Editing Fluid dynamics (section)

===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]].}}
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