I Fun question: Can this fish swim?

[DaveC426913]

You are mixing up two different, but related, topics. What makes something compressible or not is that a compressible material's normal deformation is proportional to its normal stress. What makes a fluid flow is that its shear deformation rate is proportional to its shear stress.

So the difference between a compressible gas and an incompressible liquid is in its behavior under normal stress. They behave the same under shear stress. So a fish in a sealed container can swim just fine, by applying shear stress to the water, as always.

This is good. I have not been acquainted with 'shear' stress and 'normal' stress, but now that you're described it, I can very easily understand why water can 'slide" even while being incompressible. They're two different properties.

I had been thinking of these tile puzzles:

They slide past each other easily enough but they will not compress - one tile always takes up the same area (volume).

 

[boneh3ad]

If you had a completely incompressible and inviscid flow, then I think you could make the case that a fish could not actually swim if the tank was small enough because the water would just recirculate and it would be the aquatic equivalent of a treadmill. In the real world, where viscosity will dissipate some of that energy and water actually is minutely compressible, I think a fish could swim just fine, although it would certainly get harder in the small tank because there would definitely still be some recirculation.

 

[Chestermiller]

If you had a completely incompressible and inviscid flow, then I think you could make the case that a fish could not actually swim if the tank was small enough because the water would just recirculate and it would be the aquatic equivalent of a treadmill. In the real world, where viscosity will dissipate some of that energy and water actually is minutely compressible, I think a fish could swim just fine, although it would certainly get harder in the small tank because there would definitely still be some recirculation.

What if the tank were very large in the incompressible inviscid case?

 

[DaveC426913]

If you had a completely incompressible and inviscid flow, then I think you could make the case that a fish could not actually swim if the tank was small enough because the water would just recirculate and it would be the aquatic equivalent of a treadmill.

No. The water would go to one end of the tank as the fish went to the other end - even if that end is only a few centimeters in front of its nose.

There may be some loss of efficiency due to recirculaton, but it would not prevent the fish from reaching the wall.

 

[boneh3ad]

No. The water would go to one end of the tank as the fish went to the other end - even if that end is only a few centimeters in front of its nose.

There may be some loss of efficiency due to recirculaton, but it would not prevent the fish from reaching the wall.

I am not actually sure what you are talking about, but if the flow is inviscid and incompressible (as I discussed in the statement you quoted), then there would be no such "loss of efficiency". There is no mechanism for dissipation.

The water wouldn't just go to one end of the tank either. After all, it can't go back there and just pile up; it has to go somewhere. It would be propelled backward off of the fish, it the back wall, and turn outward (stagnation point flow), then do the same when it reached the other boundaries until it came back around to the front and touched the fish. If the medium was truly incompressible (i.e. the speed of sound is infinite), then this would happen instantaneously and there is a decent chance the fish doesn't move since its potential forward velocity would be evenly matched by the oncoming flow.

The question in my mind is whether or not this is the case if you assume it starts from rest and has to accelerate to its steady speed. In that case, it's possible the fish could move before the recirculation fully sets up, but I haven't come to my own mental consensus on this yet.

What if the tank were very large in the incompressible inviscid case?

That's an interesting dilemma as far as I can tell. I am not sure that such a recirculation would necessarily be set up for an arbitrarily sized tank, so perhaps once you reach a certain size, you start instead developing vortices that form just along the back wall and the fish does swim forward. I would guess that this would be the case because certainly as the size of the tank goes to infinity, the fish can swim forward since there is no wall there to cause recirculation. it seems plausible to me, then, that for a sufficiently large tank, the actual physical situation will begin to mimic the infinite case.

 

[DaveC426913]

I am not actually sure what you are talking about, but if the flow is inviscid and incompressible (as I discussed in the statement you quoted), then there would be no such "loss of efficiency". There is no mechanism for dissipation.

I mean efficiency in the fish's movement forward. I might be slightly more difficult for the fish to move in a very small volume of water as opposed to an unbounded volume, but it will certainly be able to move.

The water wouldn't just go to one end of the tank either. After all, it can't go back there and just pile up; it has to go somewhere.

Yes. It fills the space vacated by the fish.

A one "fish-unit" volume of fish moves to one end of the tank, while a one" fish-unit" volume of water moves to the other.

 

[boneh3ad]

I mean efficiency in the fish's movement forward. I might be slightly more difficult for the fish to move in a very small volume of water as opposed to an unbounded volume, but it will certainly be able to move.Yes. It fills the space vacated by the fish.

A one "fish-unit" volume of fish moves to one end of the tank, while a one" fish-unit" volume of water moves to the other.

Except it doesn't. A fish moving doesn't just move the water behind it. It moves all of the water in front of an around it as well, so the actual "vacancy" could just as easily be argued to be out in front of the fish. Of course no such vacancy exists because this is a continuous medium, and since it is incompressible, the effects of water moving in one location are immediately felt at every other location in the fluid.

 

[DaveC426913]

Except it doesn't. A fish moving doesn't just move the water behind it. It moves all of the water in front of an around it as well, so the actual "vacancy" could just as easily be argued to be out in front of the fish.

The vacancy cannot be in front of the fish; it is the fish.

Again: A one "fish-unit" volume of fish moves to one end of the tank, while a one" fish-unit" volume of water moves to the other.

 

[boneh3ad]

The vacancy cannot be in front of the fish; it is the fish.

Again: A one "fish-unit" volume of fish moves to one end of the tank, while a one" fish-unit" volume of water moves to the other.

And that nut of fish move, and water wills in, and then the water behind that fills in, and eventually there's just a wall. So the water above and below along the wall fills that space in. Then the water along the top and bottom fills that space in. Then along the front wall. The the water in front of the fish. Now your vacancy is in front of the fish.

Of course that's not really how it all works, as it must START with the assumption that the fish moves, which is not necessarily true here. It doesn't prove that assumption at all.

Instead, the fish could flap its tail and simply propel water backward, setting up the aforementioned recirculating, and not going anywhere since that thrust would move him exactly as fast as the oncoming water stream.

 

[jbriggs444]

Instead, the fish could flap its tail and simply propel water backward, setting up the aforementioned recirculating, and not going anywhere since that thrust would move him exactly as fast as the oncoming water stream.

Newton's third law still applies, surely. If the water is propelled backward, the fish moves forward. And not just relative to the water.

 

[Chestermiller]

See this reference regarding the relative importance of viscous effects vs inertial effects in fish propulsion:

https://books.google.com/books?id=w...onepage&q=inviscid propulsion of fish&f=false

 

[Chestermiller]

Now that we have dispensed with the issue of viscous vs inertial (see my previous post), we can focus on the main issues for the inviscid case. In my judgment, there are two fluid dynamic features that need to be considered:
1. Maintaining steady movement of the fish after it has propelled itself to a certain (constant) speed
2. Propulsion (acceleration) action of the fish to generate forward thrust

Maintaining steady constant speed movement of the fish has been addressed in my post #22, where the steady inviscid solution for flow past a sphere is presented. This flow will prevail as long as the fish (sphere) is not close to the leading or trailing faces of the tank. It also illustrates DaveC436913's concept of the streamlines separating at the leading edge of the fish (to make room for the fish) and rejoining beyond the trailing edge of the fish. Note also that, for the case of inviscid flow, the drag force on the fish is zero. So, in real life, only a small forward force is required to maintain the speed against the (small) viscous drag.

So, now, the only thing left to address is generation of forward thrust. For this, imagine that the fish is held in place by an externally applied force, but that the fish is doing whatever is necessary to try to paddle forward. The question is "does the circulation flow created by this paddling create a pressure distribution on the surface of the fish (and fins) that results in (a) net forward thrust or (b) total cancellation of forward thrust?" The situation here is different from the passive case where the fish is just moving forward without any thrust. In this case, the fish itself is adding energy to the system by doing work on the water. For a very large tank, it appears to me that the backward flow in close proximity to the fish is very strong, but the return circulation from the front of the fish will be distributed over a much larger cross section of the tank, and the force of that return flow on the fish will be much less. So, I would conclude that positive forward thrust would definitely be achievable.

 

[boneh3ad]

Newton's third law still applies, surely. If the water is propelled backward, the fish moves forward. And not just relative to the water.

That is certainly a good point, and I think it does a fine job of illustrating why I need to get more sleep.