Bernouille's principle



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BERNOUILLE'S PRINCIPLE

Consider an ideal fluid moving through a pipe.


Ideal fluid

  • Non viscous  = 0

  • Incompressible  is contant thhroughout the pipe

  • No rotational motion  laminar or streamline flow

An interesting effect is that, for a fluid (e.g. air) flowing through a pipe with a constriction in it, the fluid pressure is lowest at the constriction. In terms of the equation of continuity the fluid pressure falls as the flow speed increases.



The reason is easy to understand. The fluid has different speeds and hence different kinetic energies at different parts of the tube. The changes in energy must result from work being done on the fluid and the only forces in the tube that might do work on the fluid are the driving forces associated with changes in pressure from place to place.


Fig. 20:
Application of Bernoulli's Principle

Since the fluid is driven from regions of high pressure to those of low pressure and thus increases its kinetic energy, we can write: kinetic energy/volume  work/volume is constant,

i.e. v2 p = constant.

In cases where the flow is not horizontal, we should add in the gravitational potential energy/volume also: v2 p gh = constant.



This is known as Bernoulli's equation. For the very simple cases it says what we had before - the fluid pressure is lowest where the flow speed is highest.

Venturi Effect

A fast jet of air emerging from a small nozzle will have a lower pressure than the surrounding atmospheric pressure. You can support a weight this way:


When you are in the dentist's chair, the dentist uses a device based on the venturi effect to suck saliva out of your mouth. The device is connected to a tap, as shown in the diagram. Water flows fast past the constriction, causing the pressure to drop inside the long tube. When the other end of this tube is immersed in a pool of saliva, the higher pressure outside forces the saliva up the tube, and away. In a cotton picker air flows instead of water to suck up cotton!



Fig. 22:

Fig. 23:
An "Atomizer"

The chimney effect: just the venturi effect being used to suck material up. [Note in most automobiles, petrol is sucked into the carburettor in this way.] Atomizer: This same effect makes atomizers and spray guns work. It is most important that the free surface of the liquid should be open to the atmosphere, else the high pressure outside the container and the low pressure inside will result in the container being crushed. {Fly sprays always have a small air hole.]

A spinning ball or cylinder moving through a fluid experiences a sideways force. There is a high pressure on one side (so a big force) and low pressure (small force) on the other. The ball experiences a net sideways thrust. This is one of the ways players can get cricket or ping pong balls to swerve.



Model for a knee in fluid flow:


Fig. 24:
Stream lines for a liquid flowing in streamline motion through a drain with a corner in it.

Use continuity to decide where the flow speeds up, and when it slows down.



It can be observed that the streamlines hug the sharp corner. But far enough before the corner, and far enough after it, they are parallel and equally spaced. Consider the liquid flowing between lines 1 and 2. Its cross-sectional area decreases near the corner, so the liquid speeds up there. The fluid between lines 3 and 4 has its area increased near the corner, so it slows down. So the flow velocity changes like this

Fig. 25:

How Does the pressure change as the liquid goes round the corner.



Applying Bernoulli's Law, the pressure in the liquid stream must be

Fig. 26:

Remembering that pressure can also be thought of as force per unit area, we see that the fluid must be exerting an extra large force on the tube right at the outside corner.

Bernoulli’s equation uses the equation of conservation of energy, thus it is important that there should be no energy dissipation through turbulence. Bernoulli's equation only really applies when the motion is strictly streamline. Nonetheless, provided there is not too much turbulence, the law will approximately apply.

You may not be able to apply this to fluid flowing around a bend in the system if the flow is extremely turbulent.


Blood pressure
Blood pressure is measured using a type of gauge; usually calibrated in mm Hg. The gauge is attached to a closed air-filled jacket that is wrapped around the upper arm at the level of the heart. Two values of blood pressure are measured: the maximum pressure when the heart is pumping, which is called systolic pressure; and the pressure when the heart is in the resting part of the cycle, which is called diastolic pressure. Initially, the air pressure in the jacket is increased high above the systolic pressure by means of a hand pump, and this compresses the main (brachial) artery in the arm and cuts off the flow of blood. The air pressure is then slowly reduced to the point where blood again begins to flow into the arm; it is detected by listening with a stethoscope to the characteristic tapping sound of the blood returning to the forearm. At this point, systolic pressure is just equal to the air pressure in the jacket that can be read off the gauge. The air pressure is subsequently reduced further and the tapping sound disappears when blood at low pressure can enter the artery. At this point, the gauge indicates the diastolic pressure. Normal systolic pressure is around 120 mm Hg whereas normal diastolic pressure is around 80 mm Hg.
If a person has advanced arteriosclerosis, the Bernoulli principle produces a sign called vascular flutter. In this situation, the artery is constricted as a result of an accumulation of plaque on its inner walls. In order to maintain constant flow rate through such a constricted artery, the driving pressure must increase. Such an increase in pressure requires a greater demand on the heart muscle. If the blood velocity is sufficiently high in the constricted region, the artery may collapse under external pressure, causing a momentary interruption in blood flow. At this point, there is no Bernoulli principle and the vessel, reopens under arterial pressure. As the blood rushes through the constricted artery, the internal pressure drops and again the artery closes. Such variations in blood flow can he heard with a stethoscope. If the plaque becomes dislodged and ends up in a smaller vessel that delivers blood to the heart, the person can suffer a heart attack.

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