The Bernoullis Equation for an Incompressible Inviscid flow (say water) is as follows: p + (1/2).ρV^2 + ρgh = constant where, In words, static pressure + dynamic pressure + pressure energy due to height = constant It gives the Energy Balance or Conservation Of Energy in terms of pressure, naming the Kinetic Energy of the flowing fluid as the Dynamic Pressure to unify it with the other purely pressure terms. I have found the equation always obscure in relating to the two terms Static Pressure and Dynamic Pressure especially in an Incompressible Flow. I am attatching a diagram for refernce and discussion please refer to it. It is a simple pipe of varying area of cross section and water flows within it in an enclosed volume. The volume of water is enclosed within a piston on one side and a movable solid disc made up of an elastic material such that it can slide along the inner diameter of the pipe adjusting its own periphery (under elastic force - imagine it to be under some sort of spring type force) to meet up with the pipe diameter inorder to enclose the volume of water. The piston is pushed forward with a force F and the pressure created at the piston is F/A where A is the area of the piston and section a-a. Question : does Pascal Law be applicable to the flowing enclosed volume of water ? Also note that if we consider Pascals Law or very basis of Hydraulics then pressure at section a-a must be EQUALLY transmitted at section b-b but Bernoulli suggests that pressure at section b-b must be higher than a-a. Explain this. Further on: The understanding and (lack of understanding) that I have regarding the concept behind this seemingly simple equation is as follows and please check if each of the points is correct in terms of concept and please explain the part I havent much understood. We will go in a sequence. 1. The Bernoulli Equation is based on two very basic principles of nature: a. The Law Of Conservation Of Mass - to start with b. The Law Of Conservation Of Energy - most importantly 2. The Law Of Conservation Of Mass suggests that the Flow Rate Of Mass across any cross sectional area remains the same for a flow at any instant. This leads to the important conclusion that as the area of cross section increases the velocity of the fluid must decrease in order to maintain the equal mass flow rate across the section And the velocity of the fluid must Increase at the decreasing cross section area again to conserve the mass flow rate. 3. The deduction in the second step above leads to the actual derivation of the Bernoullis Equation in an attempt to find the reason behind the observed phenomena of mass conservation. And this could be explained only by considering the fact that Energy is also always conserved. 4. The Total Energy of the flow was ofcourse the Kinetic Energy and the Potential Energy of the fluid. And as the velocity of the fluid was required to Increase and Decrease as per the reduction and increment in the cross sectional area of the flow it meant that the Kinetic Energy associated with the flow (actually streamline) must Increase and decrease accordingly. But as an Universal fact that the Energy is always conserved then the Increase in the Kinetic Energy must lead to corresponding decrease in the Potential Energy of the fluid as these are the only two types of energies associated with the flow (pressure due to height is considered negligible as elevation is negligible) 5. Thus the Static Pressure of the fluid must decrease when the velocity of the fluid increases at the smaller cros section and the Static Pressure must increase as the velociy decreases as the cross sectional area increases. Fine - as per the Law of Conservation Of Mass and the leading reason for it found in the Law Of Conservation Of Energy is considered the theory goes absolutely flawless in explaination and matches the practically observed effects of velocity anbd pressure in a flow. But the real reason behind this transformation of energies within themselves has always remained very obscure to me. My question is: a. What does Static Pressure clearly mean in an Incompressible fluid like water? and is it the same as the pressure in a gas due to thermal state? b. How actually does the pressure Increase or reduce itself almost intutively at the increasing or decreasing cross sectional area so as to let the velocity decrease or increase OR What is the ACTUAL INTERMOLECULAR CAUSE or WHAT PROMPTS the molecules in the streamline to reduce their pressure so as the velocity can increase or decrease to maintain the mass flow rate? After all the molecules dont have a mind of their own to think about this conservation laws? So can anyone please explain the actual Reason or Cause that brings about the observed change in pressure when velocity changes. This concept needs to be Actually understood at depth by me for knowing the reason that actually brings about such a change in pressure and veloity to compiment each other rather than simply muttering the fact that energy and mass are always conserved and being satisfied with the observed effects. After all we are engineers and we MUST answer all questions in terms of HOW, WHY and WHAT at the very base of anything rather than being happy with just laws and seeing them be applicable. So please answer me elaborated.
It's means the same thing in H20 or a gas, momentum change as a result of the random motion of the molecules whether gas or water. Static pressure is the measure of force from the random motion of the molecules. When the fluid is in motion, there is order introduced because of the direction of flow. Random motion still exists but the average angle at which the molecules strike the sides of something in the flow decreases as velocity increases, thus lower static pressure. At least that's how I understand it.
Hi- a)I don't understand the question. What's the difference between highly compressible and almost uncompressible fluid about pressure definition? none, as in every other substance b)if you think that pressure decrease at section where velocity increase (at same heigth) as the only way to increase the velocity with respect to some initial point, then it is physically explained. M
I skimmed your post, but hopefully we can shed some light on what you're asking. No. Yes, Bernoulli's Equation is a reduction, or simplification of Conservation of Energy. Now, often times we use Continuity (Conversation of Mass) as a second equation conveniently. What the equation actually says is that typical flows are made up of three parts: internal energy (pressure), kinetic energy (velocity), and potential energy (elevation). So for example, let's assume that we have a tall cylinder full of water. At all points in the water, the velocity is zero. If we measure our vertical datum from the bottom, the elevation (potential energy) increases linearly from the bottom. Likewise, pressure increases linearly from the bottom. So, let's assume that we have two holes in the cylinder, one at the top, and one at the bottom. The hole at the top has a tube that runs straight down the side to the bottom and exits sideways directly beside the bottom hole. Ignoring friction, which hole exits at the greater velocity? Let's find it out. We'll start with the first case where the hole is at the top, where gauge pressure is essentially 0. We know that [tex] p + \rho\frac{V^2}{2} + \rho g z = \mbox{constant} [/tex] So, our equation will look like [tex] 0 + 0 + \rho g h = 0 + \rho\frac{V^2}{2} + 0 [/tex] Solving for V gives us: [tex] V = \sqrt{2gh} [/tex] For the next case, there is no potential energy, but there is internal energy. We will use the definition of hydrodynamic pressure [tex]p = \rho g h[/tex]. So, our equation looks like [tex] \rho g h + 0 +0 = 0 + \rho\frac{V^2}{2} + 0 [/tex] You can see that ignoring friction, we've arrived at the same answer for velocity. What you can draw from this is that in a fluid there exists a dynamic between these energies. The 3 different energies can be transferred from one to the other. It doesn't matter how one arrives from point A to point B. Static Pressure is the physical molecular force that the fluid exerts on normal faces. As far as incompressible vs gas....well a fluid is a fluid. Static pressure is static pressure. I have no idea what you're trying to say here. But, as a fluid goes through a nozzle, due to continuity the fluid must increase in velocity. Because there were no outside forces acting on this, in order for this to happen, energy is transferred from internal to kinetic.
dtango - thanks for replying. Your approach to understand pressures seems an analogy from the electric current concept but does look interesting, so lets get along with the discussion and check if its really is the way you mentioned. Thanks anyway. drMS : You have simply MISSED the question - I think youve really not understood it. whats the point you are making in point b ? the question is infact - how rather what prompts this change - get it. Hi minger, Thanks for replying - though you just skimmed the post The real point Im trying to make is that - How does the Pressure change All on Its own Just like an Intutive Mind knowing that area has decreased so yeah I have to supply my part of energy to Increase Velocity and comply to the Mass conservation principle. To make it more clear, let me give you an analogy. Consider the usual example of a mass being raised to a hight h and then being released purely under the influence of gravity. We say that the mass has gained Potential Energy = m.g.h And when we drop the mass it falls towards the ground with continously Increasing Velocity. The reason for the observed inceasing velocity is given as the mass on any intermediate point ( between height h and ground) posses total energy comprising of P.E. + K.E. So as the mass continues to fall the K.E. keeps on increasing with a corresponding decrease in P.E. which in accordance to the Conservation Of Energy principle. But this is only the superficial answer to simply state the condition of the energies trying to show an energy balance saying that the P.E gets converted into K.E., it DOES NOT speak about the reason or the WHY behind the continously increasing velocity. In reality the mass DOES NOT store any kind of energy in itself (as usually said that the mass has stored P.E. = mgh) like what the capacitors do (they actually store charge) or like what a spring does(it actually stores P.E. in the form of interatomic stresses or forces) or like an insulated metal body actually storing heat energy. The means to lift the mass has actually expended energy to do so ( be it a man lifting it or a machine) The Real reason for the velocity to increase is : the acceleration due to gravity g = 9.81 acting on it continously which when acts initially exerts a downward force = mg causing the mass to accelerate and gain a certain velocity v1 at instant t1. Further the acceleration due to gravity is still acting on the mass (continously) so it will accelerate the mass further more in addition to the already exsisting velocity v1 so that the velocity increases further from v1 to v2 (v2>v1) at instant t2 and this increasing sequence is followed contionously till the mass hits the ground with a Final Velocity Vmax. This is the ACTUAL REASON for the observed increase in velocity with decreasing height of the mass and not simply the interchanging of P.E. and K.E., that is only an attempt to show compliance to the energy balance , but the actual reason that brings about these changes is the one I have explained above. SIMILARLY all that I want ot know is the ACTUAL REASON that brings about a change in the pressure to increase or decrease the velocity of flow inoreder to maintain mass flow rate. WHAT IS THE CAUSE THAT PROMPTS THE PRESSURE TO CHANGE just as the CONTINOUSLY ACTING GRAVITY was the cause of Continously Increasing Velocity of the mass. As already said, the Pressure cannot think own its own to reduce its share of energy and supply it to the molecules to raise their velocity. Do you guys get it now? So please elaborate on it.
StartlerBoy, dtango gave you the "actual reason" for why the pressure decreases when velocity goes up, you just chose to ignore him. It has nothing to do with electricity or anything else. That is how the world works. Static pressure is a result of random molecular collisions with a surface or object. In a quiescent or slowly moving fluid, that molecular motion is more random and so a given surface will experience more impacts and feel a greater force. When the fluid speeds up, it introduces some order into the system. The molecules still move randomly, but they do so as part of a mean flow in one direction. As dtango said, this means that the average angle that each molecule hits a surface with is shallower, so the surface would feels less pressure. The only other way to explain it is thermodynamically, which you apparently don't like. Still, it is the proper way to describe it, so I will go ahead and do it anyway. When the gas speeds up, that is an increase in kinetic energy. In order for energy to be conserved (and we know it is), internal energy or potential energy has to go down. In other words, in order to not violate the first law of thermodynamics, when the gas moves more quickly, the pressure (internal energy) has to go down or the gas has to move downhill so that the potential energy (gravitational) decreases. In almost every fluid system that you will see, the gravitational term is negligible, so it is almost always the static pressure that goes down. Whether you like it or not, conservation of energy explains this phenomenon. You might not consider it the "real reason", but the laws of nature do, and I doubt the laws of nature care much for whether or not you believe them.
boneh3ad : I have no where Ignored dtango, infact his view point very much appealed to me and seemed to be a satisfactory explaination which Ive mentioned to be interesting. But I just wasnt pretty much sure about it and waited for other view points. Is it ignorance?? What made you reach this conclusion ? Again the same thing - why do you draw absurd conclusions? Its not the question of liking a certain principle , its all about getting to the root of the issue and getting the right explanation behind any observed phenomena - be it the laws of nature. boneh3ad: I suggest you to read and understand what I have written my last post and read below. I have clearly mentioned that the Law Of Consevation Of Energy provides a justification for it in terms of energy. And Conservation Of Energy is an OBSERVED phenomena of nature which MANIFESTS itself through all activities involving energy exchange. Like friction between a body being rubbed by a man against a rough surface. The man expends muscular energy to overcome friction and the same energy (according to Conservation Of Energy) reappears as heat energy being dissipated from the rough surface. Bingo the energy is conserved (neglecting losses) but note that it is a mere observation. Again the Reason behind the heat being created is the breaking of bonds of the surface materials due to shear between them (remember the hills and dales or protrusions which need to be broken to cause motion). Now if you can see through it - Conservation Of Energy is absolutely correct in giving the observed effect in one way - in terms of energy BUT the reason giving rise to this energy transfromation or reappearence (muscular energy appearing or getting transformed as heat energy) is very much exsisting and evident. Try to understand the subtle difference that I am pointing at - if you can. Isnt it? I have already given thre examples in my earlier post and can give many more if you want. Just read what I have mentioned as an analogy in my earlier post, I will state it again for you anyways. Hope you understand what I mean to say at least this time. I have mentioned the mass being raised to a height h and being dropped from there under the influence of gravity. Here too the mass is said to gain P.E. and that being transformed to K.E. as the mass contines to approach the ground. As far as the Energy Conservation is concerned the explanation is right BUT note that the Energy Transformation is an OBSERVED EFFECT and the REASON for this transformation is the acceleration due to gravity acting continously on the mass and increasing its velocity. This is the REASON part that I am trying to mention and seek an answer in the Pressure Velocity relation in a flow. And yes, there is a reason to everything observed as a phenomena - be it nature or otherwise whether you like it or not. And it is very much clear and evident from the examples and explanation I have given above. So please can anyone shed some light on my Original Question and get me an proper answer. Any effort in this direction is highly appreciated guys. Please go ahead.
The reason is the same as what dtango and I have said: molecular collisions. That is the reason. Why do you feel like there should be another reason?
Much in the same way you described how an object falls, a particle physicist would describe how that happens much differently. I will presume here as I'm not 100% sure of the exact answer. However, imagine a mosh pit at a concert. Everyone has the ability to jump around and bang into anyone they want. Now, if for whatever reason, the mosh pit has to run in one direction, well then you can imagine that there's less "banging" around from person to person. Many of the chaotic collisions that happened have stopped due to the "flow". More importantly, what do you think on the subject. Remember that we're here to help, not do someone's work.
i guess the question that ur asking is why does pressure increase when velocity decreases, and vice versa and u want an explanation beyond just saying they both compliment each other for a particle thanks to conservation of energy. u want some acceleration to be formed thanks to pressure change that accelerates the particle,right? well ill give 2 explanations, 1) I am studying engineering, and as an engineer , at the undergraduate level i look at the fluid macroscopically, now i look at pressure of a fluid at a section as the resistant to flow across that section, now when looked at this way, it makes perfect sense to think that when ur resistance to flow is high across the section(pressure) ur rate of flow(velocity) is lesser. now when ur velocity decreases across a section due to inc in area thanx to continuity eq, pressure according to the abv xplanation decreases. 2) coming to the existence of force to accelerate a particle according to ur freely falling body analogy, when we take two pressures acting across two faces of a fluid element, for the fluid particle to accelerate, then there must be resultant force acting on it, and this resultant force is due to diff in pressure across two opposite faces, i.e. if i have i cylindrical element of fluid, when the pressure acting on one face of the cylinder is higher than the opposite face, then the higher pressure will push it in that direction(resultant force) thereby accelerating the flow, hence a section at lesser pressure will hav more velocity due to it being accelerated over some length of pipe. these two are macroscopic xplanations . i see that u have trouble underatandin the laws of nature and taking them for granted, i assure you even i hav my troubles with these laws and anyone whos says they dont are lying, u see these laws are the drawing board of modern science, laws that are observed, and not been disproven. we end up creating new concepts just so that energy can be conserved, such as new particles , new force fields etc. you ,i or anybody can disregard these concepts and create some other phenomenon which fits all that is observed, there is nothing wrong. but i request you to accept these laws as the rule of thumb i hope my reply has been of some use
think that the energy conservation you write about, which in your idea gives no real explanation about the cause of the motion, [tex]\frac{1}{2}\dot{x}^{2}+mgx=const[/tex], is instead exactly equivalent to the equilibrium relation [tex]m\ddot{x}=-mg[/tex] as can be proved by deriving the en. eq. in the time once. This means that saying that the gravity force is continuously accelerating a weight in a gravitational field, is the same to say that gravity force is working to increase to kinetic energy on the mass: gravitational interaction between poles can be expressed both as potential or its derivative (and in a larger thinking there is no difference between them) energy balance in a fluid tells you that pressure decrease at increasing kinetic energy. The same as before. Take the time derivative ad you will se that pressure must be reduced for an increasing velocity in a pipe. Who allows the pressure to rise in a decelarating fluid? the pipe wall reaction forces, which allow the flow to be shaped, with consequent changes in the average fluid velocity at different pipe sections(you control the displacements of the fluid, then you need reaction forces). If the pipe walls were ideally compliant, they would not be able to control the "displacement" at wall, but maybe you're controlling the pressure, as in a flow of water at sink. In that case water falls as the previous mass, and flow section are again related by mass conservation. In the first case the driving force is given by some pressure delta from inlet and outlet, in the second case by grav field. Both expressed in term of energy. But as I am quite sure that I haven't replied to you question, since it's not clear to me why talking about energy is less meaningful than talking about forces (?), and for me it is quite trivial the reason for which a fluid must decrease in velocity if its pressure is increasing. good luck! M
Madster Thanks for answering and sharing your views and importantly seconding my opinion about the laws of nature Yep people seem to have a problem to being honest enough to accept that even the laws of nature have something behind them as a cause to their manifestation and that they dont bother themselves to see through this basic cause in each case And people are simply happy to rant the established statements of the formulated laws (ofcourse they a 100% true - proven facts of nature) without bothering to ask themselves a very necessary WHY for the laws' exsistence. Also, the analogy you gave of pressure being seen as a resistance to flow and velocity being dependent on this resistance can be able to only convince oneself superficially BUT the quetion is HOW does the presure change or HOW does the resistance increase or decrease all by itself - thats what we have to look at. The second point you made This is the Newtons Law approach that is usually used to derive the Bernoullis Equation or the Eulers Equation in their most basic form starting from the newtons Laws of Motion. This approach does give you the derivation of the energy balance equation But again does not explain the meaning of the terms (static and dynamic pressure inter relation or transformations) - you just land up with the terms from mathematical steps - their interpretation physically is not seen in any book or no one has explained it anywhere. They just take it for granted as it is and move ahead. Which I cant digest or may be I couldnt see through. So if you can , please be kind enough to explain it to me. drMS thanks for replying Your approach to differentiate the P.E. + K.E. = const. equation really gave me a new insight and worked wonderfully to indicate the equivalence between energy balance and force equations - Newtons law. And as far as applying the same approach to the Energy Balance it gives the following terms: differentiate w.r.t. time : p + (1/2). ρ. v^2 + ρgh = constant (neglecting height term) gives (dp/dt) + ρ. (d^2x/dt^2) = 0 ρ. (d^2x/dt^2) = - (dp/dt) This indicates (as you have rightly said) that an increase in velocity (ie acceleration) results from a corresponding decrease in static pressure but w.r.t. time. Again this is mathematical derivation arriving at the correct observed phenomena but as you have yourself provided the reason ( the wall reactions) I am not wrong in asserting that there is a cause to this observed (derived) decrease in pressure. But can you please elaborate on the arguement you provided regarding the wall reactions and them shaping the flow rather controlling the displacement as it os a little unclear and I could not very clearly understand it. So can you please clarify further about the concept you provided - it looks pretty promising. So can you please explain what exactly you mean by the control of displacement and the role of wall reactions. Thanks in advance drMS Guys Further, Please refer to the diagram I provided in the very first post and try to explain the pressure - velocity relationship taking into consideration that - the Energy provided for the flow (though the fluid is enclosed in between a piston and a disc, but for any intermediate section like a-a or b-b the fluid does flow past it) comes from he force applied at the malle piston end and the pressure by the force F applied over the piston area A is : p = F/A. This pressure acts at the piston fluid interface and remains same till te section where the area starts to increase gradually. Now a) does Pascals Law apply to the enclosed fluid ? ( as the arrangement of enclosure is pretty much same as hydaulic pistons where fluid enclosed by smaller area on one side and larger area on the other side creates a greater force) If yes than how do we say that there is a change in pressure at any section of varying cross sectional area in the fluid ? b) If the velocity of fluid observed past the section a-a will be greater than that at section b-b due to varying areas ? So does the Pascals law or hydraulics principle apply or the bernoullis eqaution apply? c) The overall flow of the enclosed volume of fluid will be caused by the pressure diffrence between the psiton pressure ( p = F/A) on one side and he atmospheric pressure on the right side solid disc. Then how does the pressure internally go on increasing from the throat to the right hand solid disc (and it remains same as p = F/A from the piston to the throat?) Note that the energy is provided ONLY by pushing the piston on the left with a Force F so this is the only source of energy. please answer these questions. Any answer and view point is more than welcome and will be helpful in understanding the issue. So guys go ahead and help me on this. Awaiting the replies and thanks.
Madster Thanks for answering and sharing your views and importantly seconding my opinion about the laws of nature Yep people seem to have a problem to being honest enough to accept that even the laws of nature have something behind them as a cause to their manifestation and that they dont bother themselves to see through this basic cause in each case And people are simply happy to rant the established statements of the formulated laws (ofcourse they a 100% true - proven facts of nature) without bothering to ask themselves a very necessary WHY for the laws' exsistence. Also, the analogy you gave of pressure being seen as a resistance to flow and velocity being dependent on this resistance can be able to only convince oneself superficially BUT the quetion is HOW does the presure change or HOW does the resistance increase or decrease all by itself - thats what we have to look at. The second point you made This is the Newtons Law approach that is usually used to derive the Bernoullis Equation or the Eulers Equation in their most basic form starting from the newtons Laws of Motion. This approach does give you the derivation of the energy balance equation But again does not explain the meaning of the terms (static and dynamic pressure inter relation or transformations) - you just land up with the terms from mathematical steps - their interpretation physically is not seen in any book or no one has explained it anywhere. They just take it for granted as it is and move ahead. Which I cant digest or may be I couldnt see through. So if you can , please be kind enough to explain it to me. drMS thanks for replying Your approach to differentiate the P.E. + K.E. = const. equation really gave me a new insight and worked wonderfully to indicate the equivalence between energy balance and force equations - Newtons law. And as far as applying the same approach to the Energy Balance it gives the following terms: differentiate w.r.t. time : p + (1/2). ρ. v^2 + ρgh = constant (neglecting height term) gives (dp/dt) + ρ. (d^2x/dt^2) = 0 ρ. (d^2x/dt^2) = - (dp/dt) This indicates (as you have rightly said) that an increase in velocity (ie acceleration) results from a corresponding decrease in static pressure but w.r.t. time. Again this is mathematical derivation arriving at the correct observed phenomena but as you have yourself provided the reason ( the wall reactions) I am not wrong in asserting that there is a cause to this observed (derived) decrease in pressure. But can you please elaborate on the arguement you provided regarding the wall reactions and them shaping the flow rather controlling the displacement as it os a little unclear and I could not very clearly understand it. So can you please clarify further about the concept you provided - it looks pretty promising. So can you please explain what exactly you mean by the control of displacement and the role of wall reactions. Thanks in advance drMS Guys Further, Please refer to the diagram I provided in the very first post and try to explain the pressure - velocity relationship taking into consideration that - the Energy provided for the flow (though the fluid is enclosed in between a piston and a disc, but for any intermediate section like a-a or b-b the fluid does flow past it) comes from he force applied at the malle piston end and the pressure by the force F applied over the piston area A is : p = F/A. This pressure acts at the piston fluid interface and remains same till te section where the area starts to increase gradually. Now a) does Pascals Law apply to the enclosed fluid ? ( as the arrangement of enclosure is pretty much same as hydaulic pistons where fluid enclosed by smaller area on one side and larger area on the other side creates a greater force) If yes than how do we say that there is a change in pressure at any section of varying cross sectional area in the fluid ? b) If the velocity of fluid observed past the section a-a will be greater than that at section b-b due to varying areas ? So does the Pascals law or hydraulics principle apply or the bernoullis eqaution apply? c) The overall flow of the enclosed volume of fluid will be caused by the pressure diffrence between the psiton pressure ( p = F/A) on one side and he atmospheric pressure on the right side solid disc. Then how does the pressure internally go on increasing from the throat to the right hand solid disc (and it remains same as p = F/A from the piston to the throat?) Note that the energy is provided ONLY by pushing the piston on the left with a Force F so this is the only source of energy. please answer these questions. Any answer and view point is more than welcome and will be helpful in understanding the issue. So guys go ahead and help me on this. Awaiting the replies and thanks.
Pascal's law applies only to static fluids. I suppose we could broaden it to deal with moving fluids, but it would just end up looking like Bernoulli's Principle. Bernoulli's. ...that first line isn't a sentence, but has a question mark at the end. If you're asking if the velocity at a-a is greater, it is. But you don't need Bernoulli's principle for that: geometry tells you it must be. It would probably help clarify things for you if you think not in terms of "pressure" - which is uselessly vague (what kind of pressure?) - but in terms of total pressure, which is constant, as you correctly note. Like a rock in freefall that converts gravitational potential energy to kinetic energy while keeping total energy constant, this device coverts velocity pressure to static pressure while keeping total pressure constant as required by conservation of enery. You switch back and forth between characterizing it correctly and incorrectly. Ie, in the OP you say "Bernoulli suggests that pressure at section b-b must be higher than a-a" when a few lines above you show you clearly know that total pressure is constant. You can't simply drop the other forms of pressure and focus on static pressure. Then you're creating your own principle that says P1 /= P2, which is at best an incomplete description.
Total Pressure must remain constant because that is the Total Energy available or associated with the flow and we know that because of the Energy Conservation. And Total Pressure is composed of two constituents - ie static pressure and dynamic pressure and I know these two coexsist always except at stagnation point. But the point I am trying to raise is that - HOW does the inter-conversion of one type of energy take place ( ie static pressure energy and dynamic pressure energy in a flow OR the gravitiational potential energy and kinetic energy of the rock) The very basis of my thread is that we all know the Energy Balance and Conservation BUT there is ALWAYS a visible and distinct CAUSE or some PHENOMENA or SEQUENCE OF EVENTS that gives rise to the events manifesting the Energy Balance or Energy being conserved. I have very clearly shown it (ie there is a very distinct physical reason that makes occur the event showing energy conservation) with two clear examples. 1. Of a mass being raised to a height and then falling - see in my earlier post . Here the gravity acting continously on the mass gives rise to increasing velocity (Just read my post no:2 for raised mass falling down) 2. Of Friction - muscular energy expended by human hand reappearing as heat energy (refer my post no:3) 3. Or the case of a spring tuned where P.E. is said to be stored in the spring BUT there is the reason behind it i.e.e the interatomic restoring forces that cause it, saw the reason now. Or the case of the capacitors or batteries where charge is stored (electrical energy) - mentioned these examples in my previous posts. In both the cases the Energy Balance or Conservation is SEEN or APPEARS as an event which is very clearly CAUSED due to a certain clear REASON and ONLY then do we realise that energy is conserved. But there certainly exsists a distinct physical phenomena or reason which causes this. That Reason is what exactly I want to know in regards to a flow which I am very sure does exsist for a flow like for any other situation. Thus what is the cause or How does the pressure change when the area varies in a flow inoreder to observe the change in velocity ( call it to comply with COnservation of Mass or Energy Conservation). Just give me that distinct WHY or HOW that causes this pressure variation just analogous to the way there was a cause or a HOW in the cases of falling mass and friction examples that I gave. As stated elaborately in my previous posts energy conservation is an OBSERVED or SEEN EFFECT. It is not all by itself. I hope I have made my point very clear now. So please can you guys give me the answer I am looking for. And secondly - What I asked regarding the Pascals Law - How does pressure at the throat start increasing as we see But comparing the set up I gave it is similar to a enclosed fluid in hydraulic pistons or lifts where pascal law applies and says that Pressure is transmitted equally to all directions at a point -- then shouldnt the same pressure as p = F/A should be transmitted even at the increasing cross sedction area ??
When the area changes, the velocity changes in order to satisfy conservation of mass. That isn't a superficial answer, that is the cut and dried law. Mass cannot be created or destroyed. It is a fundamental law of the universe. If it didn't speed up, there would be a net loss of mass somewhere that is unaccounted for. Think of it this way: when the tube gets smaller, the particles are trying to force their way through a smaller opening. There is no way to get them all through without increasing the speed. That is why the flow speeds up. Now, the increase in velocity and thus decreased randomness of particle motion causes the static pressure to go down. Since the particles are more organized in their motion and have a much larger streamwise velocity even at the individual particle level, when they do strike the surfaces, it is with less overall force. If particles hit the surface at steep angles, the sine of that angle is large and so a lot of the force goes into the surface (high pressure, low velocity). If the particles tend to hit the surface at a shallower angle, the sine of that angle is small and not a lot of that force gets transferred to the surface (low pressure, high velocity). That explains the static pressure. I really don't understand how you can have a more basic explanation than that. Since dynamic pressure is basically representative of the kinetic energy of the air particles, it goes up because the entire flow now has more kinetic energy. If you put your hand out the window of your car, you will feel more resistance if you drive faster. This is closely related to the idea of dynamic pressure. First, understand that Pascal's law doesn't really apply to a moving fluid in the same sense that it does to a stationary fluid. Second, in the case of your drawing (if I understand what you are attempting to convey properly), the fluid really has no net velocity. It is a static system. In fact, it looks remarkably similar to the concept that drives hydraulics. You put a force on the small piston which creates a pressure, and that pressure pushes a much larger piston with greater force (but less travel).
StartlerBoy, you're trying to hard . Boneh3ad and I obviously think the same. We've both answered your question about the change in static pressure by using the kinetic molecular model to describe what's going on at a molecular level which applies for fluids (liquids and gases). I personally like that explanation because it's the most intuitive for me to understand without getting into all the maths! I like to avoid headaches as much as possible! As to the explanation to the increase in velocity I thought it was intuitive already. I guess it isn't so let's focus on that for a second. Boneh3ad points out conservation of mass. I would elaborate that to say it's because of conservation of momentum . First, a fluid in motion means that some force has been imparted to it so that it has some level of momentum. Remember that momentum = mass * velocity. Let's stick with our kinetic molecular mode of thinking. Let's pretend our liquid molecules are a like a bunch of B.B.'s (tiny ball bearings). A continuous force has already acted upon them so that now they are flowing down a channel in the same direction with some fixed velocity. Each of our little BB's have the same momentum at this point. Now the BB's run into a constriction in the channel. The constriction reduces the number of BB's that can fit through the narrower channel. Well our poor BB's in the narrower part have nowhere to go and get collided by many more BB's coming from behind in the wider part of the channel. So the momentum from the mass of BB's get's transferred to a few number of BB's. What happens? The fewer BB's in the narrow part take on the momentum from all those other BB's behind them (conservation of momentum) and thus they accelerate and go faster through the narrow part. Change in momentum = mass * change in velocity So there you go! No fancy maths needed! Hope that helps.
There isn't anything wrong with the answers already given - in particular, dtang's response is clear and concise and directly addresses your question. You didn't respond to it other than to say thanks. Since it is the correct answer and you seem to agree, I don't see what else there is to discuss. Any other answers you get can only be less relevant or wrong, so why not just accept the correct answer and end the discussion there? Anyway, for some of the secondary points: That is such a run-on sentence, I can't understand it. It sounds like you are asking that if the piston is made to move, this creates a pressure in the throat - so why doesn't this pressure get transmitted throughout the device. The answer is, it does. The total pressure is always the same throughout the system and when the piston is moving it is greater than when the piston is stationary (assuming the membrane applies a constant static pressure; just enough pressure to keep the water from just spilling out of the device). But that isn't Pascal's law anymore as Pascal's law only deals with static pressure. Again, I'll reference the OP: You're looking for a contradiction where there is none, by mixing static and total pressure. Pascal's law is about static pressure in a stationary fluid being constant; Bernoulli's equation is about total pressure in a moving flluid being constant. You could almost say that Bernoulli's principle expands and generalizes Pascal's law, as Bernoulli's principle can be reduced to Pascal's law for stationary fluids.
Greetings all, Okay, I just lost a long detailed reply, but I'll try again (shortform). I like the answers I read here - I'd just like to rephrase it how I understand it, and maybe that will aid StartlerBoy. First: There is no such thing as "dynamic" pressure, and no conversion between potential and kinetic energy occures through the Bernoulli equation. Please don't crucify me for that statment - hear me out. A fluid (or gas) is made up of many particles moving in straight-line trajectories. The average particle speed is a function of temperature, and in a stationary fluid the NET direction of all particles is zero. (This velocity is related to the speed of sound in the fluid). The boundary (pipe wall) is made up of particles vibrating within their bonds, and their vibrations also correspond to the material's temperature. If the average temperature of the boundary and the fluid are the same, the net energy exchanged during fluid particle collisions with the wall is zero, and the particles rebound with the same energy they had before the collision. The "static" pressure felt by the pipe is the average rate of momentum being reflected. This varies by density (more particles in system = more boundary collisions / second), particle mass, particle velocity (function of temp), and the angle-of-impact with the boundary. -Only the velocity component normal to the pipe surface is reflected, the parallel component is not affected. If the same fluid is flowing through the pipe now (same streamline - example of a pipe off a water tower), the average particle velocity has not changed. The "flow velocity" is the net velocity of all the fluid particles. Because the average direction of the particles' motion is parallel to the pipe wall, the average angle of impact of particles bouncing off the boundary will be lower (less "normal" to the surface). With all other variables the same, decreasing the average impact angle will decrease the normal component of momentum that must be reflected, thus reducing the percieved "Static pressure". The "dynamic pressure" is a defined quantity - the non-random component of the total fluid's particle motion, or the amount of particle momentum that must be randomized to reachive stagnant conditions. To summarize: Total pressure is a measure of average fluid particle speed and the fluid density. Static pressure is the random component of the fluid particles' momentum. Dynamic pressure is the non-random component - the net momentum after integrating. And the Bernoulli equation provides a method to calculate the change in randomness of the bulk fluid particles' motion. Neat, actually. I realize you also wanted a discussion of why laminar flow accellerates through a contracting boundary, but this post is long enough. Cheers