Fluid mechanics
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{{Continuum mechanics}}
'''Fluid mechanics''' is the study of how [[fluid]]s move and the [[force]]s on them. (Fluids include [[liquid]]s and [[gas]]es.) 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. The study of fluid mechanics goes back at least to the days of [[ancient Greece]], when [[Archimedes]] made a beginning on fluid statics, which the medieval [[Islamic physics|Muslim physicists]], [[Abū Rayhān al-Bīrūnī]] and [[Al-Khazini]], later combined with [[dynamics]] to give rise to fluid dynamics.<ref name=Rozhanskaya-642>Mariam Rozhanskaya and I. S. Levinova (1996), "Statics", p. 642, in {{Harvard reference |last1=Rashed |first1=Roshdi |last2=Morelon |first2=Régis |year=1996 |title=[[Encyclopedia of the History of Arabic Science]] |volume=1 & 3 |publisher=[[Routledge]] |isbn=0415124107 |pages=614-642}}: {{quote|"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]]."}}</ref> However, 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 that branch of physics which deals with the properties of fluid,namely liquid and gases,and their interaction with forces.
==Relationship to continuum mechanics==
Fluid mechanics is a subdiscipline of [[continuum mechanics]], as illustrated in the following table.
{| class="wikitable" border=1
|rowspan=4|[[Continuum mechanics]] the study of the physics of continuous materials
|rowspan=2|[[Solid mechanics]]: the study of the physics of continuous materials with a defined rest shape.
|colspan=2|[[Elasticity (physics)|Elasticity]]: which describes materials that return to their rest shape after an applied [[stress (physics)|stress]].
|-
|[[Plasticity (physics)|Plasticity]]: which describes materials that permanently deform after a large enough applied stress.
|rowspan=2 bgcolor=white|[[Rheology]]: the study of materials with both solid and fluid characteristics
|-
|rowspan=2 bgcolor=white|'''Fluid mechanics''': the study of the physics of continuous materials which take the shape of their container.
|bgcolor=white|[[Non-Newtonian fluid]]s
|-
|colspan=2 bgcolor=white|[[Newtonian fluid]]s
|}
In a mechanical view, a fluid is a substance that does not support [[tangential 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 fluid|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===
Fluids are composed of [[molecules]] that collide with one another and solid objects. The continuum assumption, however, considers fluids to be [[continuous function|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 (ratio)|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 [[1 (number)|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:
:<math>\rho\frac{D\mathbf{v}}{D t} = \nabla\cdot\mathbb{P} + \rho\mathbf{f}</math>
where
*<math>\rho\ </math> is the fluid density,
*<math>\frac{D}{D t}</math> is the [[substantive derivative]] (also called the material derivative),
*<math>\mathbf{v}</math> is the velocity vector,
*<math>\mathbf{f}</math> is the body force vector, and
*<math>\mathbb{P}</math> 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, <math>\mathbb{P}</math> is a symmetric tensor. In general, (in three dimensions) <math>\mathbb{P}</math> has the form:
:<math>\mathbb{P} = \begin{pmatrix}
\sigma_{xx} & \tau_{xy} & \tau_{xz} \\
\tau_{yx} & \sigma_{yy} & \tau_{yz} \\
\tau_{zx} & \tau_{zy} & \sigma_{zz}
\end{pmatrix}
</math>
where
*<math>\sigma\ </math> are normal stresses, and
*<math>\tau\ </math> 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 (physics)|drag]] of a small object being moved through the fluid is proportional to the force applied to the object. (Compare [[friction]]).
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 [[paint]]s). There are many types of non-Newtonian fluids, as they are defined to be something that fails to obey a particular property.
===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
:<math>\tau=-\mu\frac{dv}{dx}</math>
where
:<math>\tau</math> is the shear stress exerted by the fluid ("[[Drag (physics)|drag]]")
:<math>\mu</math> is the fluid viscosity - a constant of proportionality
:<math>\frac{dv}{dx}</math> 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 fluid|incompressible]] and viscosity is constant across the fluid, the equation governing the shear stress (in [[Cartesian coordinate system|Cartesian coordinates]]) is
:<math>\tau_{ij}=\mu\left(\frac{\partial v_i}{\partial x_j}+\frac{\partial v_j}{\partial x_i} \right)</math>
where
:<math>\tau_{ij}</math> is the shear stress on the <math>i^{th}</math> face of a fluid element in the <math>j^{th}</math> direction
:<math>v_i</math> is the velocity in the <math>i^{th}</math> direction
:<math>x_j</math> is the <math>j^{th}</math> direction coordinate
If a fluid does not obey this relation, it is termed a [[non-Newtonian fluid]], of which there are several types.
==See also==
{{portal|Physics}}
{{Wikibooks}}
*[[Applied mechanics]]
*[[Secondary flow]]
*[[Bernoulli's principle]]
== Notes ==
{{reflist}}
== References ==
* White, Frank M. (2003). ''Fluid Mechanics.'' McGraw-Hill. ISBN 0072402172
* Cramer, Mark. [http://www.galleryoffluidmechanics.com/ "The Gallery of Fluid Mechanics"]
* Massey, B. & Ward-Smith, J. (2005). [http://www.amazon.co.uk/Mechanics-Fluids-Bernard-Massey/dp/0415362067/ref=sr_1_5?ie=UTF8&s=books&qid=1213702113&sr=1-5 ''Mechanics of Fluids - 8th ed.''] Taylor & Francis, ISBN 978-0-415-36206-1.
==External links==
* [http://www.cfd-online.com/Wiki/Main_Page CFDWiki] -- the Computational Fluid Dynamics reference wiki.
*[http://www.interactiveflows.com/links/ Educational Particle Image Velocimetry - resources and demonstrations]
{{Physics-footer}}
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