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How fish swim according to Newtonian physics; and how to win a swimming race. Preprint · February 2021

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How$fish$swim$according$to$Newtonian$physics,$and$how$to$win$a$swimming$race.$

How$fish$swim$according$to$Newtonian$physics,$ and$how$to$win$a$swimming$race.$ $ $ $ Mr.$Nicholas$Landell-Mills$$$ 22$$November$$$2021$$ Pre-Print$DOI:$$$$10.13140/RG.2.2.26823.21921;$$$$CC$License:$CC$BY-SA$4.0$ Keywords:$Coanda$effect;$fish;$Hydrodynamics;$Newton;$physics;$swimming.$! Fig.!1a.!!Great!white!shark.!![53]!

Abstract$ Amazingly, the physics of how fish swim is still debated and unproven in the 21st Century. However, a new approach using Newtonian mechanics based on the mass-flow rate can explain how fish swim and solve the 85-year-old Grays Paradox. Newtonian physics can also be used to explain how swordfish can swim at over 100 km/hr, outpacing a cheetah running on land at about 90 km/hr. W ater is about 830x denser than air. Therefore, logically water should be harder to push out of the way and the fish should swim slower than land animals move. Applying this knowledge provides the best strategy to win swimming races and design better water propulsion. T his analysis explains why it is critical to minimize drag, and maximise the Coanda effect and the angle-of-attack (AOA) of the tail against the water. A swimming suit that mirrors fish scales would also help. The Newtonian explanation is straightforward. Fish swim through a mass of water each second (m/dt) that they accelerate to a velocity (dv) backwards with their body and tails, to create a backward force (Force BACK = ma = m/dt x dv). The fish’s scales gripping the water and the Coanda effect from the fish’s curved motion aid this process. The reaction generates an equal and opposite forward force (Thrust) that propels the fish ahead. The same logic explains as how submarine propellers accelerate a mass of water backwards to create a reactive forward force. See Fig 1b. Fig. 1b. Newtonian physics. The Newtonian explanation is significant because it provides a new method to explain how fish swim that is consistent with accepted physics, what is observed in practice, and it builds on the existing Elongated Body Theory. In turn, this approach provides new and useful insights into fish locomotion.

I. INTRODUCTION

-

“Making robot fish is hard when you don't know how they swim.” Wired Magazine 2015. [34]

-

“The swordfish is reputedly the fastest swimmer on Earth. … but how they contribute to its speed is still unknown.” 2016 [83]

-

“Exactly how fish manage this feat is something of mystery.” MIT Technology Review, 2018. [24]

A. The problems and a solution. The physics of how fish swim is still debated and unproven. There is no accepted model, equation or experiment that quantifies and explains the physics of how all fish swim, as highlighted occasionally by the media and academic articles: -

The failure to explain how fish swim for is highlighte d by the Grays Paradox remaining unexplained since 1936, despite proposed solutions in 2009 and 2014. [18] A frameworks to solve Grays Paradox based on Newtonian physics is presented in

“Understanding how fishes generate external fluid force to swim steadily and maneuver has proven to be difficult.” American Physiology Society; 2002. [33]

1

How$fish$swim$according$to$Newtonian$physics,$and$how$to$win$a$swimming$race.$ and global trade would have been significantly delayed without this technology.

another paper. [19] The key problems with resolving the debate on how fish swim include the difficulties of measuring fish movements and fluid flow around the fish. In addition, health and safety rules can limit the experimental conditions and technology used in research.

As a side note, at 56 km/hr a great white shark can swim almost three times faster than a human can run on land (about 20 km/hr). So hypothetically, even a person running on water would not escape an angry or hungry shark.

Research is limited by a focus of fluid mechanics used to explain the physics of swimming, turbulence remains a mystery for the most part, [35][36] and research does not attract big funding or attention. Also many relevant academic articles are incomprehensible, which limits the spread of knowledge. Nonetheless, these problems are surmountable with technology such as computer simulations, visualization technology, and high-speed video, which have existed for sometime. Even though experiments are relatively limited, the available data is sufficient to explain how fish swim. Another 20 years of research along existing lines of thought produces more of the same, without providing significant progress.

Contents:

I.$

Introduction$..........................................................$1$

This means that the main constraint preventing a solution is intellectual progress to create new ideas, and a willingness to consider alternatives; not a lack of evidence, data or technology. In short, a solution to explain the physics of how fish swim requires an open discussion of new ideas and alternatives. Therefore, an alternative approach using Newtonian mechanics and absolute fluid flow analysis is justified and necessary.

II.$

$New$$Insights$$–$$$Summary$....................................$3$

III.$

Newton$$Explains$$How$$Fish$$Swim$......................$4$

IV.$

Evidence$$Supporting$$Newton$..............................$7$

V.$

The$$Coanda$$Effect$...............................................$9$

Newtonian mechanics based on the mass flow rate offers a solution that is consistent with accepted physics, what is observed in practice, and other forms of locomotion (e.g. insects and birds [5][6][7]). Newtonian mechanics provides universal principles to explain how animals and boats propel themselves forward by pushing backwards to create reactive forces.

VI.$

$Fluid$$Flow$$and$$the$$Tail$....................................$11$

VII.$

$Caudal$$Tail$$Aspect$$Ratios$................................$14$

VIII.$

$Fish$$Skin$$and$$Scales$.........................................$15$

IX.$

$$Swimming$$Speeds$.............................................$16$

X.$

$Wake$Vortices$....................................................$17$

XI.$

$Fish$$Circulate$$The$$Water$..................................$18$

XII.$

$Kinetic$$Energy$$Swimming$.................................$ 20$

XIII.$

$Standard$$Equation$$For$$Drag$............................$21$

XIV.$

$Example$$Calculation$$–$$Dolphin$.......................$24$

XV.$

$Alcids$$–$$Swimming$$and$$Flying$........................$26$

$

B. Why this is significant. The higher speeds achieved by swordfish as compared to a cheetah is the reverse of what humans have achieved with technology. Airplanes (and one car) can travel faster than the speed of sound, which is approx. 1,240 km/hr (330 m/s). However, most submarines do not exceed 60 km/hr underwater, which is about 40% less than a swordfish’s top speed.

$ The Dolphin-style submarine (Seabreacher) is reported to be able to match the top speeds of a real Dolphin underwater, of 40 km/hr. But this is only with an extremely powerful engine (i.e. 1500cc 4 stroke engine, 230hp) given its mass of about 650 kg (excluding passenger). [92]

XVI.$

$Discussion$$of$$Results$........................................$27$

XVII.$ $How$$to$$Win$a$$Swimming$$Race$.......................$28$ XVIII.$ $Conclusions$........................................................$29$

This is significant because understanding how a swordfish can swim so fast provides new and useful insights. These insights could be used to design more efficient water locomotion on the surface or underwater, for manned and unmanned vehicles. Such insights could also explain why some people are faster swimmers than others who have the same fitness and muscles. Newtonian mechanics offers to provide these insights and significantly improve the understanding of swimming.

XIX.$

$$Additional$$$Information$....................................$29$

XX.$

$$References$........................................................$30$

$ Appendix$$I$$–$$Technical$$Framework$............................$32$ Appendix$$II$$–$$Current$$Theories$...................................$34$

Sea travel has been a critical global economic driver since the 16th Century, when it was discovered how to sail a boat into the wind. Sail power enabled people and goods to travel across oceans cheaply and quickly. The European industrial revolution

2

How$fish$swim$according$to$Newtonian$physics,$and$how$to$win$a$swimming$race.$

B. Key forces acting on a fish.

II. NEW INSIGHTS – SUMMARY

The key forces acting on a fish include the backward force from the tail, causing thrust, and therefore, drag. See Fig. 2b.

A. New insights summarized. Applying Newtonian physics swimming provides new and useful insight into how fish swim fast with ease. ‘Even when they glide, they don’t seem to lose any speed.’ [27] Fig. 2b. Key forces on a fish. According to Newtonian mechanics, in stable constant swimming conditions: See Fig 2c. -

The fish’s caudal fin (tail) accelerates a small amount of water each second (m/dt) to a relatively high velocity backwards (dv), to create a backward force: Force BACK

-

The reactive forward force provides thrust to push the fish ahead. Force BACK

Fig. 2a. New insights into fish locomotion.

-

The key insights into swimming based on the Newtonian approach include a better grasp of: See Fig. 2a. -

The role of the Coanda effect, which helps pull water backwards along the curved part of the fish’s body.

-

Wake vortices and the circulation of water.

-

The role played by fish scales and the tail AOA.

= ma = m/dt x dv

= Thrust = m/dt x dv

Thrust is then balanced by the drag from the large amount of water that the body of the fish swims through and accelerates to a low velocity out of its path. Drag (force) = ma = m/dt DRAG x dv DRAG

-

Drag is also described by the standard equation: Drag = 0.5 (Fish Velocity2 x Water Density x Surface Area x Drag Coefficient)

The fish scales push water backwards more effectively by gripping the water, especially at a high tail AOA, by creating friction and turbulence. In short fish scales provide better traction for their tails to push against the water. This analysis suggests that viscosity explains the fish’s ability to grip the water and how water is displaced backwards. But inertia explains the resultant forces. This aspect justifies a focus on Newtonian physics, rather than hydronamics, to explain the forces involved in how fish swim. -

The importance of buoyancy to fish locomotion.

Fig. 2c. Thrust, drag and the backward force.

The neutral buoyancy of fish also means that the strength of gravity plays almost no role in how fish swim. In a sense, fish swimming are just shifting different masses of water around. Fish are a mass of flesh and bone that accelerates a mass of water of equal density backwards, to create a backward force. The reactive forward force pushes the fish’s mass forwards, through another mass of water of equal density. -

Why the drag on a fish swimming is proportional to its velocity2.

-

The kinetic energy used in swimming and jumping.

-

An example calculation of the Newtonian forces acting on a dolphin swimming is provided in Section XIV on page 24. This offers a framework to solve Grays Paradox, which remains unexplained despite claims otherwise. [18]

-

In addition, this approach allows insight to be gained by comparing the Newtonian equation for drag above to the standard equation for drag, as shown by the equations: This can be used to explain why the drag on a fish is proportional to its velocity2. In turn, this also explains why kinetic energy (K.E. = 0.5mv2) is also proportional to a fish’s velocity2.

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How$fish$swim$according$to$Newtonian$physics,$and$how$to$win$a$swimming$race.$

III.

In the equations above the increased velocity of the water is expressed as ‘dv’, and not as acceleration (‘dv/dt’), because this action is not time dependent. It is due to a one-off force (impulse) from the fish. In contrast, the mass of water passed through by the fish is time dependent, and therefore, is expressed as the mass flow rate (m/dt).

NEWTON EXPLAINS HOW FISH SWIM

A. Forward Force = m/dt x dv.

Similar to the Elongated Body Theory, the inertia of the water provides the resistance to the backward force, allowing for the equal and opposite forward force to be created. The water pushed back creates wake vortices and is circulated.

Where: - m = Mass of water passed through and pushed back. - m/dt = Mass flow rate. - dt = Change in time (per second). - dv = Change in velocity (v) of the water displaced back. - v = Velocity of the water displaced back by the fish. - a = dv/dt = Acceleration. - Force = ma = m x dv/dt = m/dt x dv [1] - Force = ma = m x dv/dt = d(m/v)/dt [1] - Momentum = mv [1]

A key difference between undulatory and oscillatory motion, is that whole body participates more actively in undulatory motion. Whereas oscillatory motion relies on the fish’s tail swinging from side-to-side to power the fish ahead.

B. Transfer of momentum. There is no net gain or loss of momentum, energy and mass in this process of generating a forward force. Momentum and kinetic energy is transferred from the fish to the water, by accelerating the water backwards. “Momentum is transferred from the flukes (dolphin tail) to the water.” [62] See Fig. 3b. This action is expressed as the equation: Force BACK = ma = d(mv/dt)

(5)

Fig. 3a. Newtonian forces acting on a fish swimming. During undulatory and oscillatory motion the fish’s muscles contract and expand to produce ‘S’ wave-like undulations in their bodies. These waves allow the fish to exert a force through its body and tail with a positive AOA on the water, which pushes the water backwards. See Fig. 3a.

Fig. 3b. Momentum theory of a fish swimming. Combining equations (2) and (5) allows the forward force to be expressed as the change in momentum of the water:

According to Newtonian mechanics based on the mass flow rate, in both undulatory and oscillatory motion, the fish’s whole body and/or tail swims through a mass of water each second (m/dt), which it accelerates backwards to a velocity (dv). This action creates a backward force Force BACK = ma = m/dt x dv

(1)

Force BACK = Force FORWARD = d(mv)/dt

(6)

Or simply:

(7)

Units:

Force FORWARD = d(mv)/dt N

= (kg x m/s) /s

The forward side of the tail pushes the water backwards. Whereas a vacuum of low air pressure on the back-side of the tail pulls water backwards, helped by the Coanda effect.

For example, a larger tail has a greater momentum as it swings from side-to-side. Therefore it has a greater capacity to transfer momentum to the water to generate a force.

The inertia of the air provides resistance to the backward force. This dynamic allows for the generation of a reactive equal and opposite forward force:

C. Two Newtonian equations for swimming.

Force BACK = Force FORWARD

(2)

The analysis above provides two Newtonian methods and equations to calculate the forward force generated by a fish:

In short, the fish’s tail pushes water backwards, causing the fish to be propelled forwards. Equations (1) and (2) can be combined as follows:

Force FORWARD = ma = m/dt x dv

(mass flow rate)

(4)

(momentum theory)

(7)

Force BACK =

Force FORWARD = m/dt x dv

(3)

Force FORWARD = ma = d(mv)/dt

Or simply:

Force FORWARD = m/dt x dv

(4)

Both equations (4) and (7) are based on Newtons 2nd Law of Motion (Force = ma). Both are correct and produce the same values, but express the same thing slightly differently.

Units:

N

= kg/s x m/s

4

How$fish$swim$according$to$Newtonian$physics,$and$how$to$win$a$swimming$race.$

D. Undulatory v. oscillatory motion.

This means that:

According to Newtonian mechanics, the fish’s movement in the water affects the mass of water displaced each second (m/dt) and the velocity to which this water is accelerated (dv), which determines the forward force (Force = m/dt x dv). The Newtonian approach applies equally to undulatory and oscillatory motion, as well as the other fins (e.g. dorsal fins, pelvic fins, pectoral fins, …) and techniques adopted by fish to achieve forward motion by pushing water backwards.

-

A fish with a bigger tail passes though more water each second (higher m/dt). Therefore the caudal fin’s height, length, amplitude and AOA are important. The caudal fin’s impact can be assessed by its aspect ratio.

-

‘dv’ depends primarily on the tail AOA, speed, momentum, and beat frequency.

This is confirmed by research and data into dolphins, [63][64] which showed a close, positive linear relationship between:

Points relevant to both undulatory and oscillatory locomotion: -

-

‘m/dt’ and ‘dv’ are each presented as a single number in this analysis. But in practice ‘m/dt’ and ‘dv’ is hard to measure as they vary a lot across the fish’s body. In oscillatory motion, the tail moves more aggressively as compared to the total size of the fish used in undulatory motion. This is because the tail represents a smaller area that is used to generate motion and has less momentum.

-

Tail beat frequency and swimming speed. That is, fish swam faster the more frequently their tail swung from side-to-side.

-

Tail beat amplitude and swimming speed; particularly when amplitude and speeds was expressed as a function of body length, rather than meters.

An example calculation of the Newtonian forces acting on a dolphin swimming is provided in Section XIV on page 24. This calculation validates the Newtonian approach, demonstrates the importance of the fish’s tail, and provides the framework to solve Grays Paradox.

Undulatory motion In undulatory motion, size & sh...