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Why do tea leaves go to the center of the cup?
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When I stir my tea with a spoon, I observe that the tea leaves all eventually concentrate in the center of the cup. Clearly they go against the centrifugal force. Why?
Yet when I put the cup in the center of the rotating platter of my gramophone turntable, the tea leaves concentrate at the edge following the centrifugal force.
The surface of the tea is concave with the depth lower in the center in both cases. Thus the pressure in the center cannot be higher than at the edge. The tea leaves drown, so they are heavier than water and should not move to where pressure is lower.
So why do the tea leaves move to the center?
In response to the possible duplicate, the earlier question does not include the case of a rotating turntable and therefore has a lesser scope. In addition, the method of stiring is different, so the question is not a duplicate. Finally, the earlier question does not have a complete correct answer, but only a reference to the Elkman layer for a further research. Thus my question is not at all addressed earlier from any angle. The second link also does not address the case of the rotating turntable, explaining where, how, and why the leaves would collect in this case.
fluid-dynamics centrifugal-force
asked Aug 22 at 22:35
safesphere
6,37111237
add a comment |Â
up vote
11
down vote
favorite
When I stir my tea with a spoon, I observe that the tea leaves all eventually concentrate in the center of the cup. Clearly they go against the centrifugal force. Why?
Yet when I put the cup in the center of the rotating platter of my gramophone turntable, the tea leaves concentrate at the edge following the centrifugal force.
The surface of the tea is concave with the depth lower in the center in both cases. Thus the pressure in the center cannot be higher than at the edge. The tea leaves drown, so they are heavier than water and should not move to where pressure is lower.
So why do the tea leaves move to the center?
In response to the possible duplicate, the earlier question does not include the case of a rotating turntable and therefore has a lesser scope. In addition, the method of stiring is different, so the question is not a duplicate. Finally, the earlier question does not have a complete correct answer, but only a reference to the Elkman layer for a further research. Thus my question is not at all addressed earlier from any angle. The second link also does not address the case of the rotating turntable, explaining where, how, and why the leaves would collect in this case.
fluid-dynamics centrifugal-force
asked Aug 22 at 22:35
safesphere
6,37111237
4
Possible duplicate of Vortex in liquid collects particles in center
– Blue
Aug 22 at 22:50
@Blue Thanks for the link. However, please see the edit at the bottom of the question explaining why this is not a duplicate.
– safesphere
Aug 22 at 23:14
1
Possible duplicate of What forces are at work causing sand to migrate to the centre of a spinning bucket of water?
– John Rennie
Aug 23 at 4:26
@JohnRennie Thanks for the link with much better answers than in the earlier link. However, the scope of my question is broader, as it also includes the case of a rotating turntable and it is not self evident why the leaves would collect st the edge rather than, say, equally everywhere. This case is not explicitly addressed in the linked question, so it is not a duplicate. PS. Despite you masterfully dropping an ace name for a huge reward, the answer of Calmarius appears more plausible, because indeed the leaves do not go to the center until the rotation nearly ends and they don't flow up.
– safesphere
Aug 23 at 4:53
add a comment |Â
up vote
11
down vote
favorite
up vote
11
down vote
favorite
When I stir my tea with a spoon, I observe that the tea leaves all eventually concentrate in the center of the cup. Clearly they go against the centrifugal force. Why?
Yet when I put the cup in the center of the rotating platter of my gramophone turntable, the tea leaves concentrate at the edge following the centrifugal force.
The surface of the tea is concave with the depth lower in the center in both cases. Thus the pressure in the center cannot be higher than at the edge. The tea leaves drown, so they are heavier than water and should not move to where pressure is lower.
So why do the tea leaves move to the center?
In response to the possible duplicate, the earlier question does not include the case of a rotating turntable and therefore has a lesser scope. In addition, the method of stiring is different, so the question is not a duplicate. Finally, the earlier question does not have a complete correct answer, but only a reference to the Elkman layer for a further research. Thus my question is not at all addressed earlier from any angle. The second link also does not address the case of the rotating turntable, explaining where, how, and why the leaves would collect in this case.
fluid-dynamics centrifugal-force
asked Aug 22 at 22:35
safesphere
6,37111237
When I stir my tea with a spoon, I observe that the tea leaves all eventually concentrate in the center of the cup. Clearly they go against the centrifugal force. Why?
Yet when I put the cup in the center of the rotating platter of my gramophone turntable, the tea leaves concentrate at the edge following the centrifugal force.
The surface of the tea is concave with the depth lower in the center in both cases. Thus the pressure in the center cannot be higher than at the edge. The tea leaves drown, so they are heavier than water and should not move to where pressure is lower.
So why do the tea leaves move to the center?
In response to the possible duplicate, the earlier question does not include the case of a rotating turntable and therefore has a lesser scope. In addition, the method of stiring is different, so the question is not a duplicate. Finally, the earlier question does not have a complete correct answer, but only a reference to the Elkman layer for a further research. Thus my question is not at all addressed earlier from any angle. The second link also does not address the case of the rotating turntable, explaining where, how, and why the leaves would collect in this case.
fluid-dynamics centrifugal-force
asked Aug 22 at 22:35
safesphere
6,37111237
edited Aug 23 at 4:55
asked Aug 22 at 22:35
safesphere
6,37111237
asked Aug 22 at 22:35
safesphere
6,37111237
asked Aug 22 at 22:35
safesphere
6,37111237
6,37111237
4
Possible duplicate of Vortex in liquid collects particles in center
– Blue
Aug 22 at 22:50
@Blue Thanks for the link. However, please see the edit at the bottom of the question explaining why this is not a duplicate.
– safesphere
Aug 22 at 23:14
1
Possible duplicate of What forces are at work causing sand to migrate to the centre of a spinning bucket of water?
– John Rennie
Aug 23 at 4:26
@JohnRennie Thanks for the link with much better answers than in the earlier link. However, the scope of my question is broader, as it also includes the case of a rotating turntable and it is not self evident why the leaves would collect st the edge rather than, say, equally everywhere. This case is not explicitly addressed in the linked question, so it is not a duplicate. PS. Despite you masterfully dropping an ace name for a huge reward, the answer of Calmarius appears more plausible, because indeed the leaves do not go to the center until the rotation nearly ends and they don't flow up.
– safesphere
Aug 23 at 4:53
add a comment |Â
4
Possible duplicate of Vortex in liquid collects particles in center
– Blue
Aug 22 at 22:50
@Blue Thanks for the link. However, please see the edit at the bottom of the question explaining why this is not a duplicate.
– safesphere
Aug 22 at 23:14
1
Possible duplicate of What forces are at work causing sand to migrate to the centre of a spinning bucket of water?
– John Rennie
Aug 23 at 4:26
@JohnRennie Thanks for the link with much better answers than in the earlier link. However, the scope of my question is broader, as it also includes the case of a rotating turntable and it is not self evident why the leaves would collect st the edge rather than, say, equally everywhere. This case is not explicitly addressed in the linked question, so it is not a duplicate. PS. Despite you masterfully dropping an ace name for a huge reward, the answer of Calmarius appears more plausible, because indeed the leaves do not go to the center until the rotation nearly ends and they don't flow up.
– safesphere
Aug 23 at 4:53
4
4
Possible duplicate of Vortex in liquid collects particles in center
– Blue
Aug 22 at 22:50
Possible duplicate of Vortex in liquid collects particles in center
– Blue
Aug 22 at 22:50
@Blue Thanks for the link. However, please see the edit at the bottom of the question explaining why this is not a duplicate.
– safesphere
Aug 22 at 23:14
@Blue Thanks for the link. However, please see the edit at the bottom of the question explaining why this is not a duplicate.
– safesphere
Aug 22 at 23:14
1
1
Possible duplicate of What forces are at work causing sand to migrate to the centre of a spinning bucket of water?
– John Rennie
Aug 23 at 4:26
Possible duplicate of What forces are at work causing sand to migrate to the centre of a spinning bucket of water?
– John Rennie
Aug 23 at 4:26
@JohnRennie Thanks for the link with much better answers than in the earlier link. However, the scope of my question is broader, as it also includes the case of a rotating turntable and it is not self evident why the leaves would collect st the edge rather than, say, equally everywhere. This case is not explicitly addressed in the linked question, so it is not a duplicate. PS. Despite you masterfully dropping an ace name for a huge reward, the answer of Calmarius appears more plausible, because indeed the leaves do not go to the center until the rotation nearly ends and they don't flow up.
– safesphere
Aug 23 at 4:53
@JohnRennie Thanks for the link with much better answers than in the earlier link. However, the scope of my question is broader, as it also includes the case of a rotating turntable and it is not self evident why the leaves would collect st the edge rather than, say, equally everywhere. This case is not explicitly addressed in the linked question, so it is not a duplicate. PS. Despite you masterfully dropping an ace name for a huge reward, the answer of Calmarius appears more plausible, because indeed the leaves do not go to the center until the rotation nearly ends and they don't flow up.
– safesphere
Aug 23 at 4:53
add a comment |Â
2 Answers
2
active
oldest
votes
up vote
4
down vote
When I steer my tea with a spoon, I observe that the tea leaves all eventually concentrate in the centre of the cup. Clearly, they go against the centrifugal force. Why?
When you are stirring the tea in your cup you are trying to give rotational motion to the fluid layers but the layers in contact with the cup surfaces remain at zero speeds. The velocity gradient radially (as the viscous drag between the layers reduces the speed) picks up the leaves but it can not maintain the circular path in absence of required centripetal acceleration and slowly collects at the centre.
For moving along with the fluid they need necessary centripetal force and that is not being provided by the layers of fluid.
Yet when I put the cup in the centre of the rotating platter of my gramophone turntable, the tea leaves concentrate at the edge following the centrifugal force.
When you are rotating the cup filled with tea and leaves on a turntable- slowly the layers of fluid will pick up the speed and then drag the tea leaves to rotate along with layers of liquid but as its heavier than the liquid media , it again searches for required centripetal force and gets back to the rim of the cup, where it can rotate with centripetal force provided by the reaction of the
surface.
Its a centrifuge action.
There is an interesting discussion of the same event-the first part of stirring which is being quoted for added insight. the reference gives diagram of action when stirring.
Added a quote- for further explanation.
Stirring the liquid makes it spin around the cup. In order to maintain this curved path, a centripetal force in towards the center is needed (similar to the tension in a string when spinning a bucket over your head). This is accomplished by a pressure gradient outward (higher pressure outside than inside).
However, near the bottom and outer edges the liquid is slowed by the friction against the cup. There the fictitious (inertial) centrifugal force is weaker and cannot overcome the pressure gradient, so these pressure differences become more important for the water flow. This is called a boundary layer or more specifically an Ekman layer.[6]
The inertial centrifugal force due to the bulk rotation of the liquid results in the development of an outward pressure gradient within the liquid, where the pressure is higher along the rim than in the middle. This manifests itself as the formation of a concave liquid-air interface. This pressure gradient provides the necessary centripetal forces for circular motion when summed over the entirety of the rotating liquid.
However, within the boundary layers where fluid rotation is slowed by friction and viscous effects, the centripetal force due to pressure gradient is dominant over the inertial forces from rotation, and creates an inward secondary flow within the boundary layer. The flow converges at the bottom of the teacup (where the tea leaves are observed to gather) and flows upwards to the surface. Higher up, the liquid flow meets the surface and flows outwards. The leaves are too heavy to lift upwards and remain in the middle. Combined with the primary rotational flow, the leaves will be observed to spiral inwards along the bottom of the teacup.[5]
https://en.wikipedia.org/wiki/Tea_leaf_paradox
answered Aug 22 at 23:05
drvrm
1,238513
Thanks for the answer, but it seems incorrect with an unclear logic. "their speeds reduce due to its motion between layers of tea-fluid" - No, they rotate slower together with the layer of liquid, in which they are. What difference does it make to them if the upper layers rotate faster? "they can not get proper velocities to sustain the path, thereby they slowly reduce their radius and collect at the centre" - Sorry, but this makes no sense. Why do they need a higher velocity "to sustain the path"? What "path"? Finally, what is the source of the "centripetal force" you are referring to?
– safesphere
Aug 22 at 23:24
@safesphere-Let me try to explain-As one is stirring the layers of liquid are being moved but the viscous drag between the layers reduce the angular speeds and the tea leaves to maintain a radial path must have proper centripetal acceleration, otherwise it will move towards the direction of net pressure difference between the layers , which leads to collect at the center.
– drvrm
Aug 22 at 23:46
@safesphere- moreover, the action of centrifugal force(a pseudo force) can be observed only in a non-inertial frame- here one can only see the absence or unavailability of centriptal force .
– drvrm
Aug 22 at 23:53
As mentioned in the question, the leaves are heavier than water, they drown to higher pressure, so they would not move in the direction of "the net pressure difference" (to lower pressure), as you stated. It also is not clear what the pressure difference is and why. Sorry, but your answer is still short on having a clear and undisputed logical chain. It would be great ic you could edit the answer to make the explanation step by step unquestionable. Also you still did not explain the source of the "centripetal force" your answer relied upon.
– safesphere
Aug 23 at 0:02
@safesphere- well, I will try to edit the answer.thanks for the sharing.
– drvrm
Aug 23 at 0:13
 |Â
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up vote
3
down vote
Thanks for sharing the problem! Here's an explanation that might seem more intuitive.
Imagine the teacup in the center of the turntable. The cup and the fluid inside it are just rotating as a solid block; the pressure at a given height, due to the centripetal acceleration, is $P_0 + fracrho omega^2 r^22$, quadratic in $r$. This causes the quadratic surface on the top of the fluid - it can't be flat because it has to match air pressure at every $r$.
Now imagine the cup suddenly stops, but the fluid keeps rotating. This is the paradox's scenario. Viscosity slows the fluid near the surfaces of the cup. The bulk of the fluid still has approximately the same motion, so the pressure distribution isn't significantly changed, but the fluid near the bottom surface now isn't rotating circularly as fast.
The radial pressure distribution was balanced so fluid particles would follow circles when rotating at $omega$, but now the bottom fluid is rotating slower. The pressure distribution forces fluid to spiral towards the center, so the tea leaves follow.
answered Aug 23 at 3:19
mrfish
1313
Also, do you have anything denser than tea leaves to try this with? Something dense enough should go to the outside of the cup even when stirred
– mrfish
Aug 23 at 3:25
Thank you for the answer. However, the leaves do not collect at the center while the liquid is actively rotating. They seem to go to the center only when the rotation nearly stops. Can you explain why?
– safesphere
Aug 23 at 5:04
A few theories from looking at a gif of the paradox: (a) There's turbulence until flow becomes slower (b) The pressure in the center of the bottom has to be higher, redirecting water upwards. This might be keeping tea leaves out of the center when rotation is fast
– mrfish
Aug 23 at 13:07
It looks like the leaves are moving to the center only while the surface is becoming flat. Thus they appear to be moved by the flow created by the higher water level at the edge going down. This flow relates to the change in the speed of rotation, but not to the rotation at a constant soeed, as the answer describes.
– safesphere
Aug 23 at 15:47
Is it the becoming low-speed or the being low-speed that makes the leaves accumulate? Does it work if the tea starts spinning slowly? My theory was that the answer I gave works, but turbulence or the pressure causing upward flow messes with it at high $/omega$
– mrfish
Aug 24 at 2:54
 |Â
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2 Answers
2
active
oldest
votes
2 Answers
2
active
oldest
votes
active
oldest
votes
active
oldest
votes
up vote
4
down vote
When I steer my tea with a spoon, I observe that the tea leaves all eventually concentrate in the centre of the cup. Clearly, they go against the centrifugal force. Why?
When you are stirring the tea in your cup you are trying to give rotational motion to the fluid layers but the layers in contact with the cup surfaces remain at zero speeds. The velocity gradient radially (as the viscous drag between the layers reduces the speed) picks up the leaves but it can not maintain the circular path in absence of required centripetal acceleration and slowly collects at the centre.
For moving along with the fluid they need necessary centripetal force and that is not being provided by the layers of fluid.
Yet when I put the cup in the centre of the rotating platter of my gramophone turntable, the tea leaves concentrate at the edge following the centrifugal force.
When you are rotating the cup filled with tea and leaves on a turntable- slowly the layers of fluid will pick up the speed and then drag the tea leaves to rotate along with layers of liquid but as its heavier than the liquid media , it again searches for required centripetal force and gets back to the rim of the cup, where it can rotate with centripetal force provided by the reaction of the
surface.
Its a centrifuge action.
There is an interesting discussion of the same event-the first part of stirring which is being quoted for added insight. the reference gives diagram of action when stirring.
Added a quote- for further explanation.
Stirring the liquid makes it spin around the cup. In order to maintain this curved path, a centripetal force in towards the center is needed (similar to the tension in a string when spinning a bucket over your head). This is accomplished by a pressure gradient outward (higher pressure outside than inside).
However, near the bottom and outer edges the liquid is slowed by the friction against the cup. There the fictitious (inertial) centrifugal force is weaker and cannot overcome the pressure gradient, so these pressure differences become more important for the water flow. This is called a boundary layer or more specifically an Ekman layer.[6]
The inertial centrifugal force due to the bulk rotation of the liquid results in the development of an outward pressure gradient within the liquid, where the pressure is higher along the rim than in the middle. This manifests itself as the formation of a concave liquid-air interface. This pressure gradient provides the necessary centripetal forces for circular motion when summed over the entirety of the rotating liquid.
However, within the boundary layers where fluid rotation is slowed by friction and viscous effects, the centripetal force due to pressure gradient is dominant over the inertial forces from rotation, and creates an inward secondary flow within the boundary layer. The flow converges at the bottom of the teacup (where the tea leaves are observed to gather) and flows upwards to the surface. Higher up, the liquid flow meets the surface and flows outwards. The leaves are too heavy to lift upwards and remain in the middle. Combined with the primary rotational flow, the leaves will be observed to spiral inwards along the bottom of the teacup.[5]
https://en.wikipedia.org/wiki/Tea_leaf_paradox
answered Aug 22 at 23:05
drvrm
1,238513
Thanks for the answer, but it seems incorrect with an unclear logic. "their speeds reduce due to its motion between layers of tea-fluid" - No, they rotate slower together with the layer of liquid, in which they are. What difference does it make to them if the upper layers rotate faster? "they can not get proper velocities to sustain the path, thereby they slowly reduce their radius and collect at the centre" - Sorry, but this makes no sense. Why do they need a higher velocity "to sustain the path"? What "path"? Finally, what is the source of the "centripetal force" you are referring to?
– safesphere
Aug 22 at 23:24
@safesphere-Let me try to explain-As one is stirring the layers of liquid are being moved but the viscous drag between the layers reduce the angular speeds and the tea leaves to maintain a radial path must have proper centripetal acceleration, otherwise it will move towards the direction of net pressure difference between the layers , which leads to collect at the center.
– drvrm
Aug 22 at 23:46
@safesphere- moreover, the action of centrifugal force(a pseudo force) can be observed only in a non-inertial frame- here one can only see the absence or unavailability of centriptal force .
– drvrm
Aug 22 at 23:53
As mentioned in the question, the leaves are heavier than water, they drown to higher pressure, so they would not move in the direction of "the net pressure difference" (to lower pressure), as you stated. It also is not clear what the pressure difference is and why. Sorry, but your answer is still short on having a clear and undisputed logical chain. It would be great ic you could edit the answer to make the explanation step by step unquestionable. Also you still did not explain the source of the "centripetal force" your answer relied upon.
– safesphere
Aug 23 at 0:02
@safesphere- well, I will try to edit the answer.thanks for the sharing.
– drvrm
Aug 23 at 0:13
 |Â
show 2 more comments
up vote
4
down vote
When I steer my tea with a spoon, I observe that the tea leaves all eventually concentrate in the centre of the cup. Clearly, they go against the centrifugal force. Why?
When you are stirring the tea in your cup you are trying to give rotational motion to the fluid layers but the layers in contact with the cup surfaces remain at zero speeds. The velocity gradient radially (as the viscous drag between the layers reduces the speed) picks up the leaves but it can not maintain the circular path in absence of required centripetal acceleration and slowly collects at the centre.
For moving along with the fluid they need necessary centripetal force and that is not being provided by the layers of fluid.
Yet when I put the cup in the centre of the rotating platter of my gramophone turntable, the tea leaves concentrate at the edge following the centrifugal force.
When you are rotating the cup filled with tea and leaves on a turntable- slowly the layers of fluid will pick up the speed and then drag the tea leaves to rotate along with layers of liquid but as its heavier than the liquid media , it again searches for required centripetal force and gets back to the rim of the cup, where it can rotate with centripetal force provided by the reaction of the
surface.
Its a centrifuge action.
There is an interesting discussion of the same event-the first part of stirring which is being quoted for added insight. the reference gives diagram of action when stirring.
Added a quote- for further explanation.
Stirring the liquid makes it spin around the cup. In order to maintain this curved path, a centripetal force in towards the center is needed (similar to the tension in a string when spinning a bucket over your head). This is accomplished by a pressure gradient outward (higher pressure outside than inside).
However, near the bottom and outer edges the liquid is slowed by the friction against the cup. There the fictitious (inertial) centrifugal force is weaker and cannot overcome the pressure gradient, so these pressure differences become more important for the water flow. This is called a boundary layer or more specifically an Ekman layer.[6]
The inertial centrifugal force due to the bulk rotation of the liquid results in the development of an outward pressure gradient within the liquid, where the pressure is higher along the rim than in the middle. This manifests itself as the formation of a concave liquid-air interface. This pressure gradient provides the necessary centripetal forces for circular motion when summed over the entirety of the rotating liquid.
However, within the boundary layers where fluid rotation is slowed by friction and viscous effects, the centripetal force due to pressure gradient is dominant over the inertial forces from rotation, and creates an inward secondary flow within the boundary layer. The flow converges at the bottom of the teacup (where the tea leaves are observed to gather) and flows upwards to the surface. Higher up, the liquid flow meets the surface and flows outwards. The leaves are too heavy to lift upwards and remain in the middle. Combined with the primary rotational flow, the leaves will be observed to spiral inwards along the bottom of the teacup.[5]
https://en.wikipedia.org/wiki/Tea_leaf_paradox
answered Aug 22 at 23:05
drvrm
1,238513
Thanks for the answer, but it seems incorrect with an unclear logic. "their speeds reduce due to its motion between layers of tea-fluid" - No, they rotate slower together with the layer of liquid, in which they are. What difference does it make to them if the upper layers rotate faster? "they can not get proper velocities to sustain the path, thereby they slowly reduce their radius and collect at the centre" - Sorry, but this makes no sense. Why do they need a higher velocity "to sustain the path"? What "path"? Finally, what is the source of the "centripetal force" you are referring to?
– safesphere
Aug 22 at 23:24
@safesphere-Let me try to explain-As one is stirring the layers of liquid are being moved but the viscous drag between the layers reduce the angular speeds and the tea leaves to maintain a radial path must have proper centripetal acceleration, otherwise it will move towards the direction of net pressure difference between the layers , which leads to collect at the center.
– drvrm
Aug 22 at 23:46
@safesphere- moreover, the action of centrifugal force(a pseudo force) can be observed only in a non-inertial frame- here one can only see the absence or unavailability of centriptal force .
– drvrm
Aug 22 at 23:53
As mentioned in the question, the leaves are heavier than water, they drown to higher pressure, so they would not move in the direction of "the net pressure difference" (to lower pressure), as you stated. It also is not clear what the pressure difference is and why. Sorry, but your answer is still short on having a clear and undisputed logical chain. It would be great ic you could edit the answer to make the explanation step by step unquestionable. Also you still did not explain the source of the "centripetal force" your answer relied upon.
– safesphere
Aug 23 at 0:02
@safesphere- well, I will try to edit the answer.thanks for the sharing.
– drvrm
Aug 23 at 0:13
 |Â
show 2 more comments
up vote
4
down vote
up vote
4
down vote
When I steer my tea with a spoon, I observe that the tea leaves all eventually concentrate in the centre of the cup. Clearly, they go against the centrifugal force. Why?
When you are stirring the tea in your cup you are trying to give rotational motion to the fluid layers but the layers in contact with the cup surfaces remain at zero speeds. The velocity gradient radially (as the viscous drag between the layers reduces the speed) picks up the leaves but it can not maintain the circular path in absence of required centripetal acceleration and slowly collects at the centre.
For moving along with the fluid they need necessary centripetal force and that is not being provided by the layers of fluid.
Yet when I put the cup in the centre of the rotating platter of my gramophone turntable, the tea leaves concentrate at the edge following the centrifugal force.
When you are rotating the cup filled with tea and leaves on a turntable- slowly the layers of fluid will pick up the speed and then drag the tea leaves to rotate along with layers of liquid but as its heavier than the liquid media , it again searches for required centripetal force and gets back to the rim of the cup, where it can rotate with centripetal force provided by the reaction of the
surface.
Its a centrifuge action.
There is an interesting discussion of the same event-the first part of stirring which is being quoted for added insight. the reference gives diagram of action when stirring.
Added a quote- for further explanation.
Stirring the liquid makes it spin around the cup. In order to maintain this curved path, a centripetal force in towards the center is needed (similar to the tension in a string when spinning a bucket over your head). This is accomplished by a pressure gradient outward (higher pressure outside than inside).
However, near the bottom and outer edges the liquid is slowed by the friction against the cup. There the fictitious (inertial) centrifugal force is weaker and cannot overcome the pressure gradient, so these pressure differences become more important for the water flow. This is called a boundary layer or more specifically an Ekman layer.[6]
The inertial centrifugal force due to the bulk rotation of the liquid results in the development of an outward pressure gradient within the liquid, where the pressure is higher along the rim than in the middle. This manifests itself as the formation of a concave liquid-air interface. This pressure gradient provides the necessary centripetal forces for circular motion when summed over the entirety of the rotating liquid.
However, within the boundary layers where fluid rotation is slowed by friction and viscous effects, the centripetal force due to pressure gradient is dominant over the inertial forces from rotation, and creates an inward secondary flow within the boundary layer. The flow converges at the bottom of the teacup (where the tea leaves are observed to gather) and flows upwards to the surface. Higher up, the liquid flow meets the surface and flows outwards. The leaves are too heavy to lift upwards and remain in the middle. Combined with the primary rotational flow, the leaves will be observed to spiral inwards along the bottom of the teacup.[5]
https://en.wikipedia.org/wiki/Tea_leaf_paradox
answered Aug 22 at 23:05
drvrm
1,238513
When I steer my tea with a spoon, I observe that the tea leaves all eventually concentrate in the centre of the cup. Clearly, they go against the centrifugal force. Why?
When you are stirring the tea in your cup you are trying to give rotational motion to the fluid layers but the layers in contact with the cup surfaces remain at zero speeds. The velocity gradient radially (as the viscous drag between the layers reduces the speed) picks up the leaves but it can not maintain the circular path in absence of required centripetal acceleration and slowly collects at the centre.
For moving along with the fluid they need necessary centripetal force and that is not being provided by the layers of fluid.
Yet when I put the cup in the centre of the rotating platter of my gramophone turntable, the tea leaves concentrate at the edge following the centrifugal force.
When you are rotating the cup filled with tea and leaves on a turntable- slowly the layers of fluid will pick up the speed and then drag the tea leaves to rotate along with layers of liquid but as its heavier than the liquid media , it again searches for required centripetal force and gets back to the rim of the cup, where it can rotate with centripetal force provided by the reaction of the
surface.
Its a centrifuge action.
There is an interesting discussion of the same event-the first part of stirring which is being quoted for added insight. the reference gives diagram of action when stirring.
Added a quote- for further explanation.
Stirring the liquid makes it spin around the cup. In order to maintain this curved path, a centripetal force in towards the center is needed (similar to the tension in a string when spinning a bucket over your head). This is accomplished by a pressure gradient outward (higher pressure outside than inside).
However, near the bottom and outer edges the liquid is slowed by the friction against the cup. There the fictitious (inertial) centrifugal force is weaker and cannot overcome the pressure gradient, so these pressure differences become more important for the water flow. This is called a boundary layer or more specifically an Ekman layer.[6]
The inertial centrifugal force due to the bulk rotation of the liquid results in the development of an outward pressure gradient within the liquid, where the pressure is higher along the rim than in the middle. This manifests itself as the formation of a concave liquid-air interface. This pressure gradient provides the necessary centripetal forces for circular motion when summed over the entirety of the rotating liquid.
However, within the boundary layers where fluid rotation is slowed by friction and viscous effects, the centripetal force due to pressure gradient is dominant over the inertial forces from rotation, and creates an inward secondary flow within the boundary layer. The flow converges at the bottom of the teacup (where the tea leaves are observed to gather) and flows upwards to the surface. Higher up, the liquid flow meets the surface and flows outwards. The leaves are too heavy to lift upwards and remain in the middle. Combined with the primary rotational flow, the leaves will be observed to spiral inwards along the bottom of the teacup.[5]
https://en.wikipedia.org/wiki/Tea_leaf_paradox
answered Aug 22 at 23:05
drvrm
1,238513
edited Aug 23 at 7:11
answered Aug 22 at 23:05
drvrm
1,238513
answered Aug 22 at 23:05
drvrm
1,238513
answered Aug 22 at 23:05
drvrm
1,238513
1,238513
Thanks for the answer, but it seems incorrect with an unclear logic. "their speeds reduce due to its motion between layers of tea-fluid" - No, they rotate slower together with the layer of liquid, in which they are. What difference does it make to them if the upper layers rotate faster? "they can not get proper velocities to sustain the path, thereby they slowly reduce their radius and collect at the centre" - Sorry, but this makes no sense. Why do they need a higher velocity "to sustain the path"? What "path"? Finally, what is the source of the "centripetal force" you are referring to?
– safesphere
Aug 22 at 23:24
@safesphere-Let me try to explain-As one is stirring the layers of liquid are being moved but the viscous drag between the layers reduce the angular speeds and the tea leaves to maintain a radial path must have proper centripetal acceleration, otherwise it will move towards the direction of net pressure difference between the layers , which leads to collect at the center.
– drvrm
Aug 22 at 23:46
@safesphere- moreover, the action of centrifugal force(a pseudo force) can be observed only in a non-inertial frame- here one can only see the absence or unavailability of centriptal force .
– drvrm
Aug 22 at 23:53
As mentioned in the question, the leaves are heavier than water, they drown to higher pressure, so they would not move in the direction of "the net pressure difference" (to lower pressure), as you stated. It also is not clear what the pressure difference is and why. Sorry, but your answer is still short on having a clear and undisputed logical chain. It would be great ic you could edit the answer to make the explanation step by step unquestionable. Also you still did not explain the source of the "centripetal force" your answer relied upon.
– safesphere
Aug 23 at 0:02
@safesphere- well, I will try to edit the answer.thanks for the sharing.
– drvrm
Aug 23 at 0:13
 |Â
show 2 more comments
Thanks for the answer, but it seems incorrect with an unclear logic. "their speeds reduce due to its motion between layers of tea-fluid" - No, they rotate slower together with the layer of liquid, in which they are. What difference does it make to them if the upper layers rotate faster? "they can not get proper velocities to sustain the path, thereby they slowly reduce their radius and collect at the centre" - Sorry, but this makes no sense. Why do they need a higher velocity "to sustain the path"? What "path"? Finally, what is the source of the "centripetal force" you are referring to?
– safesphere
Aug 22 at 23:24
@safesphere-Let me try to explain-As one is stirring the layers of liquid are being moved but the viscous drag between the layers reduce the angular speeds and the tea leaves to maintain a radial path must have proper centripetal acceleration, otherwise it will move towards the direction of net pressure difference between the layers , which leads to collect at the center.
– drvrm
Aug 22 at 23:46
@safesphere- moreover, the action of centrifugal force(a pseudo force) can be observed only in a non-inertial frame- here one can only see the absence or unavailability of centriptal force .
– drvrm
Aug 22 at 23:53
As mentioned in the question, the leaves are heavier than water, they drown to higher pressure, so they would not move in the direction of "the net pressure difference" (to lower pressure), as you stated. It also is not clear what the pressure difference is and why. Sorry, but your answer is still short on having a clear and undisputed logical chain. It would be great ic you could edit the answer to make the explanation step by step unquestionable. Also you still did not explain the source of the "centripetal force" your answer relied upon.
– safesphere
Aug 23 at 0:02
@safesphere- well, I will try to edit the answer.thanks for the sharing.
– drvrm
Aug 23 at 0:13
Thanks for the answer, but it seems incorrect with an unclear logic. "their speeds reduce due to its motion between layers of tea-fluid" - No, they rotate slower together with the layer of liquid, in which they are. What difference does it make to them if the upper layers rotate faster? "they can not get proper velocities to sustain the path, thereby they slowly reduce their radius and collect at the centre" - Sorry, but this makes no sense. Why do they need a higher velocity "to sustain the path"? What "path"? Finally, what is the source of the "centripetal force" you are referring to?
– safesphere
Aug 22 at 23:24
Thanks for the answer, but it seems incorrect with an unclear logic. "their speeds reduce due to its motion between layers of tea-fluid" - No, they rotate slower together with the layer of liquid, in which they are. What difference does it make to them if the upper layers rotate faster? "they can not get proper velocities to sustain the path, thereby they slowly reduce their radius and collect at the centre" - Sorry, but this makes no sense. Why do they need a higher velocity "to sustain the path"? What "path"? Finally, what is the source of the "centripetal force" you are referring to?
– safesphere
Aug 22 at 23:24
@safesphere-Let me try to explain-As one is stirring the layers of liquid are being moved but the viscous drag between the layers reduce the angular speeds and the tea leaves to maintain a radial path must have proper centripetal acceleration, otherwise it will move towards the direction of net pressure difference between the layers , which leads to collect at the center.
– drvrm
Aug 22 at 23:46
@safesphere-Let me try to explain-As one is stirring the layers of liquid are being moved but the viscous drag between the layers reduce the angular speeds and the tea leaves to maintain a radial path must have proper centripetal acceleration, otherwise it will move towards the direction of net pressure difference between the layers , which leads to collect at the center.
– drvrm
Aug 22 at 23:46
@safesphere- moreover, the action of centrifugal force(a pseudo force) can be observed only in a non-inertial frame- here one can only see the absence or unavailability of centriptal force .
– drvrm
Aug 22 at 23:53
@safesphere- moreover, the action of centrifugal force(a pseudo force) can be observed only in a non-inertial frame- here one can only see the absence or unavailability of centriptal force .
– drvrm
Aug 22 at 23:53
As mentioned in the question, the leaves are heavier than water, they drown to higher pressure, so they would not move in the direction of "the net pressure difference" (to lower pressure), as you stated. It also is not clear what the pressure difference is and why. Sorry, but your answer is still short on having a clear and undisputed logical chain. It would be great ic you could edit the answer to make the explanation step by step unquestionable. Also you still did not explain the source of the "centripetal force" your answer relied upon.
– safesphere
Aug 23 at 0:02
As mentioned in the question, the leaves are heavier than water, they drown to higher pressure, so they would not move in the direction of "the net pressure difference" (to lower pressure), as you stated. It also is not clear what the pressure difference is and why. Sorry, but your answer is still short on having a clear and undisputed logical chain. It would be great ic you could edit the answer to make the explanation step by step unquestionable. Also you still did not explain the source of the "centripetal force" your answer relied upon.
– safesphere
Aug 23 at 0:02
@safesphere- well, I will try to edit the answer.thanks for the sharing.
– drvrm
Aug 23 at 0:13
@safesphere- well, I will try to edit the answer.thanks for the sharing.
– drvrm
Aug 23 at 0:13
 |Â
show 2 more comments
up vote
3
down vote
Thanks for sharing the problem! Here's an explanation that might seem more intuitive.
Imagine the teacup in the center of the turntable. The cup and the fluid inside it are just rotating as a solid block; the pressure at a given height, due to the centripetal acceleration, is $P_0 + fracrho omega^2 r^22$, quadratic in $r$. This causes the quadratic surface on the top of the fluid - it can't be flat because it has to match air pressure at every $r$.
Now imagine the cup suddenly stops, but the fluid keeps rotating. This is the paradox's scenario. Viscosity slows the fluid near the surfaces of the cup. The bulk of the fluid still has approximately the same motion, so the pressure distribution isn't significantly changed, but the fluid near the bottom surface now isn't rotating circularly as fast.
The radial pressure distribution was balanced so fluid particles would follow circles when rotating at $omega$, but now the bottom fluid is rotating slower. The pressure distribution forces fluid to spiral towards the center, so the tea leaves follow.
answered Aug 23 at 3:19
mrfish
1313
Also, do you have anything denser than tea leaves to try this with? Something dense enough should go to the outside of the cup even when stirred
– mrfish
Aug 23 at 3:25
Thank you for the answer. However, the leaves do not collect at the center while the liquid is actively rotating. They seem to go to the center only when the rotation nearly stops. Can you explain why?
– safesphere
Aug 23 at 5:04
A few theories from looking at a gif of the paradox: (a) There's turbulence until flow becomes slower (b) The pressure in the center of the bottom has to be higher, redirecting water upwards. This might be keeping tea leaves out of the center when rotation is fast
– mrfish
Aug 23 at 13:07
It looks like the leaves are moving to the center only while the surface is becoming flat. Thus they appear to be moved by the flow created by the higher water level at the edge going down. This flow relates to the change in the speed of rotation, but not to the rotation at a constant soeed, as the answer describes.
– safesphere
Aug 23 at 15:47
Is it the becoming low-speed or the being low-speed that makes the leaves accumulate? Does it work if the tea starts spinning slowly? My theory was that the answer I gave works, but turbulence or the pressure causing upward flow messes with it at high $/omega$
– mrfish
Aug 24 at 2:54
 |Â
show 2 more comments
up vote
3
down vote
Thanks for sharing the problem! Here's an explanation that might seem more intuitive.
Imagine the teacup in the center of the turntable. The cup and the fluid inside it are just rotating as a solid block; the pressure at a given height, due to the centripetal acceleration, is $P_0 + fracrho omega^2 r^22$, quadratic in $r$. This causes the quadratic surface on the top of the fluid - it can't be flat because it has to match air pressure at every $r$.
Now imagine the cup suddenly stops, but the fluid keeps rotating. This is the paradox's scenario. Viscosity slows the fluid near the surfaces of the cup. The bulk of the fluid still has approximately the same motion, so the pressure distribution isn't significantly changed, but the fluid near the bottom surface now isn't rotating circularly as fast.
The radial pressure distribution was balanced so fluid particles would follow circles when rotating at $omega$, but now the bottom fluid is rotating slower. The pressure distribution forces fluid to spiral towards the center, so the tea leaves follow.
answered Aug 23 at 3:19
mrfish
1313
Also, do you have anything denser than tea leaves to try this with? Something dense enough should go to the outside of the cup even when stirred
– mrfish
Aug 23 at 3:25
Thank you for the answer. However, the leaves do not collect at the center while the liquid is actively rotating. They seem to go to the center only when the rotation nearly stops. Can you explain why?
– safesphere
Aug 23 at 5:04
A few theories from looking at a gif of the paradox: (a) There's turbulence until flow becomes slower (b) The pressure in the center of the bottom has to be higher, redirecting water upwards. This might be keeping tea leaves out of the center when rotation is fast
– mrfish
Aug 23 at 13:07
It looks like the leaves are moving to the center only while the surface is becoming flat. Thus they appear to be moved by the flow created by the higher water level at the edge going down. This flow relates to the change in the speed of rotation, but not to the rotation at a constant soeed, as the answer describes.
– safesphere
Aug 23 at 15:47
Is it the becoming low-speed or the being low-speed that makes the leaves accumulate? Does it work if the tea starts spinning slowly? My theory was that the answer I gave works, but turbulence or the pressure causing upward flow messes with it at high $/omega$
– mrfish
Aug 24 at 2:54
 |Â
show 2 more comments
up vote
3
down vote
up vote
3
down vote
Thanks for sharing the problem! Here's an explanation that might seem more intuitive.
Imagine the teacup in the center of the turntable. The cup and the fluid inside it are just rotating as a solid block; the pressure at a given height, due to the centripetal acceleration, is $P_0 + fracrho omega^2 r^22$, quadratic in $r$. This causes the quadratic surface on the top of the fluid - it can't be flat because it has to match air pressure at every $r$.
Now imagine the cup suddenly stops, but the fluid keeps rotating. This is the paradox's scenario. Viscosity slows the fluid near the surfaces of the cup. The bulk of the fluid still has approximately the same motion, so the pressure distribution isn't significantly changed, but the fluid near the bottom surface now isn't rotating circularly as fast.
The radial pressure distribution was balanced so fluid particles would follow circles when rotating at $omega$, but now the bottom fluid is rotating slower. The pressure distribution forces fluid to spiral towards the center, so the tea leaves follow.
answered Aug 23 at 3:19
mrfish
1313
Thanks for sharing the problem! Here's an explanation that might seem more intuitive.
Imagine the teacup in the center of the turntable. The cup and the fluid inside it are just rotating as a solid block; the pressure at a given height, due to the centripetal acceleration, is $P_0 + fracrho omega^2 r^22$, quadratic in $r$. This causes the quadratic surface on the top of the fluid - it can't be flat because it has to match air pressure at every $r$.
Now imagine the cup suddenly stops, but the fluid keeps rotating. This is the paradox's scenario. Viscosity slows the fluid near the surfaces of the cup. The bulk of the fluid still has approximately the same motion, so the pressure distribution isn't significantly changed, but the fluid near the bottom surface now isn't rotating circularly as fast.
The radial pressure distribution was balanced so fluid particles would follow circles when rotating at $omega$, but now the bottom fluid is rotating slower. The pressure distribution forces fluid to spiral towards the center, so the tea leaves follow.
answered Aug 23 at 3:19
mrfish
1313
edited Aug 23 at 3:55
answered Aug 23 at 3:19
mrfish
1313
answered Aug 23 at 3:19
mrfish
1313
answered Aug 23 at 3:19
mrfish
1313
1313
Also, do you have anything denser than tea leaves to try this with? Something dense enough should go to the outside of the cup even when stirred
– mrfish
Aug 23 at 3:25
Thank you for the answer. However, the leaves do not collect at the center while the liquid is actively rotating. They seem to go to the center only when the rotation nearly stops. Can you explain why?
– safesphere
Aug 23 at 5:04
A few theories from looking at a gif of the paradox: (a) There's turbulence until flow becomes slower (b) The pressure in the center of the bottom has to be higher, redirecting water upwards. This might be keeping tea leaves out of the center when rotation is fast
– mrfish
Aug 23 at 13:07
It looks like the leaves are moving to the center only while the surface is becoming flat. Thus they appear to be moved by the flow created by the higher water level at the edge going down. This flow relates to the change in the speed of rotation, but not to the rotation at a constant soeed, as the answer describes.
– safesphere
Aug 23 at 15:47
Is it the becoming low-speed or the being low-speed that makes the leaves accumulate? Does it work if the tea starts spinning slowly? My theory was that the answer I gave works, but turbulence or the pressure causing upward flow messes with it at high $/omega$
– mrfish
Aug 24 at 2:54
 |Â
show 2 more comments
Also, do you have anything denser than tea leaves to try this with? Something dense enough should go to the outside of the cup even when stirred
– mrfish
Aug 23 at 3:25
Thank you for the answer. However, the leaves do not collect at the center while the liquid is actively rotating. They seem to go to the center only when the rotation nearly stops. Can you explain why?
– safesphere
Aug 23 at 5:04
A few theories from looking at a gif of the paradox: (a) There's turbulence until flow becomes slower (b) The pressure in the center of the bottom has to be higher, redirecting water upwards. This might be keeping tea leaves out of the center when rotation is fast
– mrfish
Aug 23 at 13:07
It looks like the leaves are moving to the center only while the surface is becoming flat. Thus they appear to be moved by the flow created by the higher water level at the edge going down. This flow relates to the change in the speed of rotation, but not to the rotation at a constant soeed, as the answer describes.
– safesphere
Aug 23 at 15:47
Is it the becoming low-speed or the being low-speed that makes the leaves accumulate? Does it work if the tea starts spinning slowly? My theory was that the answer I gave works, but turbulence or the pressure causing upward flow messes with it at high $/omega$
– mrfish
Aug 24 at 2:54
Also, do you have anything denser than tea leaves to try this with? Something dense enough should go to the outside of the cup even when stirred
– mrfish
Aug 23 at 3:25
Also, do you have anything denser than tea leaves to try this with? Something dense enough should go to the outside of the cup even when stirred
– mrfish
Aug 23 at 3:25
Thank you for the answer. However, the leaves do not collect at the center while the liquid is actively rotating. They seem to go to the center only when the rotation nearly stops. Can you explain why?
– safesphere
Aug 23 at 5:04
Thank you for the answer. However, the leaves do not collect at the center while the liquid is actively rotating. They seem to go to the center only when the rotation nearly stops. Can you explain why?
– safesphere
Aug 23 at 5:04
A few theories from looking at a gif of the paradox: (a) There's turbulence until flow becomes slower (b) The pressure in the center of the bottom has to be higher, redirecting water upwards. This might be keeping tea leaves out of the center when rotation is fast
– mrfish
Aug 23 at 13:07
A few theories from looking at a gif of the paradox: (a) There's turbulence until flow becomes slower (b) The pressure in the center of the bottom has to be higher, redirecting water upwards. This might be keeping tea leaves out of the center when rotation is fast
– mrfish
Aug 23 at 13:07
It looks like the leaves are moving to the center only while the surface is becoming flat. Thus they appear to be moved by the flow created by the higher water level at the edge going down. This flow relates to the change in the speed of rotation, but not to the rotation at a constant soeed, as the answer describes.
– safesphere
Aug 23 at 15:47
It looks like the leaves are moving to the center only while the surface is becoming flat. Thus they appear to be moved by the flow created by the higher water level at the edge going down. This flow relates to the change in the speed of rotation, but not to the rotation at a constant soeed, as the answer describes.
– safesphere
Aug 23 at 15:47
Is it the becoming low-speed or the being low-speed that makes the leaves accumulate? Does it work if the tea starts spinning slowly? My theory was that the answer I gave works, but turbulence or the pressure causing upward flow messes with it at high $/omega$
– mrfish
Aug 24 at 2:54
Is it the becoming low-speed or the being low-speed that makes the leaves accumulate? Does it work if the tea starts spinning slowly? My theory was that the answer I gave works, but turbulence or the pressure causing upward flow messes with it at high $/omega$
– mrfish
Aug 24 at 2:54
 |Â
show 2 more comments
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4
Possible duplicate of Vortex in liquid collects particles in center
– Blue
Aug 22 at 22:50
@Blue Thanks for the link. However, please see the edit at the bottom of the question explaining why this is not a duplicate.
– safesphere
Aug 22 at 23:14
1
Possible duplicate of What forces are at work causing sand to migrate to the centre of a spinning bucket of water?
– John Rennie
Aug 23 at 4:26
@JohnRennie Thanks for the link with much better answers than in the earlier link. However, the scope of my question is broader, as it also includes the case of a rotating turntable and it is not self evident why the leaves would collect st the edge rather than, say, equally everywhere. This case is not explicitly addressed in the linked question, so it is not a duplicate. PS. Despite you masterfully dropping an ace name for a huge reward, the answer of Calmarius appears more plausible, because indeed the leaves do not go to the center until the rotation nearly ends and they don't flow up.
– safesphere
Aug 23 at 4:53