero when x is negative. We find that the meniscus follows the equation

[pic] when x0 (Eq. 18)

where h is the meniscus depth. For objects that have a large diameter, the area directly above the object must be taken into account. Real water striders and the mechanical water strider have a small leg diameter compared to h, so the meniscus is well approximated by the two exponential curves alone. The cross-sectional area of the meniscus is expressed as the integral

[pic] (Eq.19)

[pic] (Eq. 20)

[pic] = [pic] = [pic] (Eq. 21)

The force per unit length of leg is then:

[pic] (Eq. 22)

Thus, in the small-angle approximation, the meniscus height and cross-sectional area are proportional to the force per unit length.

For a given total mass, one can calculate the minimum contact perimeter to support the water strider. This perimeter is the length required to support the water strider with the edges of the water-air interface pulling up vertically, as shown in Figure 3.

[pic]

Figure 6 – Maximum force per length at 90-degree angle theta

In actuality, it is nearly impossible to create a 90-degree angle, since any increase in downward force would cause the meniscus to collapse upon itself, submerging the leg. The live water striders used in this study weighed on average 2e-4 Newtons, which corresponds to a minimum contact perimeter of 3 mm (twice the minimum leg length because surface tension acts on both sides of the leg). Their four support legs have a contact perimeter of about 2 cm, which is 7 times the minimum contact length.

Forward thrust

The water strider propels itself forward by sweeping its second pair of legs backwards on the water surface. During the thrust stroke, the shape of the meniscus becomes distorted by the leg motion and would not be easily determined by simple analytical methods. However, we can use order-of-magnitude analyses to estimate the forces generated by the strider.

The forward thrust force is a combination of the hydrodynamic drag on the leg and the curvature force, which is the net horizontal force caused by a difference in the surface slopes upstream and downstream of the leg.

The hydrodynamic drag force on the leg is approximately equal to the contact area on the leg multiplied by ½ ñ V2, where the contact area is roughly the diameter of the leg multiplied by the length of the leg.

[pic] (Eq. 23)

When the strider is standing still, the net force due to surface tension is directed vertically. When the leg is moving, however, the meniscus deforms asymmetrically. When this happens, Equation (3) still holds, but surface tension pulls on the leg with both horizontal and vertical components. The horizontal component of force is the thrust due to surface tension, or the curvature force.

[pic]

Figure 7 – Deformation of meniscus caused by leg motion.

Note: this figure is in reference frame of the leg.

Bernoulli’s law can be used to explain the distortion of the meniscus. Flow separation occurs downstream of the leg due to the high Reynolds number ReH and the sharp corner caused by the leg, leaving a low-pressure wake. Because the flow is assumed to be steady, the surface tension force pulling in from the curvature of the meniscus must counteract this pressure difference at any point on the meniscus. The radius of curvature increases as the inverse of pressure. Therefore, a greater deceleration of fluid or an increase in the depth of the leg would cause the interface to have a smaller radius of curvature to match the increase in pressure.

High-speed videos of live water striders and of the mechanical water strider show that the stroking legs create many waves during propulsion. The deformation of the water surface suggests that the flow field around the leg and the meniscus shape would not be easily solved with simple mathematical models. However, it is possible to determine the approximate magnitude of the surface tension force.

By Newton’s second law, the force imparted by the leg on the water is equal and opposite to the force felt by the leg. The net force that the leg imparts on the water equals the net force on the leg, which is the sum of the hydrodynamic force on the submerged portion of the leg and the surface tension force FS.

[pic] (Eq. 24)

The meniscus itself can be seen as a bluff body traveling through the water, and because of the high Reynolds number, it experiences pressure drag due to separation behind the body. The force of the leg on the water is equal to the sum of the meniscus bluff-body drag and the hydrodynamic drag on the leg.

[pic] (Eq. 25)

Since the hydrodynamic drag acts directly on the submerged part of the leg, we see that the horizontal component of surface tension must equal to the form drag on the meniscus.

[pic] (Eq. 26)

The pressure drag on the meniscus is approximately equal to the product of stagnation pressure and the frontal area of the meniscus,

[pic] (Eq. 27)

where h, the meniscus depth, is determined by finding the average meniscus depth with the strider standing still. From the static case:

[pic] (Eq. 28)

[pic] (Eq. 29)

Surface tension places an upper limit on the force generated by the stroke. When the leg is in motion relative to the fluid, the maximum thrust results when the meniscus curves 180 degrees upstream and flattens to zero degrees downstream of the stroking leg. This rarely happens, and the horizontal component of curvature force is some finite fraction of the maximum force.

The maximum force would be equal to the twice product of surface tension and the contact perimeter. The sideways curvature force is then on the order of 2ló.

[pic]

Figure 8 – Meniscus shape at maximum thrust

Note: figure is in reference frame of leg

Deceleration

After the thrust impulse, a live water strider glides for a while on its four support legs before setting down the middle legs. During this phase, it decelerates due to the curvature force in front and behind the legs, and by the viscous drag force on the support legs.

To approximate the viscous drag, each leg can be modeled as a flat plate aligned with flow with area LD. The drag is then

[pic] (Eq. 30)

where AL is the total contact area of the legs and CD is the drag coefficient, a function of the Reynolds number of the leg based on leg length.

[pic] (Eq. 31)

The Reynolds number of the support legs is sufficiently low that flow can be assumed to be laminar. CD for laminar flow around a flat plate is

[pic] (for a flat plate in laminar flow) (Eq. 32)

The viscous drag of the body is also modeled as a flat plate, but with the properties of air used instead. The drag force on the body is:

[pic] (Eq. 33)

where AB is the surface area of the body. To calculate the Reynolds number and the force on the body, the following air properties were used.

Table 2: Properties of air

|Density, (air |1.22 kg/m3 |

|Viscosity, ( air |1.8e-5 m2/kg * s |

Aside from the drag of the body moving through in the air, there is the form drag of the two rowing legs, which are now lifted off the surface of the water. The form drag is on the order

[pic] (Eq 34)

where L is the total exposed length of the support legs, about 12 cm.

The curvature force Fs,x on the support legs is another component of drag that acts to slow down the water strider. It is equal to the pressure drag on the meniscus. The force is approximately equal to the product of stagnation pressure and the frontal area of the menisci created by the four support legs. Unlike the stroking legs, however, the support legs are aligned to the direction of the strider’s motion. The frontal area of each meniscus is then the cross-sectional area:

[pic] (Eq. 35)

[pic] (Eq. 36)

The pressure drag is roughly proportional to frontal area. Since the frontal area of the support legs is several times less than the frontal area of the stroking legs, we expect the drag force to be much smaller than the thrust force. Once again, a jump in the curvature force is expected when the water strider exceeds the wave velocity.

DESIGN

It was necessary for the device to be lightweight enough for it to be supported by surface tension. According to Equation 8, as the mass of a water strider increases, so must the surface tension force. Since the surface tension force is approximately proportional to the contact length, a large water strider will require longer legs. At a certain mass, the length of leg required becomes impractically long.

The mechanical water strider weighs 0.034 Newtons, which required a minimum contact perimeter of 4.7 cm, or a total leg contact length of 2.4 cm. The device was given a support leg contact length of 13 cm, or 5.4 times the minimum length necessary to stay afloat. The average meniscus depth h is given by Equation 20:

[pic] = 0.5 mm

A top view of the mechanical strider is shown in the next figure.

[pic]

Figure 9: Top view of mechanical water strider

In order to minimize the mass of the strider, the design was refined to be as simple as possible. Lightweight, inexpensive materials were chosen. The body frame was a 0.13 mm thick sheet of aluminum cut from a beverage can. The sheet was creased twice lengthwise to give it rigidity. The support legs and the rowing legs are pieces of stainless steel wire, 0.2 mm in diameter. This particular type of wire was chosen because it was lightweight, water resistant and thin, allowing the net force to be well approximated by the curvature force alone. The wire was rigid enough to hold its shape yet malleable enough to allow easy manipulation of form. Also, the wire was found to be non-wetting, which was an essential property that allowed the legs to press on the water surface without breaking through. Below is an image of the water strider sitting in a pan of water.

[pic]

Figure 10: The mechanical strider floating in a shallow pan of

water, 1 cm deep. The two rowing legs are lifted off the

surface. Note the shadows created by the menisci.

The shadows below the legs are caused by the refraction of light through the menisci. When submerged by force, the water strider sank, since it is denser than water. However, the four menisci created by the support legs provided enough upward force to keep the device afloat.

The actuation mechanism is composed of an elastic band attached to the rear tip of the body and wound around the nylon pulley. The elastic band was taken from a sock, and had a spring value of 3.1 e-2 Newtons per unit strain. Its maximum strain was 1.5 (it could be stretched to 250% its original length), which limited the number of leg strokes to five. The brass pulley shaft is press-fit into the pulley. Both ends of the shaft pass through the sides of the aluminum body frame.

[pic]

Figure 11: Detail of pulley (elastic band not shown)

The rowing legs are attached to the shaft with sections of plastic tube. The tension of the elastic band causes the pulley to unwind and drive the rowing legs in a circular motion when seen from the side. At the bottom of the circular path, the leg contacts the surface of the water and creates a meniscus. The backwards motion of the leg provides forward thrust. Assuming that the elastic band was stretched to its maximum, the initial tension of the band is equal to the strain of the band multiplied by its spring constant, 3.1 e-2 Newtons per unit strain:

Fband = k*åmax = 4.5e-2 N (Eq. 37)

where k is the spring constant and åmax is the maximum strain of the elastic band, 1.5. The torque on the pulley (which had a radius of 1.1 mm) is then:

Torque = Fband*r = 5.1e-5 N x m (Eq. 38)

which is equal to the force on the legs multiplied by the leg moment arm rleg:

Torque = Fleg*rleg (Eq. 39)

The maximum thrust that the band would be able to supply is then:

Fleg = Torque/rleg = 5e-3 N (Eq. 40)

which does not exceed the maximum surface tension force for a leg length of 9 cm:

Fmax = 2ól = 1.3e-2 N. (Eq. 41)

The limit on maximum leg force was an important design requirement for the strider. If the force exerted by the leg on the water were to exceed Fmax, the rowing legs would break the surface and submerge, severely decreasing the thrust force.

We did not carry out detailed analyses of the dynamics of the pulley and unraveling cord, because we assumed that there would be some mechanical losses in the pulley and cord. This experiment was primarily concerned with the fluid dynamics of the machine.

PROCEDURE

The pulley was wound by hand five revolutions, at which point the elastic band was stretched to its maximum length. The pulley was held in place against the side of the body with tweezers so that it could not rotate. The strider was then lowered into a pan of water so that the four support legs rested on the surface without breaking through. Opening the tweezers simultaneously released the water strider and freed the pulley to rotate.

To examine the deceleration of the water strider, the strider was placed in the water with the pulley unwound and with the stroking legs tilted up out of the water. The strider was given a small push and filmed until it came to an apparent standstill.

We recorded the motion of the strider at 500 frames per second with a Motionscope PCI 1000 digital camera running on the Midas motion capture software. Top and side views were captured. The videos were saved in .avi format and analyzed.

RESULTS

The mechanical water strider performed reliably in tests, as shown in the video footage. The stainless steel wire used for the legs did not break through the surface of the water, although they revolved at a rate of 4 revolutions per second. It was impossible to determine with the naked eye whether or not the leg broke the surface, but slow motion video capture shows that at the lowest point on the leg’s trajectory, the refracted image of the surface waves has a sharp discontinuity where the leg touches the surface of the water. If the leg were submerged, there would be no break in the refracted image. Because there is a break in the image, we can conclude that the rowing leg does not break through the water surface.

[pic]

Figure 12 - Mechanical water strider about 0.04 seconds after release.

Note the discontinuity in the refracted image of waves below stroking leg.

Below is a distance-time plot of the mechanical strider during one stroke-glide phase. The length of the stroke is 2.3 cm with respect to a stationary frame of reference. The thrust stroke of the mechanical water strider begins at t=0 and ends at t = 0.08 seconds. The average leg speed is then 27cm/s with respect to a stationary frame of reference. During the thrust stroke, the water strider accelerates to 0.3 m/s

[pic]

Figure 13: Distance vs. time plot of strider body during first stroke.

From these data, the thrust generated by the stroke can be calculated. The momentum of the water strider is

M*V = 3.5e-4 kg m/s (Eq. 42)

which is equal to the impulse, or the product of the average force and the time through which the force is applied. The average thrust force is equal to momentum divided by time

[pic]= 3.9e-3 N, (Eq. 43)

which turns out to be about 1.1 times the weight of the strider and is very close to the maximum thrust the strider was designed to generate, 5e-3 Newtons.

The forward thrust force is a sum of the hydrodynamic force on the leg and the curvature force from the meniscus. The relative contribution of each drag component will be estimated separately.

The hydrodynamic force on the leg is approximately equal to the contact area on the leg multiplied by ½ ñ V2, where the contact area would be the diameter of the leg multiplied by the length of the leg. On the mechanical water strider, the two stroking legs have a total of 9 cm contact length and a diameter of 0.18 mm. The velocity of the leg V was found to be about 27 cm/s. Equation 21 gives the hydrodynamic force on the leg:

[pic] 5.7e-4 N;

which is a high estimate. The drag on the meniscus of each leg manifests itself as a curvature force on the leg, and is approximately equal to the product of stagnation pressure and the frontal area of the meniscus,

[pic]

where h, the meniscus depth, is determined by finding the average meniscus depth with the strider standing still. From the static case, with all six legs down on the surface, the average depth is:

[pic]= 0.3 mm

The curvature force given by equation 27 is then on the order of 1e-3 N, on the same order of magnitude as the thrust force. The depth of the meniscus is greater in motion than 0.3 mm, since the circular motion of the pulley rotates leg into the water with enough speed that it bends outward. The force due to pressure drag on the meniscus (and hence the curvature force) is probably several multiples of 1e-3 N. A depth of 1 mm would account for the total thrust calculated from the experiment, and is a reasonable value. It was impossible, however, to determine the exact depth of the meniscus during rowing.

Gliding phase

To measure the deceleration of the strider after the stroke, the strider was given a small push, and its motion was recorded as it slowed down. The pair of stroking legs was held up off the surface of the water in order to decrease the drag and simulate the glide phase of a real water strider. The distance of the strider as a function of time is shown in the next plot.

[pic]

Figure 14. Distance vs. time plot of mechanical strider for gliding phase.

Solid line curve represents constant CD model.

Sources of drag.

The drag contributions from skin friction on the leg, the skin friction on the body, and pressure drag on the four menisci were calculated to compare their relative magnitudes.

To find the hydrodynamic drag on the support legs, we modeled each leg as a flat plate aligned with flow. Since the legs were parallel to flow, we assumed that skin friction comprised most of the drag on the leg. At the Reynolds number of the strider at top speed (based on leg length), the drag coefficient is on the order of Cd~0.001. The initial friction force is then

[pic]1e-6 N

The initial force due to air drag on the strider body was also calculated. The total surface area of the body, AB, was approximately 9 square centimeters. The length of the body was lB = 0.9 m, the Reynolds number based on lB was ReL = 1800, which corresponded to a drag coefficient of CD =0.03. The drag on the body is approximately

[pic] 1.5e-6 N

The form drag on the strider rowing legs is given by equation 34:

[pic]1.3e-6 N

The support leg menisci were modeled as bluff bodies with much smaller frontal area compared to the menisci of the rowing legs. The curvature force on the support legs is equal to the pressure drag on the menisci, and is approximated by the product of stagnation pressure and the frontal area of the menisci created by the four support legs. Because the support legs are aligned to the flow direction, the frontal area is the area above the meniscus as seen from figure 1, rather than the product lh. When the stroking legs are lifted off the surface, the meniscus depth of the support legs is an average of 0.5 mm and the frontal area of a single leg is

[pic] = 3 mm2

The initial drag force on all four menisci is then

[pic]= 2.2e-3N,

assuming a CD of about 1 (CD is usually on the order of 1 for bluff bodies and is found experimentally). The pressure drag on the meniscus is about 1000 times greater than the viscous drag on the legs and body combined; therefore the drag on the body and legs can be ignored. It is clear from these results that formation of a meniscus is a favorable way for small surface dwellers to stay afloat, since a meniscus created by a thin leg experiences very little drag from viscosity due to the small surface area of the leg. A surface swimmer can further decrease its drag by forming streamlined menisci, thereby decreasing CD.

Below is a plot of speed vs. time of the strider.

[pic]

Figure 15: Speed vs. time plot for glide phase.

Solid line curve represents constant CD model.

Since CD is only weakly dependent on Reynolds number at high Reynolds number, it can be considered constant. The velocity versus time relation can be determined by integrating the drag equation:

[pic],

where A is the total frontal area of the four menisci.

[pic] (Eq. 44)

[pic] (Eq. 45)

Integrating, we find that velocity decreases with the inverse of time:

[pic] (Eq. 46)

The velocity graph shows a theoretical velocity-time curve, calculated with a CD value of 1.35. For reference, CD of a flat plate perpendicular to flow is 2.0, so CD = 1.35 is a reasonable value for the meniscus drag coefficient. The distance traveled is the integral of velocity with respect to time:

[pic], (Eq. 47)

which evaluates to:

[pic] (Eq. 48)

Figure 14 shows the theoretical distance-time curve along with the exponential data. Both theoretical curves were calculated with a CD of 1.35. The graphs show that the constant CD model effectively predicts the motion of the decelerating strider.

CONCLUSION

The mechanical water strider performed exceptionally well, proving that it is possible to build a functional surface swimmer. It is easy to fabricate, owing to its simple design and inexpensive components. The simplicity of the device makes it ideal for demonstrations and allows for quick modifications, such as changing the cord tension, leg length, or body elevation.

We have found that the primary force responsible for propelling the water strider is the drag created by the rowing leg menisci, which is transferred to the leg as horizontal component of surface tension force when the water surface deforms. Less significant is the drag force on the leg surface in contact with water, owing to the small frontal area of the leg compared to the frontal area of the meniscus. Likewise, the main force responsible for slowing the strider is the drag force on the four menisci created by the support legs, a force proportional to the frontal area of the menisci.

Future Work

This study was primarily concerned with finding approximate figures for the surface swimmer. In the future, it would be useful to determine with more certainty the shape of the two-dimensional meniscus of the stroking leg and to calculate the drag force for a range of velocities and meniscus depths. Of particular interest is the creation of bow waves when leg speeds exceed the capillary wave speed. When leg diameter is small, a bow wave can only affect the force on the leg by a change in the meniscus shape, thus changing the surface tension force. A detailed examination of the unsteady motion of a strider leg may also reveal interesting data.

The mechanical strider will also allow us to explore the effects of varying of parameters such as body mass, leg length, and thrust force, thus giving further insight into their dynamics. Such variations would not be possible with real water striders, but would be easily done with the mechanical strider.

ACKNOWLEDGEMENTS

The author would like to thank Professor John Bush for his supervision and guidance, James Bales and the Edgerton Lab for lending us the high-speed camera, and David Hu for capturing the live water striders and for his long hours of working together.

APPENDIX

Schematics of mechanical strider

All dimensions are in cm.

[pic]

Detail of pulley

[pic]

REFERENCES

Andersen, N.M. 1976: A comparative study of locomotion on the water surface in semiaquatic bugs (Insects, Hemiptera, Gerromorpha). Vidensk, Meddr. Dansk. naturh. Foren.

Anderson, J.D. 2001: Fundamentals of Aerodynamics, Boston: McGraw-Hill

Fox, R.W. and McDonald, A.T. 1998: Introduction to Fluid Mechanics. New York: John Wiley and Sons

Keller, J.B. 1998: Surface tension force on a partly submerged body. Physics Fluids 10, 3009-3010

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