# Pressure - Velocity relation in fluids

Discussion in 'Mechanical | Automobile | Aeronautics' started by Rohan_sK, May 31, 2009.

1. ### Rohan_sK Apprentice

It is a known fact that as the velocity of a fluid increases the pressure decreases accordingly.
It can be easily explained for a varying diameter pipe.
But for a constant diameter pipe, how can we theoretically explain the decrease in pressure with an increase in flow velocity of the fluid.
Can anyone explain why this pressure - velocity varaition occurs, lets develop the fundamental concepts.

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2. ### Rohan_sK Apprentice

Whats this?? No takers still??
Where are all the brilliant Mechanical, Aeronautical, Piping Engineers??

Come on guys, I am awaiting a good explaination on the topic started. So get started.

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3. ### ronvieky Apprentice

Its not the velocity of fluids. Its the volume and pressure relation. I think you'r looking for BOYLE's law. Its an ideal gas law- applicable to all fluids- liquids and gases. Though it has certain limitations for liquids. Try to google for more info.

The velocity and area have a relation too- for calculating the Flow rate of the fluid through a conduit. Here's a simple explanation, you with me?

Force = Pressure x Area

also,

Force = Mass x Acceleration

where,

Acceleration = dv/dt, ie rate of change of Velocity.

So,

Force = M x dv/dt = P x A

Now for a particular fluid- the Mass of the fluid will remain constant.

Hence,

P x A ~ dv/dt

or

(P x A) x dv/dt = constant.

I think this is the relation you are looking for.

Boyle's law

Pressure x Volume = Constant.

Constant flowrate-
Area x Velocity = Constant flowrate.

Hope this helps you out.

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4. ### Rohan_sK Apprentice

But its not the BOYLES LAW that I am looking for.

We can mathematically derive the relation for pressure, area and change in velocity of the fluid. The constant flowrate equation is also true, but what I am looking for is the Physical interpretation of it ie WHY or rather WHAT CAUSES the pressure to change when velocity increases; or WHY or HOW does velocity adjust its value when the Area reduces.

I am looking for that WHY.

• Like x 1
#4
5. ### ronvieky Apprentice

What causes the Pressure to change when the velocity increases//

I fail to understand your question. What do you want?

You want me to explain it in molecular level? or do you want me to explain, it fromt he start?

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6. ### Rohan_sK Apprentice

I have clearly mentioned in the question that I am looking for the REASON what causes the Pressure to reduce when velocity increases( or we increase it by external means).
The formulae always give the relation part of a phenomena between the parameters involved, but they dont clarify HOW it happens.

The graphical link you gave is about Pressure-Area-Force relation, which I am not looking for. Its Pressure- Velocity relation I am interested in , rather the reason for their inter-relation.

If youwant to explain it on Molecular level, go ahead with it, not only me but many other CEans will also benefit from it.

I hope I've made my question clear.

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7. ### Munna_zama Newbie

Hi rohan,

When velocity increases flow pressure decreases due increase in pressure drop due to velocity,

if you calculate pressure drop using Darcy's Weisbach formula

Hf=pressure drop = fLV^2 / ( 2*g*D)

where f = Darcy's friction factor
L= Length of pipe
V= velocity of flow
D= Diameter of Pipe

if you observe above equation it is very much clear that if velocity increases pressure drop also increase and hence your flow pressure reduce.

• Agree x 1
#7
8. ### Rohan_sK Apprentice

The Relation given above is for the FRICTIONAL HEAD LOSS due to internal pipe surface.
The Darcys formula gives the relation between the Head Loss due to Friction in the flowing fluid wrt Flow Velocity, coefficient of friction, dia. of conduit.
This is NOT what I was looking for.
I want to know the reason WHY PRESSURE DROPS WHEN VELOCITY INCREASES.

#8
9. ### mechanicalbd Newbie

for a liquid flowing in a pipe the pressure is exerted by the liquid on the pipe. As the velocity is low there is more interaction between liquid and the pipe reasulting more amount of force on the pipe by the liquid, as the velocity is more, the attractive forces between liquid molecules is more, reasulting less force on the pipe, reasulting less pressure.
The above explanation is what i think as a logic for your questions, try to analise it.

• Agree x 1
#9
10. ### pramod.pal007 Apprentice

Engineering Discipline:
Mechanical
okay sir
you know that fluids flows only when there is pressure difference. if there is no pressure difference, no flow. this is a general fact of science as per analogy current flows from high potential to low potential water flows from high altitude to low altitude similarly when there is a pressure difference in horizontal pipe then flow occurs and the velocity is depend on pressure difference.
if not satisfied

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11. ### Rohan_sK Apprentice

The above explaination is not what I was looking for.

All Brilliant Mechanical Engineers on CE, come on , I am waiting for the correct theory and explaination behind this Inverse relation between Pressure and Velocity in the Fluid Flow.

So please get going and explain this.

#11
12. ### skipper Apprentice

Engineering Discipline:
Computer Science
It's a mechanical problem. The pressure and volume are restricted by the diameter of the channel; there's a nearly constant cross-section, for a small length of overall pipe + fluid (volume).

Assume the pipe has a constant unchanging diameter. The flow is a volume with momentum along a section of pipe (unit length); the pressure is across the pipe's diameter (the pipe 'absorbs' pressure). If the pipe is rigid material, pressure is reflected back to the fluid, perpendicular to the flow. The flow is either because the pipe is angled "downhill" or because of a feed pressure (pumping action).
Things are more complicated if the pipe is elastic (like arteries and blood vessels are), so let's stick with a fixed width container...
(well, consider that blood pressure is regulated by vasodilation - the muscles in the veins dilate and contract to maintain a safe pressure, so that the circulatory system absorbs pressure, as above, or lowers and raises it. A rigid pipe can't do this, so the flow does. Flow is what 'absorbs' the pressure against the walls of the channel a fluid is flowing "against", the reduction in pressure in the fluid is because it's moving and so carries the pressure along the walls, essentially lowering the normal force).

The relation to Boyle's law is: pressure and volume are always constant for a fixed diameter flow; PV is constant for a fixed temperature in a gas = fixed tension or normal pressure (perpendicular to the container walls) in a fluid.
Molecules flow in bulk along the pipe, so there is negligible nonlinearity in large flows (Avogadrian numbers of molecules).

Also, there's a related flow phenomenon in electric charge and electron mobility. Charging a capacitor is essentially "immediate" (ok, it takes a small amount of time, but relative to the discharge rate, it's instantanteous). This is a volume of electric charge, "statically" collected on the surface of a conductor (even though the electrons are inside the metal).
Charge "pressure" against the surface, causes the electrons to flow out of the capacitor, as if a pressure is causing the flow (it is causing the flow, it's called a voltage difference). This is exactly mechanically same as what happens when you fill a container with water, and drain it using a gravity-fed pipe.

Alrighty?

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13. ### Rohan_sK Apprentice

@ Skipper : In the first place Thank you for responding to the question and commenting on it.

There are certain things which I do not agree upon.

The pressue exerted by a Fluid is essentially due to the Height Of the Water Column and the External Pressure if any. And according to the Pascals Law it is exerted equally in all the directions.

And the Pressure exerted by the fluid always acts along the Normal to the flow or the container boundary. So the Flow is not ABSORBING any pressure but is in fact exerting it. You can at the most say that the Newtons Third Law applies here as far as the walls are concerned.

Secondly, what is the theory behind ' Reduction in pressure is because the flow carries the pressure along the walls and so Normal Force reduces' ?? On what basis can this be explained.

The Pressure Velocity Relation I was talking about equally holds TRUE for a flow through a Convergent or Divergent Nozzle attatched to a pipe, where the Pressure DROPS as the Velocity of the fluid Increases as it flows through the Converging Nozzle and vice versa in a Divergent one.

I was seekling an answer to what exatly happens on the Molecular Level which bring about this conversion in the relation between the pressure and velocity.

Thirdly, the Boyles Law gives the relation only essentially betwen te Pressure and VOLUME of a fluid and not the Velocity. Also it is applicable to Static fluids and not flows. So it is not relevant here.

Fourthly, it is a known fact that we need a Potential Difference for the flow of any Physical Quantity (like heat, current, fluidflow) across any two points. We do not need this in the Pressure Velocity Relation here.

So the question still remains unsolved. Can you all please think a little further and try to explain this phenomena.

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14. ### skipper Apprentice

Engineering Discipline:
Computer Science
Yes, in a closed pipe (assume it's full) the pressure is either because of a larger volume (at a greater height, say) 'forcing' the flow through the pipe, or equivalently because of a pump.
Yep. That's what I say above; the container or pipe has a normal reaction to the pressure, in the flow. The reaction is the absorbance (say by expanding) of flow momentum. If the container is inflexible, the normal shear force is perpendicular to the flow.
The pressure is not actually lowered in the fluid, but the normal restoring force at the container wall is less, because of transverse flow (consider that if there is no flow, pressure is maximum at at the walls); fluids are incompressible so the fluid pressure is constant, the flow rate then determines how the stress against the walls, is translated by momentum. The "flow pressure" is not the fluid (internal) pressure, it's the potential 'pushing' the fluid along.

Pressure variations are because of the reasons stated - the smaller the cross-section the less fluid there is to translate the pressure into momentum = higher nozzle pressure of fluid per volume.

Molecular versions have to account for the individual dynamics of each molecule; molecules have a shape, a distributed charge and 3 kinds of motion. Friction is very complicated (in fact, it isn't well-understood at all, like the gas laws and bulk flows are).

It's much easier to deal with a macroscopic amount of matter, as a bulk material. The relevance of Boyle's law is that it's a general formulation of pressure/volume, which relate to density/flow.
The potential difference you say is needed for flow, is equivalent to pressure. So if you're talking about "potential", the easiest potential to see, is the potential in a volume of water at a height h. This is equivalent to a volume of heat, at a temperature T, and to a volume of charge at a voltage V.
Gas/Liquid pressure is equivalent to temperature (heat potential) and voltage; as long as you use a general form of 'flow" in each case, a "transport", this is gas/liquid, heat, and charge flow, respectively.

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15. ### Rohan_sK Apprentice

The expansion or Flexibility of the container has got nothing to do with the Normal Reaction. Thre Reaction would still remain the same no matter how flexible or elastic the pipe is.

The Normal Restoring force also remains the same no matter how much the flow velocity is.

Incompressibility means that there is no volume change in the fluid on application of pressure. It does not mean that pressure remains constant.

What do you mean exactly by stress/pressure being translated by momentum

You have stated the Opposite of what the situation is; ie for smaller c/s in a convergent nozzle the pressure is LOW and the Velocity is HIGH and not the otherway round as you have stated above.

Or do you mean to say that the pressure acting on the fluid is the same but the volume of the fluid is less at a smaller c/s.

You have not made yourself clear. So if what I infer from your above quote is actually what you wanted to convey, then it must be the Force acting from an external source pushing the fluid causing flow and not the pressure as you have stated.

Is this what you meant to say :
( pressure = force/area) so MORE force acting on the SAME C/S AREA, so there are LESSER molecules to move , hence HIGHER Velocity.

The rise in velocity aspect can be explained by this but again the Drop in pressure remains unexplained ie in reality the pressure at that c/s of the pipe is REDUCED when the velocity increases.

This is even what I used to think earlier, but it did not answer my query about reduction in pressure. So I always wondered why.

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16. ### skipper Apprentice

Engineering Discipline:
Computer Science
How come? If you fill a balloon with water it reacts to the pressure by expanding; you can't expand a rigid container which is why they usually have open ends if you want them to be pipes. A pipe that was flexible would absorb flow until it reached a limit and either became rigid or burst under pressure. Fluid pressure is constant.
The pressure in a pipe is against the wall. The pressure in the flow is because of the bulk momentum in a unit volume, and because the pipe doesn't expand under pressure. The flow 'bends' the normal reaction from perpendicular to the flow, since a molecule in contact with the wall that gets a push toward the center, is moving sideways; shear is translated = the fluid flow is shear from the normal stress. The flow carries the stress at the walls away, as fluid shear.

The pressure outside the nozzle is lower than the flow pressure inside.
The velocity is the flow pressure, the fluid's internal pressure is constant.

If fluid pressure is constant, velocity of volume (flow) is the equivalent of pressure in gases.

To recap: pressure in a gas is density*temperature/volume; pressure in a fluid is density*velocity/volume; in electricity it's density*charge/volume.
There is one other thing: fluids flow faster in the center than along the edges, there's a flow gradient, so since flow in fluids = pressure in gases it looks like a pressure gradient. However a standing fluid pressure (flow velocity = 0) is because of the mass potential; flow potential is different (think about how a gas has the same mass independent of temperature).

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17. ### Rohan_sK Apprentice

How can the flow bend the direction of the Normal reaction? How can the flow CARRY the stress?

The shear in a fluid is continous between its layers and betweeen the boundary layer and the contact surface.

Is there any known source of physics theory explaining it. I would like to know about it atleast read it once.

The Velocity of flow and the pressure are two seperate parameters. And the fluids pressure cannot remain constant during a varying diameter or c/s flow or when velocity changes.

Then what would you say about all the fluid machines, equipments where the Higher velocity from a fluid is converted to higher pressure energy to get a greater pressure output from a high velocity flow.

The perfect example of this is the INVOLUTE chamber of the Centrifugal Pump, which is a tube/chamber of continously increasing cross sectional area. The increasing area reduces the velocity head and goes on inceasing the pressure head instead.

How would your theory explain this.

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18. ### skipper Apprentice

Engineering Discipline:
Computer Science
??
I think you're "keeping" the notion of gas pressure and applying it to pressure of a fluid. Try this thought experiment: take 1 bucket and make a hole in the side, near the base. Start slowly filling the bucket; if the fill rate is low, water flows out the hole and the level of water in the bucket is maintained, at the same height (fluid volume and surface pressure).

If you increase the fill rate the level in the bucket goes up, and passes the outflow rate. Pressure at the outlet is determined by the surface area of the water in the bucket, the higher it goes the higher the outflow rate. Flow pressure at the hole equals surface pressure (cross-section of container). The forces normal to the inner surface of the bucket, all pointing inwards, 'react' to the pressure by 'expelling' water from the hole. Flow out of the hole is the water 'reacting' to these forces.

Shear is not constant in fluids, e.g. water has vorticity which is nonlinear; water can move by rotating (gases do too).

Shear in laminar flows is linear, relative to vortical flow which isn't. Stress at the inside walls of the container is "absorbed" into shear by the fluid motion. Flow is slower at the edges, because there is more friction and interaction, the flow is almost linear near the center, less linear near the walls.
The centrally-directed flow 'pulls' the outer fluid along, translating normal stress into shear velocity.

If the bucket has no hole there's still the same pressures from the bucket and the surface; there's no overall flow but molecules are still subject to normal forces; the shear in a "static" fluid sums to zero = no bulk momentum.

Flow and pressure are more general than volume or density.
A gas has a volume which can vary, a liquid doesn't.
Gas pressure varies with a static volume of gas and a varying container volume, fluid pressure is static for a given fluid volume, flow varies and liquid flow is equivalent to gas pressure.
Gas flowing down a pipe has two kinds of pressure, the gas pressure which varies and the flow rate (which also varies locally, like a fluid's), which is determined by both pressures and the cross section. Liquid flowing down a pipe has only one kind of pressure, the flow rate/velocity.

p.s. the forces normal to the container, pointing directly inward don't "change direction" they always point away from the walls to the center. The fluid is translating, so the fluid "bends" not the normal force. A molecule in contact with a wall will reflect or bounce inwards, as it translates in a general direction along the flow (assuming the walls are smooth).

Here's a bender: (Physics exam) 1. Relate the partial pressure of a thin inert gas, that is converted into plasma by an oscillating electric field inside a glass container to bulk flow momentum and velocity; formulate an expression for bulk flow of charge in the plasma and outwards into the environment... In what sense is there a liquid flow-pressure? Explain your answer.
(I have a paper from a textile researcher, of all places, that does an analysis using Gauss's and Faraday's equations, if you want a peek)

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19. ### Rohan_sK Apprentice

Skipper, you have not spoken about the Centrifugal Pump Involute Casing, and the change of velocity head to pressure head there, yet.

The flow of gases is a compressible flow. And I very well understand the difference between Compressible and Incompressible flow concepts.
And I am strictly speaking about the Incompresible flows in liquids.

The pressure of water at any level would simply be the Height of Water Column, and the pressure is same for the same height and is equally tramsmitted in all directions( this the beauty of hydraulics).

And the Outflow rate is not the function of surface area, but simply the height of the water column above the hole. Higher the water column higher the pressure.

It is only the pressure acting at the particular height which causes the water to flow out as soon as it finds an opening and the constrain is removed ( the hole in this case). There is nothing as reacting to the forces, the pressure is equal in ALL directions ( it is not unidirectional as any other stresses are, thats Pascals Law).

I have never said that the shear is constant. I have clearly said that the Shear force is acting CONTINOUSLY between two adjacent layers of a fluid, and this continous shear is what distinguishes fluids from solids.

And yes you are right, shear does exsist between the layers of water in a Vortex flow too.

It is correct that the flow is slower at the edges due to friction between the fluid molecules in immediate contact with the surface, and this is exactly what causes the VELOCITY PROFILE in the fluid flow where the velocity is maximum at the center line of flow and lowest at the edges.

But there is nothing such as absorbing of the stresses being absorbed into shear. The flow is by the virtue of the driving force and the shear is merely the resistance to flow.

It is not the inner flow which pulls the outer fluid, but in fact the nature of the velocity profile is itself dictated by the friction between the outer molecules, and the velocity goes on increasing from outside to inside as the friction goes on reducing towards the centre.

Correct, there is no shear in a static fluid, except Brownian Motion in molecules.

It is wrong in saying that the flow rate is the pressure in a liquid flow. No, dont confuse in it, Pressure is a different entity and Velocity of flow is a different entity in any fluid flow.

The fluid molecule does not bend or reflect inwards. It is not this way that the flow progresses. It is the driving force ( column pressure or some other external power sorce like a pump or piston pushing or fluid already having some motion) that causes the flow.

The driving force ives the molecules the energy to overcome the shear force between the surface material and themselves and continue to flow. The flow is always in a straight line in case of a laminar flow.

For analogy, you can consider the example of a projectile which when fired in a straight horizontal direction with sufficient velocity, maintains its straight path until it has enough energy left to overcome the gravitational force acting downwards. It will bend down or change direction only after the gravity becomes dominant on the velocity in horizontal direction.
The same happens with the fluid particles in a flow, they are continously pwered by the driving force from behind, ehich keeps them in straight motion.

And I will like to read the reasearch paper you mentioned regarding the Gauss and Faradays equations. Seems interesting. Would you mind mailing them to me, ofcourse if you dont have a problem in sharing it. I will be thankful if you can email them to me.

And regarding the experiment you mentioned , I need to first get information on the topic and study and understand it first., then i will certainly comment on it. can you just give some more details on it so we can have a better idea.

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20. ### skipper Apprentice

Engineering Discipline:
Computer Science
Well, I've tried; you appear to be disagreeing in some respects, and agreeing in others.

As you say there's a distribution of the flow velocity across a fluid with momentum (I called this a gradient).
And, it is true that the flow absorbs the normal forces at the walls - the walls absorb some momentum in the fluid, if the restoring force is linear then the walls return this momentum to the fluid as a normal force. Pressure for a gas is as you say, because of compressibility. Then what is the pressure in a flow of fluid? In a standing fluid the pressure is because of the exposed surface, I mentioned that the bucket has walls too, that react to the pressure (the total pressure on the fluid is the surface pressure plus the container "pressure" or the normal forces, at the container walls).

I'm fairly sure flow pressure is equivalent to gas pressure (as long as the fluid has momentum and the gas doesn't). "Flow" is dependent on bulk properties of matter, electric current is the flow of charge in conductors, electron pressure increases with electron density. Therefore pressure in capacitors is charge pressure or density per volume, equivalent to the pressure in a static volume of gas, or a fixed rate of liquid flow.

A static fluid's pressure depends entirely on the volume of fluid and the surface area, given that the pressure from the walls is constant.

A gas with momentum will change its pressure, this is essentially what happens when a fluid flows, but since fluids do not compress, the drop is entirely a function of the flow rate -> fluid pressure is bulk flow.
This extends to solids, a solid with momentum has a flow-velocity/pressure as well. Note: this is a general extension of the idea of "pressure", "volume" is constant for solids and liquids, so there's a direct relation between gas and liquid pressure.

Why do I insist that pressure from fluid flow is equivalent to a non-moving gas volume? Because fluids that don't flow have a fixed volume and pressure. A gas not flowing has a fixed relation to temperature, a fluid will not react to temperature until BP is reached. So the potential flow in a fixed volume of gas depends on gas pressure in a fixed volume and on the temperature of the gas. Potential flow in a fluid with a fixed volume depends on the height of the fluid. So gas pressure = fluid pressure when the gas is at fixed temp and pressure, and the fluid is at a fixed height; density and gravitational potential are the real drivers but gases 'escape' by expanding = lowering pressure. A fluid 'escapes' by gaining momentum = apparent pressure drop; but, a fluid always has a density dependent pressure.
Density is the key here: a liquid has the same density at different pressures (depths), a gas's density varies at different pressures and temperatures.

I didn't say that molecules bend, fluid motion is the "bend".

p.s. dang that paper has gone missing, after all. If you want to look into plasmas it's somewhat advanced theory. Check out the Lorentz force and flow in plasmas (try wikipedia); you generally don't look at plasma physics until 3rd year or grad level, btw. If you're good at math, no worries I guess.

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