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A History and Philosophy of Fluid Mechanics
By G. A. Tokaty Dover Publications, Inc.
Copyright © 1971 G. A. Tokaty
All rights reserved.
ISBN: 978-0-486-15265-3
CHAPTER 1
Basic definitions
Science is an ordered and systematic knowledge ascertained by theoretical analysis, observation and experiment. The science, or family of sciences, which aims to study and to master the material world in the interests of mankind, used to be (and still is, here and there) called Natural Philosophy. Its methods were and are: general logical deduction, scientific hypothesis, observation, experiment, comparison, analysis, synthesis, and so on. But in all these, in one degree and form or another, Mathematics plays a role: Natural Philosophy and Mathematics are as inseparable as a living cell and water.
There are, however, two kinds of Mathematics: Pure Mathematics, which studies abstract numbers, ratios, relationships, figures, functions, and works out and formulates abstract methods and theorems of mathematical philosophy and logic, and builds up a general mathematical culture, and Applied Mathematics, which is concerned with the study of physical, technological, biological, chemical and sociological worlds. To put it differently, a system utilizing, in addition to the purely mathematical concepts of space and number, the notions of time and matter, is Applied Mathematics. It includes the mechanics of rigid and deformable bodies, the theory of elasticity and plasticity, thermodynamics and biomathematics, statistics, etc. In a restricted sense, Applied Mathematics means the use of the concepts and theorems of Pure Mathematics for the study of the problems of Mechanics, which can be defined as the mathematical theory of the motions and tendencies to motion of particles and systems of particles under the influence of constraints, and the study of motions of masses and of the effects of forces in causing or modifying these motions.
The mathematical structure of Mechanics (i.e. its integro-differential forms) is largely due to Leonhard Euler (1707—83), Joseph Louis de Lagrange (1736—1813) and William Rowan Hamilton (1805—65); in this conception, it is often called Theoretical Mechanics, or Analytical Mechanics. One should not jump to the conclusion, however, that Theoretical (Analytical) Mechanics is something different or independent of Mechanics.
The French word fluide, and its English equivalent 'fluid', means 'that which flows', i.e. a substance whose particles can move about with complete freedom (ideal fluids) or restricted freedom (real fluids). That branch of Mechanics, or Applied Mathematics, which studies the laws of motion and tendencies to motion of fluids is called Fluidmechanics. When it deals, mainly or exclusively, with liquids, that is, when 'fluid' stands for 'liquid' (in most cases meaning 'water'), Fluidmechanics becomes the mechanics of liquids, or Hydromechanics. When 'fluid' stands for 'gas' (in most cases meaning 'air'), Fluidmechanics becomes the mechanics of gases, or Aeromechanics.
Hydromechanics, in turn, may be subdivided into Hydrodynamics, Hydraulics, and Hydrostatics. The chief objective of Hydrodynamics is to establish theoretical-analytical relationships between the kinematic elements of motion, or flow, and forces which cause and maintain them. Hydraulics studies the laws of motion of liquids in tubes, pipes, channels, elbows, and other engineering devices; as a general rule, its methods are based upon the basic concepts and theorems of hydrodynamics. Hydrostatics deals with the equilibrium of liquids at rest.
When 'fluid' stands for 'air' or 'aer' ('gas', more generally), Fluidmechanics becomes Aeromechanics. The latter may be subdivided into Aerostatics, Theoretical Aerodynamics, Experimental Aerodynamics, and Mechanics of Flight. Aerostatics studies the laws of equilibrium of air and other gases at rest. Theoretical Aerodynamics is the science of motion of gases in the Eulerian-Lagrangeian-Hamiltonian sense, i.e. uses the laws, theorems, axioms and general theoretical concepts of analytical (theoretical) mechanics for the study of air-gas flows under the action of forces; its role in Aeromechanics is similar to that of Hydrodynamics in the mechanics of liquids. The main objective of Theoretical Aerodynamics is to establish theoretical-analytical relationships between the dynamic, kinematic and thermodynamic characteristics of gas flows.
Experimental Aerodynamics verifies the theories, equations and formulae of Theoretical Aerodynamics by means of laboratory experiments and corrects them through the introduction of experimentally determined coefficients. In many cases, Experimental Aerodynamics succeeds where Theoretical Aerodynamics fails.
The Mechanics of Flight works out the general equations of flight and determines the air-speed and power required, climb performance and landing-take off characteristics, range and duration of flight, stability and control criteria, and so on. In all these, it makes extensive use of the theories, theorems and fundamental concepts of almost all branches of Mechanics.
Finally, the fairly new notion, 'Aerospace', embraces and unifies Aeronautics and Astronautics, or Space Technology. Aeronautics is the general name for all the aspects and problems of flight within the atmosphere of the earth. Astronautics embraces all the aspects and problems of flight beyond the atmosphere, and may be defined as the science of motion of rockets, sputniks and spaceships beyond the atmosphere.
CHAPTER 2
Fluids and life
How important are fluids in life? Here are several examples. Human beings, animals and vegetation are literally water-based. Every living cell in your body has a fluid interior, a vital solution of various substances in water; human blood is more than nine-tenths water; our muscles average 92 per cent water; all in all, man's body contains about 71 per cent water by weight — and this water, evaporating and flowing from the surface of the body, breathed out as vapour in breath, must be continuously replenished if the body is to remain alive. Man pours down his throat five times his weight in water every year; by the time he dies, if his life span is normal, he will have drunk about 6500 gallons of it.
Consider the matter also from another angle. The desert is arid. The sun, the other source of life, kills almost every living thing there. In just one hour, it fills each square yard of the desert with more than 800 large calories — a large calorie is the amount of heat required to raise the temperature of 2·2 pounds of water by one degree centigrade. An egg can be fried on the burning sand. The Kara-Kum desert in Central Asia (USSR) covers almost 150 000 square miles. Its surface receives a stupendous amount of solar energy. But fresh water is scarce, if not absent, although whole seas of salt water lie just below thirsty pastures. Flocks of hardy sheep (three million head) roam the desert. Their numbers could increase were it not for the shortage of fresh water. It is easy to imagine the economic benefits of irrigating the desert.
It has been estimated that a mere 1 ·3 cubic yards of water is needed to produce 8 pounds of wool, nearly 17 pints of milk, up to 22 pounds of meat, or to grow 4 ounces of cotton. No wonder that the Soviet Union has decided to harness this solar energy, and a helio-installation to distil salt water is now being built in the heart of the desert. The installation will be a sizeable complex comprising distillers (each over 700 square yards in size), ferroconcrete reservoirs able to hold 740 cubic yards of water, a well, water-raising pipes, solar electric generators, water troughs, etc.
A desperate effort, indeed; but a necessary one. Perhaps efforts of this kind will convince man that water and air are far too fundamental for all forms of life for them to be treated too barbarically. It seems incredible, for example, that, in the same Soviet Union, millions and millions of cubic metres of waste water are daily being dumped into fresh sources, making them almost useless. Even in the USSR, which has a planned economy and therefore cannot be regarded as the most wasteful country in the world, each year some 3 000 000 tons of acids, 2 000 000 tons of oil products, 1 000 000 tons of fats, and hundreds of thousands of tons of salt, fibres and metal are dumped into rivers.
In the United States (undoubtedly the most wasteful country) whole rivers and lakes are being polluted to such an extent that they can no longer be used for drinking or swimming purposes. The annual American waste output (of which a large part goes into rivers and lakes) includes something like 142 million tons of smoke and fumes, 7 million discarded cars, 20 million tons of waste paper, 48 billion tin cans, 26 billion bottles and jars, 3 billion tons of waste rock and milling swarf, and 50 trillion gallons of hot water.
But man spoils not only the waters of the earth. You may have heard of the notorious 'smog' in Los Angeles and London. But in actual fact it exists in every industrial centre. Combustion of fossil fuels has increased carbon dioxide in the atmosphere by one-tenth in a century, and may attain an increase of a quarter by the year 2000: this would be catastrophic for weather and climate, and seriously damaging to every form of life.
When analysed carefully, this 'Battle of Fluids', the struggle for their protection and destruction, reveals that it, too, is based upon the laws of Fluidmechanics. We may, therefore, assume that knowledge of the history of the subject is important also from the point of view of learning how to minimize the damage to the atmosphere and water resources, if not how to prevent them from being damaged at all. But to be able to do this, that is to understand what exactly is damaged or protected, it is necessary to know, first and foremost, the compositions of the fluids themselves.
CHAPTER 3
Water and air
The composition of water has been the subject of intensive study since the emergence of chemistry and physics. But it seems to be impossible to say who exactly started it and when. The only thing we know for certain is that (already) Hero of Alexandria was anxious to know why water boiled and produced steam. We shall discuss this period of fluidmechanics later on.
E. W. Morley, of Western Reserve University, Cleveland, Ohio, in 1895, reported that the weight of Oxygen to Hydrogen in water was as 7·9395 to 1·00000, and that the volume ratio was as 1·00000 to 2·00288. F. P. Burt and E. C. Edgar, of England, in 1916, considered, on the basis of their experiments, 7·9387 to 1·00000 as the most exact weight ratio. The present value accepted by the International Union of Chemistry, Committee of Atomic Weights, is 15·9994/2 to 1·00797.
Water can be in two major states: solid (ice) and liquid (water). The first of these occurs at the so-called 'freezing point'. In other words, at a pressure of one atmosphere, ice melts (becomes water) at 0°C. If the water is pure, especially when there are in it no dissolved gases, it can be heated without boiling up to 100°C and even higher. But, in normal conditions, 100°C is the 'boiling point', at which temperature steam formation is so intense that it occupies a volume 1 700 times greater than the water itself.
By 'one atmosphere' is meant the pressure exerted by the atmosphere as a consequence of the gravitational attraction exerted upon the column of air lying directly above the point where the pressure is measured. One atmosphere = 760 mm of Hg. When the pressure is 770 mm, the boiling point occurs at 100·366°C, when it is 750 mm at 99·360°C, when it is 740 mm at 99·255°C, when it is 730 mm at 98·877°C, when it is 388 mm at 81·7°C, when it is 76 mm at 46·1°C, when it is 1520 mm at 120·6°C, when it is 7600 mm at 180·5C. Hg stands, of course, for the Latin name Hydrargyrum or Mercury (the element): atomic weight 200·59, atomic number 80, melting point — 38·87°C, boiling point 356·58°C. Mercury is a silver-white liquid metal, the only metal that is liquid at ordinary temperatures. It is widely used in Fluidmechanic measuring instruments (manometers, for instance) because of its remarkable properties; to name just one of them, it does not wet glass, which is so important in manometry.
Among the other physical characteristics of water, and of fluids generally, the most important one is density. The amount of mass per unit volume is the mass density ρ; the weight per unit volume is the weight density γ. The two are connected by the equation γ = ρg, where g is the gravitational acceleration. Both ρ and γ are different for different fluids and depend on the temperature, as shown in the table below:
[TABLE OMITTED]
Natural waters may be contaminated with insoluble suspended materials, soluble inorganic matter, soluble organic matter. In oceans, seas and salt lakes, water's principal content is sodium chloride, with small amounts of calcium, magnesium, potassium, and sulphate, carbonate, and many other elements in smaller concentrations.
And now about Air. By this is meant the whole mass of the earth's atmosphere, whose average percentage composition may be represented by the following table:
Nitrogen 78·08
Oxygen 20·95
Argon 0·93
Carbon dioxide 0·03,
and some other gases. All the figures are given for the so-called dry air, i.e. for air from which all water vapour has been removed.
Although air is composed of various gases, in most cases it is regarded as a uniform gas. How do the main physical characteristics of such a gas compare with other gases? The table below gives the answer:
[TABLE OMITTED]
It is a matter of ordinary daily observation that all fluids are capable of exerting pressure. A certain amount of effort is necessary in order to immerse your hand in water and to move it there. The effort is much less when you do the same in the atmosphere, in a gas generally, because its density is much smaller. That the atmosphere at rest exerts pressure is shown directly by means of an air pump. Amongst many experiments, a simple one is to exhaust the air within a receiver made of very thin glass; when the exhaustion has reached a certain point depending on the strength of the glass, the receiver will be shivered by the pressure of the external air. The action of wind, the motion of a wind mill, the propulsion of a boat by means of sails, and other familiar facts offer themselves naturally as instances of the pressure of the air when in motion.
All such substances as water, oil, mercury, steam, air, or any kind of gas are fluids, but in order to obtain a definition of a specific fluid, we have to find a property which is common to all these different kinds of matter, and which does not depend upon any of the characteristics by which they are distinguished from each other. This property is found in the extreme mobility of their particles and in the ease with which these particles can be separated from the mass of fluid and from each other. In other words, a fluid is a substance whose particles can be very easily separated from the whole mass.
Gases (air, for instance), by the application of ordinary force, can be easily compressed, and, if the compressing force be removed or diminished, will expand in volume. Liquids, too, are really compressible, but experiments by Canton in 1761, Perkins in 1819, Oersted in 1823, Colladon and Sturm in 1829, and others, have proved that they are only slightly compressible.
The pressure of a liquid at rest is entirely due to its weight, and to the application of some external force, while the pressure of a gas, although modified by the action of gravity, depends in chief upon its volume and temperature. The action of a common syringe will serve to illustrate the elasticity of the atmospheric air. If the syringe be drawn out and its open end then closed, a considerable effort will be required to force in the piston to more than a small fraction of the length of its range, and if the syringe be air-tight, and strong enough, it will require the application of very great power to force down the piston through nearly the whole of its range. And this experiment was good enough for our ancestors to prove that the pressure of the air increases with the compression, the air within the syringe acting as an elastic cushion.
(Continues...)
Excerpted from A History and Philosophy of Fluid Mechanics by G. A. Tokaty. Copyright © 1971 G. A. Tokaty. Excerpted by permission of Dover Publications, Inc..
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