fluid mechanics

The branch of mechanics that is concerned with the properties of gases and liquids.

Concept

The term "fluid" in everyday language typically refers only to liquids, but in the realm of physics, fluid describes any gas or liquid that conforms to the shape of its container. Fluid mechanics is the study of gases and liquids at rest and in motion. This area of physics is divided into fluid statics, the study of the behavior of stationary fluids, and fluid dynamics, the study of the behavior of moving, or flowing, fluids. Fluid dynamics is further divided into hydrodynamics, or the study of water flow, and aerodynamics, the study of airflow. Applications of fluid mechanics include a variety of machines, ranging from the water-wheel to the airplane. In addition, the study of fluids provides an understanding of a number of everyday phenomena, such as why an open window and door together create a draft in a room.

How It Works

The Contrast Between Fluids and Solids

To understand fluids, it is best to begin by contrasting their behavior with that of solids. Whereas solids possess a definite volume and a definite shape, these physical characteristics are not so clearly defined for fluids. Liquids, though they possess a definite volume, have no definite shape—a factor noted above as one of the defining characteristics of fluids. As for gases, they have neither a definite shape nor a definite volume.

One of several factors that distinguishes fluids from solids is their response to compression, or the application of pressure in such a way as to reduce the size or volume of an object. A solid is highly noncompressible, meaning that it resists compression, and if compressed with a sufficient force, its mechanical properties alter significantly. For example, if one places a drinking glass in a vise, it will resist a small amount of pressure, but a slight increase will cause the glass to break.

Fluids vary with regard to compressibility, depending on whether the fluid in question is a liquid or a gas. Most gases tend to be highly compressible—though air, at low speeds at least, is not among them. Thus, gases such as propane fuel can be placed under high pressure. Liquids tend to be noncompressible: unlike a gas, a liquid can be compressed significantly, yet its response to compression is quite different from that of a solid—a fact illustrated below in the discussion of hydraulic presses.

One way to describe a fluid is "anything that flows"—a behavior explained in large part by the interaction of molecules in fluids. If the surface of a solid is disturbed, it will resist, and if the force of the disturbance is sufficiently strong, it will deform—as for instance, when a steel plate begins to bend under pressure. This deformation will be permanent if the force is powerful enough, as was the case in the above example of the glass in a vise. By contrast, when the surface of a liquid is disturbed, it tends to flow.

Molecular Behavior of Fluids and Solids

At the molecular level, particles of solids tend to be definite in their arrangement and close to one another. In the case of liquids, molecules are close in proximity, though not as much so as solid molecules, and the arrangement is random. Thus, with a glass of water, the molecules of glass (which at relatively low temperatures is a solid) in the container are fixed in place while the molecules of water contained by the glass are not. If one portion of the glass were moved to another place on the glass, this would change its structure. On the other hand, no significant alteration occurs in the character of the water if one portion of it is moved to another place within the entire volume of water in the glass.

As for gas molecules, these are both random in arrangement and far removed in proximity. Whereas solid particles are slow-moving and have a strong attraction to one another, liquid molecules move at moderate speeds and exert a moderate attraction on each other. Gas molecules are extremely fast-moving and exert little or no attraction.

Thus, if a solid is released from a container pointed downward, so that the force of gravity moves it, it will fall as one piece. Upon hitting a floor or other surface, it will either rebound, come to a stop, or deform permanently. A liquid, on the other hand, will disperse in response to impact, its force determining the area over which the total volume of liquid is distributed. But for a gas, assuming it is lighter than air, the downward pull of gravity is not even required to disperse it: once the top on a container of gas is released, the molecules begin to float outward.

Fluids Under Pressure

As suggested earlier, the response of fluids to pressure is one of the most significant aspects of fluid behavior and plays an important role within both the statics and dynamics subdisciplines of fluid mechanics. A number of interesting principles describe the response to pressure, on the part of both fluids at rest inside a container, and fluids which are in a state of flow.

Within the realm of hydrostatics, among the most important of all statements describing the behavior of fluids is Pascal's principle. This law is named after Blaise Pascal (1623-1662), a French mathematician and physicist who discovered that the external pressure applied on a fluid is transmitted uniformly throughout its entire body. The understanding offered by Pascal's principle later became the basis for one of the most important machines ever developed, the hydraulic press.

Hydrostatic Pressure and Buoyancy

Some nineteen centuries before Pascal, the Greek mathematician, physicist, and inventor Archimedes (c. 287-212 B.C.) discovered a precept of fluid statics that had implications at least as great as those of Pascal's principle. This was Archimedes's principle, which explains the buoyancy of an object immersed in fluid. According to Archimedes's principle, the buoyant force exerted on the object is equal to the weight of the fluid it displaces.

Buoyancy explains both how a ship floats on water, and how a balloon floats in the air. The pressures of water at the bottom of the ocean, and of air at the surface of Earth, are both examples of hydrostatic pressure—the pressure that exists at any place in a body of fluid due to the weight of the fluid above. In the case of air pressure, air is pulled downward by the force of Earth's gravitation, and air along the planet's surface has greater pressure due to the weight of the air above it. At great heights above Earth's surface, however, the gravitational force is diminished, and thus the air pressure is much smaller.

Water, too, is pulled downward by gravity, and as with air, the fluid at the bottom of the ocean has much greater pressure due to the weight of the fluid above it. Of course, water is much heavier than air, and therefore, water at even a moderate depth in the ocean has enormous pressure. This pressure, in turn, creates a buoyant force that pushes upward.

If an object immersed in fluid—a balloon in the air, or a ship on the ocean—weighs less that the fluid it displaces, it will float. If it weighs more, it will sink or fall. The balloon itself may be "heavier than air," but it is not as heavy as the air it has displaced. Similarly, an aircraft carrier contains a vast weight in steel and other material, yet it floats, because its weight is not as great as that of the displaced water.

Bernoulli's Principle

Archimedes and Pascal contributed greatly to what became known as fluid statics, but the father of fluid mechanics, as a larger realm of study, was the Swiss mathematician and physicist Daniel Bernoulli (1700-1782). While conducting experiments with liquids, Bernoulli observed that when the diameter of a pipe is reduced, the water flows faster. This suggested to him that some force must be acting upon the water, a force that he reasoned must arise from differences in pressure.

Specifically, the slower-moving fluid in the wider area of pipe had a greater pressure than the portion of the fluid moving through the narrower part of the pipe. As a result, he concluded that pressure and velocity are inversely related—in other words, as one increases, the other decreases. Hence, he formulated Bernoulli's principle, which states that for all changes in movement, the sum of static and dynamic pressure in a fluid remains the same.

A fluid at rest exerts pressure—what Bernoulli called "static pressure"—on its container. As the fluid begins to move, however, a portion of the static pressure—proportional to the speed of the fluid—is converted to what Bernoulli called dynamic pressure, or the pressure of movement. In a cylindrical pipe, static pressure is exerted perpendicular to the surface of the container, whereas dynamic pressure is parallel to it.

According to Bernoulli's principle, the greater the velocity of flow in a fluid, the greater the dynamic pressure and the less the static pressure. In other words, slower-moving fluid exerts greater pressure than faster-moving fluid. The discovery of this principle ultimately made possible the development of the airplane.

Real-Life Applications

Bernoulli's Principle in Action

As fluid moves from a wider pipe to a narrower one, the volume of the fluid that moves a given distance in a given time period does not change. But since the width of the narrower pipe is smaller, the fluid must move faster (that is, with greater dynamic pressure) in order to move the same amount of fluid the same distance in the same amount of time. Observe the behavior of a river: in a wide, unconstricted region, it flows slowly, but if its flow is narrowed by canyon walls, it speeds up dramatically.

Bernoulli's principle ultimately became the basis for the airfoil, the design of an airplane's wing when seen from the end. An airfoil is shaped like an asymmetrical teardrop laid on its side, with the "fat" end toward the airflow. As air hits the front of the airfoil, the airstream divides, part of it passing over the wing and part passing under. The upper surface of the airfoil is curved, however, whereas the lower surface is much straighter.

As a result, the air flowing over the top has a greater distance to cover than the air flowing under the wing. Since fluids have a tendency to compensate for all objects with which they come into contact, the air at the top will flow faster to meet the other portion of the airstream, the air flowing past the bottom of the wing, when both reach the rear end of the airfoil. Faster airflow, as demonstrated by Bernoulli, indicates lower pressure, meaning that the pressure on the bottom of the wing keeps the airplane aloft.

Creating a Draft

Among the most famous applications of Bernoulli's principle is its use in aerodynamics, and this is discussed in the context of aerodynamics itself elsewhere in this book. Likewise, a number of other applications of Bernoulli's principle are examined in an essay devoted to that topic. Bernoulli's principle, for instance, explains why a shower curtain tends to billow inward when the water is turned on; in addition, it shows why an open window and door together create a draft.

Suppose one is in a hotel room where the heat is on too high, and there is no way to adjust the thermostat. Outside, however, the air is cold, and thus, by opening a window, one can presumably cool down the room. But if one opens the window without opening the front door of the room, there will be little temperature change. The only way to cool off will be by standing next to the window: elsewhere in the room, the air will be every bit as stuffy as before. But if the door leading to the hotel hallway is opened, a nice cool breeze will blow through the room. Why?

With the door closed, the room constitutes an area of relatively high pressure compared to the pressure of the air outside the window. Because air is a fluid, it will tend to flow into the room, but once the pressure inside reaches a certain point, it will prevent additional air from entering. The tendency of fluids is to move from high-pressure to low-pressure areas, not the other way around. As soon as the door is opened, the relatively high-pressure air of the room flows into the relatively low-pressure area of the hallway. As a result, the air pressure in the room is reduced, and the air from outside can now enter. Soon a wind will begin to blow through the room.

A Wind Tunnel

The above scenario of wind flowing through a room describes a rudimentary wind tunnel. A wind tunnel is a chamber built for the purpose of examining the characteristics of airflow in contact with solid objects, such as aircraft and automobiles. The wind tunnel represents a safe and judicious use of the properties of fluid mechanics. Its purpose is to test the interaction of airflow and solids in relative motion: in other words, either the aircraft has to be moving against the airflow, as it does in flight, or the airflow can be moving against a stationary aircraft. The first of these choices, of course, poses a number of dangers; on the other hand, there is little danger in exposing a stationary craft to winds at speeds simulating that of the aircraft in flight.

The first wind tunnel was built in England in 1871, and years later, aircraft pioneers Orville (1871-1948) and Wilbur (1867-1912) Wright used a wind tunnel to improve their planes. By the late 1930s, the U.S. National Advisory Committee for Aeronautics (NACA) was building wind tunnels capable of creating speeds equal to 300 MPH (480 km/h); but wind tunnels built after World War II made these look primitive. With the development of jet-powered flight, it became necessary to build wind tunnels capable of simulating winds at the speed of sound—760 MPH (340 m/s). By the 1950s, wind tunnels were being used to simulate hypersonic speeds—that is, speeds of Mach 5 (five times the speed of sound) and above. Researchers today use helium to create wind blasts at speeds up to Mach 50.

Fluid Mechanics for Performing Work

Hydraulic Presses

Though applications of Bernoulli's principle are among the most dramatic examples of fluid mechanics in operation, the everyday world is filled with instances of other ideas at work. Pascal's principle, for instance, can be seen in the operation of any number of machines that represent variations on the idea of a hydraulic press. Among these is the hydraulic jack used to raise a car off the floor of an auto mechanic's shop.

Beneath the floor of the shop is a chamber containing a quantity of fluid, and at either end of the chamber are two large cylinders side by side. Each cylinder holds a piston, and valves control flow between the two cylinders through the channel of fluid that connects them. In accordance with Pascal's principle, when one applies force by pressing down the piston in one cylinder (the input cylinder), this yields a uniform pressure that causes output in the second cylinder, pushing up a piston that raises the car.

Another example of a hydraulic press is the hydraulic ram, which can be found in machines ranging from bulldozers to the hydraulic lifts used by firefighters and utility workers to reach heights. In a hydraulic ram, however, the characteristics of the input and output cylinders are reversed from those of a car jack. For the car jack, the input cylinder is long and narrow, while the output cylinder is wide and short. This is because the purpose of a car jack is to raise a heavy object through a relatively short vertical range of movement—just high enough so that the mechanic can stand comfortably underneath the car.

In the hydraulic ram, the input or master cylinder is short and squat, while the output or slave cylinder is tall and narrow. This is because the hydraulic ram, in contrast to the car jack, carries a much lighter cargo (usually just one person) through a much greater vertical range—for instance, to the top of a tree or building.

Pumps

A pump is a device made for moving fluid, and it does so by utilizing a pressure difference, causing the fluid to move from an area of higher pressure to one of lower pressure. Its operation is based on aspects both of Pascal's and Bernoulli's principles—though, of course, humans were using pumps thousands of years before either man was born.

A siphon hose used to draw gas from a car's fuel tank is a very simple pump. Sucking on one end of the hose creates an area of low pressure compared to the relatively high-pressure area of the gas tank. Eventually, the gasoline will come out of the low-pressure end of the hose.

The piston pump, slightly more complex, consists of a vertical cylinder along which a piston rises and falls. Near the bottom of the cylinder are two valves, an inlet valve through which fluid flows into the cylinder, and an outlet valve through which fluid flows out. As the piston moves upward, the inlet valve opens and allows fluid to enter the cylinder. On the downstroke, the inlet valve closes while the outlet valve opens, and the pressure provided by the piston forces the fluid through the outlet valve.

One of the most obvious applications of the piston pump is in the engine of an automobile. In this case, of course, the fluid being pumped is gasoline, which pushes the pistons up and down by providing a series of controlled explosions created by the spark plug's ignition of the gas. In another variety of piston pump—the kind used to inflate a basketball or a bicycle tire—air is the fluid being pumped. Then there is a pump for water. Pumps for drawing usable water from the ground are undoubtedly the oldest known variety, but there are also pumps designed to remove water from areas where it is undesirable; for example, a bilge pump, for removing water from a boat, or the sump pump used to pump flood water out of a basement.

Fluid Power

For several thousand years, humans have used fluids—in particular water—to power a number of devices. One of the great engineering achievements of ancient times was the development of the waterwheel, which included a series of buckets along the rim that made it possible to raise water from the river below and disperse it to other points. By about 70

B.C., Roman engineers recognized that they could use the power of water itself to turn wheels and grind grain. Thus, the waterwheel became one of the first mechanisms in which an inanimate source (as opposed to the effort of humans or animals) created power.

The water clock, too, was another ingenious use of water developed by the ancients. It did not use water for power; rather, it relied on gravity—a concept only dimly understood by ancient peoples—to move water from one chamber of theclock to another, thus, marking a specific interval of time. The earliest clocks were sundials, which were effective for measuring time, provided the Sun was shining, but which were less useful form easuring periods shorter than an hour. Hence, the development of the hourglass, which used sand, a solid that in larger quantities exhibits the behavior of a fluid. Then, in about 270 B.C., Ctesibius of Alexandria (fl. c. 270-250 B.C.) used gearwheel technology to devise a constant-flow water clock called a "clepsydra." Use of water clocks prevailed for more than a thousand years, until the advent of the first mechanical clocks.

During the medieval period, fluids provided power to windmills and water mills, and at the dawn of the Industrial Age, engineers began applying fluid principles to a number of sophisticated machines. Among these was the turbine, a machine that converts the kinetic energy (the energy of movement) in fluids to useable mechanical energy by passing the stream of fluid through a series of fixed and moving fans or blades. A common house fan is an example of a turbine in reverse: the fan adds energy to the passing fluid (air), whereas a turbine extracts energy from fluids such as air and water.

The turbine was developed in the mid-eighteenth century, and later it was applied to the extraction of power from hydroelectric dams, the first of which was constructed in 1894. Today, hydroelectric dams provide electric power to millions of homes around the world. Among the most dramatic examples of fluid mechanics in action, hydroelectric dams are vast in size and equally impressive in the power they can generate using a completely renewable resource: water.

A hydroelectric dam forms a huge steel-and-concrete curtain that holds back millions of tons of water from a river or other body. The water nearest the top—the "head" of the dam—has enormous potential energy, or the energy that an object possesses by virtue of its position. Hydroelectric power is created by allowing controlled streams of this water to flow downward, gathering kinetic energy that is then transferred to powering turbines, which in turn generate electric power.

Where to Learn More

Aerodynamics for Students (Web site). <http://www.ae.su.oz.au/aero/contents.html> (April 8, 2001).

Beiser, Arthur. Physics, 5th ed. Reading, MA: Addison-Wesley, 1991.

Chahrour, Janet. Flash! Bang! Pop! Fizz!: Exciting Science for Curious Minds. Illustrated by Ann Humphrey Williams. Hauppauge, N.Y.: Barron's, 2000.

"Educational Fluid Mechanics Sites." Virginia Institute of Technology (Web site). <http://www.eng.vt.edu/fluids/links/edulinks.htm> (April 8, 2001).

Fleisher, Paul. Liquids and Gases: Principles of Fluid Mechanics. Minneapolis, MN: Lerner Publications, 2002.

Institute of Fluid Mechanics (Web site). <http://www.ts.go.dlr.de> (April 8, 2001).

K8AIT Principles of Aeronautics Advanced Text (web site). <http://wings.ucdavis.edu/Book/advanced.html> (February 19, 2001).

Macaulay, David. The New Way Things Work. Boston: Houghton Mifflin, 1998.

Sobey, Edwin J. C. Wacky Water Fun with Science: Science You Can Float, Sink, Squirt, and Sail. Illustrated by Bill Burg. New York: McGraw-Hill, 2000.

Wood, Robert W. Mechanics Fundamentals. Illustrated by Bill Wright. Philadelphia: Chelsea House, 1997.

The engineering science concerned with the relation between the forces acting on fluids (liquids and gases) and their motions, and with the forces caused by fluids on their surroundings. It is distinct from solid mechanics by virtue of the different responses of fluids and solids to applied forces. In an ideal elastic solid, the deflection or deformation is proportional to the applied stress, whereas a fluid cannot support an applied shear stress unless it is in motion. In most fluids, called simple or newtonian fluids, it is the rate of deformation of the fluid, as opposed to the amount of deformation in a solid, that is proportional to the applied stress. See also Fluid flow; Newtonian fluid.

Many substances of everyday experience and of engineering importance are found naturally in the fluid state. These include water (liquid and vapor), air (gaseous and liquid), as well as other liquids and gases of natural and industrial importance. The most common fluids are newtonian under most flow conditions.

Fluid mechanics treats the fluid as a continuum, ignoring the fact that it actually consists of individual molecules that may be, in the case of gases, widely spaced compared to molecular dimensions. Nevertheless, the continuum assumption is valid for almost all applications down to the size of bacteria. An exception occurs with gases at very low densities, such as exist in the uppermost regions of the atmosphere. At extremely high altitudes the mean free paths of air molecules—that is, the distances they travel between collisions in random thermal motion—can become as large, or even larger than, the dimensions of a space vehicle, making the assumption of a continuum invalid. See also Rarefied gas flow.

Fluid mechanics is of fundamental importance to a number of disciplines, including aerospace, chemical, civil, environmental, mechanical, and ocean engineering, as well as to climatology, geology, meteorology, and oceanography. Applications in these fields include, but are not limited to, the study of fluid forces acting on vehicles; flows in natural rivers and artificial channels and the flow of ground water; the dispersion of pollutants in the atmosphere, lakes, rivers, and oceans; the flows in the circulatory and pulmonary systems of humans and animals; the flows in pipelines that carry crude oil and natural gas over many hundreds, or even thousands, of miles from the petroleum fields of their origin to deep-water ports or refineries; the flow of molten plastics or metals filling molds in the manufacture of numerous solid parts; the flow in pumps for water distribution systems; and both hydraulic and gas turbines for power generation and propulsion. Fluid mechanics forms the basis for much of chemistry and physics, and is sometimes applied to such apparently remote fields as cosmology. The fluid mechanical behavior of gases and liquids plays an important role in the dispersion of dissolved or entrained substances. See also Aerodynamic force; Aerodynamics; Biomedical engineering; Hydraulics; Hydrodynamics.

For more information on fluid mechanics, visit Britannica.com.

A branch of mechanics concerned with the forces that fluids exert on objects that are in or moving through the fluids. The fluids most relevant to exercise and sport biomechanics are air and water.

Continuum mechanics

Navier–Stokes equations Laws Conservation of mass Conservation of momentum Conservation of energy Entropy Inequality Solid mechanics Solids · Stress · Deformation · Finite strain theory · Infinitesimal strain theory · Elasticity · Linear elasticity · Plasticity · Viscoelasticity · Hooke's law · Rheology Fluid mechanics Fluids · Fluid statics Fluid dynamics · Viscosity · Newtonian fluids Non-Newtonian fluids Surface tension Scientists Newton · Stokes · Navier · Cauchy· Hooke · Bernoulli

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Fluid mechanics is the study of how fluids move and the forces on them. (Fluids include liquids and gases.) Fluid mechanics can be divided into fluid statics, the study of fluids at rest, and fluid dynamics, the study of fluids in motion. It is a branch of continuum mechanics, a subject which models matter without using the information that it is made out of atoms. Fluid mechanics, especially fluid dynamics, is an active field of research with many unsolved or partly solved problems. Fluid mechanics can be mathematically complex. Sometimes it can best be solved by numerical methods, typically using computers. A modern discipline, called Computational Fluid Dynamics (CFD), is devoted to this approach to solving fluid mechanics problems. Also taking advantage of the highly visual nature of fluid flow is Particle Image Velocimetry, an experimental method for visualizing and analyzing fluid flow. Fluid mechanics is the branch of physics which deals with the properties of fluids, namely liquids and gases, and their interaction with forces.

Contents 1 Brief history 2 Relationship to continuum mechanics 3 Assumptions 3.1 The continuum hypothesis 4 Navier-Stokes equations 4.1 General form of the equation 5 Newtonian vs. non-Newtonian fluids 5.1 Equations for a Newtonian fluid 6 See also 7 Notes 8 References 9 External links

Brief history

Main article: History of fluid mechanics

The study of fluid mechanics goes back at least to the days of ancient Greece, when Archimedes investigated fluid statics and buoyancy. Medieval Arab natural philosophers, including Abū Rayhān al-Bīrūnī and Al-Khazini, combined that earlier work with dynamics^[1] to presage the later development of fluid dynamics. Rapid advancement in fluid mechanics began with Leonardo da Vinci (observation and experiment), Evangelista Torricelli (barometer), Isaac Newton (viscosity) and Blaise Pascal (hydrostatics), and was continued by Daniel Bernoulli with the introduction of mathematical fluid dynamics in Hydrodynamica (1738). Inviscid flow was further analyzed by various mathematicians (Leonhard Euler, d'Alembert, Lagrange, Laplace, Poisson) and viscous flow was explored by a multitude of engineers including Poiseuille and Gotthilf Heinrich Ludwig Hagen. Further mathematical justification was provided by Claude-Louis Navier and George Gabriel Stokes in the Navier-Stokes Equations, and boundary layers were investigated (Ludwig Prandtl), while various scientists (Osborne Reynolds, Andrey Kolmogorov, Geoffrey Ingram Taylor) advanced the understanding of fluid viscosity and turbulence.

Relationship to continuum mechanics

Fluid mechanics is a subdiscipline of continuum mechanics, as illustrated in the following table.

Continuum mechanics the study of the physics of continuous materials	Solid mechanics: the study of the physics of continuous materials with a defined rest shape.	Elasticity: which describes materials that return to their rest shape after an applied stress.
		Plasticity: which describes materials that permanently deform after a large enough applied stress.	Rheology: the study of materials with both solid and fluid characteristics
	Fluid mechanics: the study of the physics of continuous materials which take the shape of their container.	Non-Newtonian fluids
		Newtonian fluids

In a mechanical view, a fluid is a substance that does not support shear stress; that is why a fluid at rest has the shape of its containing vessel. A fluid at rest has no shear stress.

Assumptions

Like any mathematical model of the real world, fluid mechanics makes some basic assumptions about the materials being studied. These assumptions are turned into equations that must be satisfied if the assumptions are to hold true. For example, consider an incompressible fluid in three dimensions. The assumption that mass is conserved means that for any fixed closed surface (such as a sphere) the rate of mass passing from outside to inside the surface must be the same as rate of mass passing the other way. (Alternatively, the mass inside remains constant, as does the mass outside). This can be turned into an integral equation over the surface.

Fluid mechanics assumes that every fluid obeys the following:

• Conservation of mass
• Conservation of momentum
• The continuum hypothesis, detailed below.

Further, it is often useful (and realistic) to assume a fluid is incompressible - that is, the density of the fluid does not change. Liquids can often be modelled as incompressible fluids, whereas gases cannot.

Similarly, it can sometimes be assumed that the viscosity of the fluid is zero (the fluid is inviscid). Gases can often be assumed to be inviscid. If a fluid is viscous, and its flow contained in some way (e.g. in a pipe), then the flow at the boundary must have zero velocity. For a viscous fluid, if the boundary is not porous, the shear forces between the fluid and the boundary results also in a zero velocity for the fluid at the boundary. This is called the no-slip condition. For a porous media otherwise, in the frontier of the containing vessel, the slip condition is not zero velocity, and the fluid has a discontinuous velocity field between the free fluid and the fluid in the porous media (this is related to the Beavers and Joseph condition).

The continuum hypothesis

Main article: Continuum mechanics

Fluids are composed of molecules that collide with one another and solid objects. The continuum assumption, however, considers fluids to be continuous. That is, properties such as density, pressure, temperature, and velocity are taken to be well-defined at "infinitely" small points, defining a REV (Reference Element of Volume), at the geometric order of the distance between two adjacent molecules of fluid. Properties are assumed to vary continuously from one point to another, and are averaged values in the REV. The fact that the fluid is made up of discrete molecules is ignored.

The continuum hypothesis is basically an approximation, in the same way planets are approximated by point particles when dealing with celestial mechanics, and therefore results in approximate solutions. Consequently, assumption of the continuum hypothesis can lead to results which are not of desired accuracy. That said, under the right circumstances, the continuum hypothesis produces extremely accurate results.

Those problems for which the continuum hypothesis does not allow solutions of desired accuracy are solved using statistical mechanics. To determine whether or not to use conventional fluid dynamics or statistical mechanics, the Knudsen number is evaluated for the problem. The Knudsen number is defined as the ratio of the molecular mean free path length to a certain representative physical length scale. This length scale could be, for example, the radius of a body in a fluid. (More simply, the Knudsen number is how many times its own diameter a particle will travel on average before hitting another particle). Problems with Knudsen numbers at or above unity are best evaluated using statistical mechanics for reliable solutions.

Navier-Stokes equations

Main article: Navier-Stokes equations

The Navier-Stokes equations (named after Claude-Louis Navier and George Gabriel Stokes) are the set of equations that describe the motion of fluid substances such as liquids and gases. These equations state that changes in momentum (force) of fluid particles depend only on the external pressure and internal viscous forces (similar to friction) acting on the fluid. Thus, the Navier-Stokes equations describe the balance of forces acting at any given region of the fluid.

The Navier-Stokes equations are differential equations which describe the motion of a fluid. Such equations establish relations among the rates of change the variables of interest. For example, the Navier-Stokes equations for an ideal fluid with zero viscosity states that acceleration (the rate of change of velocity) is proportional to the derivative of internal pressure.

This means that solutions of the Navier-Stokes equations for a given physical problem must be sought with the help of calculus. In practical terms only the simplest cases can be solved exactly in this way. These cases generally involve non-turbulent, steady flow (flow does not change with time) in which the Reynolds number is small.

For more complex situations, such as global weather systems like El Niño or lift in a wing, solutions of the Navier-Stokes equations can currently only be found with the help of computers. This is a field of sciences by its own called computational fluid dynamics.

General form of the equation

The general form of the Navier-Stokes equations for the conservation of momentum is:

where

• is the fluid density,
• is the substantive derivative (also called the material derivative),
• is the velocity vector,
• is the body force vector, and
• is a tensor that represents the surface forces applied on a fluid particle (the comoving stress tensor).

Unless the fluid is made up of spinning degrees of freedom like vortices, is a symmetric tensor. In general, (in three dimensions) has the form:

where

• are normal stresses, and
• are tangential stresses (shear stresses).

The above is actually a set of three equations, one per dimension. By themselves, these aren't sufficient to produce a solution. However, adding conservation of mass and appropriate boundary conditions to the system of equations produces a solvable set of equations.

Newtonian vs. non-Newtonian fluids

A Newtonian fluid (named after Isaac Newton) is defined to be a fluid whose shear stress is linearly proportional to the velocity gradient in the direction perpendicular to the plane of shear. This definition means regardless of the forces acting on a fluid, it continues to flow. For example, water is a Newtonian fluid, because it continues to display fluid properties no matter how much it is stirred or mixed. A slightly less rigorous definition is that the drag of a small object being moved slowly through the fluid is proportional to the force applied to the object. (Compare friction). Important fluids, like water as well as most gases, behave — to good approximation — as a Newtonian fluid under normal conditions on Earth.^[2]

By contrast, stirring a non-Newtonian fluid can leave a "hole" behind. This will gradually fill up over time - this behaviour is seen in materials such as pudding, oobleck, or sand (although sand isn't strictly a fluid). Alternatively, stirring a non-Newtonian fluid can cause the viscosity to decrease, so the fluid appears "thinner" (this is seen in non-drip paints). There are many types of non-Newtonian fluids, as they are defined to be something that fails to obey a particular property — for example, most fluids with long molecular chains can react in a non-Newtonian manner.^[2]

Equations for a Newtonian fluid

Main article: Newtonian fluid

The constant of proportionality between the shear stress and the velocity gradient is known as the viscosity. A simple equation to describe Newtonian fluid behaviour is

where

τ is the shear stress exerted by the fluid ("drag")

μ is the fluid viscosity - a constant of proportionality

is the velocity gradient perpendicular to the direction of shear

For a Newtonian fluid, the viscosity, by definition, depends only on temperature and pressure, not on the forces acting upon it. If the fluid is incompressible and viscosity is constant across the fluid, the equation governing the shear stress (in Cartesian coordinates) is

where

τ_ij is the shear stress on the i^th face of a fluid element in the j^th direction

v_i is the velocity in the i^th direction

x_j is the j^th direction coordinate

If a fluid does not obey this relation, it is termed a non-Newtonian fluid, of which there are several types.

Among fluids, two rough broad dividions can be made- ideal and non- ideal fluids. An ideal fluid really does not exist, but in some calculations, the assumption is justifiable. An Ideal fluid is non viscous- offers no resistance whatsoever to a shearing force.

One can group real fluids into Newtonian and non- Newtonian. Newtonian fluids agree with Newton's law of viscosity. Non- Newtonian Fluids could either be plastic, bingham plastic, pseudoplastic, dilatant, thexotropic, rheopectic, viscoelatic

See also

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Wikibooks has more on the topic of Fluid mechanics

• Applied mechanics
• Secondary flow
• Bernoulli's principle

Notes

1. ^ Mariam Rozhanskaya and I. S. Levinova (1996), "Statics", p. 642, in Rashed, Roshdi & Régis Morelon (1996), Encyclopedia of the History of Arabic Science, vol. 1 & 3, Routledge, 614-642, ISBN 0415124107:

   "Using a whole body of mathematical methods (not only those inherited from the antique theory of ratios and infinitesimal techniques, but also the methods of the contemporary algebra and fine calculation techniques), Arabic scientists raised statics to a new, higher level. The classical results of Archimedes in the theory of the centre of gravity were generalized and applied to three-dimensional bodies, the theory of ponderable lever was founded and the 'science of gravity' was created and later further developed in medieval Europe. The phenomena of statics were studied by using the dynamic apporach so that two trends - statics and dynamics - turned out to be inter-related withina single science, mechanics. The combination of the dynamic apporach with Archimedean hydrostatics gave birth to a direction in science which may be called medieval hydrodynamics. [...] Numerous fine experimental methods were developed for determining the specific weight, which were based, in particular, on the theory of balances and weighing. The classical works of al-Biruni and al-Khazini can by right be considered as the beginning of the application of experimental methods in medieval science."

2. ^ ^{a b} Batchelor (1967), p. 145.

References

• Batchelor, George K. (1967), An Introduction to Fluid Dynamics, Cambridge University Press, ISBN 0521663962 
• Kundu, Pijush K.; Cohen, Ira M. (2008), Fluid Mechanics (4th revised ed.), Academic Press, ISBN 978-0-123-73735-9 
• Massey, B.; Ward-Smith, J. (2005), Mechanics of Fluids (8th ed.), Taylor & Francis, ISBN 978-0-415-36206-1 
• White, Frank M. (2003), Fluid Mechanics, McGraw-Hill, ISBN 0072402172 

External links

Wikimedia Commons has media related to: Fluid mechanics

• Free Fluid Mechanics books
• Annual Review of Fluid Mechanics
• CFDWiki -- the Computational Fluid Dynamics reference wiki.
• Educational Particle Image Velocimetry - resources and demonstrations
• prof. DSc Ivan Antonov, Technical University of Sofia, Bulgaria.

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