- #1
fezzik
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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...
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...