Fluid mechanics is the branch of physicswhich involves the study of fluids (liquids,gases, and plasmas) and the forces on them. Fluid mechanics can be divided into fluid statics, the study of fluids at rest; and fluid dynamics, the study of the effect of forces on fluid motion. It is a branch of continuum mechanics, a subject which models matter without using the information that it is made out of atoms; that is, it models matter from amacroscopic viewpoint rather than frommicroscopic. Fluid mechanics, especially fluid dynamics, is an active field of research with many problems that are partly or wholly unsolved. Fluid mechanics can be mathematically complex, and can best be solved by numerical methods, typically using computers. A modern discipline, calledcomputational fluid dynamics (CFD), is devoted to this approach to solving fluid mechanics problems. Particle image velocimetry, an experimental method for visualizing and analyzing fluid flow, also takes advantage of the highly visual nature of fluid flow.
Brief historyEdit
Main article: History of fluid mechanics
The study of fluid mechanics goes back at least to the days of ancient Greece, whenArchimedes investigated fluid statics andbuoyancy and formulated his famous law known now as the Archimedes' principle, which was published in his work On Floating Bodies – generally considered to be the first major work on fluid mechanics. Rapid advancement in fluid mechanics began withLeonardo da Vinci (observations and experiments), Evangelista Torricelli (invented the barometer), Isaac Newton (investigatedviscosity) and Blaise Pascal (researchedhydrostatics, formulated Pascal's law), 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, Jean le Rond d'Alembert, Joseph Louis Lagrange,Pierre-Simon Laplace, Siméon Denis Poisson) and viscous flow was explored by a multitude of engineers including Jean Léonard Marie Poiseuille and Gotthilf Hagen. Further mathematical justification was provided byClaude-Louis Navier and George Gabriel Stokes in the Navier–Stokes equations, andboundary layers were investigated (Ludwig Prandtl, Theodore von Kármán), while various scientists such as Osborne Reynolds, Andrey Kolmogorov, and Geoffrey Ingram Tayloradvanced the understanding of fluid viscosity and turbulence.
Main branchesEdit
Fluid staticsEdit
Main article: Fluid statics
Fluid statics or hydrostatics is the branch offluid mechanics that studies fluids at rest. It embraces the study of the conditions under which fluids are at rest in stable equilibrium; and is contrasted with fluid dynamics, the study of fluids in motion.
Hydrostatics is fundamental to hydraulics, theengineering of equipment for storing, transporting and using fluids. It is also relevant to geophysics and astrophysics (for example, in understanding plate tectonics and the anomalies of the Earth's gravitational field), to meteorology, to medicine (in the context of blood pressure), and many other fields.
Hydrostatics offers physical explanations for many phenomena of everyday life, such as why atmospheric pressure changes withaltitude, why wood and oil float on water, and why the surface of water is always flat and horizontal whatever the shape of its container.
Fluid dynamicsEdit
Main article: Fluid dynamics
Fluid dynamics is a subdiscipline of fluid mechanics that deals with fluid flow—thenatural science of fluids (liquids and gases) in motion. It has several subdisciplines itself, including aerodynamics (the study of air and other gases in motion) and hydrodynamics(the study of liquids in motion). Fluid dynamics has a wide range of applications, including calculating forces and moments onaircraft, determining the mass flow rate ofpetroleum through pipelines, predictingweather patterns, understanding nebulae ininterstellar space and modelling fission weapon detonation. Some of its principles are even used in traffic engineering, where traffic is treated as a continuous fluid, and crowd dynamics.
Fluid dynamics offers a systematic structure—which underlies these practical disciplines—that embraces empirical and semi-empirical laws derived from flow measurement and used to solve practical problems. The solution to a fluid dynamics problem typically involves calculating various properties of the fluid, such as velocity, pressure, density, andtemperature, as functions of space and time.
Relationship to continuum mechanicsEdit
Fluid mechanics is a subdiscipline ofcontinuum mechanics, as illustrated in the following table.
Continuum mechanics
The study of the physics of continuous materialsSolid mechanics
The study of the physics of continuous materials with a defined rest shape.Elasticity
Describes materials that return to their rest shape after applied stresses are removed.Plasticity
Describes materials that permanently deform after a sufficient applied stress.Rheology
The study of materials with both solid and fluid characteristics.Fluid mechanics
The study of the physics of continuous materials which deform when subjected to a force.Non-Newtonian fluids do not undergo strain rates proportional to the applied shear stress.Newtonian fluids undergo strain rates proportional to the applied shear stress.
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.
AssumptionsEdit
Balance for some integrated fluid quantity in a control volume enclosed by a control surface.
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 be held true.
For example, consider a fluid in three dimensions. The assumption that mass is conserved means that for any fixed control volume (for example a sphere) – enclosed by a control surface – the rate of change of the mass contained is equal to the rate at which mass is passing from outside to insidethrough the surface, minus the rate at which mass is passing the other way, from inside tooutside. (A special case would be when the mass inside and the mass outside remain constant). This can be turned into an equation in integral form over the control volume.[1]
Fluid mechanics assumes that every fluid obeys the following:
Conservation of massConservation of energyConservation of momentumThe continuum hypothesis, detailed below.
Further, it is often useful (at subsonicconditions) to assume a fluid isincompressible – that is, the density of the fluid does not change.
Similarly, it can sometimes be assumed that the viscosity of the fluid is zero (the fluid isinviscid). 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).
Continuum hypothesisEdit
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. However, 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 one are best evaluated using statistical mechanics for reliable solutions.
Navier–Stokes equationsEdit
Main article: Navier–Stokes equations
The Navier–Stokes equations (named afterClaude-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 pressureand 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 of 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 theReynolds number is small.
For more complex situations, involvingturbulence, such as global weather systems, aerodynamics, hydrodynamics and many more, solutions of the Navier–Stokes equations can currently only be found with the help of computers. This branch of science is called computational fluid dynamics.
General form of the equationEdit
The general form of the Cauchy momentum equation is:
where

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