Vacuum or pressure to move spaghetti through a hole

[Chestermiller]

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.

This system is mechanistically very much different from the system involving a noodle (although it can readily be modeled using solid mechanics). The only need for an experiment would be to confirm the model predictions quantitatively. In post #102, I solved the solid mechanics of the noodle problem without invoking any noodle tension variation caused by viscous resistance at the lips (i.e., it was assumed that the lips are frictionless).

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.

In post #97, we already confirmed that, rather than driving the noodle into the mouth, the viscous forces actually do the opposite, providing a small amount of additional "frictional" drag to resist the noodle movement into the mouth.

Please read the entire thread to see the evolution of our thinking on this problem, rather than focusing on incorrect assessments early on in the thread. We can thank @PAllen for finally getting us going on the right track in posts #78, 80, 82, 85, 89, and 94.

 

[Andreas C]

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

Someone has to try sucking a really really long "noodle" with a vacuum pump or a vacuum cleaner or something to test it. I estimate the density of boiled spaghetti is something like... 1500-2000kg/m^3? They don't float in water but I don't think they're very dense. Let's pick 2*10^3 kg/m^3. If g=10, and Pa=101*10^3 N/m^2, then you wouldn't be able to suck noodles about... 5 meters long... Yeah, that's a pretty long noodle... I'm not sure how you test that...

 

[Chestermiller]

Someone has to try sucking a really really long "noodle" with a vacuum pump or a vacuum cleaner or something to test it. I estimate the density of boiled spaghetti is something like... 1500-2000kg/m^3? They don't float in water but I don't think they're very dense. Let's pick 2*10^3 kg/m^3. If g=10, and Pa=101*10^3 N/m^2, then you wouldn't be able to suck noodles about... 5 meters long... Yeah, that's a pretty long noodle... I'm not sure how you test that...

How about that "someone" be you? How about we make that your assignment?

 

[Andreas C]

How about that "someone" be you? How about we make that your assignment?

Uh... Sure, but... I don't really have any of the right tools for that... Where do you find a 5 meter long noodle?

 

[Baluncore]

Uh... Sure, but... I don't really have any of the right tools for that... Where do you find a 5 meter long noodle?

I do however notice that “two minute noodles” come in a pack that is often extruded, dried and then packed as a single length. By identifying and tagging the free end it would be possible to rehydrate the noodles without stirring, then to draw a single noodle thread back from the bundle.

But my concern here becomes one of quality control, that of the Chinese noodle product versus the Italian pasta product.

 

[Chestermiller]

Uh... Sure, but... I don't really have any of the right tools for that... Where do you find a 5 meter long noodle?

Get yourself a pasta press.

 

[Andreas C]

Baluncore, I'm afraid they don't sell that in my country. In fact even finding noodles isn't easy. Spaghetti is where it's at.

 

[Andreas C]

Get yourself a pasta press

Nah I'll skip on that. 100-something euros are more than I intend to spend for noodle experiments... If anyone can find massive pasta, he should try it. Or we could find a more readily available analogue.

 

[PAllen]

Someone has to try sucking a really really long "noodle" with a vacuum pump or a vacuum cleaner or something to test it. I estimate the density of boiled spaghetti is something like... 1500-2000kg/m^3? They don't float in water but I don't think they're very dense. Let's pick 2*10^3 kg/m^3. If g=10, and Pa=101*10^3 N/m^2, then you wouldn't be able to suck noodles about... 5 meters long... Yeah, that's a pretty long noodle... I'm not sure how you test that...

You would also need to do this from high railing of some kind, with the noodle hanging off. Otherwise most of the noodle's weight would be supported, e.g., a long noodle coiled in a plate could still readily be sucked in.

 

[BillTre]

You would also need to do this from high railing of some kind, with the noodle hanging off. Otherwise most of the noodle's weight would be supported, e.g., a long noodle coiled in a plate could still readily be sucked in.

Or try it with a variety of weights on the end of the noodle.

 

[Andreas C]

Or try it with a variety of weights on the end of the noodle.

Yes, I had a little self facepalm moment when I realized the solution could be so simple... But experiments involving meter long pasta are definitely more exciting than hanging weights on noodles.

 

[Andreas C]

I guess I'll try it then. I'll start by finding the density of spaghetti and take it from there. Probably not this week though.

 

[Andreas C]

Oh wait no, that's not going to work... It's not consistent with the assumptions, is it?

 

[Stephen Tashi]

With all due respect, this is silly

All physical models are silly in the sense of omitting details. For example, it's "silly" to model friction in a practical situation as a constant force. Textbook problems in elementary mechanics (which frequently involving frictionless contact) are silly in that sense. The interesting question, to me, whether elementary mechanics , applied to this particular problem, is "silly" in the sense that it cannot explain why the noodle is sucked into the mouth.

In the case of a rod (instead of a curved strand of spaghetti) elementary mechanics (meaning the mechanics that explains resultant forces in terms of other specific forces) seems unable to explain the component of force that pushes the sphagetti into the mouth except by saying that the force is the resultant of the forces on the ends of the rod. The forces on the side surface of the rod make no contribution.

If we now consider the curved shape of a strand of spaghetti, the original post can be interpreted as asking how pressure acting perpendicular to the side of the of spaghetti and the ends of the spaghetti can explain why there is is a component of force that moves the spaghetti into the mouth. (In this case, neither end of the strand of spaghetti need be in a vertical plane, so forces on those faces aren't entirely in the direction of "out of" or "into" the mouth.)

The original post asked:

I have learned that pressure is acting angular to any surface. With the spaghetti, that surface will cause the pressure to act 90° to it, and (in my thoughts) not be able to create a force that pulls or push the spaghetti into my mouth.

If we just assert the force that pushes the spaghetti into the mouth is the cross section of the strand times the pressure differential, this is reasonable model, but it doesn't explain how that model is deduced. To explain that, we need to go beyond elementary mechanics into fluid mechanics and the mechanics of materials.

Of course, there is a long internet tradition of answering a different question that was asked!

 

[Baluncore]

Is there an increase in diameter of the pasta as it passes between the lips ?
If so, how much of that increase is due to the differential pressure and how much due to tension ?

 

[Chestermiller]

All physical models are silly in the sense of omitting details. For example, it's "silly" to model friction in a practical situation as a constant force. Textbook problems in elementary mechanics (which frequently involving frictionless contact) are silly in that sense. The interesting question, to me, whether elementary mechanics , applied to this particular problem, is "silly" in the sense that it cannot explain why the noodle is sucked into the mouth.

In the case of a rod (instead of a curved strand of spaghetti) elementary mechanics (meaning the mechanics that explains resultant forces in terms of other specific forces) seems unable to explain the component of force that pushes the sphagetti into the mouth except by saying that the force is the resultant of the forces on the ends of the rod. The forces on the side surface of the rod make no contribution.

If we now consider the curved shape of a strand of spaghetti, the original post can be interpreted as asking how pressure acting perpendicular to the side of the of spaghetti and the ends of the spaghetti can explain why there is is a component of force that moves the spaghetti into the mouth. (In this case, neither end of the strand of spaghetti need be in a vertical plane, so forces on those faces aren't entirely in the direction of "out of" or "into" the mouth.)

The original post asked:If we just assert the force that pushes the spaghetti into the mouth is the cross section of the strand times the pressure differential, this is reasonable model, but it doesn't explain how that model is deduced. To explain that, we need to go beyond elementary mechanics into fluid mechanics and the mechanics of materials.

Of course, there is a long internet tradition of answering a different question that was asked!

Have you examined my post #102?

 

[PAllen]

..

In the case of a rod (instead of a curved strand of spaghetti) elementary mechanics (meaning the mechanics that explains resultant forces in terms of other specific forces) seems unable to explain the component of force that pushes the sphagetti into the mouth except by saying that the force is the resultant of the forces on the ends of the rod. The forces on the side surface of the rod make no contribution.

This is a simplification based on rigidity, with rigidity always being a simplification, in principle. This was posed because even the analysis of this case was going astray at points in the thread.

If we now consider the curved shape of a strand of spaghetti, the original post can be interpreted as asking how pressure acting perpendicular to the side of the of spaghetti and the ends of the spaghetti can explain why there is is a component of force that moves the spaghetti into the mouth. (In this case, neither end of the strand of spaghetti need be in a vertical plane, so forces on those faces aren't entirely in the direction of "out of" or "into" the mouth.)

The original post asked:If we just assert the force that pushes the spaghetti into the mouth is the cross section of the strand times the pressure differential, this is reasonable model, but it doesn't explain how that model is deduced. To explain that, we need to go beyond elementary mechanics into fluid mechanics and the mechanics of materials.

Of course, there is a long internet tradition of answering a different question that was asked!

I believe chestermiller addressed this in his detailed post on a hanging, flaccid noodle. Trying to formulate this result conceptually, I would say as follows:

All objects are really fluid (rigidity is the the limit of extreme viscosity). In this sense, pressure all around the noodle is significant and the noodle exterior portion treated as fluid propagates the outside pressure to the aperture, while the noodle interior portion propagates the interior pressure. There is thus a pressure difference across the aperture within the noodle. This leads to 'flow' of the noodle, with cohesion propagating this force as tension over the length of the noodle.

Note, this more complete model even explains various dynamics. For example, the rate of acceleration of a rigid rod would be given by pressure difference times aperture area divided by mass of the whole rod, and this acceleration would remain constant in the absence of friction until the whole rod crossed the aperture. In the case of a noodle, the acceleration would be greater because the whole of interior noodle does not have to be moved, and as the portion moving into the cavity increases in speed, the noodle inside will buckle; thus not all of it needs to be moved. Thus the acceleration will be pressure times aperture area divided by (mass of noodle outside the aperture plus some fractional part of mass inside the noodle that would be complex to model). Thus, until friction becomes significant, the rate of acceleration of noodle will tend to increase (because mass of exterior portion is decreasing).

Consider also a hanging noodle with weight attached such that the whole thing is static (similar to chestermiller's analysis except that the weight outside isn't all due to noodle). In this static case, the exterior noodle is under tension such that at the aperture the tension matches the force from pressure difference. When the weight is cut, a tension release wave propagates at the speed of sound to the aperture at which point the noodle begins to move.

 

[Chestermiller]

This is a simplification based on rigidity, with rigidity always being a simplification, in principle. This was posed because even the analysis of this case was going astray at points in the thread.I believe chestermiller addressed this in his detailed post on a hanging, flaccid noodle. Trying to formulate this result conceptually, I would say as follows:

All objects are really fluid (rigidity is the the limit of extreme viscosity). In this sense, pressure all around the noodle is significant and the noodle exterior portion treated as fluid propagates the outside pressure to the aperture, while the noodle interior portion propagates the interior pressure. There is thus a pressure difference across the aperture within the noodle. This leads to 'flow' of the noodle, with cohesion propagating this force as tension over the length of the noodle.

Note, this more complete model even explains various dynamics. For example, the rate of acceleration of a rigid rod would be given by pressure difference times aperture area divided by mass of the whole rod, and this acceleration would remain constant in the absence of friction until the whole rod crossed the aperture. In the case of a noodle, the acceleration would be greater because the whole of interior noodle does not have to be moved, and as the portion moving into the cavity increases in speed, the noodle inside will buckle; thus not all of it needs to be moved. Thus the acceleration will be pressure times aperture area divided by (mass of noodle outside the aperture plus some fractional part of mass inside the noodle that would be complex to model). Thus, until friction becomes significant, the rate of acceleration of noodle will tend to increase.

Consider also a hanging noodle with weight attached such that the whole thing is static (similar to chestermiller's analysis except that the weight outside isn't all due to noodle). In this static case, the exterior noodle is under tension such that at the aperture the tension matches the force from pressure difference. When the weight is cut, a tension release wave propagates at the speed of sound to the aperture at which point the noodle begins to move.

I would add that, in my analysis (post #102), I did not consider the noodle to be a very viscous fluid, but rather an elastic Hookean solid. And I considered how the isotropic gas pressure outside the noodle interacted with the solid mechanical stress distribution within the noodle.

 

[Swamp Thing]

Please read the entire thread to see the evolution of our thinking on this problem, rather than focusing on incorrect assessments early on in the thread. We can thank @PAllen for finally getting us going on the right track in posts #78, 80, 82, 85, 89, and 94.

My apologies. I should have taken more care to read what had already been talked about.

This system is mechanistically very much different from the system involving a noodle (although it can readily be modeled using solid mechanics).

Perhaps it is somewhat similar to the noodle system at the moment when the noodle is just about to overcome the static friction with the lips and is just about to move? I would be interested in a vector map showing the internal displacement of the pasta/RTV before the noodle begins to slither.

Also (in an unrelated activity that I did in the past) I was able to extrude very long threads of RTV with noodle-like thickness and flaccidity, so this might be the way to mechanize some of the experiments that are being discussed. (Rather than hanging weights onto one's limp noodle )

 

[Chestermiller]

Perhaps it is somewhat similar to the noodle system at the moment when the noodle is just about to overcome the static friction with the lips and is just about to move? I would be interested in a vector map showing the internal displacement of the pasta/RTV before the noodle begins to slither.

My analysis assumes that there is no friction (either static or kinetic) at the lips.

 

[Andreas C]

Also (in an unrelated activity that I did in the past) I was able to extrude very long threads of RTV

What is RTV?

 

[Swamp Thing]

What is RTV?

Room Temperature Vulcanizing Rubber. https://en.wikipedia.org/wiki/RTV_silicone

 

[Andreas C]

Room Temperature Vulcanizing Rubber. https://en.wikipedia.org/wiki/RTV_silicone

Ooooh silicone! It's not really as flacud as noodles, but it doesn't sound bad.

 

[Stephen Tashi]

Trying to formulate this result conceptually, I would say as follows:

All objects are really fluid (rigidity is the the limit of extreme viscosity).

That's a hard explanation to "swallow" ! :)

In this sense, pressure all around the noodle is significant and the noodle exterior portion treated as fluid propagates the outside pressure to the aperture, while the noodle interior portion propagates the interior pressure. There is thus a pressure difference across the aperture within the noodle. This leads to 'flow' of the noodle, with cohesion propagating this force as tension over the length of the noodle.

Granting the noodle is a fluid, that is a clear explanation. The situation would be similar to sucking on a balloon filled with water.

Note, this more complete model even explains various dynamics. For example, the rate of acceleration of a rigid rod would be given by pressure difference times aperture area divided by mass of the whole rod, and this acceleration would remain constant in the absence of friction until the whole rod crossed the aperture. In the case of a noodle, the acceleration would be greater because the whole of interior noodle does not have to be moved, and as the portion moving into the cavity increases in speed, the noodle inside will buckle; thus not all of it needs to be moved.

Good point.

 

[PAllen]

That's a hard explanation to "swallow" ! :).

It's quite literally true. Granite can flow over very long time scale.

 

[DaTario]

Hi All,

What about the time reversal of this situation (it happened to me this reveillon btw): a closed and not shaken bottle of champagne is taken from the refrigerator having no metalic protection for the stopper to pop. As the temperature rises the stopper spontaneously pop. Isn´t this stopper sufficiently similar to a noddle as it is in the OP, but in time reversal?

I guess the role of dissipation in this case is not essencial for the understanding of the movement itself.

 

[Chestermiller]

It's quite literally true. Granite can flow over very long time scale.

While the rheological behavior of granite may feature a viscous component at very long times of load application (e.g., thousands of years), under ordinary circumstances, its rheological response is overwhelmingly linearly elastic. The rheological behavior of an uncooked noodle is almost certainly going to be linearly elastic. In the case of a cooked noodle, its rheological response may be more nearly viscoelastic, but the absence of creep elongation under its own weight (at laboratory time scales) strongly suggests primarily elastic response.

 

[Andy Resnick]

In the case of a rod (instead of a curved strand of spaghetti) elementary mechanics (meaning the mechanics that explains resultant forces in terms of other specific forces) seems unable to explain the component of force that pushes the sphagetti into the mouth except by saying that the force is the resultant of the forces on the ends of the rod. The forces on the side surface of the rod make no contribution.

I think Chestermiller has provided a clear model for both- in equilibrium, the suction force F and weight of the spaghetti are balanced: F = mg → ΔP/A = ρ A H g →ΔP = ρgH, where A is the cross-sectional area of the noodle and H the length. This model replaces the lips by a planar aperture and balances the forces normal to that plane. within the plane, the in-plane pressure differential between the interior and exterior of the noodle results in no net force by symmetry. Adding viscous friction here makes no difference- this is an equilibrium condition, no net motion. If additional suction is applied, the lubricating sauce exerts a drag force (proportional to relative velocity) and the force balance equation has a frictional term analogous to air resistance. The equation of motion for a point along the moving noodle can be written analogously to that of rolling up a chain, since H = H(t), with an extra viscous term. I think it would look like [Edit]: ΔP'(t) - C dH/dt- ρgH(t) = ρ H(t) d²H/dt². If we adjust ΔP(t) to result in a constant velocity 'v' then ΔP(t) = ρgH₀-Cv-ρgvt ≈ ρgH₀-ρgvt if viscous drag is neglected.

If we now consider the curved shape of a strand of spaghetti, the original post can be interpreted as asking how pressure acting perpendicular to the side of the of spaghetti and the ends of the spaghetti can explain why there is is a component of force that moves the spaghetti into the mouth.

I'm not sure I understand what you mean here.

If we just assert the force that pushes the spaghetti into the mouth is the cross section of the strand times the pressure differential, this is reasonable model, but it doesn't explain how that model is deduced. To explain that, we need to go beyond elementary mechanics into fluid mechanics and the mechanics of materials

In some sense- yes, that's true. There are assumptions built into this model (as every model) homogeneous isotropic incompressible pasta, for example. Knowing what minority parts of reality are responsible for the majority of observed behavior (measured as a fractional error, for example) does require knowing more than elementary physics.

 

[Chestermiller]

I
In some sense- yes, that's true. There are assumptions built into this model (as every model) homogeneous isotropic incompressible pasta, for example. Knowing what minority parts of reality are responsible for the majority of observed behavior (measured as a fractional error, for example) does require knowing more than elementary physics.

In my post, I briefly alluded to how the solid mechanics of the noodle (and the state of stress within the noodle, as captured by the stress tensor) is related to the isotropic pressure loading of the air acting on the noodle. I think @Stephen Tashi's question related to how things would change if the end of the noodle were curved (rather than flat) or if it were oriented at an angle to the noodle axis. The answer is that nothing would change.

Below is a figure I developed of an object with rounded ends immersed in a fluid that is applying isotropic pressure to the object.

If we ask what the axial stress would be at the location where the hypothetical "split line" is indicated, we would conclude from a force balance on either portion of the object (each of which is in static equilibrium) that the internal tensile stress would have to be equal to minus the pressure (-p) surrounding the object. This would apply to the normal stress on any cross section located anywhere within the object.

Within the portion of the noodle inside our mouth, the noodle is immersed in an isotropic pressure environment and there is no other loading on the noodle, so the same basic stress state applies. For the portion of the noodle dangling from our mouth, the axial stress along the noodle is changing as a result of the gravitational loading on the noodle, but the radial component of the stress tensor is constant and still determined by the isotropic room air pressure loading on the surface. So, in this region, the state of stress within the noodle is not isotropic (except at the very bottom of the dangling section where both the radial and axial components are equal to one another and to minus the room air pressure).

I hope that this description helps with @Stephen Tashi's doubts.

 

[Andy Resnick]

In my post, I briefly alluded to how the solid mechanics of the noodle (and the state of stress within the noodle, as captured by the stress tensor) is related to the isotropic pressure loading of the air acting on the noodle. I think @Stephen Tashi's question related to how things would change if the end of the noodle were curved (rather than flat) or if it were oriented at an angle to the noodle axis. The answer is that nothing would change.

Agreed. Out of random curiosity, what is the internal stresses and strains within the curved 'droopy' section of the noodle? I understand that one side is in tension and the other in compression, does that result in deformation of a circular cross section to an ellipse? and how does the nonlinear effects: kinking, wrinkling or tearing, get incorporated?

 

[Chestermiller]

Agreed. Out of random curiosity, what is the internal stresses and strains within the curved 'droopy' section of the noodle? I understand that one side is in tension and the other in compression, does that result in deformation of a circular cross section to an ellipse? and how does the nonlinear effects: kinking, wrinkling or tearing, get incorporated?

What an interesting question!

Fortunately for us, since we are neglecting the bending rigidity of the noodle, the system is statically determinate. If we include the normal loading on the noodle in our analysis, we can determine the normal load per unit length N at the lower lip. We do this by including the curvature within the 'droopy' noodle section in a way similar geometrically to the way we treat a curved trajectory of a particle in motion (although, in our case, the loading is static). The differential force balances on the noodle in the tangential and normal directions are straightforward to obtain, and are given by:
$$\frac{dT}{ds}=\rho g A\frac{dz}{ds}\tag{1}$$
$$N=\frac{T}{R}+\rho g A\sqrt{1-\left(\frac{dz}{ds}\right)^2}\tag{2}$$
where T is now the gauge tension and N is the gauge normal force per unit length, R(s) is the local radius of curvature of the lower lip contour, and A is the cross sectional area of the noodle, and z is the elevation above the bottom of the noodle. (Henceforth, for convenience, we will be working in terms of gauge quantities, rather than absolute quantities. Thus, $\sigma_s$ is now the gauge axial stress, which is equal to T/A). If we integrate Eqn. 1 with respect to s (starting at s=0 at the bottom of the noodle), we obtain:$$T(s)=\sigma_s A=\rho g A z(s)\tag{3}$$Combining Eqns. 2 and 3 yields:
$$N(s)=\rho g A\left[\frac{z(s)}{R(s)}+\sqrt{1-\left(\frac{dz}{ds}\right)^2}\right]\tag{4}$$
I think this is where I'll stop for now, to give you a chance to digest this. But start thinking about how these results can be used to determine the stresses and strains within the noodle in the curved 'droopy' section. Consider the possibility of starting out simple by analyzing the case of a square noodle cross section rather than a circular noodle.

 

[Andy Resnick]

The differential force balances on the noodle in the tangential and normal directions are straightforward to obtain, and are given by:
$$\frac{dT}{ds}=\rho g A\frac{dz}{ds}\tag{1}$$
$$N=\frac{T}{R}+\rho g A\sqrt{1-\left(\frac{dz}{ds}\right)^2}\tag{2}$$
where T is now the gauge tension and N is the gauge normal force per unit length, R(s) is the local radius of curvature of the lower lip contour, and A is the cross sectional area of the noodle, and z is the elevation above the bottom of the noodle.

Thanks! I think I understand so far, but shouldn't there be an azimuthal component to N(s)? Did you write an expression for the neutral line? I'm thinking about if/how the cross section of a flexible cylinder varies through a bend.

 

[Chestermiller]

Thanks! I think I understand so far, but shouldn't there be an azimuthal component to N(s)? Did you write an expression for the neutral line? I'm thinking about if/how the cross section of a flexible cylinder varies through a bend.

Terminology question: What is "aximuthal" component? Regarding the cross section, can you elaborate?

 

[Andy Resnick]

Terminology question: What is "aximuthal" component? Regarding the cross section, can you elaborate?

By 'azimuthal', I mean the stress components within a planar cross-section of the noodle at some value of 's'. For example, use coordinate labels for that stress tensor σ_rθ. I expect, for parts of the noodle hanging vertically, σ_rθ=0, while for a noodle placed on a horizontal surface I expect something like σ_rθ=rcosθ ρg (ρgz, in the lab frame). Does that make sense?

 

[Chestermiller]

By 'azimuthal', I mean the stress components within a planar cross-section of the noodle at some value of 's'. For example, use coordinate labels for that stress tensor σ_rθ. I expect, for parts of the noodle hanging vertically, σ_rθ=0, while for a noodle placed on a horizontal surface I expect something like σ_rθ=rcosθ ρg (ρgz, in the lab frame). Does that make sense?

At this point, we're not looking at the components of the stress tensor yet. We're just looking at the external forces acting on the noodle, which, fortunately, we can do without considering the stress distribution (because the system is statically determinate). And, of course, at the surface of the lip, we are assuming no friction, so the external shear stress on that surface is taken to be zero.