Does Decreasing the Volume of a Pipe Increase Fluid Flow Pressure According to Bernoulli's Equation?

[azaharak]

One form of bernoullis equation (fluid flow) incompressible is

the change in pressure plus [the change in (velocity squared)*density/2] = 0

when the height of the fluid remains the same.

This along with the continuity of mass A1v1=A2v2 allows you to infer that decreasing the volume of a given pipe would result in a higher speed of fluid flow (incompressible).


This would in turn result in a pressure drop if you plugged it into bernoullis equation?

Is this valid reasoning? Just because the fluid is moving faster doesn't mean the pressure is greater in the thinner pipe?

If this is so, then there are many bad claims on the internet that squeezing a hose results in a larger pressure? Or blood through thinner veins results in higher pressure?


Thanks for any comments

Best

 

[azaharak]

Another argument would be that if the resulting pressure was greater in the smaller exiting pipe, than wouldn't the fluid flow backwards into the larger pipe?

 

[Studiot]

If this is so, then there are many bad claims on the internet that squeezing a hose results in a larger pressure? Or blood through thinner veins results in higher pressure?

If you squeeze a flexible pipe there will be a temporary increase in pressure at the constriction as the fluid is pushed sideways. Most will end up downstream but some will end up upstream. The relative amounts will depend upon the degree of constriction, the timescale, the initial flow rate etc.

Very likely the act of constriction will change the flow regime from laminar to turbulent which obeys a different set of equations. After the fluid involved in the initial pressure increase has dissipated and the flow settles back down to its new regime your A₁v₁=A₂v₂ formula will be reasserted so the constriction will control the flow rate.

The statement about pressure increase refers to 'pressure drop' along the pipe for both hoses and blood vessels. This is because the circumference of a pipe is proportional to the diameter, whereas the xsection is proportional to the square of the diameter. So the volumetric flow rate passing through a given longitudinal surface area is proportional to the diameter. This is along winded way of saying that large bore pipes offer proportionately less resistance to flow so impose a proportionately lower pressure drop.

go well

 

[azaharak]

So then the correct wording should be...


smaller pipes create larger pressure drops (resitive effects)... however the fluid in the smaller pipe (when returned to steady lamina flow) has a lower pressure.

And if one made that argument that squeezing a hose results in a higher exit speed and thus higher pressure would be technically inaccurate?

Thank you

 

[arildno]

1. The continuity equation tells you, all by itself, that the velocity MUST be greater in the thinner pipe than in the wider pipe.

2. Thus, it means that a fluid particle traveling from the wider section into the thinner section must experience some sort of acceleration.
In the inviscid case, such an acceleration can only be provided to the fluid particle IF the pressure in the wider section is greater than the pressure in the thinner section.

Is that clear?

 

[azaharak]

Yes it makes sense...

I'm just having trouble with the following statement posted on Yahoo answers"Describe two ways that increase the water pressure in a hose that is analogous with increasing blood pressure in the veins"

answer:
"use a smaller diameter hose or only have the nozzle partially open or partially obstuct the end of the hose with your finger...same volume thru a smaller diameter will increase pressure...vasoconstiction of the veins decreases their diameter, increasing BP"

Thanks again!
http://answers.yahoo.com/question/index?qid=20071205090744AAuljnJ

 

[arildno]

The critical thing to understand is that the stationary flow we have as our premise for the validity of our equation is NOT present, say, in the immediate aftermath of a constriction of the pipe!I'll make an idealized "explanation" here, that possibly might be of some use:

If you squeeze together a pipe, making one section thinner, this will cause an IMMENSE increase of local pressure at the constriction, most likely causing som backflow of water into the wider section.
(Clearly, as we constrict our pipe, we are no longer dealing with a stationary flow pattern!).
As the backflow continues, we generate two effects:
1. There will be a partial [/I]emptying[/I] of the pipe beyond the constriction, radically lowering the pressure there.
2. The backflowing water will collide with the oncoming water, increasing the local pressure prior to the construction.

(Probably bulging out the pipe, i.e, changing pipe geometry, but let us neglect that in the following)

If the driving force is strong enough, the backflow will be pushed back (no more "clogging" at the constriction), and after a while, the flow will settle down in a new stationary manner, namely with higher pressure prior to the constriction, and lower pressure beyond.Thus, your two equations provides a fairly accurate snapshot of the situation you'll have once stuff has settled down a bit.
(Assuming of course, that effects due to viscosity are negligible)

 

[Manhar]

Bernoulii Equation applied to blood pressure. Pinching the tube or closing the tube does not increase pressure. Here the applied pressure or available pressure in form static height is the source of total pressure. If you pinch the tube or close the tube, pressure in the tube will be equal to the applied pressure. Flow in the tube or velocity in the tube is created by pressure drop. If the end of tube is open to the atmosphere and you pinch the end, velocity will increase because the pressure available before pinch point will be convereted into velocity.
In our blood flow system, when the inner diameter of our arteries is reduced by cholestrol lining, velocity will increase, there by pressure drop will increase. Quantity of blood in the system remains same for practical purpose, as a result, heart will generate little higher pressure (working hard) to circulate a fixed quantity of blood. Higher pressure generated by heart to keep all blood in circulation is known as high blood pressure. Heart can generate to the maximum limit of its capacity, if need goes beyond the capacity yo know what happens.

More explanation is required to explain in details to those who do not have knowledge of hydraulics, but for our purpose it will be enough here.

 

[Studiot]

"Describe two ways that increase the water pressure in a hose that is analogous with increasing blood pressure in the veins"

What a pity you didn't start with this in your first post.

It really is a daft question.

Pressure and flow in hoses is not the same as P and F in blood vessels.

This is because there is a pump as part of the system in blood vessels. There are also additional 'pumps' otherwise known as the venous return.
edit: Bernoulli's equation does not directly apply across pumps or compressors.

A hosepipe is conected to a supply tank or water main which offers a fixed head or static pressure.
In the absence of flow, the pressure at the other end will be the same throughout and the same as the fixed head.
When there is flow there will be a friction loss of head down the length of the hose. The pressure upstream of the exit point will be the pressure head, minus this loss minus the pressure head that converted to velocity head, by Bernoulli's equation. The loss will depend upon the flow rate and the physical characteristics of the hosepipe. If the pressure loss is total, ie the end of the hose is open to atmosphere then all the available tank pressure head is converted to velocity. You cannot get any more than this.

 

[Ken G]

I think perhaps a more concise way to put it is that blood flowing in narrow blood vessels acts like a viscous fluid, and Bernoulli is intended for ideal (friction-free) fluids.

 

[Studiot]

Bernoulli does not apply because a pump does work on the fluid. This is true whether the fluid is viscous or inviscid.

 

[Ken G]

Bernoulli still applies downstream of a pump. It doesn't matter the source of the pressure, what matters is whether or not energy is conserved in the steady flow. Granted, blood flow isn't steady because the pump isn't steady, but the lessons of a steady flow still apply on the appropriate timescales.

 

[rcgldr]

My attempt to answer the OP:

Just because the fluid is moving faster doesn't mean the pressure is greater in the thinner pipe?

The assumption is that the pipe does no work on the fluid and doesn't generate any forces parallel to the flow of an ideal fluid (zero viscosity or friction). If the fluid accelerates or decelerates, the only source for this acceleration would have to be internal pressure within the fluid itself. In places where the fluid accelerates, the pressure would be decreasing and vice versa.

One issue with this is the concept of an inviscid fluid, one with zero vicosity. In such an ideal (not realistic) fluid, there's no interaction between streamlines so the actual flow in a pipe of varying diameters wouldn't be predictable. A second assumption is made that mass flow within the pipe is uniform across any cross sectional area of the pipe.

If this is so, then there are many bad claims on the internet that squeezing a hose results in a larger pressure?

The pressure at the tap is greater than the ambient pressure outside. The pressure within the hose decreases with distance from the tap and the rate of the pressure decrease is related to the rate of flow. If there's no flow, the pressure is the same everywhere. If there is maximum flow, the pressure differential between tap and exit point of the hose is greatest. Squeezing the hose reduces the flow rate and the amount of pressure drop within the hose.

Or blood through thinner veins results in higher pressure?

Think of the entire circulation system as a balloon containing a fixed amount of fluid. Constricting the vascular system is like squeezing the balloon, and the overall pressure increases.

 

[Ken G]

That's an interesting point you raise, there's a difference between pressure differences (which Bernoulli applies to for ideal steady flows) and the overall pressure. There's that whole diastolic/systolic business, but I don't know if constricting the vessels affects both the same or not.

 

[Studiot]

Let us set out the differences in more detail.

A hosepipe, with sufficient supply to run full bore, is a length of single pipe, usually of sufficient constant diameter and short enough that the pressure drop down its length is small compared to the piezometric head imposed.
The piezometric head imposed arises from an essentially infinite source and may be considered constant. There are no interruptions to the flow.
In normal use there is neither mass nor heat transport across the pipe walls.

So the flow can be considered steady and from a constant head. There are no branching effects. So Bernoulli's equation may be applied directly.

The cardio-vascular system is not one pipe but many, with a system of steadily reducing diameters directly coupled to a system of directly increasing diameters.

The bloodstream is used to transport both heat and substances around the body. There is both mass and heat transport across the pipe walls.

There is not one source of pressure head, but many. The main pump is a positive displacement type, not a pressure head (impeller) type so the energy input is converted not to piezometric head, but to pulses of fluid movement. Many smaller displacement pumps are distributed round the system where muscles rhythmically squeeze and relax the vessels. This is known as the venous return mechanism.

Of course downstream of one pump is upstream of the next.

So the flow is not steady, mass and thermal energy are exchanged across the system boundaries in an unsteady manner. The pumping system cannot maintain a cosntant piezometric head. There are many branch effects.

The diameters and lengths of the pipes are such that considerable friction pressure loss is experienced.

So Bernouilli's equation, even in its thermodynamic flow form, cannot be directly applied except to very short individual length of pipe.

 

[sophiecentaur]

Bernoulli applies, of course but it describes the situation when the system has settled down. I hate to introduce the analogy but think of Kirchoff II.
There hasn't been enough emphasis on the 'source impedance' and the whole flow system in this discussion. A pump and supply pipe, like any source of power, has resistance. The greater the flow, the more energy is expended in getting the water through to the supply outlet and the lower the available pressure.
Venting a pump directly from the outlet port will result in the lowest pressure (static) measured at its output.
With a very small bore hose connected to the outlet, the flow is reduced and this will increase the pressure at the outlet port because less energy is used up in getting less water through the system. If your heart is pumping at the same rate, thinner blood vessels will result in higher pressure than for wider blood vessels but I think that homeostasis may well kick in and the heart may well pump harder when CO2 levels are higher, due to constriction, at a 'normal' rate. I think it's a double whammy, here. I reckon high blood pressure is not so much a direct problem, in itself, but it is an indicator that the heart is having to work too hard.

 

[Studiot]

Bernoulli applies, of course but it describes the situation when the system has settled down.

How do you apply Bernouilli to a positive displacement pump?

This type of pump disconnects inlet and outlet once every stroke.

It is a condition of Bernouilli that you can follow a streamline through - in fact Bernoulli applies along streamlines.

Talking of steamlines, it is a condition of streamlines that no fluid or heat energy crosses a streamline. But I have noted that both enter and leave the arterial system at many points.

 

[sophiecentaur]

OK. But my point about pressures and the supply impedance still applies. I would say, in fact, that Bernoulli is not relevant to explaining most of these situations. You could say that it applies within devices but not the whole circuit. Dang, there goes the old electrical analogy again.

 

[Studiot]

But my point about pressures and the supply impedance still applies. I would say, in fact, that Bernoulli is not relevant to explaining most of these situations. You could say that it applies within devices but not the whole circuit. Dang, there goes the old electrical analogy again.

So Bernouilli's equation, even in its thermodynamic flow form, cannot be directly applied except to very short individual length of pipe

Yeah that's cool.

 

[Ken G]

So Bernouilli's equation, even in its thermodynamic flow form, cannot be directly applied except to very short individual length of pipe.

This I agree with, and your general description of the blood-flow system. Which of the breaks from the assumptions of Bernoulli's equation that is the most serious I do not know, but I still suspect the assumption that is most seriously amiss in the capillaries is the neglect of forces between the fluid and the walls, hence the term "capillary action". Increases in blood pressure when capillaries constrict may be primarily due to these forces, I would suggest, although the point about a total decrease in volume has merit too, and their relative importance might be separable by comparing the diastolic to the systolic pressure.

 

[Studiot]

Yes my treatment of the CV system is very broad brush. You are also right to observe that the pipe walls are semi permeable, especially at the smaller sizes and their dimensions are such that their wall thickness is comparable to their bore.

 

[Ken G]

It's probably another example of the rather checkered history of the Bernoulli principle as applied to real systems, for which the devil is in the details!