Vacuum or pressure to move spaghetti through a hole

[Spinnor]

Just for reference I got up on my porch roof this morning and sucked on a long clear plastic hose stuck in a gallon of water. My sucking abilities were able to lift a column of water about 10 feet. I think that translates to a nearly 5 psi air pressure difference between the inside of my mouth and outside air pressure. Your abilities may vary.

Cooked spaghetti is also very slippery as we all know.

Assume no friction, assume that cooked spaghetti has the same density as water, assume a pressure difference of 5 psi the smart guys here should be able to calculate the theoretically longest piece of hanging cooked spaghetti that a person could suck into their mouth. I think it should be of order 10 feet.

 

[Andreas C]

I'm impressed, but I see nowhere in the equation the optimal cooking time for spaghetti at STP??

Spaghetti doesn't really cook at STP...

 

[Chestermiller]

In this post, I present the results of an analysis of hydrodynamic lubrication flow in the gap between the lips and the spaghetti noodle. The figure below shows the steady flow of a Newtonian viscous fluid situated between two infinite parallel plates, driven by a combination of an axial pressure gradient, and drag flow produced by axial movement of the upper plate at a velocity V .

By applying the methodology alluded to by Andy Resnick in post #68, we obtain the following equation for the axial velocity profile of the fluid (as a function of the cross-channel coordinate y):
$$v_x=\frac{(p_1-p_2)}{2\eta L}y(h-y)+V\frac{y}{h}\tag{1}$$where $\eta$ is the fluid viscosity. The velocity gradient at the upper boundary (aka the fluid shear rate at the upper boundary) is obtained by differentiation of Eqn. 1 with respect to y to yield:
$$\left(\frac{dv_x}{dy}\right)_{y=h}=-\frac{(p_1-p_2)h}{2\eta L}+\frac{V}{h}\tag{2}$$
From Newton's law of viscosity, the shear stress $\tau$ in the x-direction exerted by the upper plate on the fluid in the gap between the plates is equal to fluid viscosity times the velocity gradient:
$$\tau=\eta \left(\frac{dv_x}{dy}\right)_{y=h}=-\frac{(p_1-p_2)h}{2L}+\eta \frac{V}{h}\tag{3}$$
By Newton's 3rd law, this is equal and opposite to the shear stress in the x-direction exerted by the fluid on the upper plate: $$\frac{(p_1-p_2)h}{2L}-\eta \frac{V}{h}$$
In our problem, the upper (moving) plate is representative of the noodle, and the lower plate is representative of our lip. If we use the above relationship for the shear stress to carry out an axial force balance on the noodle, we obtain:$$\pi r^2(\sigma_2-\sigma_1)=2\pi rL\left(-\frac{(p_1-p_2)h}{2L}+\eta \frac{V}{h}\right)\tag{4}$$or equivalently,
$$(\sigma_2-\sigma_1)=-\frac{(p_1-p_2)h}{r}+ \frac{2\eta LV}{rh}\tag{5}$$where $\sigma$ is the axial tensile stress within the noodle.
For a horizontal rigid noodle, from analysis of the solid mechanics behavior of the portions of the noodle outside the lips, we find that:$$(\sigma_2-\sigma_1)=-(p_2-p_1)\tag{6}$$Combining Eqns. 5 and 6 then yields:
$$V=\frac{rh}{2\eta L\left[1+\frac{h}{r}\right]}(p_1-p_2)\tag{7}$$
In the case of a vertical rigid rod or a flaccid noodle containing vertical sections (or curved sections with vertical components along the contour), solid mechanics analysis must be modified to include the effect of gravity on the tensile stress variations. Therefore, for these cases, Eqn. 6 does not apply.

 

[DaTario]

I would say that the viscosity of fluids, which are present at the surface of spaghetti, plays an important role in forcing the spaghetti itself.

 

[Chestermiller]

I would say that the viscosity of fluids, which are present at the surface of spaghetti, plays an important role in forcing the spaghetti itself.

What are you saying that's different from what we have already said?

 

[jartsa]

Let us consider a soft but straight string of spaghetti. If the opposite ends are at different pressures, then there simply must be a force pushing the string: pressure difference multiplied by area.

As the spaghetti is soft, it can not transmit a pushing force. That may seem like a problem, but consider this: When we suddenly decrease the pressure at our end of a very long rigid rod, the far end can not immediately exert a pushing force on our end, but still our end starts to move immediately.

 

[Chestermiller]

Let us consider a soft but straight string of spaghetti. If the opposite ends are at different pressures, then there simply must be a force pushing the string: pressure difference multiplied by area.

As the spaghetti is soft, it can not transmit a pushing force. That may seem like a problem, but consider this: When we suddenly decrease the pressure at our end of a very long rigid rod, the far end can not immediately exert a pushing force on our end, but still our end starts to move immediately.

This is an incorrect assessment of what is happening (both for an uncooked noodle and for a cooked noodle). If you are convinced that the solid mechanical behavior of the noodle is much more important than the fluid mechanics behavior (discussed by several Science Advisers and Mentors), please provide a valid solid mechanics analysis of the situation (including stresses, strains, external loads, and deformations) based either on the Theory of Elasticity or on the (more straightforward) Strength of Materials approach. Your analysis should include actual equations and their solutions.

Chet

 

[PAllen]

I'm not convinced the solid noodle case needs any explanation except pressure. Consider replacing your lips with an airtight frictionless aperture. The solid noodle will clearly be sucked in. There is no fluid involved at all. The flaccid noodle is the interesting case.

Also, gravity can be made irrelevant by imagining the experiment in a pressurized cabin of a rocket in orbit. I do not think there would be any difference in behavior for this experiment.

 

[Andy Resnick]

Consider replacing your lips with an airtight frictionless aperture. The solid noodle will clearly be sucked in.

With all due respect, this is silly- linear slide and fluid bearing companies would love to hear from you. Please provide some evidence, as opposed to a fertile imagination, that an "airtight aperture" is frictionless, and that an "airtight frictionless aperture" allows one object to slide freely through another.

 

[PAllen]

With all due respect, this is silly- linear slide and fluid bearing companies would love to hear from you. Please provide some evidence, as opposed to a fertile imagination, that an "airtight aperture" is frictionless, and that an "airtight frictionless aperture" allows one object to slide freely through another.

It's an idealization, as often done for problem simplification. The idea is to separate what can be explained by pressure from what cannot. How well it could be approximated in the real world is a separate question e.g. I have no idea whether a precisely machined aluminum noodle and graphite apperture would approximate this.

 

[Chestermiller]

I'm not convinced the solid noodle case needs any explanation except pressure. Consider replacing your lips with an airtight frictionless aperture. The solid noodle will clearly be sucked in.

Not so fast. What do you think the stress distribution within the noodle looks like in this situation? If the part of the noodle outside the mouth is surrounded by atmospheric pressure $p_a$, the axial tensile stress within the part of the noodle outside the mouth is $-p_a$ (compression). And, if the the part of the noodle inside the mouth is surrounded by vacuum pressure $p_m$, the axial tensile stress within the part of the noodle inside the mouth is $-p_m$. So, across the aperture (the lips), the tensile stress within the noodle has to change from $-p_a$ to $-p_m$. The only way this can happen is if there is friction (shear stress) at the surface of the aperture. If there is no friction, the noodle will have to be constantly accelerating, with an acceleration equal to $(p_a-p_m)A/m$, where m is the mass of the noodle and A is its cross sectional area. However, I don't think that constant acceleration approaching this is observed in practice.

 

[PAllen]

Not so fast. What do you think the stress distribution within the noodle looks like in this situation? If the part of the noodle outside the mouth is surrounded by atmospheric pressure $p_a$, the axial tensile stress within the part of the noodle outside the mouth is $-p_a$ (compression). And, if the the part of the noodle inside the mouth is surrounded by vacuum pressure $(p_m)$, the axial tensile stress within the part of the noodle inside the mouth is $(-p_m)$. So, across the aperture (the lips), the tensile stress within the noodle has to change from $-p_a$ to $(-p_m)$. The only way this can happen is if there is friction (shear stress) at the surface of the aperture. If there is no friction, the noodle will have to be constantly accelerating, with an acceleration equal to $(p_a-p_m)A/m$, where m is the mass of the noodle and A is its cross sectional area. However, I don't think that constant acceleration approaching this is observed in practice.

But then the friction only serves to slow the rod down, producing an eventual constant velocity similar to terminal velocity of a body falling in atmosphere.

 

[Chestermiller]

But then the friction only serves to slow the rod down, producing an eventual constant velocity similar to terminal velocity of a body falling in atmosphere.

Yes. In my judgment, the terminal velocity is attained very rapidly in the noodle situation. At least, that seems to me to be a reasonable approximation. That is, the noodle advances much closer to constant velocity than to constant acceleration.

 

[Andy Resnick]

It's an idealization, as often done for problem simplification. The idea is to separate what can be explained by pressure from what cannot. How well it could be approximated in the real world is a separate question e.g. I have no idea whether a precisely machined aluminum noodle and graphite apperture would approximate this.

Ugh- aside from aluminum noodles being unappetizing, you haven't even done a zeroth-order approximation, But, fine. Let's pretend.

Consider dry lubricants- graphite powder, for example. That's a real-life version of dry water, so we eliminated your odd insistence that the fluid can be abstracted away. And indeed, even though the powder acts as a fluidized bed, the viscosity is very low- let's go with your simplification of a frictionless interface- and make it zero viscosity. We physicists could also suggest using superfluid He, but there's the non-simple problem of what happens if you aspirated liquid He.

Now what happens? Most likely, you can't suck up the noodle- unless you are an astronaut orbiting the Earth (as you also suggested), because then the tiny ΔP will indeed create an imbalanced force, and you could indeed aspirate the noodle. [Edit]: a frictionless interface means the noodle will move freely under any imbalanced, momentary axial force and the housing cannot arrest the motion. So the device must be in free-fall (true free-fall, not attached to an orbiting craft).

[Edit]: So, by removing viscosity, your simplified problem does not describe a broadly applicable class of behavior, it is actually restricted to a highly specific (and useless) physical situation.

 

[PAllen]

Now consider the flaccid noodle, with the apperture having frictionless lubricant. Being frictionless, it cannot exert any force on the noodle by flowing. Yet, I believe, in such a case, the noodle would still be sucked in - very fast. As an approach to understanding this, consider replacing the noodle with a long balloon at above standard pressure, standard pressure on one side of the apperture, and vacuum on the other. The balloon would be sucked in by expansion on the vacuum side pulling the other side in by tension and flow of the balloon material. Now, get from here to a more plausible model of a noodle. I am thinking the noodle effectively flows in, and boundary flow is irrelevant or secondary in nature.

 

[Chestermiller]

Now consider the flaccid noodle, with the apperture having frictionless lubricant. Being frictionless, it cannot exert any force on the noodle by flowing. Yet, I believe, in such a case, the noodle would still be sucked in - very fast. As an approach to understanding this, consider replacing the noodle with a long balloon at above standard pressure, standard pressure on one side of the apperture, and vacuum on the other. The balloon would be sucked in by expansion on the vacuum side pulling the other side in by tension and flow of the balloon material. Now, get from here to a more plausible model of a noodle. I am thinking the noodle effectively flows in, and boundary flow is irrelevant or secondary in nature.

Again, without friction, you could not maintain the tensile stress difference in the balloon between inside and outside the mouth without an unrealistically huge acceleration of the balloon. In the case of a limp noodle, the same basic principle applies. The tensile stress difference can only be maintained by acceleration of the noodle or by friction at the surface of the aperture.

 

[DaTario]

What are you saying that's different from what we have already said?

Dear Chestermiller, once I have figured out what seemed to be the relevant physical notion in OP (imo) I immediately decided to contribute. I have noticed that you and Andy Resnik have given similar answers (but more specific than mine) just after having published my comment. If appologizing is necessary here, I am for it.
It was an interesting question, which is associated to some topics of my liking. By the way, I would like to be explicit in saying that am not disputing authorship here.

 

[DaTario]

As a somewhat experienced spaghetti eater I have the impression that, if we could see the movement of the spaghetti in slow motion, some transversal vibration could be observed. I guess the phenomena seems to involve an intake of pasta, sauce and air. If this empirical consideration shows up to be correct under a controlled experimental test it may become likely that the air intake is to cause the cylindrical shaped pasta to vibrate transversally while entering the mouth.
DaTario

 

[PAllen]

Is my understanding correct that it should be possible to distinguish models experimentally, as follows (unfortunately, I have no relevant equipment or access to a lab):

Have a chamber with a rubber orifice which can be lubricated with different fluids, and is also connected to a vacuum pump. Then testing against rods and noodles of different thickness, ideally measuring the force exerted when the pump is running and the outside end of noodle and rod is attached to a force measuring device:

- boundary flow model would suggest that for a given lubricant, the effect is proportional radius (really circumference, but same difference for proportionality); and that different lubricants (high viscosity oil, low viscosity oil, sugar syrup, etc.) could make a significant difference.

- pressure dominant case would predict force proportional to radius squared, with minimal difference by lubricant, especially for radius not too small (only impact is friction, which would be less relevant as radius got larger).[edit: and friction would be zero for a rod held static by attachment to a force measuring device]

Unfortunately, while using my mouth, I can easily determine there is more force for bigger radius, I have nowhere near the precision to distinguish proportional to r versus r².

 

[jartsa]

This is an incorrect assessment of what is happening (both for an uncooked noodle and for a cooked noodle). If you are convinced that the solid mechanical behavior of the noodle is much more important than the fluid mechanics behavior (discussed by several Science Advisers and Mentors), please provide a valid solid mechanics analysis of the situation (including stresses, strains, external loads, and deformations) based either on the Theory of Elasticity or on the (more straightforward) Strength of Materials approach. Your analysis should include actual equations and their solutions.

Chet

Sorry but I prefer energy considerations:

Let's say a big tank is filled with air at 1 atm and also some cooked spaghetti, and another big tank is filled with air at 0.9 atm. These tanks are connected by a tube, and at the middle of the tube there are some kind of mechanical lips.

Now a physicist goes into the tank, applies dry ice on the end of a noodle and sticks the frozen part through the "lips". Let's say there's only one long noodle.

Now if one liter of spaghetti is sucked from tank1 to tank2, then the energy of tank1 decreases by: 100000 N/m² * 0.001 m³ = 100 J
And the energy in tank2 increases by: 90000 N/m² * 0.001 m³ = 90 J
Total change of energy is 10 J (Tanks are big, so pressures do not change)

If one liter of air goes from tank1 to tank2 that also causes a 10 J decrease of pressure energy in the tanks.

What I'm getting at here is that some amount of energy is used to propel the spaghetti from tank1 to tank2, the energy is generated by x liters of spaghetti leaving tank1 and y liters of air leaving tank1.

If the volume of the spaghetti entering tank2 is much larger than the volume of air entering tank2, then the air was not important energy source for the motion of spaghetti.

So the question is: When sucking spaghetti, does more spaghetti or air enter the mouth?

Oh yes, let's put both tanks in a heat bath, so that the total energy in the tanks can actually decrease when the pressure energy does some work. ... Correction, let's just wait till tank2 reaches the same temperature as the surroundings.

 

[Chestermiller]

Is my understanding correct that it should be possible to distinguish models experimentally, as follows (unfortunately, I have no relevant equipment or access to a lab):

Have a chamber with a rubber orifice which can be lubricated with different fluids, and is also connected to a vacuum pump. Then testing against rods and noodles of different thickness, ideally measuring the force exerted when the pump is running and the outside end of noodle and rod is attached to a force measuring device:

- boundary flow model would suggest that for a given lubricant, the effect is proportional radius (really circumference, but same difference for proportionality); and that different lubricants (high viscosity oil, low viscosity oil, sugar syrup, etc.) could make a significant difference.

- pressure dominant case would predict force proportional to radius squared, with minimal difference by lubricant, especially for radius not too small (only impact is friction, which would be less relevant as radius got larger).

Unfortunately, while using my mouth, I can easily determine there is more force for bigger radius, I have nowhere near the precision to distinguish proportional to r versus r².

I'm a modeller, and, as a modeller, I am very confident that these issues can be resolved without resorting to experiments (at least initially to get the lay of the land).

I feel very frustrated. I've been trying to get responders to focus exclusively on the noodle, using simple free body diagrams and force balances (and perhaps a smidgen of stress analysis), but no one seems willing. I contend that the uncooked noodle can be analyzed as a rigid- or slightly deformable rod, and that the limp noodle can be analyzed as either an inextensible- or slightly extensible rope.

I'm prepared to present such analyses myself, but I wanted to give others a chance. Anybody?

Please excuse my frustration.

 

[Baluncore]

Examination of Chestermiller's equation (1) in post #73 shows that there is a critical sauce viscosity above which the maximum sauce flow velocity within the channel is always lower than or equal to the velocity of the upper plate = spaghetti surface velocity. It suggests that with a thick viscous sauce the spaghetti must be propelled by the axial pressure difference within the spaghetti rather than by the drag of the moving sauce. It raises the question of the relative strength of the two propulsion forces. Indeed, for a thick sauce, the sauce would be drawn through the channel by drag from the spaghetti.

 

[Chestermiller]

I have carried out the solid mechanics analyses of the noodle I alluded to in my previous post, and have come to the following conclusions:

1. For a horizontal rigid noodle, it is not possible to suck the noodle into the mouth using a significant pressure difference and with a constant noodle velocity (i.e., no acceleration) without friction at the lips. At constant velocity, the frictional resistance is necessary to balance the pressure difference. Otherwise, the noodle will be constantly accelerating.

2. For a vertical rigid noodle or a dangling limp noodle, it actually is possible to suck the noodle into the mouth using a significant pressure difference and with a constant noodle velocity even without friction at the lips. The gravitational force on the noodle compensates for the acceleration of the noodle required in case 1. This does not mean that there is no significant viscous frictional drag at the lips in actual practice (with both the frictional drag and the weight of the noodle balancing the pressure difference). We simply need to approximately quantify the magnitude of the frictional drag (using reasonable estimates of the parameters involved, such as viscosity, noodle velocity, and annular gap) to compare it with the contribution of gravity to the overall pressure difference.

P.S., If anyone is interested in the solid mechanics analysis for these cases, I can provide it.

Chet

 

[PAllen]

I have carried out the solid mechanics analyses of the noodle I alluded to in my previous post, and have come to the following conclusions:

1. For a horizontal rigid noodle, it is not possible to suck the noodle into the mouth using a significant pressure difference and with a constant noodle velocity (i.e., no acceleration) without friction at the lips. At constant velocity, the frictional resistance is necessary to balance the pressure difference. Otherwise, the noodle will be constantly accelerating.
Chet

I'm not sure why you keep stressing the constant velocity case. Crude experiments I can do readily establish that for the rigid cylinder case with constant pressure difference, and common lubricants, the cylinder does steadily accelerate. I did not reach the point where terminal velocity was reached due to friction increasing proportional to speed, but presumably such point is readily reached for a long cylinder.

To me, this (plus your gravity example) establishes that the rigid case is primarily a matter of pressure difference modified by obvious factors like friction.

My mind is still open on the flaccid noodle case, especially for practical parameters.

 

[Andy Resnick]

I'm a modeller, and, as a modeller, I am very confident that these issues can be resolved without resorting to experiments (at least initially to get the lay of the land).

I feel very frustrated. I've been trying to get responders to focus exclusively on the noodle, using simple free body diagrams and force balances (and perhaps a smidgen of stress analysis), but no one seems willing. I contend that the uncooked noodle can be analyzed as a rigid- or slightly deformable rod, and that the limp noodle can be analyzed as either an inextensible- or slightly extensible rope.

I'm prepared to present such analyses myself, but I wanted to give others a chance. Anybody?

Please excuse my frustration.

FWIW, I very much appreciate the time and effort you have invested in this thread.

 

[Chestermiller]

I'm not sure why you keep stressing the constant velocity case.

I just had the sense that, when I suck in a noodle (through my actual lips), it travels close to constant velocity. Maybe I'm wrong, or maybe I'm subconsciously continuously adjusting the suction so that the velocity is close to constant.

Crude experiments I can do readily establish that for the rigid cylinder case with constant pressure difference, and common lubricants, the cylinder does steadily accelerate. I did not reach the point where terminal velocity was reached due to friction increasing proportional to speed, but presumably such point is readily reached for a long cylinder.

To me, this (plus your gravity example) establishes that the rigid case is primarily a matter of pressure difference modified by obvious factors like friction.

My mind is still open on the flaccid noodle case, especially for practical parameters.

I'm going to present my solid mechanics analysis of the limp noodle case to show how it works. I'm also going to make a rough calculation of the viscous drag force at the lips and compare it to the dangling weight.

 

[Chestermiller]

I'm reporting back on some calculations I've done to estimate the drag force on the noodle by the lips, and compare this with the weight of the noodle dangling from the lips. The calculations also give the minimum level of vacuum required to suck the noodle in. To get the dangling weight of the noodle, my results are similar to those of @OmCheeto in a previous post. The calculated weight is based on the following:

Noodle radius = 0.1 cm
Noodle density = 1 gm/cc
Noodle length = 30 cm

This gives a noodle mass of about 1 gram, and a noodle dangling weight of about 1000 dynes (0.01 N).

To crudely estimate the drag force by the lips, I used the following formula:
$$F=2\pi r L \eta \frac{V}{h}$$
where L is the length of the lip channel and h is the annular gap between the lips and the noodle. The calculation was based on the following parameter values:

L = 1 cm
$\eta=0.01\ \frac{gm}{cm\ sec}$ (i.e., water at room temperature, 0.01 Poise)
V = 6 cm/sec (noodle suction speed)
h = 0.001 cm

Using these values, I get a drag force of about 40 dynes. Thus, the drag force would be a small fraction of the dangling weight of the noodle and, in practice, can be neglected (This confirms @PAllen's contention).

The suction required to draw the noodle into the mouth (in cm of water) would be about equal to the length of the noodle, and would thus be equal to about 30 cm of water. This would correspond to about 3000 Pa, or 12" of water. This would be the minimum vacuum to suck the noodle in.

 

[BillTre]

I just read this thread and since I was making some spaghetti, so I did some experiments.

This may partly dis-entangle some variables:
wetness
diameter
water vs. sauce
horizontal vs. vertical noodle
straight vs, bent flexible noodle

Dry noodles (diameter = 1.93mm):
Using noodles that are about 1.93 mm diameter (dry):
1) I can not suck in a dry noodle with dry lips. It felt like too much friction.
2) I can suck in a "dry noodle if I wet my lips and the noodle with saliva (very thin film on the noodle, most likely less than 0.1 mm). Much of the wetness seems to get sucked into the noodle. Could only suck in an inch or two (2.5-5.0 cm).
This was most easily done in a vertical orientation (head tilted forward so mouth opening is approximately horizontal).
Horizontal dry noodles have to propped up outside the mouth or it runs into things inside the mouth.
Vertical dry noodles also work.
3) If I only wet a short distance of the noodle I can suck it in until my lips hit the dry part of the noodle

Wet noodles (diameter = 2.8 mm):
4) wet only from the cooking water; works well in either a horizontal or vertical orientation.
5) Wet with sauce (sauce from a jar, quick dinner), sucked (seemingly) more easily than with just water. Warning: In the vertical orientation I almost got some sauce in my eye (remember to wear your googles when experimenting!)
6) noodle, cooked but laying around and tacky on the surface: could suck it in, but much more difficult (in either horizontal or vertical orientation). Similar to the dry noodle

Cooked, wet hot dog (diameter = ~19.4 mm):
7) These sucked in very easily (horizontal or vertical).
They were very pliable and had a somewhat greasy surface.

I guess in the vertical orientation I could put a weight on the end to get a real idea of the applied force, but food is now gone.

Conclusions:
Horizontal vs. vertical (noodle or head) orientation does not seem to matter much.
Straightness of noodle doesn't seem to matter much.
Larger diameter may help.
Minimal wetness is needed to reduce friction (at least).
Sauce coating seems to work best.

I'll let you guys figure out the meaning of all this.

 

[Chestermiller]

NEWS FLASH FROM FOX NEWS: TODAY IS NATIONAL SPAGHETTI DAY

 

[Andy Resnick]

I just read this thread and since I was making some spaghetti, so I did some experiments.
<snip>

Gasp- perform experiments instead of endless noodling? :)

 

[BillTre]

NEWS FLASH FROM FOX NEWS: TODAY IS NATIONAL SPAGHETTI DAY

Yumm!

 

[Chestermiller]

The figure below shows a schematic for the Limp Noodle case. This is the situation we will be analyzing.

The noodle extends from inside the mouth, through the lips, over the lower lip, and then dangles vertically downward. The pressure outside the mouth is atmospheric pressure $p_a$, and the pressure inside the mouth is $p_v<p_a$. Both pressures act strictly perpendicular to surfaces that they contact.

Now let's consider the state of stress within the noodle. In this development, we will be working with absolute pressures and stresses, rather than gauge pressures and gauge stresses. We also adopt the usual sign convention that tensile stresses are regarded as positive and compressive stresses are regarded as negative.

Inside the mouth, the noodle is in a state of isotropic compression throughout, caused by the surrounding air at pressure $p_v$. Both the radial and axial components of tensile stress within the noodle are equal to $-p_v$. In the figure, only the axial component of tensile stress is shown. We use the symbol $\sigma$ to represent the axial component of tensile stress in our analysis. So, $\sigma=-p_v$ for the entire portion of the noodle inside the mouth.

Outside the mouth, the axial tensile stress within the noodle varies with elevation z in order to support the weight of the noodle. However, at the very bottom of the noodle at zero elevation (z = 0), the axial tensile stress is equal to $\sigma =-p_a$, because the noodle is in contact with room air acting normal to its bottom surface.

We will next carry out a differential force balance on the noodle to quantify how the axial tensile stress $\sigma$ within the noodle varies along the contour coordinate s (see figure); the contour coordinate s is the cumulative distance measured along noodle contour, starting at the very bottom of the noodle.

In this development, we treat the noodle in the same manner that we frequently treat ropes in many mechanics problems. The bending rigidity of the noodle is neglected, and the primary focus of the force balance is in the tangential direction along the noodle contour; the forces acting normal to the contour (e.g., at the bottom lip) are of less interest here, and are not considered. Unlike mechanics problems, the weight of the noodle is included in the force balance. Our objective is to determine the minimum pressure difference required to just support the weight of the noodle, such that the noodle will be in static equilibrium. Any additional pressure difference will cause the noodle to accelerate along the contour into the mouth.

If we perform a differential force balance in the tangential direction on the section of noodle between contour locations $s$ and $s+\Delta s$ (see figure), we obtain: $$A[\sigma(s+\Delta s)-\sigma(s)]=(\rho A \Delta s)\left(g\frac{dz}{ds}\right)\tag{1}$$where A is the cross sectional area of the noodle, $\rho A \Delta s$ is the mass of noodle between contour locations $s$ and $s+\Delta s$, $\rho$ is the density of the noodle, and $g\frac{dz}{ds}$ is the component of gravity in the local tangential direction along the contour. If we divide Eqn. 1 by A$\Delta s$ and take the limit as $\Delta s$ approaches zero, we obtain:$$\frac{d\sigma}{ds}=\rho g\frac{dz}{ds}\tag{2}$$This equation can immediately be integrated between the two ends of the noodle (subject to the boundary conditions on $\sigma$) to yield:
$$-p_v+p_a=\rho g H\tag{3}$$where H is the maximum elevation of the noodle (relative to the bottom of the dangling portion, see the figure). Therefore, for static equilibrium of the noodle, $$\Delta p=(p_a-p_v)=\rho g H\tag{4}$$

This completes the description on how to determine the pressure difference required for the Limp Noodle case.

 

[Andreas C]

The figure below shows a schematic for the Limp Noodle case. This is the situation we will be analyzing.View attachment 111083
The noodle extends from inside the mouth, through the lips, over the lower lip, and then dangles vertically downward. The pressure outside the mouth is atmospheric pressure $p_a$, and the pressure inside the mouth is $p_v<p_a$. Both pressures act strictly perpendicular to surfaces that they contact.

Now let's consider the state of stress within the noodle. In this development, we will be working with absolute pressures and stresses, rather than gauge pressures and gauge stresses. We also adopt the usual sign convention that tensile stresses are regarded as positive and compressive stresses are regarded as negative.

Inside the mouth, the noodle is in a state of isotropic compression throughout, caused by the surrounding air at pressure $p_v$. Both the radial and axial components of tensile stress within the noodle are equal to $-p_v$. In the figure, only the axial component of tensile stress is shown. We use the symbol $\sigma$ to represent the axial component of tensile stress in our analysis. So, $\sigma=-p_v$ for the entire portion of the noodle inside the mouth.

Outside the mouth, the axial tensile stress within the noodle varies with elevation z in order to support the weight of the noodle. However, at the very bottom of the noodle at zero elevation (z = 0), the axial tensile stress is equal to $\sigma =-p_a$, because the noodle is in contact with room air acting normal to its bottom surface.

We will next carry out a differential force balance on the noodle to quantify how the axial tensile stress $\sigma$ within the noodle varies along the contour coordinate s (see figure); the contour coordinate s is the cumulative distance measured along noodle contour, starting at the very bottom of the noodle.

In this development, we treat the noodle in the same manner that we frequently treat ropes in many mechanics problems. The bending rigidity of the noodle is neglected, and the primary focus of the force balance is in the tangential direction along the noodle contour; the forces acting normal to the contour (e.g., at the bottom lip) are of less interest here, and are not considered. Unlike mechanics problems, the weight of the noodle is included in the force balance. Our objective is to determine the minimum pressure difference required to just support the weight of the noodle, such that the noodle will be in static equilibrium. Any additional pressure difference will cause the noodle to accelerate along the contour into the mouth.

If we perform a differential force balance in the tangential direction on the section of noodle between contour locations $s$ and $s+\Delta s$ (see figure), we obtain: $$A[\sigma(s+\Delta s)-\sigma(s)]=(\rho A \Delta s)\left(g\frac{dz}{ds}\right)\tag{1}$$where A is the cross sectional area of the noodle, $\rho A \Delta s$ is the mass of noodle between contour locations $s$ and $s+\Delta s$, $\rho$ is the density of the noodle, and $g\frac{dz}{ds}$ is the component of gravity in the local tangential direction along the contour. If we divide Eqn. 1 by A$\Delta s$ and take the limit as $\Delta s$ approaches zero, we obtain:$$\frac{d\sigma}{ds}=\rho g\frac{dz}{ds}\tag{2}$$This equation can immediately be integrated between the two ends of the noodle (subject to the boundary conditions on $\sigma$) to yield:
$$-p_v+p_a=\rho g H\tag{3}$$where H is the maximum elevation of the noodle (relative to the bottom of the dangling portion, see the figure). Therefore, for static equilibrium of the noodle, $$\Delta p=(p_a-p_v)=\rho g H\tag{4}$$

This completes the description on how to determine the pressure difference required for the Limp Noodle case.

This post should be on the home page of physicsforums: "Physicsforums.com: the ONLY place on the internet you'll find this sort of analysis on noodles"

 

[Andy Resnick]

The figure below shows a schematic for the Limp Noodle case. This is the situation we will be analyzing.Therefore, for static equilibrium of the noodle, $$\Delta p=(p_a-p_v)=\rho g H\tag{4}$$

This completes the description on how to determine the pressure difference required for the Limp Noodle case.

That makes sense- it's the same hydrostatic pressure required to draw a 'fluid rope' up a height H. Nice!

 

[Swamp Thing]

Here is a thought experiment that is a bit different from the noodle situation, but may help to validate or falsify a given model.

Take a rigid transparent plastic tube about one centimeter in diameter. Connect it to a suitable large syringe. Use the syringe to extrude a "rope" of clear RTV rubber through the plastic tube so that the tube is full of the RTV, plus you have some rope hanging out from one end of the tube. Allow the RTV to cure and set into a rope with one end stuck inside the plastic tube.

Before extruding, you need to add some colored particles to the RTV, so that you can observe any internal deformations that might happen inside the RTV during the actual experiment.

Now pass the tube through a hole in a plastic jar and seal it in place with part of the tube inside and part outside the jar. The extruded RTV rope hangs outside the jar. Note that the RTV is stuck in place within the tube, so there is no fluid flow at the junction - in fact, there is no gap at all between the RTV and the tube. The RTV is stuck firmly in place.

At this point we create a vacuum within the jar. As the pressure falls, we watch the marker particles that we embedded in the RTV. Although the RTV won't slide through the tube, it is quite possible that the pressure differential would distort the RTV such that particles in the tube would deflect towards the inside of the jar. The displacement would likely be proportional to the pressure differerence. Furthermore, particles near the center would probably deflect the most, while particles near the wall of the tube would remain essentially fixed.

If this does happen, then it confirms that there is indeed a static force that is trying to push the RTV inwards, without needing to invoke any fluid flow at the interface. On the other hand, if the particles show absolutely no deflection, then we are left with no option but the "fluid flow --> shear force" theory.

For what it's worth, my personal intuition expects to see the RTV being deformed when we apply the pressure differential. If the pressure is large enough, this would be enough to shear the bond and propel the rope into the jar. But the "sauce flow model" would predict that the RTV would just not budge, no matter how high a pressure we applied.