Fluid Mechanics

I

Introduction

Fluid Mechanics, physical science dealing with the action of fluids at rest or in motion, and with engineering applications and devices using fluids. Fluid mechanics is basic to such diverse fields as aeronautics (see Aeroplane), chemical, civil, and mechanical engineering, meteorology, naval architecture (see Ships and Shipbuilding), and oceanography.

Fluid mechanics can be subdivided into two major areas: fluid statics, or hydrostatics, which deals with fluids at rest, and fluid dynamics, concerned with fluids in motion. The term hydrodynamics is applied to the flow of liquids or to low-velocity gas flows in which the gas can be considered as being essentially incompressible. Aerodynamics or gas dynamics is concerned with the behaviour of gases when velocity and pressure changes are sufficiently large to require inclusion of the compressibility effects.

Applications of fluid mechanics include jet propulsion, turbines, compressors, and pumps (see Compressed Air). The utilization of water and oil pressure in engineering is the field of hydraulics.

II

Fluid Statics or Hydrostatics

A fundamental characteristic of any fluid at rest is that the force exerted on any particle within the fluid is the same in all directions. If the forces were unequal, the particle would move in the direction of the resultant force. It follows that the force per unit area, the pressure, exerted by the fluid against the walls of an arbitrarily shaped containing vessel is perpendicular to the walls at every point. If the pressure were not perpendicular an unbalanced tangential force component would exist and the fluid would move along the wall.

This concept was first formulated in a slightly extended form by the French mathematician and philosopher Blaise Pascal in 1647. Known as Pascal's law, it states that the pressure applied to an enclosed fluid is transmitted equally in all directions and to all parts of the enclosing vessel, if pressure differences due to the weight of the fluid can be neglected. This law has extremely important applications in hydraulics.

The top surface of a liquid at rest in an open vessel will always be perpendicular to the resultant forces acting on it. If gravity is the only force, the surface will be horizontal. If other forces in addition to gravity act, then the “free” surface will adjust itself. For instance, if a glass of water is spun rapidly about its vertical axis, both gravity and centrifugal forces will act on the water and the surface will form a parabola that is perpendicular to the resultant force.

If gravity is the only force acting on a liquid contained in an open vessel, the pressure at any point within the liquid is directly proportional to the weight of a vertical column of that liquid above that point. This, in turn, is proportional to the depth of the point below the surface and is independent of the size or shape of the container. Thus the pressure at the bottom of a vertical pipe 2.5 cm (1 in) in diameter and 15 m (about 50 ft) high that is filled with water is the same as the pressure at the bottom of a lake about 15 m (about 50 ft) deep. Similarly, if a pipe 30 m (100 ft) long is filled with water, and slanted so that the top is only 15 m (50 ft) above the bottom vertically, the water will exert the same pressure at the bottom of the pipe, even though the distance along the pipe is much greater than the height of the vertical pipe. The weight of a column of fresh water 30 cm (12 in) high and with a cross section of 6.5 sq cm (1 sq in) is 195 g (0.435 lb) and this will be the force exerted at the bottom. A column of the same height but 12 times the diameter will have 144 times the volume and will weigh 144 times as much, but the pressure, which is force per unit area, will remain identical. The pressure at the bottom of a mercury column of the same height will be 13.6 times as great, as mercury is 13.6 times as dense as water. See also Atmosphere; Barometer; Capillarity.

The second important principle of fluid statics was discovered by the Greek mathematician and philosopher Archimedes. Archimedes' principle states that a submerged body is subject to a buoyancy force that is equal to the weight of the fluid displaced by that body. This explains why a heavily laden ship floats; its total weight equals exactly the weight of the water that it displaces, and this weight exerts the buoyant force supporting the ship.

The point at which all forces producing the buoyant effect, or upthrust, may be considered to act is called the centre of buoyancy and is the centre of gravity of the displaced fluid. When the ship is upright, this point is on the vertical centre line of the ship; similarly the centre of gravity of the ship structure is also on this line. When the ship rolls, the centre of buoyancy moves sideways and the line of action of the buoyancy force, which is always vertical, intersects the centre line of the ship at a point called the metacentre. For stability, the metacentre must be above the centre of gravity of the ship, irrespective of the position of the centre of buoyancy. In the vast majority of ships the centre of buoyancy is actually below the centre of gravity. See Stability.

Archimedes' principle also makes possible the determination of the density of an object that is so irregular in shape that its volume cannot be measured directly. If the object is weighed first in air and then in water, the difference in weights will equal the weight of the volume of the water displaced, which is the same as the volume of the object. Thus the density of the object (weight divided by volume) can readily be determined. In very high-precision weighing, the weight of the displaced air also has to be accounted for in arriving at the correct volume and density.

III

Fluid Dynamics or Hydrodynamics

This branch of fluid mechanics deals with the laws of fluids in motion; these laws are considerably more complex and, in spite of the greater practical importance of fluid dynamics, only a few basic ideas can be discussed here.

Interest in fluid dynamics dates from the earliest engineering application of fluids in machines. Archimedes made an early contribution by his invention of the screw pump, if the tradition ascribing it to him is true. The pushing action of the Archimedes screw is similar to that of the corkscrewlike device in a meat grinder. Other hydraulic machines and devices were developed by the Romans, who not only used Archimedes' screw for irrigation and mine pumping but also built extensive aqueduct systems, some of which are still in use. The Roman architect and engineer Vitruvius invented the horizontal waterwheel during the 1st century bc, which revolutionized corn milling.

Despite the early practical applications of fluid dynamics, little or no understanding of the basic theory existed, and development lagged accordingly. After Archimedes, more than 1,800 years elapsed before the next significant scientific advance was made by the Italian mathematician and physicist Evangelista Torricelli, who invented the barometer in 1643, and formulated Torricelli's law, which related the efflux velocity of a liquid through an orifice in a vessel to the liquid height above it. The next great advance in the development of fluid mechanics had to await the formulation of the laws of motion by the English mathematician and physicist Isaac Newton. These laws were first applied to fluids by the Swiss mathematician Leonhard Euler, who derived the basic equations for a frictionless, or inviscid, fluid.

Euler first recognized that dynamical laws for fluids can only be expressed in a relatively simple form if the fluid is assumed incompressible and ideal, that is, if the effects of friction or viscosity can be neglected. Because, however, this is never the case for real fluids in motion, the results of such an analysis can only serve as an estimate for those flows in which viscous effects are small.

A

Incompressible and Frictionless Flows

These flows follow Bernoulli's principle, named after the Swiss mathematician and scientist Daniel Bernoulli. The principle states that the total mechanical energy of an incompressible and inviscid (frictionless) flow is constant along a streamline. Streamlines are imaginary flow lines that are always parallel to the local direction of the flow, and that for steady flow are also the lines followed by individual fluid particles. Bernoulli's principle leads to an interrelationship between pressure effects, velocity effects, and gravity effects, and indicates that the velocity increases as the pressure decreases. This principle is important in nozzle design and in flow measurements, and it can also be used to predict the lift of a wing in flight (see Aeroplane).