Hydrodynamic Levitation

[jewish_scientist]

I watched this video on a stable system of a ball being held up by a stream of water. I do not like the explanation given. When the water hits the right side, most of the water goes to the upper right quadrant. The upper left quadrant gets the second most spray. Barely any water is left by the time we get to the lower left quadrant.

[gmalivuk]

Water in the upper right quadrant isn't being deflected in either direction on average, so is mostly irrelevant. Water in the upper left is going to the left so it pushes the ball right. That balances out the push to the left from the water stream itself.

The upward force is mostly applied by the water stream directly, so the amount of spray in the lower left doesn't seem important.

[jewish_scientist]

Water in the upper right quadrant isn't being deflected in either direction on average...

What do you mean by this? Water in the upper right quadrant is going up and right on average, meaning that it should apply a force toward the lower left quadrant. If you say that the vertical component of this force is negated by the water stream, then I would buy that; but to say that the upper and lower left quadrants negate the horizontal component is dubious. Because so much more water is going to the upper right quadrant than the left quadrants,I doubt that the horizontal components of their forces are significant in comparison. I am not saying that you are wrong; I am saying that I do not understand.

[speising]

I'd like to see how far he could angle the stream without loosing the ball.

[gmalivuk]

Water in the upper right quadrant isn't being deflected in either direction on average...

What do you mean by this? Water in the upper right quadrant is going up and right on average, meaning that it should apply a force toward the lower left quadrant. If you say that the vertical component of this force is negated by the water stream, then I would buy that; but to say that the upper and lower left quadrants negate the horizontal component is dubious. Because so much more water is going to the upper right quadrant than the left quadrants,I doubt that the horizontal components of their forces are significant in comparison. I am not saying that you are wrong; I am saying that I do not understand.

Spray in the upper right is going mostly up and a bit right. In the upper left it's mostly left and a bit up. In the lower left it's mostly down and a bit left. In the lower right it's mostly right and a bit down.

Up and down plus the stream and gravity balance based on the height (if the ball is higher the stream is weaker and it falls). Laterally, there's more water going slightly right and less water going directly left, which also balance because if the ball drifts too far right, spray in the upper right is deflected further right and the ball moves left, while if the ball drifts too far left then the upper right spray goes more straight up, which is no longer sufficient to balance the spray going left and the ball moves right..

[Eebster the Great]

Right, the total amount of water hitting the side of the ball in the stream pushing it away may indeed be greater than the amount of water spraying off the other side pushing it back into the stream, but the horizontal component of the velocity of the water on the outside is clearly much greater than the horizontal component of the velocity of the water on the inside.

[morriswalters]

My problem is, what process causes a force vector through the center of the ball? I've given it some time and end up with a weird conceptual picture. Mass of the ball is negligible with respect to the water. The net velocity vector for the water in the vertical axis is upwards. So what holds the water to the ball? Low pressure on the balls surface as compared to the waters surface. Made possible because the mass is low and the surface area high so the predominant energy source is water tension. I like it intuitively. But I wouldn't give odds.

[gmalivuk]

Water sticks to most surfaces, and you can't say the ball's mass is negligible because its weight is precisely the thing balancing out the upward force of the water.

Also, there's not a single force through the center of the ball, which is why the ball is spinning so fast.

[Eebster the Great]

Have you ever taken a physics class? What prevents the ball from accelerating is the balance of forces. The upward force due to the water is balanced by the downward force of gravity, so the ball's mass is clearly non-negligible. The outward force due to the fact that the water jet is off-center is balanced by the inward force due to the water spraying off the outside at high speed. This is a stable configuration, because if the ball moves slightly one way or the other, the forces become unbalanced, accelerating it back toward equilibrium. For instance, if the ball rises too high, the water pressure drops, making gravity dominate, accelerating it back down. If it drops too low, the water pressure increases, lifting it back up. If the ball moves too far outside the jet, the outward component of the force due to the water decreases, causing the inward force due to the spray to dominate, pushing it back in. If the ball moves too far into the jet, the outward force will dominate, pushing it back out. So as long as you don't disturb it too badly, the ball will continue to bob around the equilibrium point.

[Zamfir]

Spoiler:

I thought a picture might help.

There are three (groups of) forces acting on the ball.

1.the weight, downwards from its centre of gravity.

2. The absorption of momentum where the water hits the ball, resulting in an upwards force at a point offset from the middle, but less than a radius away from from the middle. This force is larger than the weight.

3.And last, the emission of water momentum at the edge, resulting in distributed forces pointing against the direction of rotation, all roughly offset from the c.g. by 1 radius.

There's three degrees of freedom in the picture. Vertical, horizontal and rotation.

In vertical z direction, the mechanism is clear. The vertical component of the incoming water is larger than that of the outgoing rays, and the difference is the weight of the ball. If the ball is perturbed downwards, then the incoming jet hits faster and more coherent. Thus, all the water forces increase but the weight does not, so here's a net force back up again.

Second, rotation. Let's look at torque around the c.g. The weigh by has no torque around t he c.g. The incoming jet produces a counterclockwise torque. The outgoing rays produce a clockwise torque. This latter torque increases as the ball rotates faster, and this brings stability around an equilibrium rotation speed. Faster rotation causes a torque that slows rotation, and vice versa. The equilibrium speed depends on the offset of the incoming jet - a larger offset requires faster rotation

Last, horizontal direction. This is the most complex one, and I couldn't predict that it works, without the video. At one position relative to the jet (and at one associated rotation speed) the leftward and rightward rays balance. If the ball is perturbed to the left, the rotation speeds up, more water is sprayed leftward, causing a rightward restoring force.

As a bonus: equilibrium out of the drawn plane (y-axis, and rotation around the x-axis). I am not sure on that one. The balls and disks are surprisingly stable in theze diirections too. They don't fall off the jet sideways, and the axis of the 'hanging disk' effect doesn't precess or wobble much either. The video guy talks about spinning tops, and he might be right. But tops do show precession in these timescales, and changing precession. Even though tops are heavier than a styrofoam ball.

[morriswalters]

Have you ever taken a physics class?

A couple.

Also, there's not a single force through the center of the ball, which is why the ball is spinning so fast.

I understand that. There is a moment arm caused by the force of the water. I'm still trying to understand how that gets translated to a force vector equal in magnitude and opposite in direction to gravity.

In vertical z direction, the mechanism is clear. The vertical component of the incoming water is larger than that of the outgoing rays, and the difference is the weight of the ball. If the ball is perturbed downwards, then the incoming jet hits faster and more coherent. Thus, all the water forces increase but the weight does not, so here's a net force back up again.

Yes. Maybe you can clear it up. What physical principle says that the rays must balance the forces.

[gmalivuk]

In most configurations, they don't balance, but obviously they do in one, because the thing observably works, and Zamfir and I both explained why that equilibrium is stable (at least, why it's stable in the directions and axes we discussed, he one more than me).

[Xanthir]

They're also all using styrofoam, which water adheres to pretty readily. I've done enough screwing around with water and plastic balls in my youth that I don't think it works nearly as well for standard smooth toy plastic; the water wouldn't be carried around the ball nearly as much and would mostly just fling upwards, thus not providing enough counterforce inwards to keep the ball in the stream.

[ucim]

Consider a single glob of water that hits the styrofoam ball at its edge. It sticks to the ball for a while, and then lets go. In this time, the ball rotates some amount, taking that glob with it. (The impact of the glob imparts some of this rotation energy, but that's not important right now). Swinging that glob around changes its velocity - moving it to the left (assuming CCW spinning), but conservation of momentum means that the ball needs to move to the right (further into the water jet) to compensate, so the center of mass stays in the same place. Then the glob is released. The release of the glob does nothing but end this dance. The outgoing spray does nothing - the force has already been applied to the styrofoam ball by the swinging-around of the temporarily sticking water.

Think of a dance, where a line of boys run (from the south) towards one side of a spinning girl, who grabs a boy and whirls him to the west before letting him go. This brings the girl closer to the center of the stream of boys. But the next boy needs to be swung out (along the path of the girl's arms) before she can swing him back behind her to the west to let him go. This pushes the girl away from the flow, before pulling her back into it. So there's less net force from that boy. Thus the girl swings all the boys to the west, maintaining her position.

Jose

[morriswalters]

In most configurations, they don't balance, but obviously they do in one, because the thing observably works, and Zamfir and I both explained why that equilibrium is stable (at least, why it's stable in the directions and axes we discussed, he one more than me).

Yeah, I've watched the video a few times myself. So I know it works, but I wanted to know why. Something somewhere is converting rotational energy to a positive vector along the axis formed by the center of gravity. I think the key is low mass and high surface area, but I'm not going to try and defend that thought.

@ucim
Thanks, I never thought about linear or angular momentum in quite that fashion.

@Eebster the Great
On further thought. Statics, Dynamics, Electromagnetism, Thermodynamics, Materials Science, and Astronomy, in there somewhere my academic train derailed.

[Zamfir]

Yes. Maybe you can clear it up. What physical principle says that the rays must balance the forces.

Regarding the vertical direction only: keep in mind that the water jet from the hose is moving against gravity. The water slows down as it gets higher, until it falls down again.

So at lower height, the water jet is too strong. It accelerates the ball upwards, when the ball gets too low. At higher height, the jet is too weak to keep the ball up. There is a sweet spot in between. The ball moves to that spot, then bobs around it.

The same applies to other directions. The rays are not balanced in every combination of position and rotating speed. They are only balanced in one such situation. In every other situation, there is a net force towards that equilibrium situation.

[Soupspoon]

Something somewhere is converting rotational energy to a positive vector along the axis formed by the center of gravity.

Remember that something is also converting a different linear vector into rotational energy. Given the conversion from an arbitrary linear force into rotation, a reconversion of rotational force to our more contested linear one is actually obvious. And there thus may well exist (for certain conditions) particular pairings of linear vectors that produce a stable (or metastable) configuration.

The calculation of what incident and resultant forces might even so levitate a cannonball is going to be difficult (perhaps the degree of hydrophilia of its surface needs to be dialled up so high as to be impossible as we seem to expect it to be with a more hydrophobic coating on the existing polystyrene sphere), but maybe possible. And at the limits of possibility the metastability of a given situation might be close enough to its tipping maxima to 'bounce' over and away from this particularly privileged point of inflection through undampened oscillations.

If you see what I mean.

[morriswalters]

Water sticks to most surfaces, and you can't say the ball's mass is negligible because its weight is precisely the thing balancing out the upward force of the water.

Negligible in the sense that the ball doesn't have much mass weight compared to the forces acting on it. My apologies this was incorrect.

If you see what I mean.

Yeah I see what you mean. Them little tiny rays can't support a cannonball. But you answered my question indirectly.

perhaps the degree of hydrophilia of its surface needs to be dialled up so high as to be impossible as we seem to expect it to be with a more hydrophobic coating on the existing polystyrene sphere

So what holds the water to the ball?

[ucim]

So what holds the water to the ball?

Hydrogen bombs. I mean... bonds.

James Bonds.

(Sorry)

Jose

[morriswalters]

I wouldn't know, I passed Chemistry by fortune and a good roll of the RNG.

[Eebster the Great]

Water sticks to most surfaces, and you can't say the ball's mass is negligible because its weight is precisely the thing balancing out the upward force of the water.

Negligible in the sense that the ball doesn't have much mass compared to the forces acting on it.

The weight of the ball balances the upward force. It cannot be negligible compared to the force acting on it if it is equal to that force. Moreover, mass is not the same as weight and cannot be compared to a force. If there were an unbalanced force on the ball, it would accelerate in inverse proportion to its mass. So mass can't be negligible.

[morriswalters]

I'm using it incorrectly then. In any case Soupspoon answered my question.

Moreover, mass is not the same as weight and cannot be compared to a force.

Ouch! Mea culpas. I retract it.

The weight of the ball balances the upward force.

Yes. I thank my floor for holding me up every day.
edit

A spinning ping pong ball is held in a diagonal stream of air by the Coandă effect. The ball "sticks" to the lower side of the air stream, which stops the ball from falling down. The jet as a whole keeps the ball some distance from the jet exhaust, and gravity prevents it from being blown away.

[ucim]

So what holds the water to the ball?

In ordinary cases (like a ping pong ball), surface tension is what does it. Molecules are attracted to each other by various forces whose ultimate origin is the relative positioning of electrons, making parts of molecules more positively charged and other parts more negatively charged. Liquids tend to cling together more easily than they cling to gasses (such as air); this is what causes water droplets to form. Sometimes they are attracted to solids more, sometimes less, depends on the liquid and the solid. This is what leads to the meniscus and to capillary action.

But for the styrofoam ball, I think there's something else at play - all the nooks and crannies trap water that hits it, at least for a while... long enough to make part of a revolution. To test this theory, try it with a rough styrofoam ball and a smooth styrofoam ball, and see if there's a difference in the behavior.

Jose

[morriswalters]

Thanks, but the Coandă effect explains it sufficiently for me. The effect will actually work on glass. You can see it by holding a round water glass in a stream. Here is a link to a paper out of Fermilab, a pdf file by the way. The paper is amusing. I gather it is a couple of very well educated guys making an argument about how we should talk about lift. The Mathematician is a pilot evidently. He attributes the effect to viscosity.

I'll be honest with you, this whole conversation has been a mystery to me. It certainly didn't help with my question. Do you ask questions about things you understand? I found the answer in the comments section of the YouTube video. Sheer luck. Zanfir's model of the process seems cogent enough. He never said anything about how water got grippy fingers and grabbed a Styrofoam ball though. I also think that people have a different take on the rooster tails then I do.

[Zamfir]

Thanks, but the Coandă effect explains it sufficiently for me. The effect will actually work on glass. You can see it by holding a round water glass in a stream.

Neither of those are the Coanda effect, though. These are surface tension effects, something unrelated.

The Coanda effect is about turbulent jets of air in air (or jets of water in water). A water jet in air will have a surrounding air-jet, and that jet might see the Coanda effect, but I am fairly sure that it is not strong enough to have much effect on the (heavy!) water jet.

Note that the ball has to spin, in order to bend the water - that's not needed for Coanda flow.

[morriswalters]

I actually suggested surface tension as the driver in my first post. Whatever effect is causing it it needs to adhere well enough to carry the forces exerted by the water. The jet from below isn't lifting anything directly. That jet can translate the ball and does rotate it, but it can't lift it against gravity. No vector which the stream can produce ever passes through the center of mass. Given no other opposing force the ball would fall. Even if the stream was directed directly upward through the center of mass it wouldn't be stable. Sooner or later the ball would move away from the stream. The video guy actually demonstrates this.

The force holding up the ball has to be the resultant of all the vertical forces on the upper hemisphere. The restoring force has to be the sum of all the horizontal forces meaning that it must always point to the stream. And both of those need to act through the center of mass. This is the condition that makes the ball stable. Is this controversial?

My current question would be what the rooster tails represent? You could model this as a weight on a string. That may be simple minded on my part, but the direction of the bulk of that spray would then be indicative of the tangential motion introduced when the water droplets separate from the surface. Indicating that the bulk of the force is in the right upper quadrant, directed up and towards the stream.

Note that the ball has to spin, in order to bend the water - that's not needed for Coanda flow.

What? No. Call the effect whatever you like, but water will bend around the ball even if it doesn't rotate. The glass in my sink indicates that.

I need to see this better, so I'm off to the hardware store.

[Eebster the Great]

There is no reason a force must or ought to be applied directly to the center of mass of anything. All that matters are the sums of forces and moments.

[Soupspoon]

Hold a tray of drinks by its rim. You support it against gravity. The grip you give it represents no single force that passes through your hand, vertically upwards through the CoG of the tray, as it might if you supported it by an underslung hand.

The water in the jet does something akin to the grip of the overslung fingers that provides a counter-couple to the off-centre lift beneath the tray, to shift the combined lifting force out into 'virtual' space beyond your actual hand. Only dynamic, rather than (usually) static...

[ucim]

My current question would be what the rooster tails represent? You could model this as a weight on a string. That may be simple minded on my part, but the direction of the bulk of that spray would then be indicative of the tangential motion introduced when the water droplets separate from the surface.

What does a rocket's exhaust push against? The answer is subtle for those who have not thought it through - the exhaust pushes against the rocket. The subtlety is that at first sight people think of the exhaust as part of the rocket (it looks like it's attached), but it's not. Once the exhaust leaves the nozzle, it's no longer part of the system.

Ditto here. The rooster tails are no longer part of the system. Like rocket exhaust, they carry away momentum, and require an equal and opposite momentum to be imparted to the rest of the system (the ball and its attached water), but that momentum was transferred by the rotation of the ball changing the velocity of the (then) attached water that just became a rooster tail.

The net force on the ball is the vector sum of the upwards stream of water and the negative of the rooster tails.

Jose

[morriswalters]

It's a condition of stability. Any force that doesn't pass through the center creates a moment. Zamfir uses torque, I was taught in term of both torque and moment arms. In the case of a sphere free in space look at the example of a ball on a string. You can create transient moment but it's self damping. The ball comes to a rest state with the axis represented by the string passing through the center of mass or gravity. This is the reason that plumb bobs work. Forces from beneath are effectively hinged at the point of contact.

As a bonus: equilibrium out of the drawn plane (y-axis, and rotation around the x-axis). I am not sure on that one. The balls and disks are surprisingly stable in theze diirections too. They don't fall off the jet sideways, and the axis of the 'hanging disk' effect doesn't precess or wobble much either. The video guy talks about spinning tops, and he might be right. But tops do show precession in these timescales, and changing precession. Even though tops are heavier than a styrofoam ball.

I went back to this so I could use this line.

The tail doesn't wag the dog.

The water would show precession around its axis, not the balls, if it applies in this situation at all. For a disc you have an additional element. At its face the disc moves closer to the stream if it rotates in either direction since it's a rectangle in the plane.

@ucim
Yes I know. Or at least I think I do. That force is normal to the center of mass. My pitiful physics tell me that the drops are moving along an axis perpendicular to that normal.

I have to ask, does anyone ever feel a moment of terror when they press the post button?

[ucim]

Yes I know. Or at least I think I do. That force is normal to the center of mass.

Yes, the net force is... well, "normal" isn't the right word. (Normal means perpendicular, and the center of mass is a point). The net force acts through the center of mass, mostly. There's still a net torque that keeps the ball spinning (against air friction) but it's tiny.

My pitiful physics tell me that the drops are moving along an axis perpendicular to that normal.

None of the drops do, but the net force represented by all the drops together does... mostly (as above).

Follow one glob of water from impact to release. What happens? For simplicity, consider the blob that impacts the edge, sticks for 90 degrees of rotation, and exits horizontally.

I have to ask, does anyone ever feel a moment of terror when they press the post button?

Only on second thought.

Jose

[Eebster the Great]

Consider a table with four legs. Each of the legs supports the weight of the table with an upward force near the corner. None of them come anywhere close to the center of mass. Therefore all of them produce a torque ("moment of force" is a synonym for "torque"). However, the sum of the torques is zero, and therefore there is no net torque. That's why the table has zero rotational acceleration.

[Soupspoon]

Unless it is a lazy susan and you have a curious cat....

[morriswalters]

Consider a table with four legs. Each of the legs supports the weight of the table with an upward force near the corner. None of them come anywhere close to the center of mass. Therefore all of them produce a torque ("moment of force" is a synonym for "torque"). However, the sum of the torques is zero, and therefore there is no net torque. That's why the table has zero rotational acceleration.

Ok, And? With all due respect I have been told this multiple times. Balanced forces. It's a product of symmetry in the horizontal plane. But I'm looking for the glue which lets the water lift the ball. We could go on. But I don't see that as being in either of our best interests. Given the imperfections of my knowledge about physics I think I'll let it lay.

@ucim
Put a plane tangent to the surface, the vector is normal to that plane. Symmetry puts the line through the center. I would accept any alternate way of expressing it.

What follows in spoilers is me handing out intuition, guesses, and idle speculation about things that I don't have a clear understanding of.

Spoiler:

I can't do that, I don't know that I understand it well enough. I can describe what I see in the video. And why I think it happens. But I don't have any confidence that this is correct in any respect.
As the water rotates the forces of rotation accelerate the water away radially. As it moves away from the center conservation of momentum tries to slow down the rotation to make up for the mass moving away from the center while at the same time accelerating the mass. Constant energy being added keep the rotation steady. The water accelerates up and away. The water starts to neck as surface tension reaches its limit. The mass continues to accelerate until surface tension is overwhelmed and the stream breaks. Repeating as it rotates until there is no water left with the energy to break the surface tension. The rooster tails are a marker for how much mass is lost and when. They also an indicator of how the radial force is distributed around the ball. My guess would be that whatever holds the water to the surface of the ball exerts a stronger force than surface tension. I would also speculate the rotation will be constant assuming that surface tension is the predominant force.

And with that I'm out. Thanks everyone.

[ucim]

Put a plane tangent to the surface, the vector is normal to that plane. Symmetry puts the line through the center. I would accept any alternate way of expressing it.

"Normal to the ball's surface, therefore passing through the ball's center."

As to following a blob going around, here's my version:

Spoiler:

Setup: The ball's already in play. A blob of water hits a rotating styrofoam ball at the edge of the ball. Surface tension plus the nooks and crannies serve to capture this water. The blob has the same velocity of the edge of the ball (previous blobs have spun the ball up to this rate, so the stream's impact does not impart any upwards force upon "impact"). At this point the "system" consists of the ball plus the blob, treated as a unit.

As the blob is carried around by the ball, centrifugal forces try to pull the blob away from the ball. They eventually succeed; in this case after 90 degrees of rotation the blob separates and is now traveling at the same speed, but horizontally, tracing out its part of the rooster tail. All of the vertical momentum of the blob must therefore have been transferred to the ball; this is what keeps the ball up. The blob however has acquired an equal amount of sideways momentum, since the speed is the same and this particular blob lasted 90 degrees. It was imparted by the spinning of the ball; conservation of momentum requires that the ball acquire an equal and opposite momentum - this is what pushes the ball further into the stream. Another way of looking at the process is to see the system rotating around the center of mass, which is not the center of the ball, but partway towards the edge that has the blob. The offset rotation pushes the ball into the stream. The center of mass is also rising (although the ball is not), because the blob is rising, at least at first.

Now, for any blob we are following, I've simplified the "rest of the system" to be just the styrofoam ball. In reality, it's that plus all the other blobs that are attached. They let go at different times, not all at (the example) 90 degrees. Nonetheless, the overall idea (which is clearer in the simplified scenario) is still correct.

Hope this helps. No need to disappear if you're still puzzled. We're patient.

Jose

[morriswalters]

I watched this video on a stable system of a ball being held up by a stream of water. I do not like the explanation given. When the water hits the right side, most of the water goes to the upper right quadrant. The upper left quadrant gets the second most spray. Barely any water is left by the time we get to the lower left quadrant.

I've done a paper on this since you posted. Most of it can be solved through geometry and a free body diagram. You have seen something similar to this if you've been to a playground, it's called a teeter totter or a see saw. What is more difficult to see is the lever that connects the stream to the ball and the pivot point. To begin with think of the stream as a special case of a pipe. Where the walls are formed by water tension.

I'll make a statement without supporting it. The maximum size of the Styrofoam ball is the is equivalent to the maximum size of a ball of the same mass that could be levitated by the Bernoulli Effect in the quarter inch stream that is lifting the ball. I offer this with a data point, the density of polystyrene and water are approximately equal.

I then offer that this graphic describes the water flow over the top of the ball.And that the top plate is represented by the interaction of water and air. If it helps, consider a paint roller leaving an area of paint behind. The center of flow carries the flow to the front and deposits it. In the case of a paint roller, the paint is in the brush. In the case of the top of the ball, the moving water is between the surface and and the fixed plate.

Now consider a ceiling with a leak. If you pay attention you will see a pattern in how leaks propagate across a ceiling. Very small leaks will wet the surface, larger leaks will spread and drip. If you put the equivalent amount of water on a floor, made like the ceiling, you find that the difference is in the energy required to hold the water on the ceiling against gravity. So you can increase surface area, decrease mass, or both. So you can model the forces around the ball by drilling holes along one side of a pipe along its length, with the condition that the water doesn't overflow the top.(Torricelli's law)

Next I will describe the stable point. It is the point the from the edge of the ball to a plumb line drawn though a point 45 degrees from the horizontal downwards. that is the neutral position, the angle will vary depending up the forces acting on the center of mass. The point is a local maxima for the horizontal forces. The vertical forces act in a different frame of reference. It's uphill to that maxima from the edge and downhill from the maxima to the origin at the center of the ball. Between the ball and the origin the ball will accelerate and leave the point of stability. The only way to get in to that point of stability is to be placed there.

You can build a toy to illustrate this. Anchor two pulleys in a horizontal line. Run a cable between them. Let a third pulley represent the ball, place it between the other two. To make your life simple keep the center of gravity below the third pulley. Attach weights to each end equal to half the weight of the center pulley. If you did this the way I did, you will see that the center will always seek to go to that center, no matter how you displace it as long as the center pulley never crosses the top. This is the zone of stability. Everywhere else be Dragons.

Two further data points. A sphere is symmetric in all three axis. Assuming that the its mass is evenly distributed its center of mass and gravity are the same. This is true for a disk in only in the plane it exists in. In yaw the center of mass and gravity diverge and the disk will wobble and fall. In the roll axis the ball doesn't doesn't exist except in the plane.

Now I'll connect the dots. The ball is lifted by a combination of the the up force at the bottom and the forces at the top of the ball as described. This will work only as long as gravity can counter balance any extra forces at the top of the ball, in the horizontal or the vertical direction. It would be an interesting experiment to see how fast you can approach the stream without overshooting.

edited to add a link