Vortex Physics Explained: Bernoulli's Principle & Beaker Resistance

[STOKER]

Hi Guys,
Please excuse my ignorance. I was wondering about the physics involved in the vortex created by using a stirrer bar in a beaker of water. Is this Bernoulli's Principle, of balancing speeds and pressures? And does the resistance offered by the sidewall of the beaker completely change the principles involved in this case, to what happens in a tornado?
If you could dumb it right down for me it would be appreciated.
Thanks in advance.
STOKER

 

[gsal]

I am no fluids guy, and don't quite know how sophisticated an answer you are looking for, but, for what is worth, here are my 2 cents...being shapeless, water simply tries to flatten down in the direction of strongest force/gravity...when spinning inside a beaker, it would be a balance between gravity in one direction and centrifugal force in the other...does this help?

 

[STOKER]

I am no fluids guy, and don't quite know how sophisticated an answer you are looking for, but, for what is worth, here are my 2 cents...being shapeless, water simply tries to flatten down in the direction of strongest force/gravity...when spinning inside a beaker, it would be a balance between gravity in one direction and centrifugal force in the other...does this help?

Hi gsal,
thanks for the reply, it's much appreciated. I'm game for a greater degree of complexity in your explanation. The vortex formed is funnel- shaped, and I was wondering why the radius diameter at the base is narrow, compared to the greater width at the top, and would the water speed vary with increasing distance (height) from the stirrer bar (or as a result of the curved path [angular velocity?] of the water body from the central vertical rotating axis), and therefore would this bring into play Bernoulli's principle, with a balancing of speeds and pressures. Also would the increase in pressure at the bottom of the beaker due simply to the height of the water column be significant to affect the phenomenon. I hope I have conveyed my questions with clarity. Thanks again.

 

[gsal]

Yes and no ;-)

Yes, I would say that the small funnel radius at the bottom has to do with the higher column of water above that point, that keeps the water from "falling down" over other water as it is being pulled by centrifugal force. So it is a balancing act between those the two forces and I am thinking this exercise can be solved simply with free body diagram approach.

But, no, I don't particularly think Bernoulli comes into play here as I think the same funnel can be produced in an alternate manner with "fluid velocity" of zero...I propose a slightly different exercise where there is no stirrer bar and instead the entire beaker and water system is placed on top of a rotating base and the entire beaker-water system is brought into final rotating steady state in which there is no relative motion between water and beaker or between water and water...heck, you can even use a non-fluid, like small slippery beads, and yet, mass would flow to minimize potential energy and balance weight vs centrifugal forces.

 

[STOKER]

Yes and no ;-)

Yes, I would say that the small funnel radius at the bottom has to do with the higher column of water above that point, that keeps the water from "falling down" over other water as it is being pulled by centrifugal force. So it is a balancing act between those the two forces and I am thinking this exercise can be solved simply with free body diagram approach.

But, no, I don't particularly think Bernoulli comes into play here as I think the same funnel can be produced in an alternate manner with "fluid velocity" of zero...I propose a slightly different exercise where there is no stirrer bar and instead the entire beaker and water system is placed on top of a rotating base and the entire beaker-water system is brought into final rotating steady state in which there is no relative motion between water and beaker or between water and water...heck, you can even use a non-fluid, like small slippery beads, and yet, mass would flow to minimize potential energy and balance weight vs centrifugal forces.

Thanks again gsal,
I don't doubt that your understanding is far superior to mine, but I'll need to think further about it as my understanding of it is still a bit fragile

 

[stedwards]

It's something like an inverted tornado.

The rotor flings fluid at the bottom toward the beaker sides. This flows upward along the sides of the container that must go somewhere. At the surface the fluid has angular momentum, and is forced to the center. As it converges to the center, the angular momentum or angular velocity doesn't go away. This generates a dip or funnel shape at the center.

If the fluid where only driven in circulation by friction or drag by the container walls, the surface shape would be concave, or bowl shaped. But the surface curvature is opposite; it is convex.

 

[STOKER]

It's something like an inverted tornado.

The rotor flings fluid at the bottom toward the beaker sides. This flows upward along the sides of the container that must go somewhere. At the surface the fluid has angular momentum, and is forced to the center. As it converges to the center, the angular momentum or angular velocity doesn't go away. This generates a dip or funnel shape at the center.

If the fluid where only driven in circulation by friction or drag by the container walls, the surface shape would be concave, or bowl shaped. But the surface curvature is opposite; it is convex.

Thanks for your input stedwards.My brain is quite slow at processing this sort of stuff

 

[gsal]

o.k., I will try to take another shot at an explanation in keeping up with your idea of a stirrer bar and a stationary beaker...but let's change the stirrer bar for something more steady and symmetric...just a simple, smooth rod being used as a drill bit in a drill...as if you were making a milkshake.

In as much as I am going to try to stay in your frame of reference, I do want you to visualize and keep up with a few of things though...

First, cylindrical coordinate system with origin at the bottom and center point of the cylindrical beaker. r radially out, of course, phi around and z up along the axis of the beaker.

Then, I am sure you agree, Earth gravitational force is acting in the -z direction; then, I want you to think of the centrifugal force as another gravitational force acting (perpendicular to the Earth one) in the r direction, radially out all the way around. For Earth gravity, the bottom of the beaker is the floor, ground and the surface where things tend to fall down to and stop there; then, rotate your head 90 degrees and think for the centrifugal-force (CF) "gravity" that the walls of the beaker are the floor and ground and the surface things tend to fall down to due to this CF "gravity"...getting this?

Hopefully, just thinking of these two forces acting on the same mass, you can start visualizing a net force resulting from these two components that starts pointing towards the bottom corner of the beaker...and that alone should start giving you a sense of why you get a "ramp" made of water (look only at left or right half of the cylinder) from the top of the beaker to the bottom center.

O.k., boundary condition: As you know, because of friction, there is no slip between beaker and water and so, the velocity of water along the walls of the beaker is zero. But the velocity is also zero along the bottom of the beaker...it is very important not to forget about this because it tends to eliminate the ability to produce angular momentum on the water as you approach the bottom of the beaker...so, between this condition and the column of water above this water...the radius of the funnel is necessarily smaller down here.

Getting back to the source of angular momentum...let's put the smooth rod in a drill and place down the center of the beaker, half way down for example, and rotate at, say, 500 rpm.

Boundary condition: Again, the slip between rod and water is zero and so, the water in contact with the rod moves just as fast and because of friction within water, the angular momentum starts spreading radially out (but it is being held down at r = wall of beaker). Velocity of water would be continuous, though, highest at the rod, zero along beaker surfaces.

Once water starts also experiencing CF gravity in the radial direction, it will start separating from the rod and start falling down in this direction, too, and there is no wall (lid) there along the top of the beaker to stop it...so, it starts rolling down a ramp "down" in the r direction and also "sideways" (up) in the z direction (you need to tilt your head here!).

For a given finite volume of water, its final position would be a balancing act between Earth gravity (and column of water above this point) and CF gravity due to the angular velocity at the same point.

gsal

 

[stedwards]

I don't understand, gsal. The question was about a rotor (probably a rotating magnet) at the bottom of a beaker rather than a rotating rod.

 

[STOKER]

o.k., I will try to take another shot at an explanation in keeping up with your idea of a stirrer bar and a stationary beaker...but let's change the stirrer bar for something more steady and symmetric...just a simple, smooth rod being used as a drill bit in a drill...as if you were making a milkshake.

In as much as I am going to try to stay in your frame of reference, I do want you to visualize and keep up with a few of things though...

First, cylindrical coordinate system with origin at the bottom and center point of the cylindrical beaker. r radially out, of course, phi around and z up along the axis of the beaker.

Then, I am sure you agree, Earth gravitational force is acting in the -z direction; then, I want you to think of the centrifugal force as another gravitational force acting (perpendicular to the Earth one) in the r direction, radially out all the way around. For Earth gravity, the bottom of the beaker is the floor, ground and the surface where things tend to fall down to and stop there; then, rotate your head 90 degrees and think for the centrifugal-force (CF) "gravity" that the walls of the beaker are the floor and ground and the surface things tend to fall down to due to this CF "gravity"...getting this?

Hopefully, just thinking of these two forces acting on the same mass, you can start visualizing a net force resulting from these two components that starts pointing towards the bottom corner of the beaker...and that alone should start giving you a sense of why you get a "ramp" made of water (look only at left or right half of the cylinder) from the top of the beaker to the bottom center.

O.k., boundary condition: As you know, because of friction, there is no slip between beaker and water and so, the velocity of water along the walls of the beaker is zero. But the velocity is also zero along the bottom of the beaker...it is very important not to forget about this because it tends to eliminate the ability to produce angular momentum on the water as you approach the bottom of the beaker...so, between this condition and the column of water above this water...the radius of the funnel is necessarily smaller down here.

Getting back to the source of angular momentum...let's put the smooth rod in a drill and place down the center of the beaker, half way down for example, and rotate at, say, 500 rpm.

Boundary condition: Again, the slip between rod and water is zero and so, the water in contact with the rod moves just as fast and because of friction within water, the angular momentum starts spreading radially out (but it is being held down at r = wall of beaker). Velocity of water would be continuous, though, highest at the rod, zero along beaker surfaces.

Once water starts also experiencing CF gravity in the radial direction, it will start separating from the rod and start falling down in this direction, too, and there is no wall (lid) there along the top of the beaker to stop it...so, it starts rolling down a ramp "down" in the r direction and also "sideways" (up) in the z direction (you need to tilt your head here!).

For a given finite volume of water, its final position would be a balancing act between Earth gravity (and column of water above this point) and CF gravity due to the angular velocity at the same point.

gsal

Thank you gsal. That was most kind of you to go to so much trouble to explain it to me. And yes stedwards, I was referring to a magnetic rod at the bottom of the beaker (stirrer bar). The principle would be similar though wouldn't it?

 

[gsal]

Sorry, guys, it is true...the concept of magnetic stirrer bar did not enter my mind at all on the above commentaries...laboratory equipment is not part of my every-day life or even part of my every-10year life; words like beaker do send me several decades back to my high school chem-lab days, where, by the way, we did have a stirring rod (bar, whatever ) to stir substances the same way one uses a spoon to stir cream into coffee...so, this is what I imagined was the source of angular momentum; hence the drill bit substitution.

I think this misunderstanding happens too often around here, where a post author writes a minimal title and some description that sounds just fine given this other great context inside his/her mind. It may be enough for certain people, with similar mindset or similar job...but posts that are more self-contained and self-explanatory open the door to a lot more respondents from other disciplines and other knowledge which just may have the answer the OP is looking for.

Needless to say, the word magnet/ic did not appear anywhere in the OP and only in post #9 as a clarification. Magnetic stirrer bar, eh? I understand now why stedwards was talking about flinging. It's funny, I thought about bringing a mixer and a blender into my descriptions...I guess a blender would have been pretty close.

Anyway, hopefully most of my commentaries still apply regardless of the source of angular momentum, far enough from the stirrer bar, that is. Sure, there is probably some flinging (mostly radial) right at the bar and some eddy currents as such mass needs to be replaced by some other flowing in the -z direction and entering the plane of the stirrer bar for some more flinging.

Oh well...hope the whole thing still helps in visualizing what may be happening with a spinning fluid in a container.

gsal

 

[stedwards]

Ah, you were thinking of a bar oriented vertically rather than horizontally.

 

[STOKER]

I think this misunderstanding happens too often around here, where a post author writes a minimal title and some description that sounds just fine given this other great context inside his/her mind. It may be enough for certain people, with similar mindset or similar job...but posts that are more self-contained and self-explanatory open the door to a lot more respondents from other disciplines and other knowledge which just may have the answer the OP is looking for.

Needless to say, the word magnet/ic did not appear anywhere in the OP and only in post #9 as a clarification. Magnetic stirrer bar, eh? I understand now why stedwards was talking about flinging. It's funny, I thought about bringing a mixer and a blender into my descriptions...I guess a blender would have been pretty close.gsal

Hi gsal, guilty as charged. I agree fully with your criticisms, and am chastened by them. Your points are so valid, they are probably worth incorporating into the forum rules/guidelines, whereby all experiments described must include measurements/dimensions/scale, and an adequately detailed description. Apologies for wasting your time through my inadequate description.

 

[stedwards]

Don't worry about it stoker. Formulating a question is a difficult art.

I'm at the level of finger painting, but plan to move up to paint-by-the-numbers real soon. Currently I'm giving a mentor a difficult time over topological issues for my poor questioning performance.

 

[STOKER]

Don't worry about it stoker. Formulating a question is a difficult art.

I'm at the level of finger painting, but plan to move up to paint-by-the-numbers real soon. Currently I'm giving a mentor a difficult time over topological issues for my poor questioning performance.

Thanks for the moral support stedwards, it's appreciated

 

[STOKER]

It's something like an inverted tornado.

The rotor flings fluid at the bottom toward the beaker sides. This flows upward along the sides of the container that must go somewhere. At the surface the fluid has angular momentum, and is forced to the center. As it converges to the center, the angular momentum or angular velocity doesn't go away. This generates a dip or funnel shape at the center.

If the fluid where only driven in circulation by friction or drag by the container walls, the surface shape would be concave, or bowl shaped. But the surface curvature is opposite; it is convex.

Hi stedwards,
So does this mean that there is a velocity gradient the further you move from the magnetic stirrer bar, both horizontally and vertically. But if there is a greater volume of water in the lower half of the beaker due to the weight of the overlying water column, wouldn't that mean that less energy would be required to rotate the water in the top half of the beaker at the same speed as the lower half? So wouldn't this be a balancing of speed and pressure i.e. Bernoulli's principle?

 

[stedwards]

Um. Just reread the rest of my post #6. I thought it was pretty clear, but maybe not.

If it were me, i might be asking "why is there a funnel shape at the top. Is this your focus?

 

[STOKER]

Um. Just reread the rest of my post #6. I thought it was pretty clear, but maybe not.

If it were me, i might be asking "why is there a funnel shape at the top. Is this your focus?

Hi stedwards,
Sorry for having such a thick skull, but yes, that is my focus