How Do Molecules Behave Near a Faucet According to the Continuity Principle?

[arashmh]

From the viewpoint of the continuity principle, we know that the stream of water is fatter near the mouth of the faucet and skinner lower down.

The question is how single molecules understand when/how they should deviate from their perpendicular free fall to a deviated one ?

 

[I like Serena]

Welcome to PF, arashmh!

The molecules in the middle have no choice, they're just along for the drive, mostly driven by gravity through which they have to accelerate.

The molecules on the edge are drawn in by the adhesion force.

 

[arashmh]

Welcome to PF, arashmh!

The molecules in the middle have no choice, they're just along for the drive, mostly drive by gravity in which they have to accelerate.

The molecules on the edge are drawn in by the adhesion force.

Thanks Serena , I know that the only available force for molecules at the edge is adhesion force , but why the adhesion force gets stronger as the molecules go down (and get more speed ) ?

 

[I like Serena]

Thanks Serena , I know that the only available force for molecules at the edge is adhesion force , but why the adhesion force gets stronger as the molecules go down (and get more speed ) ?

Uhh... The adhesion force get weaker as the molecules go down.

It has to, since the molecules on the edge can never reach the center.

 

[rcgldr]

The mass flow across any horizontal plane is constant, so the cross sectional area of the flow decreases as the flow speeds up. If there's enough adhesion, the flow just narrows, if not, the flow breaks up.

 

[klimatos]

From the viewpoint of the continuity principle, we know that the stream of water is fatter near the mouth of the faucet and skinner lower down.

The question is how single molecules understand when/how they should deviate from their perpendicular free fall to a deviated one ?

One factor you might consider is Bernoulli's Principle. As the speed of the falling water increases, the pressure differential of the surrounding air increases. This forces a smaller diameter column.

 

[cjl]

One factor you might consider is Bernoulli's Principle. As the speed of the falling water increases, the pressure differential of the surrounding air increases. This forces a smaller diameter column.

This isn't really relevant to the problem - in this case, the static pressure of the water won't change (even though it is speeding up). The water is speeding up by converting gravitational potential energy to kinetic, so no loss of static pressure is necessary (and in fact the pressure inside the column will be effectively equal to the ambient pressure the whole time).

Instead, the reason is simple continuity. The mass (and volumetric) flow rate of the water must be the same near the bottom of the column of water as it is near the top, since clearly water isn't being generated out of thin air in the middle of the column. Since the water is speeding up due to gravity, the cross sectional area must decrease to keep the flow rate the same. The surface tension of the water is what keeps it as a single column, rather than splitting up into several separate streams or drops.

 

[arashmh]

One factor you might consider is Bernoulli's Principle. As the speed of the falling water increases, the pressure differential of the surrounding air increases. This forces a smaller diameter column.

klimatos , how about when we repeat this with all surrounding air removed ? I guess it's not related to the air velocity or air pressure surrounding the water stream

 

[arashmh]

This isn't really relevant to the problem - in this case, the static pressure of the water won't change (even though it is speeding up). The water is speeding up by converting gravitational potential energy to kinetic, so no loss of static pressure is necessary.

Instead, the reason is simple continuity. The mass (and volumetric) flow rate of the water must be the same near the bottom of the column of water as it is near the top, since clearly water isn't being generated out of thin air in the middle of the column. Since the water is speeding up due to gravity, the cross sectional area must decrease to keep the flow rate the same. The surface tension of the water is what keeps it as a single column, rather than splitting up into several separate streams or drops.

Dear Cjl, we all have the same knowledge about a mathematical principle called continuity or mass conservation. i c what u mean but I'm asking about what happens at the molecular level. all molecules at one horizontal cross section of the water stream have the same velocity at the beginning of the fall. The effect of gravity continues to affect all molecules. the point is about the molecules near the edge being deviated from their path to be more closer to the center to make a cone. How they "know" that they have to be deviated ? u know we are not talking about the rules, instead the basic behaviour of molecules is of interest .. now can u help in this manner?

 

[klimatos]

klimatos , how about when we repeat this with all surrounding air removed ? I guess it's not related to the air velocity or air pressure surrounding the water stream

There would be no stream. Instead, all of the water would instantly vaporize.

 

[klimatos]

This isn't really relevant to the problem - in this case, the static pressure of the water won't change (even though it is speeding up). The water is speeding up by converting gravitational potential energy to kinetic, so no loss of static pressure is necessary (and in fact the pressure inside the column will be effectively equal to the ambient pressure the whole time).

I'm not talking static pressure. I'm talking dynamic pressure. The water is moving with regard to the surrounding air. Hence, the dynamic pressure of the moving water against the static surrounding air is reduced. This pressure differential causes the surface water to move inward. It will not do so without the application of some force. What force do you suggest causes this movement?

[/QUOTE]Instead, the reason is simple continuity. The mass (and volumetric) flow rate of the water must be the same near the bottom of the column of water as it is near the top, since clearly water isn't being generated out of thin air in the middle of the column. Since the water is speeding up due to gravity, the cross sectional area must decrease to keep the flow rate the same. [/QUOTE]

This is certainly good classical fluid dynamics. It explains what happens, but it doesn't explain why. The question remains what force causes the water to move inward? Surface tension? I don't think the magnitude is sufficient, but I'm willing to be convinced.

 

[arashmh]

I'm not talking static pressure. I'm talking dynamic pressure. The water is moving with regard to the surrounding air. Hence, the dynamic pressure of the moving water against the static surrounding air is reduced. This pressure differential causes the surface water to move inward. It will not do so without the application of some force. What force do you suggest causes this movement?

Instead, the reason is simple continuity. The mass (and volumetric) flow rate of the water must be the same near the bottom of the column of water as it is near the top, since clearly water isn't being generated out of thin air in the middle of the column. Since the water is speeding up due to gravity, the cross sectional area must decrease to keep the flow rate the same. [/QUOTE]

This is certainly good classical fluid dynamics. It explains what happens, but it doesn't explain why. The question remains what force causes the water to move inward? Surface tension? I don't think the magnitude is sufficient, but I'm willing to be convinced.[/QUOTE]

uhu Klimatus! that's it! I'm looking for the force that causes the molecules to move inward !

 

[Drakkith]

Two forces combine to cause this effect. First, there is an attractive force between water molecules due to hydrogen bonds. Then we have gravity of course. As the water exiting the tap begins to accelerate due to gravity, it pulls on the water behind it through the intermolecular forces. Imagine you have two balls attached by a piece of rope. Set the balls as far apart on the ground as the rope will allow and then go and pull the middle of the rope in perpendicular direction to the rope. What happens? The balls don't move parallel to your pull, they move inward until the contact each other! A similar effect is happening here.

 

[cjl]

I'm not talking static pressure. I'm talking dynamic pressure. The water is moving with regard to the surrounding air. Hence, the dynamic pressure of the moving water against the static surrounding air is reduced. This pressure differential causes the surface water to move inward. It will not do so without the application of some force. What force do you suggest causes this movement?

Unfortunately, you seem to be mixing up dynamic and static pressure here. As the speed of motion of a fluid is increased, the dynamic pressure increases. In fact, this is the reason behind the typical bernoulli relation - in the absence of external forces, the total or stagnation pressure (which is static pressure plus dynamic pressure) is constant. So, if the flow speeds up, the dynamic pressure is increased, which necessarily causes a reduction in static pressure. However, in this case, there is an external force - gravity. The work done by gravity is the cause for the increasing dynamic pressure, and the static pressure stays constant.

As for the force? I already said - the reason is the surface tension of the water.

 

[mrspeedybob]

Water molecules are polar, they have a + charged end and a - charged end. If you put 2 of them next to each other they will try to align so the + end of 1 sticks to the - end of the other. If you put a bunch of them together they will all try to align into positions that cause them to attract, this is what causes water to have surface tension. A water molecule on the edge of the stream doesn't separate because it is electrostaticly attracted to the next one in, which attracted to a few others further in, and so forth.

 

[PatrickPowers]

From the viewpoint of the continuity principle, we know that the stream of water is fatter near the mouth of the faucet and skinner lower down.

The question is how single molecules understand when/how they should deviate from their perpendicular free fall to a deviated one ?

The water falls faster, so the same amount of water has to stretch to fill more length. That is, the amount of water per second stays the same, but the length goes up with the square of time.

You see the same thing in a river. Rapids are always shallower and/or narrower than the rest of the river, and slow-moving portions are deeper and/or wider.

 

[PatrickPowers]

From the viewpoint of the continuity principle, we know that the stream of water is fatter near the mouth of the faucet and skinner lower down.

The question is how single molecules understand when/how they should deviate from their perpendicular free fall to a deviated one ?

Aha, I haven't answered your question. How does a single molecule know? It has to do with

1) water molecules attract one another so the water doesn't "want' to turn into a spray.

2) Other molecules are moving away downward and upward so there is space opening up toward the center. The molecules "have to" move in there to prevent spray.

In other words, it's minimal energy. It would take energy to spread out or compress the stream any more than it is. But that is sort of tautological.

 

[arashmh]

Two forces combine to cause this effect. First, there is an attractive force between water molecules due to hydrogen bonds. Then we have gravity of course. As the water exiting the tap begins to accelerate due to gravity, it pulls on the water behind it through the intermolecular forces. Imagine you have two balls attached by a piece of rope. Set the balls as far apart on the ground as the rope will allow and then go and pull the middle of the rope in perpendicular direction to the rope. What happens? The balls don't move parallel to your pull, they move inward until the contact each other! A similar effect is happening here.

Dear Drakkith
Thanks for your interesting example. I assume that in this analogy, the rope plays the role of gravity force , right ? why should we just pull the middle rope while gravity acts on all molecules? If we pull every rope upward , all balls will displace horizontally for a little and then they all move parallel to my pull .. isn't it the case ?

 

[arashmh]

Aha, I haven't answered your question. How does a single molecule know? It has to do with

1) water molecules attract one another so the water doesn't "want' to turn into a spray.

2) Other molecules are moving away downward and upward so there is space opening up toward the center. The molecules "have to" move in there to prevent spray.

In other words, it's minimal energy. It would take energy to spread out or compress the stream any more than it is. But that is sort of tautological.

Thats a good point, can u explain how preventing from being a spray bring the energy to its minimal level ?

 

[Drakkith]

Dear Drakkith
Thanks for your interesting example. I assume that in this analogy, the rope plays the role of gravity force , right ? why should we just pull the middle rope while gravity acts on all molecules? If we pull every rope upward , all balls will displace horizontally for a little and then they all move parallel to my pull .. isn't it the case ?

No, the rope acts like the intermolecular force. Your pulling acts like gravity.

 

[PatrickPowers]

Thats a good point, can u explain how preventing from being a spray bring the energy to its minimal level ?

The molecules attract one another. That means it would take work/energy to move the molecules apart to make a spray. Anything that takes work/energy by definition takes the water out of its minimal energy state, so you can expect that to not happen spontaneously.

 

[arashmh]

No, the rope acts like the intermolecular force. Your pulling acts like gravity.

I c , by the rope i meant the pulling of the rope. now , in your analogy we pull the central rope only or all ropes ?

 

[arashmh]

The molecules attract one another. That means it would take work/energy to move the molecules apart to make a spray. Anything that takes work/energy by definition takes the water out of its minimal energy state, so you can expect that to not happen spontaneously.

uhu, can we formulate the problem to c when the stream finally divides into multiple streams and sprays?

 

[klimatos]

uhu Klimatus! that's it! I'm looking for the force that causes the molecules to move inward ![/QUOTE]

Exactly. And that force is the pressure differential brought about by the water moving with respect to the surrounding air.

This Bernoulli Effect is often ignored or overlooked. A force that can pluck a forty-pound sheet of plywood out of the bed of a pickup truck, can lift a fully-loaded 747 into the skies, and can suck two passing ships together so strongly that considerable steering effort is required to avoid a collision will find constricting a column of tap water to be mere child's play.

 

[arashmh]

uhu Klimatus! that's it! I'm looking for the force that causes the molecules to move inward !

Exactly. And that force is the pressure differential brought about by the water moving with respect to the surrounding air.

This Bernoulli Effect is often ignored or overlooked. A force that can pluck a forty-pound sheet of plywood out of the bed of a pickup truck, can lift a fully-loaded 747 into the skies, and can suck two passing ships together so strongly that considerable steering effort is required to avoid a collision will find constricting a column of tap water to be mere child's play.[/QUOTE]

can we formulate the problem to c when the stream finally divides into multiple streams and sprays?

 

[cjl]

uhu Klimatus! that's it! I'm looking for the force that causes the molecules to move inward !

Exactly. And that force is the pressure differential brought about by the water moving with respect to the surrounding air.

This Bernoulli Effect is often ignored or overlooked. A force that can pluck a forty-pound sheet of plywood out of the bed of a pickup truck, can lift a fully-loaded 747 into the skies, and can suck two passing ships together so strongly that considerable steering effort is required to avoid a collision will find constricting a column of tap water to be mere child's play.

Except that it isn't relevant in this case for the reason that I've already explained. Twice.

The bernoulli effect works just fine, but in this particular case, there is no change in static pressure as the water falls. Thus, there is no pressure force to constrict the flow.

 

[russ_watters]

Agreed: gravitational head is converted to velocity pressure. Static pressure remains constant at zero (gauge pressure)...

...and reducing atmospheric pressure will have virtually no effect on the shape of the cone. An actual vacuum makes life tough, as the water would boil as it falls, but I suspect for a while it would remain similar.

 

[arashmh]

Agreed: gravitational head is converted to velocity pressure. Static pressure remains constant at zero (gauge pressure)...

...and reducing atmospheric pressure will have virtually no effect on the shape of the cone. An actual vacuum makes life tough, as the water would boil as it falls, but I suspect for a while it would remain similar.

so , up to now, the summary is this : the conic form of the water stream has nothing to do with the air surrounding it , so bernulli goes out of the consideration.

we know that a force pulls the molecules toward the center . if it's not the pressure of air surrounding it , the only remaining choice is the adhesion force between molecules, right ? now assume that in a fictionary case no molecule is pulled toward center and the water stream continues to flow like a perfect column in rectangular shape . put aside the continuity , what will be wrong then in "molecular" level ? do we have violated something in this case ?

 

[cjl]

The problem is that for that to be true, either the density of the flow would have to decrease or the velocity of the flow would have to be constant. Otherwise, more water would be flowing past a given point near the bottom of the column than near the top, which is clearly not possible.

 

[klimatos]

can we formulate the problem to c when the stream finally divides into multiple streams and sprays?

All real-world laminar flows contain irregularities. As the water velocity increases, these irregularities amplify to become turbulence. In falling water, these turbulent flows tend to break the stream apart. Eventually, you get spray.

The forces that turn laminar flows into spray are internal to the flow--not usually external, although external forces can add to the turbulence.

 

[klimatos]

Let us put this issue to rest with a simple experiment that we all can perform.

Take a small piece of waxed paper and press it against the side of the faucet outlet with your thumb. Let it hang straight down. Now turn on the water. You will note that the waxed paper moves inward as soon as the water flows down the stream side of the paper.

This illustrates the Bernoulli Principle that the flow of a fluid across a surface creates a drop in pressure on that surface. Since the pressure on the dry side of the membrane remains the same, the membrane move inward. On the opposite side of the stream, the pressures differential is directly on the air-water interface. Once again, that surface moves inward. Haven't you all noticed that the shower curtain moves toward the shower spray, and not away from it?

Addendum: I suggest using waxed paper to avoid any chance of using surface tension to explain the movement. Wax and water repel one another, not attract.

 

[arashmh]

Let us put this issue to rest with a simple experiment that we all can perform.

Take a small piece of waxed paper and press it against the side of the faucet outlet with your thumb. Let it hang straight down. Now turn on the water. You will note that the waxed paper moves inward as soon as the water flows down the stream side of the paper.

This illustrates the Bernoulli Principle that the flow of a fluid across a surface creates a drop in pressure on that surface. Since the pressure on the dry side of the membrane remains the same, the membrane move inward. On the opposite side of the stream, the pressures differential is directly on the air-water interface. Once again, that surface moves inward. Haven't you all noticed that the shower curtain moves toward the shower spray, and not away from it?

Addendum: I suggest using waxed paper to avoid any chance of using surface tension to explain the movement. Wax and water repel one another, not attract.

Thats a clever point u pulled in :)
but i feel it has to do something with molecules of water , not the air surrounding it.

 

[klimatos]

Except that it isn't relevant in this case for the reason that I've already explained. Twice.

The bernoulli effect works just fine, but in this particular case, there is no change in static pressure as the water falls. Thus, there is no pressure force to constrict the flow.

cjl: Saying something multiple times does not make it true. Why are you referring to static pressures in what is obviously a dynamic situation?

From the standpoint of kinetics, there is no static pressure in moving water. There are only dynamic pressures that vary from time to time at any given place and from place to place at any given time. These variations allow streams to pluck heavy rocks from their beds.

Finally, the Bernoulli Principle is always in effect whenever you have a fluid in motion or an object in motion relative to and in contact with a fluid.

 

[Drakkith]

I c , by the rope i meant the pulling of the rope. now , in your analogy we pull the central rope only or all ropes ?

Ok, instead of you pulling on the rope, put a third ball in the middle of the rope. You would then act as gravity by pulling that middle ball. In the falling water each ball is connected to multiple balls (molecules) so by pulling that one, you pull others which then pull even more molecules along, stretching the falling stream out.

 

[arashmh]

cjl: Saying something multiple times does not make it true. Why are you referring to static pressures in what is obviously a dynamic situation?

From the standpoint of kinetics, there is no static pressure in moving water. There are only dynamic pressures that vary from time to time at any given place and from place to place at any given time. These variations allow streams to pluck heavy rocks from their beds.

Finally, the Bernoulli Principle is always in effect whenever you have a fluid in motion or an object in motion relative to and in contact with a fluid.

OK, bernoulli works always . but we have to keep in mind that this experiment with paper leads to the same result even if there is no curve in water stream , i.e when the flow rate of water is so tremendous!
So i c that this bernoulli effect , affects the water stream , but it;s just one of contributers , i want to know others and the reason they contribute in pulling the molecules towards the center.

 

[cjl]

cjl: Saying something multiple times does not make it true. Why are you referring to static pressures in what is obviously a dynamic situation?

Because the relevant pressure here is the static pressure. Perhaps you're confusing it with the stagnation pressure? The static pressure is absolutely relevant in a dynamic situation - in fact, the static pressure is really the only pressure you care about directly in a dynamic situation. The static pressure describes the pressure exerted by the fluid. Dynamic pressure is merely a method of measuring the kinetic energy of the fluid, and the dynamic pressure is always equal to 1/2*ρv².

To go into somewhat more detail, since the entire fluid column is exposed to the atmosphere, a force balance requires that the static pressure in the entire fluid column is equal to the atmospheric pressure surrounding it. The increase in velocity is coming from potential energy (or, as russ correctly put it using slightly different terminology, it comes from the gravitational head). This is causing an increase in the stagnation or total pressure, as the static pressure is staying constant and the dynamic pressure is increasing. Often, the Bernoulli equation is written in such a way as to reflect this term, specifically:

P+1/2ρv²+ρgh = constant

In this particular case, h is decreasing and v is increasing, while P is constant throughout the column.

From the standpoint of kinetics, there is no static pressure in moving water. There are only dynamic pressures that vary from time to time at any given place and from place to place at any given time. These variations allow streams to pluck heavy rocks from their beds.

Perhaps you should brush up on your fluid dynamics terminology.

To go back to the bernoulli equation as I have written it above, for an inviscid, incompressible, lossless flow, P+1/2ρv²+ρgh = constant.

This is typically broken down into three separate components. P is the static pressure, and it describes the actual pressure felt by an area at that location in the flow. 1/2ρv² is the dynamic pressure, and it describes the kinetic energy of the flow (or, to look at it another way, it is the amount of pressure that would be gained if the flow were slowed down to a stop with no losses). Finally, ρgh is an external force term describing the influence of an external force (in this case, gravity). If other external forces are present, this term can be generalized to include them as well.

When people using the bernoulli relationship talk about pressure dropping as velocity increases, they are typically referring to the simplified case where there is no external force term. In this case, the equation simplifies to P + 1/2ρv² = constant. In this case, as the velocity of the flow is increased (perhaps by running it through a constriction in a pipe), it is clear that the dynamic pressure term increases. Since the stagnation (or total) pressure is constant, this requires that the static pressure decrease. Since static pressure is in fact what most people are referring to when they refer to pressure, this allows for the rough generalization that pressure decreases as velocity increases.

Since this simplification relies on no external force however, it does not apply in this case.

Finally, the Bernoulli Principle is always in effect whenever you have a fluid in motion or an object in motion relative to and in contact with a fluid.

Well, the Bernoulli principle always applies when you have an inviscid, incompressible, lossless flow. However, in this case, those approximations are fairly valid. However, that doesn't mean that you are applying the Bernoulli principle correctly. In this case, you are neglecting the presence of the external force term, which is why you are getting incorrect results.


Chris
(Graduate student in aerospace engineering focusing on fluid mechanics and propulsion)

 

[arashmh]

Because the relevant pressure here is the static pressure. Perhaps you're confusing it with the stagnation pressure? The static pressure is absolutely relevant in a dynamic situation - in fact, the static pressure is really the only pressure you care about directly in a dynamic situation. The static pressure describes the pressure exerted by the fluid. Dynamic pressure is merely a method of measuring the kinetic energy of the fluid, and the dynamic pressure is always equal to 1/2*ρv².

To go into somewhat more detail, since the entire fluid column is exposed to the atmosphere, a force balance requires that the static pressure in the entire fluid column is equal to the atmospheric pressure surrounding it. The increase in velocity is coming from potential energy (or, as russ correctly put it using slightly different terminology, it comes from the gravitational head). This is causing an increase in the stagnation or total pressure, as the static pressure is staying constant and the dynamic pressure is increasing. Often, the Bernoulli equation is written in such a way as to reflect this term, specifically:

P+1/2ρv²+ρgh = constant

In this particular case, h is decreasing and v is increasing, while P is constant throughout the column.


Perhaps you should brush up on your fluid dynamics terminology.

To go back to the bernoulli equation as I have written it above, for an inviscid, incompressible, lossless flow, P+1/2ρv²+ρgh = constant.

This is typically broken down into three separate components. P is the static pressure, and it describes the actual pressure felt by an area at that location in the flow. 1/2ρv² is the dynamic pressure, and it describes the kinetic energy of the flow (or, to look at it another way, it is the amount of pressure that would be gained if the flow were slowed down to a stop with no losses). Finally, ρgh is an external force term describing the influence of an external force (in this case, gravity). If other external forces are present, this term can be generalized to include them as well.

When people using the bernoulli relationship talk about pressure dropping as velocity increases, they are typically referring to the simplified case where there is no external force term. In this case, the equation simplifies to P + 1/2ρv² = constant. In this case, as the velocity of the flow is increased (perhaps by running it through a constriction in a pipe), it is clear that the dynamic pressure term increases. Since the stagnation (or total) pressure is constant, this requires that the static pressure decrease. Since static pressure is in fact what most people are referring to when they refer to pressure, this allows for the rough generalization that pressure decreases as velocity increases.

Since this simplification relies on no external force however, it does not apply in this case.




Well, the Bernoulli principle always applies when you have an inviscid, incompressible, lossless flow. However, in this case, those approximations are fairly valid. However, that doesn't mean that you are applying the Bernoulli principle correctly. In this case, you are neglecting the presence of the external force term, which is why you are getting incorrect results.


Chris
(Graduate student in aerospace engineering focusing on fluid mechanics and propulsion)

Chris , that was awsome! now can u explain what forces the molecules to form a conic shape ?

 

[russ_watters]

Chris , that was awsome! now can u explain what forces the molecules to form a conic shape ?

Again, surface tension is the force. It acts against the differing velocity along its length.

This is similar to why a rubber band narrows when you stretch it.

The Bernoulli's explanation just doesn't fit: there isn't anything for static pressure to do and it doesn't add anything useful. Just consider what would change if you ignore the issue of pressure (answer: nothing). And consider that since pressure is a square function of velocity, it adds problems, such as why isn't the shape a parabola and is there enough pressure to move the water when it is only a few cm from the tap.

 

[klimatos]

OK, bernoulli works always . but we have to keep in mind that this experiment with paper leads to the same result even if there is no curve in water stream , i.e when the flow rate of water is so tremendous!
So i c that this bernoulli effect , affects the water stream , but it;s just one of contributers , i want to know others and the reason they contribute in pulling the molecules towards the center.

Since the Bernoulli Effect fully explains the tapering, what makes you think that there are other factors at work?

Let me add another argument to the hypothesis. The tapering of the stream is not uniform, but increases with an increase in the water velocity. This means that the force or forces that cause the constriction are functions of some power of the velocity. This automatically eliminates any inter-molecular force, since hydrogen bonding is independent of fluid flow. With the exception of the Van der Waals attraction (which is also independent of fluid flow) and possible ionic attractions (same independence) I know of no other inter-molecular bonding forces that you might offer to support an argument for internal molecular attractive forces.

The Bernoulli Effect, in contrast, is a function of the square of the velocity. It thus fully explains the increase in constriction with the increase in flow velocity.

By the way, the molecules are not being pulled toward the center, they are being pushed.

 

[russ_watters]

I'll respond in more detail later, but a clear counterexample: sand. If Bernoulli's principle were strongly at work here, a poured column of sand should form itself into a coherent stream.

 

[arashmh]

Again, surface tension is the force. It acts against the differing velocity along its length.

This is similar to why a rubber band narrows when you stretch it.

The Bernoulli's explanation just doesn't fit: there isn't anything for static pressure to do and it doesn't add anything useful. Just consider what would change if you ignore the issue of pressure (answer: nothing). And consider that since pressure is a square function of velocity, it adds problems, such as why isn't the shape a parabola and is there enough pressure to move the water when it is only a few cm from the tap.

ok, and what would have been violated if the stream didn't make a conic form with the presence of tension ?

 

[cjl]

Since the Bernoulli Effect fully explains the tapering, what makes you think that there are other factors at work?

Except that it doesn't. Read my last post for details (I'm not typing all of that out again).

Let me add another argument to the hypothesis. The tapering of the stream is not uniform, but increases with an increase in the water velocity. This means that the force or forces that cause the constriction are functions of some power of the velocity. This automatically eliminates any inter-molecular force, since hydrogen bonding is independent of fluid flow. With the exception of the Van der Waals attraction (which is also independent of fluid flow) and possible ionic attractions (same independence) I know of no other inter-molecular bonding forces that you might offer to support an argument for internal molecular attractive forces.

Actually, if you work out the details, I think you'll find that the tapering is such that the area follows a 1/v profile (actually, this must be true for continuity to be satisfied in an incompressible flow). Thus, the diameter should follow a 1/sqrt(v) profile. Since v is proportional to sqrt(h) for a freefalling object (or liquid in this case), the diameter should actually follow a 1/h^1/4 profile, and this only follows from the fact that the flow is incompressible, the flow is in freefall, and mass is conserved.

The Bernoulli Effect, in contrast, is a function of the square of the velocity. It thus fully explains the increase in constriction with the increase in flow velocity.

If your proposed mechanism increases with the square of the velocity, why does the actual profile go as the inverse square root (or directly as the inverse, if you want to look at area instead of diameter)?

By the way, the molecules are not being pulled toward the center, they are being pushed.

You really haven't shown that at all. You've simply asserted it extensively.

I'll give one more (very good) reason why your argument can't be true. If the real cause were that the static pressure decreased with velocity (which is the standard bernoulli effect), the water could never exceed a speed of approximately 14 meters per second. At 14 meters per second, the static pressure of the water would be zero, the water would boil off, and you'd be left with a cloud of water vapor (seriously). If the water were exchanging its static pressure for velocity (which is the entire idea behind the bernoulli effect), and it had a stagnation pressure of 1 atmosphere, the static pressure would be zero at just over 14 meters per second (and the dynamic pressure would be exactly 1 atmosphere). Since this clearly doesn't happen - water falls faster than 14 meters per second quite frequently (though not usually in most people's houses), the velocity must be coming from some other source (namely, gravitational potential energy or gravitational head).

 

[cjl]

ok, and what would have been violated if the stream didn't make a conic form with the presence of tension ?

Conservation of mass. The surface tension keeps the stream as a single stream rather than breaking up into droplets or several separate streams, and the conservation of mass requires that at any point, the product of the density of the fluid, the cross sectional area of the flow, and the velocity is constant (to put it more mathematically, ρ₁A₁V₁ = ρ₂A₂V₂). Since we're assuming water is incompressible, the density won't be changing, and the velocity is obviously increasing, so the only way for this to be satisfied is if the area has a corresponding decrease.

If you're wondering where that relation comes from? The cross sectional area of a flow multiplied by its velocity is the volumetric flow rate past that area (I can go into more detail if this doesn't make sense to you). So, if you take the rate of volume flow past an area, and then you multiply it by the density, you get the rate of mass flow past that area. Since mass is conserved, the rate of mass flow past an area near the top of the stream must be the same as the rate of mass flow past an area near the bottom, and that is where the relation arises.

 

[arashmh]

If you're wondering where that relation comes from? The cross sectional area of a flow multiplied by its velocity is the volumetric flow rate past that area (I can go into more detail if this doesn't make sense to you). So, if you take the rate of volume flow past an area, and then you multiply it by the density, you get the rate of mass flow past that area. Since mass is conserved, the rate of mass flow past an area near the top of the stream must be the same as the rate of mass flow past an area near the bottom, and that is where the relation arises.

Dear Cjl, i am a phd student of chemical engineering and i know every single detail about conservation of mass and I'm sure u know it too and in detail .

but my point is that , the molecules do not know any thing about conservation of mass as we do :) the thing that makes me confused is that , which forces exactly acts upon the molecules (and under what mechanism) that makes molecules behave in such a way that we can (from a higher level) describe the totality of their behaviour as conservation of mass (of water stream).

in other words, i want to know the exact mechanism in molecular level, so that if i run a molecular simulation of water molecules, I as an "observer" watch them forming a cone, did u get my point ?

 

[cjl]

But that has been explained to you several times. What keeps the water as a single, connected stream is the intermolecular attractive forces. The example of pouring sand is a great example to show what would happen in their absence - the column would remain at a constant diameter, and the density would decrease with height (you'll notice this is also a valid solution to the mass conservation relation given above). The intermolecular forces keep the water stream at a constant density, and thus they are the direct cause of the narrowing of the stream.

 

[256bits]

The flow of water exiting the end of the tube (faucet) has a velocity profile, and one has to take that into account.

http://www.engineersedge.com/fluid_flow/flow_velocity_profiles.htm

At the walls of the pipe the velocity of the water is basically zero, and depending upon laminar or turbulent flow, the velocity profile is as shown.

For laminar flow the streamlines do not mix when the flow is within the pipe, and from one section to next there is a pressure drop. At exit the pressure drop is non-existant, or we can say that the fluid surface is now at atmospheric pressure ( if exiting into the atmosphere ).

This is the part that you all have been missing:
There is a slight pressure gradient from the surface to the center due to the surface tension of the water. As the center is traveling faster than the outer surface, and due to continuity and the fact that the flow in incompressable, the flow constricts. This is most noticable on immediate exit from the pipe, where the velocity profile from the pipe continues on into the free liquid flow.

For a gravity situation, the liquid accelerates, and the flow further constricts due to continuity, the interior always at a faster velocity than that of the outer surface, until a state farther down in the flow is reached where the surface tension is able to pinch off droplets.

 

[klimatos]

I'll respond in more detail later, but a clear counterexample: sand. If Bernoulli's principle were strongly at work here, a poured column of sand should form itself into a coherent stream.

Sand is not a fluid. There are no hydrogen bonds between the grains of sand. In the water column it is the surface tension (hydrogen bonds) that keeps the stream coherent and the pressure differential that constricts the column.

If the Bernoulli Effect were not in play, how do you explain the inward movement of the waxed paper in my previous post? How do you explain a shower curtain moving toward the shower spray? Absent the Bernoulli Effect, there is no other way to explain these movements.

 

[256bits]

How do you explain a shower curtain moving toward the shower spray? Absent the Bernoulli Effect, there is no other way to explain these movements

If you are taking a hot shower, the air within the shower stall is heated and rises - colder air moving in by pushing the shower curtain inward, would be my best explanation.

 

[klimatos]

From his OP onward, arashmh has indicated that he is interested in why the phenomenon of water column tapering occurs. He has indicated that he is conversant with continuity equations, but does not want a macroscopic description of the phenomenon nor any mathematic equations explaining the parameter relationships. As he said in his OP, he wants to know WHY a surface molecule moves inward. What force or forces cause it to move? And he wants the explanation in molecular terms.

Many of the responding posts have contained equations. Some of these equations were verbal and some were in notation. Equations never explain why something happens, they only explain what happens. (“To make the equation come out right” is not an explanation. It is a cop out.) For example, Boyle’s Law doe not explain why the pressure doubles when we halve the volume of gas in a container (temperature kept constant). It only describes what happens. We must bring in kinetic gas theory and statistical mechanics to explain why.

Along this same line, continuity equations and other fluid dynamic equations and statements of principle do not explain why the surface molecules in a column of water move inward as the water velocity increases. They don’t mention molecules at all. And I have been as guilty as everyone else.

Let me correct that omission. The ambient air pressure (frequency of molecular impact times the mean impulse per impact) of the air on the air-water interface is greater than the pressure of the flowing water on that same interface. Consequently the interface moves inward until the two pressures equalize. This interface is curved and receives two forces from the water. The first is from the parallel flow which diminishes the pressure on the surface in keeping with the Bernoulli Effect. This diminution allows the interface to be pushed inward. The second force is the direct impact of the water molecules on the upper part of the curved interface. This increases the pressure on the interface and tends to push the interface outward. This second force increases the closer you get to the center of the column because interior flow is faster than surface flow. The interface comes to rest when all three of these forces balance one another.

The waxed paper experiment (Post #31) demonstrates that the Bernoulli Effect is in operation. Explanations that ignore the Bernoulli Effect must still account for the inward movement of the waxed paper. If you carefully vary the flow rate, you will note that the faster the flow the greater the deviation. No explanation involving surface tension will produce this result.

 

[256bits]

From the viewpoint of the continuity principle, we know that the stream of water is fatter near the mouth of the faucet and skinner lower down.

The question is how single molecules understand when/how they should deviate from their perpendicular free fall to a deviated one ?

Your situation is similar to flow from an orifice.

2 other situations you may find interesting:

1. Oscillatory patterns in the flow. If you have ever noticed the flow seems to be twisted or having a noticable pattern, then this is caused by the surface tension attempting to bring the flow to a minimum surface ie a circular cross section. A harmonic is set up as the inertia of the water overcompensates and overshoots the circular form. A square exit will have the sides become the corners and the corners become the sides ( not exactly , but just to expalin what happens ), and this continues down the stream.

2. A flow exiting horizontally from will have the bottom part traveling faster than the top due to the pressure difference caused by the height of the orifice. The stream will flatten out as the bottom interferes with the upper layers.

Even though the pressure differences are small, the effects are visually noticable.