The mystery of the tangent and the radius fluid in spiral pipe on rotating disk

[fezzik]

Here's something simple but also a bit puzzling, let me know if you have any ideas... For clarity, I'll describe three cases before asking the final question (skip ahead if you like).

1. Mounted on a disk are two curved fences, just a few centimetres apart from each other. Between the fences is a narrow path curved in the shape of a spiral that starts at the centre of the disk and heads to its circumference. The disk rotates at a constant angular velocity while a frictionless puck slides on the disk. The challenge is to slide the puck from the centre of the disk to its circumference, between the fences, without the puck touching either of the fences. Luckily, the spiral is an Archimedean spiral, so the puck, traveling at a constant "absolute" linear velocity, starting from the centre of the disk and traveling outward to its circumference, can avoid touching the fences. That is, the "absolute" path of the puck, i.e., its path as viewed by a stationary observer, is a straight line, and its absolute velocity is also constant, while the path the puck traces out on the disk is an Archimedean spiral, which happens to be the same curvature as the fences.

2. Mounted on a similar disk is a curved pipe that follows the same spiral path that was described by the puck in the case described above. The diameter along the frictionless pipe is such that fluid flowing along a section of pipe will always have the correct speed so an element of that fluid will travel in a straight line (according to a stationary observer) from the centre of the disk to its circumference. So, the fluid element is not "pushed sideways" onto the walls of the pipe.

3. Back to the puck: the puck still travels in a straight line absolute path, but halfway out from the centre of the disk, it suddenly accelerates to infinite velocity. Luckily, the fences are redesigned so, halfway from the centre, the fences begin to follow a straight path pointing radially outward. So, the puck can still travel in a straight line without touching either of the fences.

4. And finally, back to the pipe. Halfway from the centre, the fluid enters a second section of pipe which has a diameter so small that the velocity of the fluid becomes infinite (let's just pretend). In what direction should this second section of pipe be pointed so an element of fluid flowing down the pipe is never "pushed sideways" against a pipe wall? (I.e., what direction should the second section be pointed in so the fluid, at all points between the centre of the disk and its circumference, including at the transition to the second section of pipe, experiences zero total acceleration normal to its direction of flow?)

Should the second section of pipe be aligned (a) tangentially to the curvature of the spiral of the pipe (i.e., tangential to the curve of the first section of pipe), or (b) radially?

The path taken by a fluid element in case (a) is not a straight line, but two lines at an angle to each other. Anyway, (a) seems to me the right answer, but I don't have a good explanation yet for why it wouldn't be (b), as example 3 above might suggest. (It has to do with the fluid element's direction changing as it flows down the pipe, I guess...)

Thanks for any feedback...

 

[UndeniablyRex]

If I understand this correctly, it's either neither or both. When the fluid is accelerated, it's going to put a force on something in the opposite direction, this is Newtons laws. The only thing to push against is a wall, and therefore will always have a normal component.

There is one exception I can think of. That is if the disk is spinning very fast, and there was a pocket of air between the transition section(space between section 1 and 2) and the liquid currently exiting this device. That is to say that the liquid at the "nozzle" is unable to push on the transition section because there's nothing tangible connecting the two.
If the disk is spinning fast and you have a drop of liquid at the very skinny nozzle, the liquid at the exit point would be forced out by all the liquid, the drop, behind it because of the centripetal force. This would give answer (b).

 

[fezzik]

Thanks for the feedback... It's true, when it accelerates, the fluid element does have to push against something. However, unless the fluid is accelerating while rounding an actual bend, this doesn't necessarily need to include the sidewalls of the pipe -- it could be the fluid behind it. E.g., consider a perfectly straight pipe that becomes narrower at a certain point -- the fluid accelerates, but the total force normal to the flow adds up to zero.

So, I can describe the spiral pipe situation another way -- while a person in a rotating reference frame would say a fluid element traveling outward is taking a curving, spiral path, a person in an inertial reference frame knows better -- the fluid is actually taking a straight path during the first (spiral) section of pipe. However, while the path of the fluid element is radial, the direction of the streamline of the fluid element has to be parallel to the pipe, which means that the streamline is not radial; rather, it is tangential to the curve of the spiral. So, while the fluid element is traveling radially, you could say it's not "pointed" in a radial direction -- it actually points further and further away from radial as it travels along the Archimedean spiral further away from the centre of the disk.

This is in contrast to the simpler story of the puck I described -- it has a straight path radially outward, and, it doesn't "turn" around its own axis to align itself with the curvature of the spiral like the fluid element does. That is, while the fluid element points in a direction tangential to the spiral, if you drew a line on the puck, it would always face in the same direction -- the puck doesn't follow a streamline along the spiral, it simply slides straight out radially. (Well, you could impart a spin to the puck, but that's not necessary.)

Does that clarify/mystify things a bit?

 

[Cleonis]

[..]
what direction should the second section be pointed in so the fluid, at all points between the centre of the disk and its circumference, including at the transition to the second section of pipe, experiences zero total acceleration normal to its direction of flow?

The key here is the transition from small velocity to extreme velocity. (Let's say a jump to a velocity so high that compared to it the tangential velocity of the disk is negligable.)

As the example of the puck demonstrates:
- Initially the puck has a modest radial velocity, small in comparison to the tangential velocity of the disk. Under those circumnstances the motion of the puck with respect to the disk is tangential to the disk.
- then the jump to a velocity that makes the disk's angular velocity insignificant. Then the motion of the puck with respect to the disk is radial.

So yes, for the motion with respect to the disk the jump in velocity makes for a transition from moving tangentially (along the archimedean spiral), to moving close to radially.

 

[UndeniablyRex]

it could be the fluid behind it. E.g., consider a perfectly straight pipe that becomes narrower at a certain point -- the fluid accelerates, but the total force normal to the flow adds up to zero.

This is mostly true. Say you have a pipe with a small aperture at one end(the nozzle) and the other end is the source of incoming water. I would argue that the accelerating force behind the exiting jet of water is the pressure created by the source of incoming water on the opposite site.
To relate this to your problem, if the fluid is continuous, no matter how you point the second section, the accelerating force will always be the pressure inside the spiral. Since liquid is exiting one side, it means that the pressure on that wall will be less than the pressure on the opposite wall. This tells me that there is no good answer to your problem.

I think in order to answer this question we need to know why it is accelerating.


There is another way to approach it that may work. If the second section is pointed tangentially, it will put a torque on the disk, and the only place to put a force on the disk is one of the walls.
If the second section is pointed tangentially, then there is no net torque and therefore the difference in force between two opposite sections of wall is zero. That is to say that the pressure on one side of the spiral is equal to the pressure of the section of wall directly opposite.

 

[zeebek]

could anyone make a sketch? really hard to imagine what is being asked

 

[UndeniablyRex]

I imagine it to be something like this:http://www.malcams.com/admin/assetmanager/images/maldoc2a.gif

where there is a groove on a disk, and the disk spins. The fluid or puck is traveling in the groove.
In the photo, the groove is continuous, but in this discussion the groove starts at the center and gets bigger and bigger.

 

[fezzik]

Thanks for the posts everyone, I have to read them carefully, but in the meantime I've attached a sketch and another description here:

The fluid travels along the spiral pipe, which is fixed to the disk, which rotates at constant angular velocity in a clockwise direction. An observer in the rotating reference frame sees the fluid element moving in a spiral path from A to B along the spiral pipe, while an observer in the inertial frame sees the fluid element moving from A to B in a straight line along the y-axis. As the observer in the inertial frame can see, since the fluid element moves at constant speed in a straight line, it experiences no acceleration, and in particular, zero total acceleration in a direction normal to its direction of movement. I.e., it experiences zero tangential and zero radial acceleration, its radial velocity is constant, and its (absolute) tangential velocity is always zero. (Its relative tangential velocity, i.e. that seen by the rotating observer, is increasing.)

When the fluid element reaches B, the local tangential direction of the spiral pipe is described by the orange line BC, while its path, as seen by the observer in the inertial frame, is still following along the y-axis. It enters a second section of pipe and accelerates instantly to infinity (if this abstraction is confusing, perhaps I should just say it instantly accelerates to a very high velocity along the pipe). In order for the fluid element to experience no acceleration normal to its direction of motion anywhere along its path from A to the outer edge of the disk (in particular, including at point B), should the second section of pipe follow BC or BD? (Or, if we consider the case of a very high velocity rather than infinite, should the second section of pipe be closer to BC or BD?)

That is, the fluid element should not be "pushed up" against one of the sidewalls of the pipe more than the other -- the fluid should experience no centripetal or Coriolis acceleration, the only acceleration it experiences should be parallel to the direction of the pipe. (Note two things: at B, the motion of the fluid element (as seen by the stationary observer) is along the y-axis -- it's all radial, there is no tangential component -- while the direction of the fluid streamline is not radial, it's along the tangent described by BC.)

 

[fezzik]

(Below, I've added one more case that may be easier to analyze -- skip ahead if you like.)

Having thought about it a bit more, it does seem like the second section of pipe should follow the radial line BD (referring to the sketch I attached above), but I'm still on the fence, so to speak. Consider what happens when you give the fluid element a higher radial velocity between A and B -- it inscribes a "wider" Archimedean spiral, i.e. a spiral with larger distances between the turnings. And when the velocity becomes very high, the spiral becomes so wide as to approach the radial direction. Also, obviously, if the fluid element travels in a straight line from A to B, then unless it is subjected to a tangential acceleration at B, it will continue to travel in a straight line from B to D.

So it seems intuitive that, if the fluid element travels at a certain radial velocity between A and B, and then travels at a very high radial velocity between B and the edge of the disk, that it should take a very wide spiral path approaching the radial line BD. (When the velocity is infinite, the spiral would be "infinitely wide" -- i.e., it would be the line BD.)

But what's unclear to me is how taking the path BD satisfies the objective of the fluid element not being accelerated against the sidewalls of the pipe at any point, including B. Recalling the example of the sliding puck, the objective was to slide the puck between the two spiral fences without it touching either of the fences that are on either side of it; at B, the puck accelerates to a very high velocity (or infinite, if you prefer) and so the fences take a path closer to BD. So, all along its path from the centre of the disk to the radius, including at B, it never touches the fences -- the puck experiences zero total tangential acceleration along its path.

The goal with the fluid is similar -- to send the fluid element down the pipe without it being pushed up against one sidewall of the pipe more than the other. But at B, the direction of the streamline is tangential to the pipe, so that's why it also seems to make sense that the fluid should accelerate in a tangential direction BC...

A case considering continuous rather than instantaneous radial acceleration
In case it helps shed light on what's going on here, I'll describe one more case that may be easier to analyze, where acceleration is continuous rather than instantaneous. The fluid element travels outward from the centre to the outer edge of the disk at an increasing radial velocity, so it inscribes on the rotating disk a widening spiral, let's say a logarithmic spiral. As in the case previously discussed, the diameter of the spiral pipe that the fluid flows down narrows at just the right rate to allow the fluid to travel in a straight line, according to a stationary observer. So, the fluid element experiences an outward radial acceleration along its path, but experiences zero acceleration normal to the radial direction (I said it that way to avoid confusing the tangent of the disk with the tangent of the spiral).

Question:
Can the fluid element travel along this logarithmic spiral pipe (but in a straight line, according to the stationary observer) without being forced up against one of the sides of the pipe more than the other? Sorry about the sloppy phrasing of the question, but I think you know what I mean.

(It seems to me the answer to this question is no, which would suggest that the answer to my first question, that described in the sketch, can't be BD.)

 

[fezzik]

This is a sketch of a situation that I think has some similarities with my last example (which referred to a logarithmic spiral pipe). A fluid element flows along a curving, narrowing pipe. In order to maintain a constant velocity in the x direction, it has to accelerate with respect to the pipe, meaning it has to accelerate in the y direction.