Why Does Fluid Pressure Decrease and Velocity Increase in a Tapering Pipe?
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- Опубликовано: 27 сен 2024
- Bernoulli's Equation vs Newton's Laws in a Venturi
Often people (incorrectly) think that the decreasing diameter of a pipe correlates to a smaller volume and thus higher pressure. But the ideal gas law has no place in INCOMPRESSIBLE fluid flow.
Others try to explain the decrease in static pressure using Bernoulli's Principle, however, unless you have a VERY good handle on Bernoulli's, that is mostly a self supporting argument.
Back all the way up to Newton's 2nd Law to understand why there is a decrease in pressure in a pipe taper.
I don't think it's the same kind of pressure we are discussing here. There's static pressure and there's dynamic pressure. Static pressure is the pressure exerted on the walls of the pipe (what pressure gauges read). Dynamic pressure is the force at which a fluid is moving per the area of the pipe. When we take a closer look at the equation again we find that the dynamic pressure is a function of the flow velocity: the higher the velocity, the higher the dynamic pressure.
What I have personally experienced from my few years as a mechanical engineer in the oil and gas industry is that. When flow is stationary, readings on pressure gauges increase and are almost nearly the same irrespective of pipe diameter. But once there's movement, the smaller diameter pipe records a lower reading on a pressure gauge than that of the bigger one. This is because in the smaller pipe fluids move faster so there's less time for molecules to stay at point to be read by a pressure gauge. But in the bigger pipe flow velocity is lower therefore molecules stay longer at a point and hence are picked up by pressure gauges. If pressure gauges could be installed parallel at the centre of a pipe's diameter we will see that pressure gauges in a small pipe will read higher than a bigger pipe, because fluid will rush with more speed into the pressure gauge.
So the confusion is really about understanding static pressure and dynamic pressure. In a smaller pipe, static pressure is lower and dynamic pressure is higher due to a higher flow speed. But in a bigger pipe, static pressure is higher whilst dynamic pressure is lower due to a lower flow speed.
That you are saying indirectly in big diameter and small diameter pipe will have same pressure but gauges are not showing in small pipe due to its moving high velocity
I think video is quiet clear. unbalacne forces due to staic pressure is (P1 and P2 here) is causing the water molecule to accelerate in all directions on right side but molecule is moving in direction where resultant of all falls .due to accelration we have velocity and that leads to dynamic pressure .
so the force acting on fluid molecule in all direction causes static pressure and there unbalanced nature cause acceleartion and hence velocity leading to dynamic pressure.
I hope this clears your doubt.
The explanation via Newton's 2nd law is a good one. However, what it clearly indicates, is that due to increasing velocity, there is necessarily a (positive to the right) acceleration, and therefore there must be a corresponding pressure gradient to explain this acceleration (to produce the necessary force). It does not state why the velocity (and hence acceleration) changed in the first place. That links you back to conservation of mass, or continuity.
So: Continuity explains why the velocity increases, and then via F= m(dv/dt) we can infer the necessity of a pressure gradient.
Well put!
YES, YES. However, that is STILL a deductive INFERENCE, NOT a cause and effect explanation. I give full detail 2 weeks after this comment (above).
@@Observ45er I am also unsatisfied with this video's answer. Feels like a chicken and egg scenario. I am interested in your answer, however I can't find it.
@@INTEGRALPHYSICS Are you working on a corrected version?
@@rever4217from am intuitive mechanics perspective look at it this way.
The constricting pipe has water bouncing off it, these water molecules bounce backwards into the flow. Increasing the pressure of the oncoming flow.
This increased pressure results in a pressure gradient.
By the time the pipe becomes straight again there is no more water bouncing back increasing pressure, and thus the pressure remains steady (and net lower than all sections to the left as shown) but all the water now has a higher velocity due to the previous sections of pipe having accelerated it due to the pressure gradient.
Fun fact. In 1965 the Wood brothers used a fuel holding tank with this design which allowed them to dump 58 gallons of fuel into the tank of Jim Clark's Indy car in 15 seconds, while everyone else took 45 seconds to a minute. Thereby helping him secure the 1965 Indy 500 victory.
Fluid pressure DOEN NOT REDUCE in a venturi.
It becomes directional.
Pressure only appears to decrease tangentially to the flow.
If you measure along the flow, pressure actually INCREASES in a venturi.
The overall pressure remains much the same.
R.
@@RalphEllis That is what static vs dynamic pressure means. In Bernoulli's Equation (shown in this video), static pressure + dynamic pressure is constant in a system. If there is an increase in dynamic pressure (what you are calling "directional" pressure) then there must be a corresponding decrease in static pressure (what you refer to as "tangentially to the flow".
How is that related to Bernoulli? Isn't the flow always the same because of continuity?
Venturi effect. Does not increase or decrease pressure. It only increases velocity much like placing your thumb over the water hose.
Thanks for that, never heard that before but that is cool
To think about this question intuitively, the pressure will stack up when water flows from a wide diameter to a smaller diameter. So the Pressure will be higher at the low-velocity part but remain unpressurized at the high-velocity part.
It's like pinching the soft water hose will let the water spray further. When doing so you will feel the force to pinch the hose, which will lead to pressurizing the original water flow.
The video made me understand it. Your explanation made me GET it.
I think there's something to do with the difference b/n STATIC and DYNAMIC pressure. You're example helped me understand something about "pressurized" because I was always taught that a resistance to flow (ie, smaller diameter pipe or something) INCREASES pressure
I have some googling to do lol
Good point 👌
What about pressure in 2 different pipes of different diameters originating from same source. Or when we introduce pressurized air to water is it only the velocity gain or also the pressure gain.
I really don't know why nobody ever explains it like this, love it!!
Thanks!
@@INTEGRALPHYSICS hey, what is still confusing me is the pipe diameter and pressure. Won't the smaller pipe have greater pressure as water flows through?
I came here looking for a scientific reasoning to a magical ability im writing and I learned alot more then I thought I would lol.
oh crap magical pressure regions, it's Stormlight Archive all over again!
Outstanding. One step further would be to realize that it’s not the fluid going from slow to fast that causes the pressure to fall, but rather that the pressure difference accelerates the fluid from slow to fast. Same thing, but more clarity.
I could be wrong, but...
I don't buy that explanation for the phenomenon: it seems circular. If I accept that explanation then I have to ask why the pressure difference exists in the first place. We appear to have a cause and effect problem: did the pressure difference cause the change on velocity, or did the change in velocity cause the pressure difference? Put another way, just because the equation balances doesn't tell us cause and effect. I see it that the most fundamental fact is energy conservation. The facts are as follows: 1) the fluid isn't compressible; 2) the mass flow past any point in the pipe must be the same as the mass flow past any other point (before or after the taper); 3) the energy flow past any two points must also be equal. So if the cross sectional area after the taper is smaller then the velocity simply must be higher. For any molecule, therefore, energy conservation mandates that if velocity increases then pressure must drop. Newton's Second Law allows the equation to balance, yes, because it too relies on energy conservation, but it does not explain WHY the phenomenon occurs. This is a problem with math - or rather the misuse of math. The math doesn't incorporate the sequence, the process. It is a language for expressing the activity around the phenomenon, yes, but it doesn't explain the phenomenon in terms of cause & effect.
@@julianopificius6910 agreed, the maths simply explains how it works in terms of energy and stuff like that. But there is a deeper fundamental physical explanation that is missing. F=ma and all other rules like that are observations to transcribe what we see, they don't actually explain why and how a specific mechanism works
This may be right or wrong but, imagine a pipe 3 water molecules wide, which then converges into a pipe which is one molecule wide. There is pressure in the system as we know, so the water in the wider part is being forced into the thinner part. All of a sudden, 3 molecules which could stand side by side are now forced into a pipe too narrow, so they have to queue up behind each other in a line. This basically thins and elongates the flow of water, and because the 1 molecule wide flow cant extend backwards into the 3 molecule wide section, it has to move forwards faster. Imagine squeezing toothpaste out of a tube, you may move your fingers a little but a greater length of toothpaste will come out than how much you displaced your fingers, because its being extruded through a narrower section. So basically, the narrowing of the tube squeezes the molecules towards the centre of the pipe. This causes them to spread out lengthways, which is where the mass flow rate is maintained. The reason why they dont spread out backwards and only extend forwards, is because the pressure from the water behind is forcing them to only travel one way
@@julianopificius6910 I've got the same issue as you, which I made a post about as its own post; and only then, I saw this.
@@julianopificius6910 YES! YES! YES! YES! YES! YES!
Did I say YES!!!
Equations NEVER show / explain cause and effect!.
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I have a full, corrected explanation 2 weeks after yours (above), but in short:
The walls decreasing in diameter is a restriction that RAISES the pressure in the LEFT section AND up stream.!. The right section is like a hole in a pressure tank partly relieving the pressure.
There is a conversion of some dynamic pressure to static pressure at the sloping wall that is 'communicated up-stream [that is a characteristic of fluids to communicate pressures gradually away from a new pressure region - Pressure Gradient.
It is JUST LIKE the small nozzle on a garden hose shooting farther. Here is a demo showing it being measured:
*ruclips.net/video/hZ5fZ3K4_mE/видео.html*
@@nade5557you just explained this whole video for me😂. I'm sitting here thinking he didn't explain why necessarily just proved that it does. (At least to me anyway as I am not fluent in the language of maths) Your explanation is the why this does what it does. If there wasn't a drop in pressure, backflow could or would occur. Thank you for this, my scalp was getting sore from scratching my head😂
I wish my physics and hydraulics profs had explained this in so much depth. Brilliant!
It finally clicked for me! Thanks a lot :)
Worth mentioning is another (wrong-ish) explanation that the increase in velocity decreases random motion, thus decreasing the pressure exerted on the pipe. However, Bernolli's principle only applies in streamline flow, i.e. no random motion.
Thanks!
Right! Many people try to find a reason for the reduced pressure by thinking of the molecule vibration and always make wrong assumptions. It is common to assume that the random motion decreases toward the surface and increases down stream making it no longer random about a center in all directions - bad assumption.
Finally, the response to this question after all these years! :) :) Thanks!
Clever explanation. Better than I got in school 😊
Thank you for this clear answer !
It was not intuitive for me but after I saw the video I feel like it's intuitive :
It's logic that the fluid goes to a place where there is some resistance of the flow which generates an increase of pressure, and when it passes this place, there is no resistance anymore so the fluid is less under pressure.
This explanation is actually very useful to understand physiology and blood pressure. There is a lot of tapering in blood vessels... Thanks!
that was nice from you. that is what I got from you" imagine if you and your friend pushing an object to each other. Now the stronger one will push the object to the weak one" . Now just substitute your self with pressure. if the pressure on narrow side of the pipe was higher that particle wouldn't have moved to the narrow side
Rotate it 90 degrees clockwise. Now we have a better intuition of pressure because of gravity. The key is "what is ahead?" The molecules at the top part (wide) are almost stuck since they have a small sink at the bottom, so they are pressing the walls. On the other hand, the molecules in the lower part (narrow) are almost free, because they have nothing ahead (below), so they are almost not pressing the walls.
No, this example is poor, because changing the height of the fluid ALSO changes the pressure in the fluid.
A vertical pipe has increased pressure as the depth increases adding confusion into the mix.
@@meateawthe example I propose is an open tube on both sides, like a funnel. The pressure only comes from gravity
I’m a practical tool and die designer and I build machine tools.. really enjoyed that which I already knew but just couldn’t explain it. “Work” is a concept many don’t understand. “Work” hardening is also another concept many can’t fathom. Unrelated here but then again aren’t they ?? Thank you.. I just subscribed.. Encore!!!
Thank you for the great explanation brother, I was in need of a quick brush up and luckily this video confirmed that I can retain SOME information. Just a heads up to anyone with epilepsy, just skip the the 11 second mark.
All the physics and all Engineering can be trace back to the 3 laws of Newton and the 3 laws of Thermodynamics
Why is he allowed to use idea of cross sectional area of a particle?!! I thought they dont Have area!
The easiest way to explain it is by looking at what changes when the pipe gets narrower. The point where the pipe becomes narrower causes water molecules to impact the wall of the pipe at an angle, which will deflect the molecules to the middle. This will increase the pressure in the middle of the pipe. Higher pressure in the middle effectively funnels water molecules into the narrow pipe. Another way to think of it: "Take 3 marbles, and line them up next to each other. Sqeeze the 2 marbles at the end and see what happens to the middle one. It is flung outwards.
Even most engineers i know cant explain this concept as well as u have done. Thank you
That's because their job is to use it, not explain it; They're two different things.
...and thanks!
I've been thinking about this a lot, and if you take it down to the molecular level, in air, the molecules are moving in all directions at one speed and when we say that the gas is really moving is that there are more molecules moving in one direction than another, but the individual speed of each molecule is still the same (or so I think) plus the pressure is nothing more than the number of times per second that the molecules hit a surface and how fast they hit it. And that complicates these mental experiments even more. 😅😅
A good representation of the fact is how we can use water jets to cut through solid steel in a matter of seconds.
@gabrielhacecosas, I like your explanation, but my head’s hurting trying to intuit the pressure in the next bit of pipe, if the narrow section was followed by pipe at the original diameter. It seems to me like it would drop further but the flow would slow but I think I’m mixing behaviour of a gas (cools as it expands) and liquids (less compressible). It’s got me thinking though, which is fun.
@@michaellinner7772 it's in the name i guess if i had to explain it to a kid or dumb it dowmn I'd say : water jet , meaning pressured water moving really reallly fast so it cuts through almsot anything 😂,like the dc comic flash 😂
Thanks for this video. I have realized that I was always wrong to think that the pressure in the smaller part of the pipe, was going to increase. Thinking to watering plants with a plastic pipe, if you put a finger partially closing the pipe you feel higher pressure. But isn’t so. It’s higher speed of the water.
That higher speed is more kinetic energy and it is converted to increased static pressure by the partial blockage. I give full details 7 days after this (above).
The way I see it intuitively, is that since the particles are accelerating during the taper, for any particle the one further is faster and the one behind is slower. That means by the time they reach the shorter pipe, they are more spaced out. The reverse is true for a outward taper. Same thing happens to cars in traffic!
Hey yes, I think intuitively, this makes more sense... we need to imagine that everything is flowing... and not just stationary stuck there... I think understanding that everything is flowing is the pivotal part of fluid dynamics as compared to simple newtonian motion understanding where bodies flow as a whole solid block... which is much more straight forward to visualise... If fluids are flowing, then pressure will build up and velocity within the fluid changes... personally, this is hard to intuitively grasp because my brain does not process both velocity flow and pressure distribution at the same time... Here's an explanation, because, once u block up the end and the fluid stops flowing, then the pressure throughout the pipe becomes equal, no matter the cross section..
One mistake that many do is to consider the velocity variation as the cause of a pressure variation, while it is the opposite. Indeed, a change in velocity means an acceleration, which means a force applied. Usually, in a fluid the "forces", so what can cause a velociry variation, are mainly relered to pressure, viscosity, gravity.
The same for the aerodynamic of a wing: the profile of the wing imposes the bend of the fluid lines (since the air cannot compenetrate the solid body of the wing), which causes the modification of the pressure around the wing itself, which determines the forces on the wing (lift and drag) and the change in fluid velocity around the wing. Babinsky gave a good explanation for aerodynamics in his paper "how wings work". Obviously the viscosity plays also an important role, keeping the streamlines attached to the wing body (otherwise they would simply deflect at the wing nose and then remaining straight instead of curving).
What I suggest for flows in simple pipes is always to solve the continuity eq and the momentum eq of the NS equations in their integral form. Help you visualize the physics of the problem.
i have watched so many videos on betnoullis principle but this does justice to it ...thank you
Bernoulli’s Principle may be just Newton’s Second Law, but it is not intuitively obvious. What would happen if we started with a flow where static pressure was constant throughout, but fluid necessarily accelerated into the constriction? We could not set this up physically, but it could be the initial condition of a computer simulation.
The answer is that the fluid would decompress in the constriction until Bernoulli’s Principle was re-established. Waves of decompression would travel at the speed of sound in the fluid both upstream and downstream. However, if the fluid is almost incompressible, then there is hardly any energy associated with the overpressure, so decompression is an insignificant process. In the resulting steady state after decompression, we can then fall back on the association between Bernoulli and Newton.
Yes, and it can be stated more generally that the momentum equation of the Navier Stokes equations, rho Du/Dt = - delta p + div tau + rho g is derived by (per unit volume) Newton's 2nd law. The density rho times the substantial or material derivative of the velocity vector u is the mass x du/dt = ma portion, and the pressure gradient and divergence of shear forces (tau) + gravitational force are the applied net force. Bernoulli's equation is a particular case of the Navier Stokes momentum equation and is therefore likewise derived via Newton.
Very good and simplified explanation.
i was wondering about this for so long and this video was the most reasonable one i find in the net but stil not completely convinced cuz realistic feeling convince me that pressure increase with the increase of velocity
Finally a video that helped in clearing the concept. Good work.
Thanks!
Thanks @INTEGRAL PHYSICS. Like @nozack5612 mentioned, the explanation through Newton's second is good but still left me missing an understanding of why the nozzle creates the pressure/velocity change. To round it out I offer this explanation (after researching more and thinking this through). My aha moment relies on considering the following: 1) static pressure fundamentally is a measure the fluid particles change in momentum to a surface (think walls of a container or more commonly the cross sectional area of a shape), and 2) continuity of mass flowrates between point 1 and 2 and the relation to incompressibility (i.e. the average number of particles in a given volume cannot change). Try to ignore the physical nozzle and imagine a setup where steady, incompressible flow goes from a larger diameter pipe to a smaller diameter pipe--the same fluid, the same density, the same mass flowrate. Take a cross section of the larger diameter pipe, there are more particle collisions (high static pressure) there because there are more particles flowing through that cross section at an instant in time. Now in a cross section of the smaller diameter pipe, with the same flow (i.e. flowrate) there are less particle collisions (low static pressure) because there are less particles flowing through that cross section at an instant in time. And since there are less particles in the cross section, to match the same mass flowrate as in the larger section the velocity must increase (i.e. the particles have to move faster through the pipe, otherwise that would mean the density is changing somewhere in the pipe!).
Okay so working under that knowledge, how does a steady, incompressible fluid flow go from a larger diameter pipe section to a smaller diameter pipe section? A nozzle! I think the counter-intuitive part is that we expect the fluid to be squeezed by the nozzle getting smaller and that means a larger pressure right? Nope. That only happens if we take the same mass of non-moving fluid from a larger volume into a smaller volume. However, I think that does happen when talking about supersonic nozzles or fluids that are moving REALLY fast (i.e. Mach numbers greater than 1) because compressibility changes, but I haven't studied enough yet.
That's my current understanding which still feels incomplete, but I hope that helps someone. I've been re-studying fluids for the FE exam which is why I'm here.
A simpler way to think of this: is because the taper creates resistance to the airflow, so therefore the system must exert higher pressure to maintain the air flow. Higher pressure requires more energy. More energy means more heat. This carries over to electrical resistance too.
yes - rarely is the requirement for an energy input mentioned...
That's almost close.
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Sounds like you have it, but to be sure:
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In this case, the narrowing of the pipe is a restriction (whether by a nozzle or finger). The fluid is, in effect, 'hitting' the inward-sloping wall, thus pushing more and increasing the pressure along the sloping wall, AND because pressure acts in all directions, this increases the pressure in the FAT section.
Now you can view the narrow section as an opening in the now pressurized fat section, letting fluid escape at a higher speed because the pressure there is lower there. It is the pressure difference ALONG the flow that Accelerates the fluid mass - This is Newton.
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So. . .
Using your view, it is the P and therefore the F in the fat section that has increased above that of the narrow section. This is a NET P in the direction of the narrow section.
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It is analogous to the narrow garden hose shooting farther. The narrowing causes the pressure inside the hose to increase. This makes it even higher above the atmosphere than it was, thus accelerating the water even more and shooting farther.
It is a difference in NET PRESSURE that Accelerates fluid toward the lower pressure. Newton in fluids.
See it in this demo:
ruclips.net/video/hZ5fZ3K4_mE/видео.html
Great intuitive explanation! Your graphics and handwriting is absolutely Formidable! Thank you!
Glad you think so!
I was always more interested when the flow is in the opposite direction.
I, a lowly operator, could not convince the engineer at a plant I used to work at that pressure increases at an enlargement. He was not having it.
You should have explained the operation of a feedwater injector to him.
@@robertmurdock9750 never dealt with those but I know how a volute on a centrifugal pump works!
It's possiblle that he was unable to quantify "pressure" many people have difficulty with this.
As a plumber I explain this to apprentices like this
" what is the pressure of the water after it comes out of your shower head?"
They ponder this a bit and usually get it wrong ( it's 14.7psia if you are at sea level )
I then ask this question
"Would you rather have a shower with 1 gallon per minute ( gpm ) and 100 pound square inch ( psi IN the pipe feeding the head ) or 100 gpm and 1 psi?
The pressure is only there to get the medium to its point of use, after its out of the pipe it is at atmospheric pressure.
Velocity does play a role here but as mentioned FLoW is the thing you're after
@@MrDmadness Can you explain why a narrow nozzle results in higher force? That is, when a hose has a narrow nozzle, you feel the push-back or recoil force far more. Is it that once it exits the pipe it's back at atmospheric pressure but still has the velocity? Where is the "extra" force or energy in the equation coming from?
@@DiscoFang it results in a higher rate of flow, force is area x pressure. What youre feeling is the motion if the flowing water encountering a restriction. Essentully you are feeling water hammer.
similar to a decrease in lanes on a freeway. before the decrease there are more lanes, the cars are moving very slow and all packed tightly together but after the decrease in lanes the cars are moving fast but also all spread out from each other
It is so nice to see someone able to apply the fundamental principles that so many others pass right by. . . . . However . . . .
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That said. you have a few rough spots there. You are SO CLOSE, but still missed something important. Marked thus below if you can't wait ******
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Yes, at time 1:27, using only Bernoulli's Equation is a c--p answer. It only says WHAT happens, but NOT why [[however, I show here that you do the SAME THING below [deductive reasoning] yet call THAT [below] "the" explanation.
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First, at time 2:18. The work MUST come from the Pressure Gradient.
{Are you familiar with that common fluids term? It is a difference in pressure between two locations}
A net force IS REQUIRED to accelerate the mass. You can't get around that first principle. That is something that CLEARLY explains a cause and effect -- it tells us the WHY there is acceleration.
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Side issue: Unfortunately, you say that it is the same as the "explanation" from Bernoulli's equation ... BUT ... Bernoulli's Equation does _NOT_ "explain" anything. It only shows _WHAT_ happens [and how much], but not WHY!.
In fact you even say: "Remember, Bernoulli's Equation is derived from the work-Energy Theorem." Therefore the B-Equation is not a fundamental law of physics; the energy stuff is.
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Back to the main topic:
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Your talk about Newton's Second Law at 2:45 to 4:50 is spot on. [[This actually just repeats the part about your 'work' talk at 2:18]]
Namely a force is the _CAUSE_ of acceleration [of a mass] of the fluid. That is an real explanation! [[Actually, the First Law is the one that tells us that a force causes acceleration, but your path is ok too - except that equation also does not tell us that force CAUSES acceleration - deductive reasoning. It just verifies that it agrees with Newton.]]
.However, you have NOT explained WHY that Pressure Gradient occurs - ONLY that it must be present [because we see some acceleration --> there must be a force]. Deductive reasoning, but not an explanation of the physics cause and effect. . .
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So ^WHY does the Pressure Gradient occur?* What is the CAUSE of this EFFECT.??.
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First, people are fooled into focusing on the static pressure decrease on the right, instead of the higher pressure on the left.
[[You appear to understand that you are talking about what is commonly called "Static Pressure"]]
For starters:
You totally missed the lesson of the finger over the garden hose demonstration.
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Do you see? When you say: "If we see B , then there 'must be' an A; this deductive, not a definitive cause-effect explanation.
SO. . . All you've done is shown with the various "physics laws" is that the pressure decreases, but still NOT WHY it decreases.
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BTW: At time 2:35 there is nothing to click to see the Work-Energy explanation.
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****** So, here is THE _why_ EXPLANATION: ******
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The lower static pressure on the right is NOT the result.
The higher pressure on the left is a result!.
Here is the CAUSE of that INCREASED pressure and, THEREFORE, the cause of the Pressure Gradient:
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The diameter decrease is a restriction of the flow. THAT CAUSES a pressure INCREASE on the left section.!.
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HOW a.k.a. WHY?
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Fluid approaching the pipe walls as they narrow, increases the pressure on that section of wall. Fluid moving toward a wall is the cause of an increased pressure against the wall. Just like a wind increases pressure on any surface it approaches - blows against. We easily feel this when wind blows on us. If a surface stops the flow we have stagnation and ALL the kinetic energy [dynamic pressure] is converted to Potential energy: a.k.a. "stagnation Pressure".
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But we have sloping walls with fluid approaching them, so only some of the kinetic energy is converted to potential energy - static pressure. This is an increase in the pressure ON the sloping wall. This pressure "communicates" inward and UP-stream into the large section, thus INCREASING the pressure there.
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To prove this, we can simply observe that if we add a narrow section [a small nozzle \to a garden hose the water shoots farther. There is a smaller cross section and, therefore less water, BUT had the pressure NOT changed, it would shoot the same distance. In other words Newton--> less water, same pressure -->less mass --> smaller area=less force, THERFORE same acceleration. SO: Proof the pressure increased upstream IN the hose by the addition of the smaller diameter.
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The analogy is that the left hand section is like a pressurized tank and the narrow section is a hole letting the pressure out. . .
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Here is a video showing the manometer-MEASURED pressure increasing upstream when a restricting nozzle is added [the garden hose demo]:
ruclips.net/video/hZ5fZ3K4_mE/видео.html
This is the most well thought out comment I have ever seen on RUclips. Spot on. Thank you.
@@INTEGRALPHYSICS I got some error and YT won't let me post more..?..
SO. . . . You're not going to correct your errors..?..?..
As a hvac student theres a much simpler awnser for this. There is ALWAYS a pressure drop due to RESTRICTION. Restriction in a tapering pipe, restriction in a evaporator coil you name it
Just came up with this now. I think a good way to visualize it is a sand timer...when the sand hits the constriction, the sand packs together and there is high pressure with all the sand packed together and the sand is moving slowly. The sand that manages to pass through the constriction moves quickly as it drops and has few sand particles around it... low pressure.
it's the same principle as a planes wing...the fluid (air) flow is faster over the top and therefore has less air pressure that the slower flow under the wing and thus gives it lift...
and if we look at it in terms of vectors, the faster you go in the horizontal direction, the less of an effect the vertical direction has...so the 'direction of flow' pressure will be greater than the vessel wall pressure...the faster it goes, the less wall pressure...
This is an accurate way to describe it for sure
If i had this kind of explanation in my fluid mechanics class, i would have aced all my exams 😅
I'm not going to watch the video, I just came here to say that I'm sure you are wrong and I have 0 evidence to back up my claim
. . . and that is OH SO important for everybody here to know.
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NOT!
😂😂😂 that made me laugh thank you that made my night
@extrarice365 correctly understands how to use the RUclips comment section. Lol
Thank you for this elegant explainer.
Glad it was helpful!
thank you for not just hand waving it to be due to Bernoulli's principle. Ive watched like adozen videos that discuss preasure etc. and ALL of them just say "due to Bernoulli's principle thing XYZ happens"
Now explain why fluid has pressure
Seeing that the pressure is only lower when the water is moving is a point. Ie static pressure would be equalised along the pipe but as soon as the end is opened the water has somewhere to go but cant continue to carry the extra force with is as now more water gets restricted by the funnel and as the water ahead of it is moving towards a lower energy state ie the pressure drops but the speed increases.
Another good way to visualize this is that, if we look at the larger diameter hose; it will require more pressure to push the particle through the smaller diameter, thus after the pressure has increased to push the particle "smaller" then the pressure will decrease in the smaller hose due to "work" diverging to zero. Because work is being most performed when the hose diameter is changing. This is what I had gotten out of my Physics class haha.
if there is only one thing you get out of physics it should be work.... work isthe underlayment of virtually everything.
@JustBeingHonest813 You've actually got it a bit better than the author. . . The walls decreasing in diameter is a restriction that RAISES the pressure in the LEFT section. The right section is like a hole in a pressure tank.
*ruclips.net/video/hZ5fZ3K4_mE/видео.html*
THANKYOU HAVE BEEN LOOKING FOR THIS SINCE FOREVER
Yeah, I get this question a lot... I figured it would help someone out there in RUclips land.
If the velocity on the left side is forced, then the taper part in the middle acts as resistance. Hence the pressure on the left side increases. Since the velocity is forced and remains unchanged, thr velocity on the right side of the taper must be higher
YES! That IS correct! The author is SO CLOSE to getting to the true explanation by focusing on fundamentals, but misses some important things. I posted a full comment on his errors just above this.
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In short, The walls decreasing in diameter is a restriction that RAISES the pressure in the LEFT section. The right section is like a hole in a pressure tank. Here it is measured:
*ruclips.net/video/hZ5fZ3K4_mE/видео.html*
That makes a lot of sense, but could that be the whole explanation?
A thought experiment to illustrate my question (for which I sincerely don’t know the answer): Let’s imagine a long tube with a constant thickness along its length. It suffers a pressure p1 from the fried it is carrying. Now, let’s cut the first half and make it thicker, with a transition area. If we now measure the pressure in the second half of the tube (the one with the same thickness as before), would it still be p1?
@@AntonioReis640 if you mean the fluid in that pipe is stationary, the pressure on both sides would be equal. But if you increase the internal volume of the tube and don't add fluid to occupy that additional volume, I guess the pressure would decrease a bit. But, the pressure would be equal in the entire tube. In order to flow, there has to be a pressure difference. The flow direction will be from high pressure to low pressure
Gas and Fluid dynamics are pretty hard to grasp to me. I'm not educated on the subject so I can be fully wrong. My experience on the subject comes from me building a flow bench once. I had to calculate the air flow by measuring the pressure drop indicated by water column. I got the calculation from a book but in fact it applied Bernoulli's equation. I used the flowbench to test a cilinder head I ended up butchering, which ended the project =)
@@AntonioReis640 I'm sorry, but I am unable to understand your mechanical configuration is with "carrying fried" and what the diameters (not thickness) are and where you're measuring pressure. . .
@@what9418 Be careful because the _FLOW_ does not have to be from high-to-low pressure. *Acceleration* is in the direction from high to low pressure. A flow headed toward a higher pressure will be slowing a.k.a. deceleration or negative acceleration.
Newton applies.
I've always used the thought that "faster fluids have less time to put pressure on the same area as a slower moving fluid, thus, the static pressure is lower" as a mnemonic.
But that is just not correct. Pressure is the combined effect of particle collisions on a given surface. Those collisions 'i.e. the time they have to put pressure' is determined by the speed of sound in the fluid, it's completely independent of the flow velocity, especially for water.
@@Michallote If a mnemonic serves its purpose, you can't call it incorrect. A mnemonic is not the thing.
That's an absolutely fantastic explanation. Why did no one expalin it this way at uni? Thanks
But what is the logic behind this decrease in pressure? Explain by logic not by equations, please. this is a request from me.
Would that explain why the fluid would gain higher pressure if we put an enlarged pipe after that small one?
The pressure goes higher but the fluid would still flow forward.
I was wondering the same thing but I think going small pipe into large pipe your pressure would go down upon entering your larger pipe. I believe it has to do with the flow direction. An example traffic jam on two lanes suddenly hitting a 6 lane freeway, velocity could increase and pressure decrease.
cant lie, that ellipse ruler seems so freakin' useful, i might get one
I'm not sure if I'm understanding correctly, but it seems to me like a more intuitive understanding of the pressure relationship would be to look at the situation in reverse. I see the funneling as increasing pressure on the input side rather than decreasing pressure on the output 🤔
Very good and scientific analysis behind the Bernoulli equation. My visualization of it was that as the individual molecules accelerate, they create more distance between themselves, and if the change in velocity was great enough, the distance between particles will be greater than at atmospheric, thus the vacuum.
That isn't correct because water is constant density and with air, the pressure change is SO SMALL at these speeds, that the density change is irrelevantly small.
The pressure changes for this and flight are amazingly small and so is the density change of air.
What you and the author are missing is that the sloping sides as the diameter decreases is the cause of a pressure INCREASE in the fat section. That slope converts some dynamic pressure kinetic energy to static pressure in the wide section.
Think of that fluid running toward, or a glancing blow to the slope as if it ids a "partial" stagnation pressure.
@@Observ45er Well, fluids may not compress but they do still translate potential energy into kinetic energy both during compression and expansion as can be seen during cavitation. My idea was that even though there is no density change there is the "lack of propensity" to compress relative to the liquid towards the wider end of the tube, and if that situation were to be extrapolated, it would even translate into "less compression" than at atmospheric.
@@ChuddleBuggy I'm not following all of that.
RE: "they do still translate potential energy into kinetic energy"
Yes. As Euler showed, a Pressure Gradient [potential] is the cause of fluid acceleration [kinetic].
ok. An analogy is a compressed spring accelerating some mass.
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RE: "there is the "lack of propensity" to compress relative to the liquid towards the wider end "
If I understand this, it is a directional preference for pressure to act, but a fundamental property of fluids is the omni-directional property of pressure. So you are violating that first principle.
Also, if there is this 'resistance to compress', why would the pressure be higher? Isn't this 'lack' a resistance to an increase in pressure? That would prevent pressure from increasing, if true. . .no?
I'm not seeing the reasoning. But in any case, that's not it.
There are well-understood first principles that apply.
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I have a full comment explaining, but in short:
The sloping walls at the pipe narrows is a restriction for flow.
This surface with fluid moving toward it, is easily seen as a cause of increase in pressure in the wide section, on the left.
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This is no different than a wind blowing on us and we can directly feel the pressure increase. The decreasing diameter is a restriction that is the cause of increased pressure on the left.
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Focusing on the pressure decrease to the right is where people go astray.
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This is something we easily see in our every day, layman experience when we use the garden hose nozzle to shoot that long stream, or put a finger over the end to shoot it far.
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Here's a demo that clearly shows the pressure rise caused by a restriction, measured with a manometer. ruclips.net/video/hZ5fZ3K4_mE/видео.html
@@Observ45er I understand the part where the pressure increases due to the increase in resistance down the line. That's intuitive enough. Where we might be misunderstanding each other is why the pressure after the constriction decreases even to the point of becoming less than atmospheric.
As i stated in my earlier comment about "translating potential energy into kinetic energy", what i meant is that even though the liquid may not have increased or decreased in density by an appreciable amount, the liquid will still have possessed the potential for creating an effect due pressure difference it would impose.
Another way to look at it would be that since the velocities of the flow between the wide are of the pipe and the narrow area then there must be a pressure difference between the 2. Of course, the flow volume would be the same, but the velocities would be vastly different, which would introduce the reasoning behind the increase in velocity as a response to the unity of the flow volume throughout the entire system, thus causing the analogy to become a bit more unwieldy.
At any rate, the miniscule change in density of the liquid as it passes the restriction into the narrower tube is in fact the reasoning upon which my analogy is based.
@@ChuddleBuggy I'd have to analyze that further, but your original comment is incorrect and I see no foundation for a tiny compressibility explaining r5he venturi's behavior.
Thank you for this amazing education! You are a god send. Wondered abiut this for years
Pressure in pipe is momentum transfered by the fluid molecule with pipe wall normal to the direction of flow
As per continuity if area decrease to maintain same flow rate velocity has to increase in the direction of flow which means particle has resultant velocity more towards flow direction reducing the momentum sharing time aka impuse which means force exterted on pipe wall reduce which reduce the pressure
Very well explained and something I used to deal with on a daily basis and within valves.
Thank you!
Why does pressure drop in the globe valve?
Bernoulli equation applied at 2 points along a streamline is a conservation of energy equation: total pressure stays conserved, so an increase in dynamic pressure means a decrease in static pressure. That’s a perfectly valid and non-crappy explanation.
Yes, but is is non specific as to WHY.
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In this case, the narrowing of the pipe is a restriction (whether by a nozzle or finger). The fluid is, in effect, 'hitting' the inward-sloping wall, thus pushing more and increasing the pressure along the sloping wall, AND because pressure acts in all directions, this increases the pressure in the FAT section.
Now you can view the narrow section as an opening in the now pressurized fat section, letting fluid escape at a higher speed because the pressure there is lower there. It is the pressure difference ALONG the flow that Accelerates the fluid mass - This is Newton.
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So. . .
Using your view, it is the P and therefore the F in the fat section that has increased above that of the narrow section. This is a NET P in the direction of the narrow section.
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It is analogous to the narrow garden hose shooting farther. The narrowing causes the pressure inside the hose to increase. This makes it even higher above the atmosphere than it was, thus accelerating the water even more and shooting farther.
It is a difference in NET PRESSURE that Accelerates fluid toward the lower pressure. Newton in fluids.
See it in this demo:
ruclips.net/video/hZ5fZ3K4_mE/видео.html
@@Observ45er "The fluid is, in effect, 'hitting' the inward-sloping wall". Not so, the flow will always be parallel with the wall near the wall, and in fact has zero velocity at the wall (no-slip condition), so the flow is definitely not 'hitting' the wall. This is continuum mechanics, not a rarefied gas where you can treat the fluid like billiard balls and have to rely on statistical mechanics.
The WHY is explained simply by the fact that the flow obeys the Navier-Stokes equations, from which the Bernoulli explanation follows.
Analogies are of course useful and a great tool to explain some things, but real world physical phenomena don't always have to follow some simplified analogy.
@@frikkied2638 It appears I am unable to respond any more.
Very good video. In addition it might be worthwhile to watch Eugene Khutoryansky's great graphic explanation of the Bernoulli effect demonstrating what happens at an atomic or molecular scale.
ruclips.net/video/TcMgkU3pFBY/видео.html
Bernoulli make some assumptions, but using just purely Newtonian analysis can give you an intuitive understanding why. Nature is very thrifty, and she loves to conserve everything. In this case it's mass flow rate that needs to be conserved. Due to the conservation of mass flow rate (ρAv), the fluid accelerates as it enters the narrower section. According to Newton's Second Law, this acceleration is driven by a net force, which is related to the pressure difference between the wider and narrower sections. Newton's Third Law tells us that the forces exerted by the fluid particles on the pipe wall are reciprocated with equal and opposite forces. The average of these forces per unit area is what we measure as pressure. In the narrower section, less net force is needed to maintain the higher velocity, which leads to a corresponding decrease in pressure.
Newton got almost everything right ;) He was right about gravity at the macro scale, so we'll give him that. I prefer Leibniz's treatment of Calculus and I think that's what we are taught mostly in school. Leibniz is the one that came up with the symbol for integration (he modified the summation symbol), and he came up with the dy/dx notation we are all familiar with today. Plus he invented binary notation, and as a programmer I have to say that gives him an automatic lead ;)
Oh the velocity is used as kinetic energy to thst is converted into potencial energy that is used to pump the water trough smaller pipe so it slows down
What confuses me also, as Area decreases Pressure increases, so how come in the narrower section of the pipe the Pressure is less; not trying to pinch holes in your explanation but rather introducing a new reason behind the confusion over this concept. Good video.
You're confused because you're not making the distinction between Dynamic Pressure and Static Pressure. The Gauge measures Static Pressure. Because the particles are accelerated parallel to the pipe, the dynamic pressure goes up along that vector. At the same type, the Static Pressure on the Pipe goes down. Think of it this way. If the water is pushing more in the right (along the direction of flow) it's pushing less on the pipe (perpendicular to the direction of flow).
find the flow velocities v1.v2. v3 in the conduit shown in fig belw. the flow rate q is 800l/min and diameter d1.d2.d3. at section 1.2 and 3 are 50,60 and 100mm repectively
beautifully explained
You’re trading the potential energy of the pressure in the fluid ahead of the nozzle for kinetic energy in the fluid after the nozzle. That’s what nozzles do. ;)
Excellent sketch skills!
Thank you! Cheers!
Bernoulli is just conservation of total entergy combine with conservation of mass flow.
wow just wow. What an explanation.
Bernoulli was an engineering condition, ranging from Steam Turbine numbers to meet conditional probabilities to daily carburetor replaced by fuel injection for momentum-obsessed.
For easier understanding, since water has a mass, I imagine that the water is like a bubble gum that when you pinch one portion and stretch it out, it would be like going into the narrower pipe being stretch out and consequently with lesser weight, or lesser pressure.
May be a more intuitive or comprehensible way of saying what you said at 4:37 is; Since we know F1 is larger than F2, the pressure P1 would also be higher than P2. And since, P1 and F1 being larger than P2 and F2, the fluid accelerates from left to right from a lower velocity higher pressure to a higher velocity lower pressure region.
Also, may be another way to look at the Bernoulli's Equation is; since the dynamic pressure increases as the fluid gets higher velocity, it stores more kinetic energy into it. And, at the same time, due to having a higher acceleration, the fluid particles are more or less less static and thus have low static pressure. Correct me please, if I am wrong.
I always enjoy thinking through this problem because you can look at the problem from the perspectives of Bernoulli's, Energy or Force. Each way of thinking is reinforced by the other. IMO, in order to fully understand this problem you really need to be able to see it through the lens of all three.
Because power is the same at both sides and power is force × velocity. If velocity increases then force decreases. Lower force-> lower pressure.
well put!
You should try to use this for 2 stroke expansion pipes. It includes the speed of sound. And how the air flow pressure differences. Just and idea
Super Explanation. Thanks a ton!
It is in fact bernouli principal, ivs been a plumber my entire life. An increase in velocity causes a decrease in pressure, this is taught in 1st year. Something you're not equating properly is pipe wall friction, though newton's 2nd law does apply here. Break everything down to joules if you want to know where the friction losses are and how much is lost.
Just like an electrical transformer. Whenever you reduce the Volts and the Amps will go up. The two are INVERSELY related.
In water pipes as pressure drops velocity goes up, and vice versa. The two are inversely related.
I'm just a pipefitter, but when the fluid passes through the reducer, the least resistance of that flow, is causing some vacuum from the opposite side of that flow. Thus dropping the pressure. But I'm not a scientist or matheticision.
This might be the observation of an ignorant - but in the drawing shown above, the assumed flow is from left to right i.e., from the large diameter pipe to the small diameter pipe. This lets one argue that the pressure HAS to be lower in the small diameter pipe otherwise the fluid couldn't flow from left to right. If the flow is reversed i.e. the fluid flows from the small diameter pipe to the large diameter pipe what happens then? If the pressure situation remains the same then the fluid must flow from an area of low pressue to an area of higher pressure. What are the forces that provide the energy to allow this?
Perfect presentataion. Thank you. raphael nyc
good explanations I haven't seen before, relating i back to newtons 2 nd law, but cant we go further modeling multiple particles to explain the increase in speed.
Wow great tech drawing!!! I’m a drafter use auto cad and Solidworks a lot very cool drawing and breakdown, insta sub
Awesome! Thank you!
I am currently learning the basics of electronics. I’ve heard my dad describe this with regard to current and voltage and amps. I definitely still don’t have the straight, but I have a feeling this explanation will go along way toward me making solid progress.
I have a plumbing and electrical ticket. They are very very similar
Plumbing has
1. Pressure
2. Flow
3. Friction
Electricity ( the flow of electrons through a conductor )
1. Voltage ( electrical pressure )
2. Amps ( flow of electrons )
3. Ohms ( resistance to flow )
The formulas are even basically the same for calculations.
Plumbing has valves, electrical has switches.
Both stop or regulate the flow of medium.
:)
@MrDmadness I agree. However, I'm confused by the fact that a reduction in wire size does not reduce voltage (pressure) as it does in a fluid. Although the current (flow) is reduced.
@@roberta.6399 wait, a reduction in wire cross area increases resistance, which does decrease voltage. This is why thin wires overheat, because current does not change. No?
@@roberta.6399 The more correct analogy is: Fluid pressure drop (for a given fluid distance displacement) is analogous to voltage drop per unit length of wire.
@@roberta.6399 Yes, it is a bad analogy. because the current is reduced in the WHOLE circuit, not just a thinner wire section.
Nice try, but it fails.
Because it has to.
Going even deeper, how would Feynman explain the pressure drop based on the atomic theory? One of his most famous explanations is why liquid in a container gets cooler when the lid is removed. Most of us just say, "oh well it's the latent heat of vaporization etc., etc. Feynman: "The fast ones leave".
I find there are few things in physics more interesting than trying to figure out how Feynman would explain something.
@@INTEGRALPHYSICS I have tried to think about what he might say several times. Something about collisions per unit time per unit area on the pipe wall. But I can't finish it.
The "work energy theorem" comes from the Newton 2nd law (by integration) so the two explanations are equivalent.
So it mainly depends on how you're trained in physics (force or energy approach). Ideally, you've got to master the two approaches (and the link between them).
Helped a lot, thank you
You're welcome!
I think you should include special relativity in your answer. We just can't understand it without it.
Ok, now please explain why reducing the pipe diameter of a nozzle lowers the pressure but increases the recoil force. That is: why a fire-hose with a narrower diameter nozzle is far harder to hold than a larger nozzle?
Mythbusters actually made a contraption out of 4 fire-hoses that worked like a drone/jetpack to lift a man off the ground. They only got the lift when they reduced the nozzle size. It had a DRAMATIC effect.
'Ok, now please explain why reducing the pipe diameter of a nozzle lowers the pressure but increases the recoil force.'
This is easy enough.
Open a second copy of this, put on 11 seconds to reference the picture easily.
Ignore whatever units may have been in the original video, we're going to use our own and make any math very easy so we can concentrate on what is happening instead of math.
A is the left, larger pipe. B is the reduction or neck. C is the right smaller pipe.
Say A is 1 inch pipe, B is the reduction, and C size is chosen to be 1/2 the volume of A per length.
Since C is 1/2 volume of A per length, then by definition 1 inch of water in A squeezes down into 2 inches long, but half the volume for each inch. But it's still the WHOLE volume in the 2 inches, and has to move twice as fast for it to flow out as fast as the water is flowing in A.
So we have mass 1 in pipe A moving at speed 1, and we have 2 times mass 1/2 moving twice as fast in C for that same total 1 mass to move through the smaller pipe in the same time.
If you throw the same mass object at twice the speed, the force it exerts back on you from throwing it is twice as much.
And realize, that 'extra speed' in the water in C that's now moving twice as fast is coming from the extra pressure you had to exert on the water in pipe A to make it squeeze down and accelerate into pipe C. Realize B is a wedge or inclined plane, and you're doing work pushing the water in A harder to make it squeeze down and accelerate as it enters into C. The work and changes don't come from nowhere, it's being exerted on the water and causes the changes in motion and pressure.
Really clear explanation. Kudos.🎉
Glad it was helpful!
You should explain it without using formulas at all:
Without a greater pressure in the thick pipe, water wouldn't move at all because the cone acts as a restriction. Restrictions tend to slow things down to a halt. Therefore if water is moving through it anyway, as with any restriction, pressure must be greater in the thick pipe.
Or
For the water to accelerate through the cone, caused by the narrowing of the pipe, it needs greater pressure on one end. So, since the water keeps flowing despite the acceleration, that needs a pressure difference to occur, pressure must be higher in the thick pipe.
What you did was really just using different equation. A simpler one, yes, but not ultimately enlightening for people not good with numbers.
Energy equations often have the advantage and elegance of not having to deal with the nitty gritty as Newton Second law. I appreciated the mental gymnastics you present, but even more the value of energy equations ;-)
How do we apply the same logic in reverse scenario in which the size increases?
I think that something is missing in the P=F/A explanation.
In that scenario: Continuity requires the fluid slow down. To slow down the fluid in the taper, there must be more pressure in front of the fluid than behind.
The way my brain intuits it is to think first of the particles that move from the larger area to the smaller area without any interaction with other particles on the way, then factor in every particle that interacts with the walls or other particles on the way, those particles impart energy of non parallel vectors before being inside the small area and therefore the remaining particles simply have more parallel vectors and on average must then be 'faster'.
You're getting close. The sloped walls is a restriction. Fluid approaching a wall increases the pressure - even when approaching at an angle. Think of wiong blowing at you - you feel the pressure increase.
This *INCREASES* the pressure on the *LEFT* fat section.!. The right section is like a hole in a pressure tank, where the pressure escapes to a lower pressure place.
After all ,You did not say why the presser decreases either ! . You just proved this event .
beautiful handwriting
Thank you.
Forces are just gradients in the conserved fields (gradients in potential trajectories through state space) so the invocation of newtons laws is also just a complicated rehashing of conservation of energy.
To "really" explain this in this form, I think you'd need a particle based answer where it's interacting with the walls. Some explanation of "how" the force is applied that's more than just "pressure" which is a statistical concept. What happens to particles that hit the pipe wall and how does that rob them of vertical momentum and direct it forward.
Wait! You just used Bernoulli (which is derived from Work-Energy) to support Newton. And then used Newton to justify Bernoulli in turn. How does that work?
Simple. The fluid particles have energy. They are moving in all directions. You move them without adding energy, the translational energy has to go down. That reduces the pressure and the temperature because some of that energy is used up to make the whole mass move.
Does this explain why when you squeeze the end of a hose pipe to shoot the water further it's because the water will go further not because of any increase in pressure but the increase in the velocity?
Sort of. That's actually the next video I'm working on.
Nice explanation, but. I Think you fudged a little right at the end. In your last line bottom right, you have “A” for both P1 and P2. But P1 is under the left “High Pressure” gauge, and P2 is under the right “Low Pressure” gauge. The area of the first is A1, and the area of the second is A2, and A1 is NOT equal to A2. What you need here I believe are differentials at your “M” and then that needs to be integrated from P1 & A1 to P2 & A2. One gets the same answer, but then the maths are correct, which is necessary in physics.
I absolutely did. I made this for my 10th grade intro to engineering class so things got simplified quite a bit... I didn't really think it was going to get north of a quarter million views.