In this lecture the following are introduced:
• Density and pressure
• Pressure at a depth in a static fluid
• Gauges and gauge pressure
• Pascal's principle
• Air pressure

Introduction

Fluids keep us alive, help us move, and give us energy.
Generally,
A solid holds its shape.
A fluid flows to take the shape of its container.
A plastic can be moulded or shaped (clay).
A plasma is an ionised gas (eg flame, stars).

Lagrange and fluid particles

J.L.Lagrange (1736 - 1813) established a method of dealing with fluids. In his method the fluid is divided into small masses by taking volume elements. The small masses are called fluid particles. The resultant forces on each particle are found. By solving for the position and velocity of all particles as functions of time, the fluid can be quantified. This is very tedious!

Since fluids flow through volumes and interact with areas, it is more convenient to use the intensities of these parameters. Density is the intensity of mass and pressure is the intensity of force.

Density

The density of a fluid is its mass per volume. It is a scalar quantity and its SI unit is kilogram per cubic metre.
The average density of a small volume is given by:  

The density at a point is the limit as the volume goes to zero.  

This may be a problem if individual atoms have to be taken into account, but not in the normal macroscopic environment.

Example densities

Material or Object	Density in kg.m-3
Interstellar space	10-20
Air (200C, 1 Atm)	1.21
Nitrogen (00C, 1 Atm)	1.25
Oxygen (00C, 1 Atm)	1.43
Carbon Dioxide (00C, 1 Atm)	1.98
Ethanol	791
Water (200C, 1 Atm)	998
Whole blood	1,060
The Sun (average)	1,400
Iron	7,900
Lead	11,340
Mercury	13,600
Gold	19,280
White dwarf star	10+10
Black hole	10+20

Pressure

The pressure exerted by a fluid on its surroundings is its force per area. In a fluid the force is always at right angles to the surface involved because the fluid cannot sustain a shear.
The average pressure on a small area is given by:  

The pressure at a point is the limit as the area goes to zero.  

Pressure is a scalar quantity and doesn't depend on the orientation of the area. The SI unit for pressure is the Pascal, named after Blaise Pascal, inveterate gambler, philosopher, mathematician, and scientist, born in Clermont-Ferrand, France in 1623 and died in 1662 (age 39).

Example pressures

Place	Pressure
Best laboratory vacuum	10 pPa
Faintest sound	30 μPa
Loudest tolerable sound	30 Pa
Blood pressure	16 kPa
Atmosphere (00C, 1 Atm)	101.3 kPa
Deepest part of ocean	110 MPa
Centre of Sun	20 PPa

Euler's method for fluids

L. Euler (1707 - 1783) also established a method of dealing with fluids. In his method the focus is on a point in space rather than a particle of fluid. The density ρ (x, y, z ,t) and the velocity v (x, y, z, t) are specified for the particular volume of space, and the effects of the fluid can then be quantified.

Static Fluids

A fluid at rest is the simplest situation.

The difference in pressure between the top and the bottom of a column of fluid is found by the weight of the column pressing down on the bottom area.

The pressure exerted by a liquid that is not moving, increases linearly with depth below the surface.
Note that ρgh is a pressure difference. If the surface of the fluid column is at atmospheric pressure P₀, then the pressure at a depth h below the surface is:

Pascal's paradox

Even though the weight above each crossectional area at the bottom is different, the pressure is the same. This is because the walls can provide upward or downward force

In the diagram below, the liquid does not move, so the pressure along the hoizontal line in the diagram is the same at each point. The heights of liquid are the same in each tube, so the pressure in a connected fluid depends only on depth, it does not depend on the shape or volume of water above it.

Force on a rectangular dam wall

A dam wall is a width "w", height "h" and thickness "t". It has water to a depth "d" in front of it. Find the force exerted by the water on the dam.

A force is a pressure times an area, and this force is the average water pressure (½×ρgd), times the area of water in contact with the dam wall (wd). Notice that this depends on the depth of the water and not the volume of water behind the dam.

Torque due to weight of dam

The weight of the dam wall provides a torque about the bottom edge away from the water.

The thin slice shown produces the small force dF from its weight, i.e. dF = dm×g = dV×ρ×g, where ρ is the density of the wall material. dV will be the volume of the slice, i.e. dV = hw·dx. The small torque it produces is the force dF, times the perpendicular distance, x, from the fulcrum point.

A torque is a force times a distance, and this torque is the weight of the dam wall ρg(hwt), times the average distance of the wall from the fulcrum (½t).

Pressure gauges

The Open Arm Mercury Manometer

The Open Arm Mercury Manometer is a "U tube" (really a "J" tube) filled with Mercury and open to the atmosphere on the longer side. A tube connects the shorter side to the pressurised vessel (or pipe) to be measured.
At D, the pressure is atmospheric.
At C, the pressure is .
At B, the pressure at the interface of the Mercury is the same as at C because it is at the same depth in a connected fluid.
At A, above the interface, the pressure will decrease with height, to be:

Usually the density of the fluid is much less than Mercury and its contribution can be neglected. The difference in the height of Mercury thus gives the pressure above atmospheric pressure.

The Bourdon gauge

In 1849 Eugene Bourdon invented the Bourdon tube, which is an oval tube bent into a circular arc. Pressure in the tube will tend to straighten it out. This movement is usually then translated by gears and linkages into a 3000 rotation. Nowadays a helically wound elastic element is used that is attached directly to the pointer shaft (as shown in the green picture). This eliminates the need for any movement-amplifying gears or linkages, which are the parts that fail most frequently.

Usually a series of pressure sensing devices are used to protect against too much pressure and sudden returns to the zero. In contrast to the Mercury manometer, a Bourdon gauge can measure "absolute" pressure, i.e. pressure above vacuum.

Gauge pressure

The gauge pressure is the pressure above atmospheric, as opposed to the absolute pressure above vacuum.

Example

The pressure just under the surface of the ocean is 101.3 kPa. If sea water has a density of 1030 kg.m^-3, find the depth where the pressure has doubled.

A diver 10m underwater will not be crushed because the pressure on their body has doubled. What happens is that their internal body pressures also increase, thus pushing outwards. More internal pressure means that more air will dissolve into the blood stream. A sudden decrease in pressure will cause the dissolved air to come out of solution in the form of bubbles. When Nitrogen bubbles are caught in the joints, this causes people to bend over and double-up in pain, hence this is called "the bends". The person needs to be re-pressurised so that the Nitrogen dissolves again, then they can be slowly de-pressurised.

What happens if you step (or are pushed) out of a space shuttle into the vacuum of space when you haven't got any protective gear on?

Pascal's principle

Pascal's principle states that a change in pressure is transmitted throughout a liquid without loss.
The Cartesian diver

Any increase in pressure on the side of the bottle is added to the pressure on the diver so that the diver descends.

"Hydraulics" are applications of this principle.

The Hydraulic jack

Compressed air forces an increase in the oil pressure in the smaller tube. This increase in pressure is transmitted through the fluid. The ratio of the areas of the cylinders gives the ratio of the forces.

Hydraulic brakes

The force of the foot on the pedal increases the pressure in the brake lines (pipes). This is transmitted through the liquid without loss to magnify the force on the brake pads. Note: if there is any air in the brake lines, the foot pressure will compress that air without increasing the pressure at the brake pads.    

Talk about nothing

In the early Renaissance, the argument about whether or not a vacuum could exist, reappeared. The Scholastics insisted (with Aristotle) that a vacuum was logicaly impossible. They said the Universe would sooner fall apart than to permit an abhorred Nothing in its midst! For example, Descartes said if everything were removed from a vessel, then the sides must immediately touch for a vessel cannot contain "Nothing"! The problem was that they identified vacuum (emptiness) with "nothing" (Latin: nihil). Galileo noticed that "it was not possible, either by pump or any other machine working on the principle of attraction, to lift water a hair's breath above 18 cubits (10.3 m)". He is said to have commented that Nature's abhorrence of a vacuum was limited to this height.    

Air Pressure

The air, in which we are immersed, has mass and weight so it also exerts a pressure on us, not only downwards but also sideways.

The inertia of the air pushing down.

How do you get an egg into a bottle?

The collapsing can

Measuring air pressure

In 1644 Evangelista Torricelli (one of Galileo's pupils) suggested an experiment of inverting a tube of mercury into a bowl of mercury. He noticed that the Mercury fell a little bit and left a void above it. He said the void at the top was vacuum and the mercury was held up by outside air pressure. The height of the Mercury would then give the air pressure relative to vacuum and as above, Pressure=ρgH.

Pascal, in Rouen (1647), was convinced by Torricelli's conclusions and leaped into the controversy when he saw he could make some money. Pascal performed experiments with tubes of various diameters, in which the space for the void now either bulging or narrow. He inclined his tubes till the void disappeared and then reappeared. He put a bladder inside the top. Every time the height remained constant while the empty space varied enormously. Arguing against a vacuum in the Torricelli tubes, the "Plenists" in the town said the apparent void was simply Mercury vapour, which expanded to fill the space. To put things to the test, Pascal constructed two tubes of glass, each 14m long. He bound them to ships' masts and made pivots at the centres. He filled one with water and stoppered it. He then inverted it into a tub of water and removed the stopper. The water fell to 10.3m. He asked the Plenists what would happen if wine should be used in the test? They said that wine was more volatile than water and would release more vapours and so would fall further in the tube. After bets were organised, Pascal filled the second tube with red wine.

The wine stood higher than the water, not lower. air pressure=density × gravity × height. Wine is less dense than water, so to keep the product of density and height the same, the less dense wine must stand higher.

Standard Air Pressure

Because of density differences, 10.3 m of water is equivalent to 760 mm of Mercury. Standard atmospheric pressure is set equal to this.

Standard Air Pressure	=760 mm Hg = 760 torr = 1 Atmosphere
In other units:

Air pressure is not constant. Higher pressures usually indicate fine weather and the pressure falls as storms approach.

Cautionary Tale

One man made a barometer to check on weather changes. He made it with water in a glass tube, and it was so long that it stuck out through the roof of his house. He put the small outline figure of a woman in it, so that he could easily see the height of the water. The figure rose out of the roof and disappeared back inside as the weather changed. He was quite proud of this until the locals were so spooked by the mysterious flying figure, that they burned him as a witch.

Summarising:

The gauge pressure is the pressure above atmospheric, as opposed to the absolute pressure above vacuum.
Pascal's principle states that a change in pressure is transmitted throughout a liquid without loss.
A vacuum is space that is emptied of matter, but it is not "nothing" because it has properties.
Standard Air Pressure is: P₀=  1 Atmosphere = 760 mm Hg = 760 torr = 101.3 kPa = 1013 millibars