Editing Fluid dynamics (section)

==Equations==
{{See also|Transport phenomena}}
The foundational axioms of fluid dynamics are the [[Conservation law (physics)|conservation law]]s, specifically, [[conservation of mass]], [[conservation of momentum|conservation of linear momentum]], and [[conservation of energy]] (also known as the ''[[first law of thermodynamics]]''). These are based on [[classical mechanics]] and are modified in [[quantum mechanics]] and [[general relativity]]. They are expressed using the [[Reynolds transport theorem]].

In addition to the above, fluids are assumed to obey the [[continuum assumption]]. At small scale, all fluids are composed of molecules that collide with one another and solid objects. However, the continuum assumption assumes that fluids are continuous, rather than discrete. Consequently, it is assumed that properties such as density, pressure, temperature, and flow velocity are well-defined at [[infinitesimal]]ly small points in space and vary continuously from one point to another. The fact that the fluid is made up of discrete molecules is ignored.

For fluids that are sufficiently dense to be a continuum, do not contain ionized species, and have flow velocities that are small in relation to the speed of light, the momentum equations for [[Newtonian fluid]]s are the [[Navier–Stokes equations]]—which is a [[non-linear]] set of [[differential equations]] that describes the flow of a fluid whose stress depends linearly on flow velocity gradients and pressure. The unsimplified equations do not have a general [[Solution in closed form|closed-form solution]], so they are primarily of use in [[computational fluid dynamics]]. The equations can be simplified in several ways, all of which make them easier to solve. Some of the simplifications allow some simple fluid dynamics problems to be solved in closed form.{{citation needed|date= May 2014}}

In addition to the mass, momentum, and energy conservation equations, a [[thermodynamics|thermodynamic]] equation of state that gives the pressure as a function of other thermodynamic variables is required to completely describe the problem. An example of this would be the [[Ideal gas law|perfect gas equation of state]]:

:<math>p= \frac{\rho R_u T}{M}</math>

where {{mvar|p}} is [[pressure]], {{mvar|ρ}} is [[density]], and {{mvar|T}} is the [[absolute temperature]], while {{mvar|R<sub>u</sub>}} is the [[gas constant]] and {{mvar|M}} is [[molar mass]] for a particular gas. A [[Constitutive equation|constitutive relation]] may also be useful.

===Conservation laws===
Three conservation laws are used to solve fluid dynamics problems, and may be written in [[integral]] or [[Differential (infinitesimal)|differential]] form. The conservation laws may be applied to a region of the flow called a ''control volume''. A control volume is a discrete volume in space through which fluid is assumed to flow. The integral formulations of the conservation laws are used to describe the change of mass, momentum, or energy within the control volume. Differential formulations of the conservation laws apply [[Stokes' theorem]] to yield an expression that may be interpreted as the integral form of the law applied to an infinitesimally small volume (at a point) within the flow.

{{glossary}}
{{term|[[Continuity equation#Fluid dynamics|Mass continuity]] (conservation of mass)}}{{defn
| The rate of change of fluid mass inside a control volume must be equal to the net rate of fluid flow into the volume.  Physically, this statement requires that mass is neither created nor destroyed in the control volume,<ref name="J.D. Anderson 2007">{{cite book |last=Anderson |first=J. D. |title=Fundamentals of Aerodynamics |location=London |edition=4th |publisher=McGraw–Hill |year=2007 |isbn=978-0-07-125408-3 }}</ref> and can be translated into the integral form of the continuity equation:
: <math>\frac{\partial}{\partial t} \iiint_V \rho \, dV = - \, {} </math> {{oiint|preintegral = |intsubscpt =<math>{\scriptstyle S}</math>|integrand = <math>{}\,\rho\mathbf{u}\cdot d\mathbf{S}</math>}}

Above, {{mvar|ρ}} is the fluid density, {{math|'''u'''}} is the [[flow velocity]] vector, and {{mvar|t}} is time.  The left-hand side of the above expression is the rate of increase of mass within the volume and contains a triple integral over the control volume, whereas the right-hand side contains an integration over the surface of the control volume of mass convected into the system.  Mass flow into the system is accounted as positive, and since the normal vector to the surface is opposite to the sense of flow into the system the term is negated. The differential form of the continuity equation is, by the [[divergence theorem]]:
<math display="block">\ \frac{\partial \rho}{\partial t} + \nabla \cdot (\rho \mathbf{u}) = 0 </math>}}

{{term|[[Momentum|Conservation of momentum]]}}
{{see also|Cauchy momentum equation}}{{defn
| [[Newton's second law of motion]] applied to a control volume, is a statement that any change in momentum of the fluid within that control volume will be due to the net flow of momentum into the volume and the action of external forces acting on the fluid within the volume.
: <math> \frac{\partial}{\partial t} \iiint_{\scriptstyle V} \rho\mathbf{u} \, dV = -\, {} </math> {{oiint|preintegral = |intsubscpt = <math>_{\scriptstyle S}</math> |integrand}} <math> (\rho\mathbf{u}\cdot d\mathbf{S}) \mathbf{u} -{}</math> {{oiint|intsubscpt = <math>{\scriptstyle S}</math>|integrand = <math> {}\, p \, d\mathbf{S}</math>}} <math>\displaystyle{}+ \iiint_{\scriptstyle V} \rho \mathbf{f}_\text{body} \, dV + \mathbf{F}_\text{surf}</math>

In the above integral formulation of this equation, the term on the left is the net change of momentum within the volume. The first term on the right is the net rate at which momentum is convected into the volume. The second term on the right is the force due to pressure on the volume's surfaces. The first two terms on the right are negated since momentum entering the system is accounted as positive, and the normal is opposite the direction of the velocity {{math|'''u'''}} and pressure forces. The third term on the right is the net acceleration of the mass within the volume due to any [[body force]]s (here represented by {{math|'''f'''<sub>body</sub>}}). [[Surface force]]s, such as viscous forces, are represented by {{math|'''F'''<sub>surf</sub>}}, the net force due to [[Stress (mechanics)|shear forces]] acting on the volume surface. The momentum balance can also be written for a ''moving'' control volume.<ref>{{ cite journal | last1 = Nangia | first1 = Nishant | last2 = Johansen | first2 = Hans | last3 = Patankar | first3 = Neelesh A. | last4 = Bhalla | first4 = Amneet Pal S. | title = A moving control volume approach to computing hydrodynamic forces and torques on immersed bodies | journal = Journal of Computational Physics | volume = 347 | pages = 437–462 | year = 2017 | doi = 10.1016/j.jcp.2017.06.047 | arxiv = 1704.00239 | bibcode = 2017JCoPh.347..437N | s2cid = 37560541 }}</ref>

The following is the differential form of the momentum conservation equation.  Here, the volume is reduced to an infinitesimally small point, and both surface and body forces are accounted for in one total force, {{math|'''F'''}}.  For example, {{math|'''F'''}} may be expanded into an expression for the frictional and gravitational forces acting at a point in a flow.
<math display="block"> \frac{D \mathbf{u}}{D t} = \mathbf{F} - \frac{\nabla p}{\rho} </math>

In aerodynamics, air is assumed to be a [[Newtonian fluid]], which posits a linear relationship between the shear stress (due to internal friction forces) and the rate of strain of the fluid. The equation above is a vector equation in a three-dimensional flow, but it can be expressed as three scalar equations in three coordinate directions. The conservation of momentum equations for the compressible, viscous flow case is called the Navier–Stokes equations.<ref name="J.D. Anderson 2007"/>}}

{{term|[[Conservation of energy]]}}
{{see also|First law of thermodynamics (fluid mechanics)}}{{defn
|Although [[energy]] can be converted from one form to another, the total [[energy]] in a closed system remains constant.
<math display="block"> \rho \frac{Dh}{Dt} = \frac{Dp}{Dt} + \nabla \cdot \left( k \nabla T\right) + \Phi </math>

Above, {{mvar|h}} is the specific [[enthalpy]], {{mvar|k}} is the [[thermal conductivity]] of the fluid, {{mvar|T}} is temperature, and {{math|Φ}} is the viscous dissipation function. The viscous dissipation function governs the rate at which the mechanical energy of the flow is converted to heat. The [[second law of thermodynamics]] requires that the dissipation term is always positive: viscosity cannot create energy within the control volume.<ref>{{cite book |last=White |first=F. M. |title=Viscous Fluid Flow |location=New York |publisher=McGraw–Hill |year=1974 |isbn=0-07-069710-8 }}</ref> The expression on the left side is a [[material derivative]].}}
{{glossary end}}