A Treatise on Hydrodynamics with Numerous Examples, 2 vols. An Introduction to Fluid Dynamics. Cambridge University Press, Dynamics of Fluids in Porous Media. A Treatise on Hydromechanics, 2 vols. A Study in Logic, Fact, and Similitude, rev. Princeton University Press, The Phenomena of Fluid Motions.

Topics in Fluid Mechanics.

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## Fluid Mechanics -- from Eric Weisstein's World of Physics

A Mathematical Introduction to Fluid Mechanics, 3rd ed. An Introduction to Computational Fluid Mechanics, corr. Perspectives in Fluid Mechanics: Introduction to Geophysical Fluid Dynamics. Fluid Dynamics for Physicists. Introduction to Fluid Mechanics.

Computational Techniques for Fluid Dynamics, Vol. Fundamental and General Techniques, 2nd ed. Specific Techniques for Different Flow Categories, 2nd ed. Introduction to Fluid Mechanics, 4th ed. Wiley and Sons, Topics in Geophysical Fluid Dynamics. Modern Developments in Fluid Dynamics: Eds Incompressible Computational Fluid Dynamics: Gordon and Breach, Fluid Mechanics, 2nd ed. Volume 6 of Course of Theoretical Physics.

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Elements of Gas Dynamics. Introduction to Mathematical Fluid Dynamics. The Science of Fluids. Van Nostrand Reinhold, Theoretical Hydrodynamics, 5th ed. Mathematical Theory of Compressible Fluid Flow. The foundations of hydrodynamics, on the other hand, were not laid until the 18th century when mathematicians such as Leonhard Euler and Daniel Bernoulli began to explore the consequences, for a virtually continuous medium like water, of the dynamic principles that Newton had enunciated for systems composed of discrete particles.

Their work was continued in the 19th century by several mathematicians and physicists of the first rank, notably G. Stokes and William Thomson. By the end of the century explanations had been found for a host of intriguing phenomena having to do with the flow of water through tubes and orifices, the waves that ships moving through water leave behind them, raindrops on windowpanes, and the like. There was still no proper understanding, however, of problems as fundamental as that of water flowing past a fixed obstacle and exerting a drag force upon it; the theory of potential flow, which worked so well in other contexts , yielded results that at relatively high flow rates were grossly at variance with experiment.

This problem was not properly understood until , when the German physicist Ludwig Prandtl introduced the concept of the boundary layer see below Hydrodynamics: Boundary layers and separation. Since that time, the flow of air has been of as much interest to physicists and engineers as the flow of water, and hydrodynamics has, as a consequence, become fluid dynamics. The term fluid mechanics , as used here, embraces both fluid dynamics and the subject still generally referred to as hydrostatics.

One other representative of the 20th century who deserves mention here besides Prandtl is Geoffrey Taylor of England. Taylor remained a classical physicist while most of his contemporaries were turning their attention to the problems of atomic structure and quantum mechanics , and he made several unexpected and important discoveries in the field of fluid mechanics.

The richness of fluid mechanics is due in large part to a term in the basic equation of the motion of fluids which is nonlinear— i. It is characteristic of systems described by nonlinear equations that under certain conditions they become unstable and begin behaving in ways that seem at first sight to be totally chaotic. In the case of fluids, chaotic behaviour is very common and is called turbulence. Mathematicians have now begun to recognize patterns in chaos that can be analyzed fruitfully, and this development suggests that fluid mechanics will remain a field of active research well into the 21st century.

For a discussion of the concept of chaos , see physical science, principles of. Fluid mechanics is a subject with almost endless ramifications, and the account that follows is necessarily incomplete. Some knowledge of the basic properties of fluids will be needed; a survey of the most relevant properties is given in the next section.

For further details, see thermodynamics and liquid.

Fluids are not strictly continuous media in the way that all the successors of Euler and Bernoulli have assumed, for they are composed of discrete molecules. The molecules, however, are so small and, except in gases at very low pressures, the number of molecules per millilitre is so enormous that they need not be viewed as individual entities.

There are a few liquids, known as liquid crystals, in which the molecules are packed together in such a way as to make the properties of the medium locally anisotropic , but the vast majority of fluids including air and water are isotropic. A word perhaps is needed about the difference between gases and liquids , though the difference is easier to perceive than to describe.

In gases the molecules are sufficiently far apart to move almost independently of one another, and gases tend to expand to fill any volume available to them. In liquids the molecules are more or less in contact, and the short-range attractive forces between them make them cohere; the molecules are moving too fast to settle down into the ordered arrays that are characteristic of solids, but not so fast that they can fly apart.

Thus, samples of liquid can exist as drops or as jets with free surfaces, or they can sit in beakers constrained only by gravity, in a way that samples of gas cannot. Such samples may evaporate in time, as molecules one by one pick up enough speed to escape across the free surface and are not replaced. The lifetime of liquid drops and jets, however, is normally long enough for evaporation to be ignored.

There are two sorts of stress that may exist in any solid or fluid medium, and the difference between them may be illustrated by reference to a brick held between two hands. If the holder moves his hands toward each other, he exerts pressure on the brick; if he moves one hand toward his body and the other away from it, then he exerts what is called a shear stress. A solid substance such as a brick can withstand stresses of both types, but fluids, by definition, yield to shear stresses no matter how small these stresses may be.

This property, about which more will be said later, is a measure of the friction that arises when adjacent layers of fluid slip over one another. These three quantities are linked together by what is called the equation of state for the fluid. For other fluids knowledge of the equation of state is often incomplete. When an element of fluid is compressed, the work done on it tends to heat it up. In reversible adiabatic processes for such gases, however, the temperature rises on compression at a rate such that. For liquids the ratio between the isothermal and adiabatic compressibilities is much closer to unity.

The molar specific heat is the amount of heat required to raise the temperature of one mole through one degree. This is greater if the substance is allowed to expand as it is heated, and therefore to do work, than if its volume is fixed. The principal molar specific heats, C P and C V , refer to heating at constant pressure and constant volume, respectively, and.

Solids can be stretched without breaking, and liquids, though not gases, can withstand stretching, too. Water owes its high ideal strength to the fact that rupture involves breaking links of attraction between molecules on either side of the plane on which rupture occurs; work must be done to break these links.

However, its strength is drastically reduced by anything that provides a nucleus at which the process known as cavitation formation of vapour- or gas-filled cavities can begin, and a liquid containing suspended dust particles or dissolved gases is liable to cavitate quite easily.

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Work also must be done if a free liquid drop of spherical shape is to be drawn out into a long thin cylinder or deformed in any other way that increases its surface area. Here again work is needed to break intermolecular links. The surface of a liquid behaves, in fact, as if it were an elastic membrane under tension, except that the tension exerted by an elastic membrane increases when the membrane is stretched in a way that the tension exerted by a liquid surface does not.

Surface tension is what causes liquids to rise up capillary tubes, what supports hanging liquid drops, what limits the formation of ripples on the surface of liquids, and so on. Mathematically, the rate at which the pressure in a stationary fluid varies with height z in a vertical gravitational field of strength g is given by. Applied to the atmosphere, equation would imply that the pressure falls to zero at a height of about 10 kilometres.

Compressible flow in gases. Instruments for comparing pressures are called differential manometers, and the simplest such instrument is a U-tube containing liquid, as shown in Figure 1A. It is a consequence of that.

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A barometer for measuring the pressure of the atmosphere in absolute terms is simply a manometer in which p 2 is made zero, or as close to zero as is feasible. The barometer invented in the 17th century by the Italian physicist and mathematician Evangelista Torricelli , and still in use today, is a U-tube that is sealed at one end see Figure 1B.

It may be filled with liquid, with the sealed end downward, and then inverted. On inversion, a negative pressure may momentarily develop at the top of the liquid column if the column is long enough; however, cavitation normally occurs there and the column falls away from the sealed end of the tube, as shown in the figure. Between the two exists what Torricelli thought of as a vacuum, though it may be very far from that condition if the barometer has been filled without scrupulous precautions to ensure that all dissolved or adsorbed gases, which would otherwise collect in this space, have first been removed.

Even if no contaminating gas is present, the Torricellian vacuum always contains the vapour of the liquid, and this exerts a pressure which may be small but is never quite zero. The liquid conventionally used in a Torricelli barometer is of course mercury, which has a low vapour pressure and a high density. The high density means that h is only about millimetres; if water were used, it would have to be about 10 metres instead. Figure 1C illustrates the principle of the siphon. The top container is open to the atmosphere, and the pressure in it, p 2 , is therefore atmospheric.

If the bottom container is also open to the atmosphere, then equilibrium is clearly impossible; the weight of the liquid column prevails and causes the liquid to flow downward. The siphon operates only as long as the column is continuous; it fails if a bubble of gas collects in the tube or if cavitation occurs.

### Surface tension of liquids

Cavitation therefore limits the level differences over which siphons can be used, and it also limits to about 10 metres the depth of wells from which water can be pumped using suction alone. Consider now a cube of side d totally immersed in liquid with its top and bottom faces horizontal. As Archimedes must have realized, there is no need to prove this by detailed examination of the pressure difference between top and bottom. It is obvious because, if the solid body could somehow be removed and if the cavity thereby created could somehow be filled with more fluid instead, the whole system would still be in equilibrium.

## Fluid mechanics

The extra fluid would, however, then be experiencing the upthrust previously experienced by the solid body, and it would not be in equilibrium unless this were just sufficient to balance its weight. He understood that the pure metal and the alloy would differ in density and that he could determine the density of the crown by weighing it to find its mass and making a separate measurement of its volume. Perhaps the inspiration that struck him in his bath was that one can find the volume of any object by submerging it in liquid in something like a measuring cylinder i.