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{{about|Euler equations in classical fluid flow||List of topics named after Leonhard Euler}}
{{dablink|This page assumes that [[classical mechanics]] applies; For a discussion of compressible fluid flow when velocities approach the [[speed of light]] see [[relativistic Euler equations]].}}

In [[fluid dynamics]], the '''Euler equations''' are a set of equations governing [[inviscid flow]]. They are named after [[Leonhard Euler]]. The equations represent conservation of mass (continuity), momentum, and energy, corresponding to the [[Navier–Stokes equations]] with zero [[viscosity]] and [[heat conduction]] terms. Historically, only the continuity and momentum equations have been derived by Euler. However, fluid dynamics literature often refers to the full set – including the energy equation – together as "the Euler equations".<ref>Anderson, John D. (1995), Computational Fluid Dynamics, The Basics With Applications. ISBN 0-07-113210-4</ref>

Like the Navier-Stokes equations, the Euler equations are usually written in one of two forms: the "[[conservation law|conservation]] form" and the "non-conservation form". The conservation form emphasizes the physical interpretation of the equations as conservation laws through a control volume fixed in space. The non-conservation form emphasizes changes to the state of a control volume as it moves with the fluid.

The Euler equations can be applied to [[compressible flow|compressible]] as well as to [[incompressible flow]] – using either an appropriate [[equation of state]] or assuming that the [[divergence]] of the [[flow velocity]] field is zero, respectively.

==History==

The Euler equations first appeared in published form in Euler's article "Principes généraux du mouvement des fluides," published in ''Mémoires de l'Academie des Sciences de Berlin'' in 1757 (in this article Euler actually published only the ''general'' form of the continuity equation and the momentum equation;<ref>[http://www.math.dartmouth.edu/~euler/pages/E226.html E226 -- Principes generaux du mouvement des fluides]</ref> the energy conservation equation would be obtained a century later). They were among the first [[partial differential equations]] to be written down. At the time Euler published his work, the system of equations consisted of the momentum and continuity equations, thus it was underdetermined except in the case of an incompressible fluid. An additional equation, which was later to be called the [[Adiabatic process|adiabatic condition]], was supplied by [[Pierre-Simon Laplace]] in 1816.

During the second half of the 19th century, it was found that the equation related to the conservation of energy must at all times be kept, while the adiabatic condition is a consequence of the fundamental laws in the case of smooth solutions. With the discovery of the [[special theory of relativity]], the concepts of energy density, momentum density, and stress were unified into the concept of the [[stress-energy tensor]], and energy and momentum were likewise unified into a single concept, the [[Four-momentum|energy-momentum vector]].<ref name=Christodoulou>{{cite journal|doi=10.1090/S0273-0979-07-01181-0|last=Christodoulou|first=Demetrios|title=The Euler Equations of Compressible Fluid Flow|journal=Bulletin of the American Mathematical Society|volume=44|issue=4|date=October 2007|url=http://www.ams.org/bull/2007-44-04/S0273-0979-07-01181-0/S0273-0979-07-01181-0.pdf|accessdate=June 13, 2009|pages= 581–602}}</ref>

==Conservation and component form==
In differential form, the equations are:

:<math>
\begin{align}
&{\partial\rho\over\partial t}+
\nabla\cdot(\rho\bold u)=0\\[1.2ex]
&{\partial(\rho{\bold u})\over\partial t}+
\nabla\cdot(\bold u\otimes(\rho \bold u))+\nabla p=\bold{0}\\[1.2ex]
&{\partial E\over\partial t}+
\nabla\cdot(\bold u(E+p))=0,
\end{align}
</math>

where 
*''ρ'' is the fluid [[mass density]],
*'''''u''''' is the fluid [[velocity]] [[Vector (geometric)|vector]], with components ''u'', ''v'', and ''w'',
*''E = ρ e + ½ ρ ( u<sup>2</sup> + v<sup>2</sup> + w<sup>2</sup> )'' is the total [[energy]] per unit [[volume]], with ''e'' being the [[internal energy]] per unit mass for the fluid,
*''p'' is the [[pressure]],
*''<math>\otimes</math>'' denotes the [[tensor product]], and
*'''0''' is the [[zero vector]].

These equations may be expressed in subscript notation. The second equation includes the [[divergence]] of a [[dyadic product]], and may be clearer in subscript notation:

:<math>
{\partial\rho\over\partial t}+
\sum_{i=1}^3
{\partial(\rho u_i)\over\partial x_i}
=0,
</math>

:<math>
{\partial(\rho u_j)\over\partial t}+
\sum_{i=1}^3
{\partial(\rho u_i u_j)\over\partial x_i}+
{\partial p\over\partial x_j}
=0,
</math>

:<math>
{\partial E\over\partial t}+
\sum_{i=1}^3 {\partial((E+p) u_i)\over\partial x_i}
=0,
</math>

where the ''i'' and ''j'' subscripts label the three Cartesian components: ''( x<sub>1</sub> , x<sub>2</sub> , x<sub>3</sub> ) = ( x , y , z )'' and ''( u<sub>1</sub> , u<sub>2</sub> , u<sub>3</sub> ) = ( u , v , w )''. These equations may be more succinctly expressed using [[Einstein notation]], in which matched indices imply a sum over those indices and <math>\partial_t=\frac{\partial}{\partial t}</math> and <math>\partial_i=\frac{\partial}{\partial x_i}</math>:

:<math>
\partial_t \rho+\partial_i(\rho u_i)=0\,
</math>

:<math>
\partial_t(\rho u_j)+\partial_i(\rho u_i u_j)+\partial_j p=0\,
</math>

:<math>
\partial_t E+\partial_i((E+p)u_i)=0.\,
</math>

The above equations are expressed in [[conservation law|conservation form]], as this format emphasizes their physical origins (and is often the most convenient form for [[computational fluid dynamics]] simulations).  By subtracting the velocity times the mass conservation term, the second equation (momentum conservation), can also be expressed as:

:<math>
[\partial_t(\rho u_j)+\partial_i(\rho u_i u_j)+\partial_j p] - u_j[\partial_t \rho+\partial_i(\rho u_i)]=
\rho \partial_t u_j+\rho u_i \partial_i u_j+\partial_j p=0\,
</math>

or, in vector notation:

:<math>
\rho\left(
\frac{\partial}{\partial t}+{\bold u}\cdot\nabla
\right){\bold u}+\nabla p=\bold{0}
</math>

but this form for the momentum conservation equation obscures the direct connection between the Euler equations and [[Newton's second law of motion]]. Similarly, by subtracting the velocity times the above momentum conservation term, the third equation (energy conservation), can also be expressed as:

:<math>
\partial_t (\rho e) +\partial_i(\rho e u_i) + p\partial_i u_i=0\,
</math>

or

:<math>
\frac{\partial \rho e}{\partial t}+\nabla\cdot(\rho e \bold u)+p\nabla\cdot \bold u=0
</math>

==Conservation and vector form==
In vector and conservation form, the Euler equations become:

:<math> 
\frac{\partial \bold m}{\partial t}+
\frac{\partial \bold f_x}{\partial x}+
\frac{\partial \bold f_y}{\partial y}+
\frac{\partial \bold f_z}{\partial z}={\bold 0},
</math>

where

:<math>
{\bold m}=\begin{pmatrix}\rho  \\  \rho u  \\  \rho v  \\ \rho w  \\E\end{pmatrix};
</math>

:<math>
{\bold f_x}=\begin{pmatrix}\rho u\\p+\rho u^2\\  \rho uv \\ \rho uw\\u(E+p)\end{pmatrix};\qquad
{\bold f_y}=\begin{pmatrix}\rho v\\  \rho uv \\p+\rho v^2\\ \rho vw \\v(E+p)\end{pmatrix};\qquad
{\bold f_z}=\begin{pmatrix}\rho w\\  \rho uw \\  \rho vw \\p+\rho w^2\\w(E+p)\end{pmatrix}.
</math>

This form makes it clear that '''''f'''<sub>x</sub>'', '''''f'''<sub>y</sub>'' and '''''f'''<sub>z</sub>'' are [[flux]]es.

The equations above thus represent [[conservation of mass]], three components of [[conservation of momentum|momentum]], and [[conservation of energy|energy]].  There are thus five equations and six unknowns.  Closing the system requires an [[equation of state]]; the most commonly used is the [[ideal gas law]] (i.e. ''p&nbsp;=&nbsp;ρ&nbsp;(γ−1)&nbsp;e'', where ''ρ'' is the density, ''γ'' is the [[adiabatic index]], and ''e'' the internal energy).

Note the odd form for the energy equation; see [[Rankine–Hugoniot equation]].  The extra terms involving ''p'' may be interpreted as the mechanical work done on a fluid element by its neighbor fluid elements.  These terms sum to zero in an incompressible fluid.

The well-known [[Bernoulli's equation]] can be derived by integrating Euler's equation along a [[Streamlines, streaklines and pathlines|streamline]], under the assumption of constant density and a sufficiently stiff equation of state.

==Non-conservation form with flux Jacobians==
Expanding the [[flux]]es can be an important part of constructing [[numerical solution|numerical solvers]], for example by exploiting ([[approximation|approximate]]) solutions to the [[Riemann problem]]. From the original equations as given above in vector and conservation form, the equations are written in a non-conservation form as:

:<math>
\frac{\partial \bold m}{\partial t}
+ \bold A_x \frac{\partial \bold m}{\partial x} 
+ \bold A_y \frac{\partial \bold m}{\partial y} 
+ \bold A_z \frac{\partial \bold m}{\partial z} 
= {\bold 0}.
</math>

where '''A'''<sub>''x''</sub>, '''A'''<sub>''y''</sub> and '''A'''<sub>''z''</sub> are called the flux [[Jacobian matrix and determinant|Jacobian]]s, which are [[matrix (mathematics)|matrices]] equal to:

:<math>
  \bold A_x=\frac{\partial \bold f_x(\bold s)}{\partial \bold s}, \qquad
  \bold A_y=\frac{\partial \bold f_y(\bold s)}{\partial \bold s} \qquad \text{and} \qquad
  \bold A_z=\frac{\partial \bold f_z(\bold s)}{\partial \bold s}.
</math>

Here, the flux Jacobians '''A'''<sub>''x''</sub>, '''A'''<sub>''y''</sub> and '''A'''<sub>''z''</sub> are still functions of the state vector '''''m''''', so this form of the Euler equations is nonlinear, just like the original equations. This non-conservation form is equivalent to the original Euler equations in conservation form, at least in regions where the state vector '''''m''''' varies smoothly.

===Flux Jacobians for an ideal gas===
The [[ideal gas law]] is used as the [[equation of state]], to derive the full Jacobians in matrix form, as given below:<ref>See Toro (1999)</ref>

:{| class="toccolours collapsible collapsed" width="80%" style="text-align:left"
! Flux Jacobians in matrix form for an ideal gas
|- 
|The ''x''-direction flux Jacobian:

:<math>
\bold A_x= \left[ 
\begin{array}{c c c c c}
0 & 1 & 0 & 0 & 0 \\
\hat{\gamma}H-u^2-a^2 & (3-\gamma)u & -\hat{\gamma}v & -\hat{\gamma}w & \hat{\gamma} \\
-uv & v & u & 0 & 0 \\
-uw & w & 0 & u & 0 \\
u[(\gamma-2)H-a^2] &  H-\hat{\gamma}u^2 & -\hat{\gamma}uv & -\hat{\gamma}uw & \gamma u
\end{array}
\right].
</math>

The ''y''-direction flux Jacobian:

:<math>
\bold A_y= \left[ 
\begin{array}{c c c c c}
0 & 0 & 1 & 0 & 0 \\
-vu & v & u & 0 & 0 \\
\hat{\gamma}H-v^2-a^2 &  -\hat{\gamma}u & (3-\gamma)v & -\hat{\gamma}w & \hat{\gamma} \\
-vw & 0 & w & v & 0 \\
v[(\gamma-2)H-a^2] &  -\hat{\gamma}uv & H-\hat{\gamma}v^2 & -\hat{\gamma}vw & \gamma v
\end{array}
\right].
</math>

The z-direction flux Jacobian:

:<math>
\bold A_z= \left[ 
\begin{array}{c c c c c}
0 & 0 & 0 & 1 & 0 \\
-uw & w & 0 & u & 0 \\
-vw & 0 & w & v & 0 \\
\hat{\gamma}H-w^2-a^2 &  -\hat{\gamma}u  & -\hat{\gamma}v & (3-\gamma)w& \hat{\gamma} \\
w[(\gamma-2)H-a^2] &  -\hat{\gamma}uw  & -\hat{\gamma}vw & H-\hat{\gamma}w^2 & \gamma w
\end{array}
\right].
</math>
Where <math>\hat{\gamma}=\gamma-1</math>.
|}

The total [[enthalpy]] ''H'' is given by:

:<math>
H = \frac{E}{\rho} + \frac{p}{\rho},
</math>

and the [[speed of sound]] ''a'' is given as:

:<math>
a=\sqrt{\frac{\gamma p}{\rho}} = \sqrt{(\gamma-1)\left[H-\frac{1}{2}\left(u^2+v^2+w^2\right)\right]}.
</math>

===Linearized form===

The linearized Euler equations are obtained by linearization of the Euler equations in non-conservation form with flux Jacobians, around a state '''''m''''' = '''''m'''''<sub>0</sub>, and are given by:

:<math>
\frac{\partial \bold m}{\partial t}
+ \bold A_{x,0} \frac{\partial \bold m}{\partial x} 
+ \bold A_{y,0} \frac{\partial \bold m}{\partial y} 
+ \bold A_{z,0} \frac{\partial \bold m}{\partial z} 
= {\bold 0},
</math>

where '''A'''<sub>''x,0''</sub>, '''A'''<sub>''y,0''</sub> and '''A'''<sub>''z,0''</sub> are the values of respectively '''A'''<sub>''x''</sub>, '''A'''<sub>''y''</sub> and '''A'''<sub>''z''</sub> at some reference state '''''m''''' = '''''m'''''<sub>0</sub>.

===Uncoupled wave equations for the linearized one-dimensional case===

The Euler equations can be transformed into uncoupled [[wave]] equations if they are expressed in [[method of characteristics|characteristic variables]] instead of conserved variables. As an example, the one-dimensional (1-D) Euler equations in linear flux-Jacobian form is considered:

:<math>
\frac{\partial \bold m}{\partial t}
+ \bold A_{x,0} \frac{\partial \bold m}{\partial x} = {\bold 0}.
</math>

The matrix '''A'''<sub>''x,0''</sub> is [[Diagonalizable matrix|diagonalizable]], which means it can be decomposed into:

:<math>
 \mathbf{A}_{x,0} = \mathbf{P} \mathbf{\Lambda} \mathbf{P}^{-1},
</math>

:<math>
\mathbf{P}= \left[\bold r_1, \bold r_2, \bold r_3\right] =\left[ 
\begin{array}{c c c}
1 & 1 & 1  \\
u-a & u & u+a \\
H-u a & \frac{1}{2} u^2 & H+u a \\
\end{array}
\right],
</math>

:<math>
\mathbf{\Lambda} 
= \begin{bmatrix}
\lambda_1 & 0 & 0  \\
0 & \lambda_2 & 0 \\
0 & 0 & \lambda_3 \\
\end{bmatrix}
= \begin{bmatrix}
u-a & 0 & 0  \\
0 & u & 0 \\
0 & 0 & u+a \\
\end{bmatrix}.
</math>

Here '''''r'''<sub>1</sub>'', '''''r'''<sub>2</sub>'', '''''r'''<sub>3</sub>'' are the [[right eigenvector]]s of the matrix '''A'''<sub>''x,0''</sub> corresponding with the [[eigenvalue]]s ''λ<sub>1</sub>'', ''λ<sub>2</sub>'' and ''λ<sub>3</sub>''.

Defining the ''characteristic variables'' as:

:<math>\mathbf{w}= \mathbf{P}^{-1}\mathbf{m},</math>

Since '''A'''<sub>''x,0''</sub> is constant, multiplying the original 1-D equation in flux-Jacobian form with '''P'''<sup>−1</sup> yields:

:<math>
\frac{\partial \mathbf{w}}{\partial t} + \mathbf{\Lambda} \frac{\partial \mathbf{w}}{\partial x} = \mathbf{0}
</math>

The equations have been essentially [[Linear independence|decoupled]] and turned into three wave equations, with the eigenvalues being the wave speeds.  The variables ''w''<sub>i</sub> are called ''Riemann invariants'' or, for general hyperbolic systems, they are called ''characteristic variables''.

==Shock waves==
The Euler equations are [[nonlinear]] [[hyperbolic partial differential equation|hyperbolic]] equations and their general solutions are [[waves]]. Much like the familiar oceanic [[waves and shallow water|waves]], waves described by the Euler Equations [[breaking wave|'break']] and so-called [[shock waves]] are formed; this is a nonlinear effect and represents the solution becoming [[multi-valued function|multi-valued]]. Physically this represents a breakdown of the assumptions that led to the formulation of the differential equations, and to extract further information from the equations we must go back to the more fundamental integral form. Then, [[weak solution]]s are formulated by working in 'jumps' (discontinuities) into the flow quantities – density, velocity, pressure, entropy – using the [[Rankine–Hugoniot equation|Rankine–Hugoniot shock conditions]]. Physical quantities are rarely discontinuous; in real flows, these discontinuities are smoothed out by [[viscosity]]. (See [[Navier–Stokes equations]])

Shock propagation is studied – among many other fields – in [[aerodynamics]] and [[rocket|rocket propulsion]], where sufficiently fast flows occur.

==The equations in one spatial dimension==
For certain problems, especially when used to analyze compressible flow in a duct or in case the flow is cylindrically or spherically symmetric, the one-dimensional Euler equations are a useful first approximation. Generally, the Euler equations are solved by [[Riemann|Riemann's]] [[method of characteristics]]. This involves finding curves in plane of independent variables (i.e., ''x'' and ''t'') along which [[partial differential equation]]s (PDE's) degenerate into [[ordinary differential equation]]s (ODE's). [[Numerical analysis|Numerical solutions]] of the Euler equations rely heavily on the method of characteristics.

== Steady flow in streamline coordinates {{anchor|Steady flow in streamline coordinates}} ==
In the case of steady flow, it is convenient to choose the [[Frenet–Serret frame]] along a [[Streamlines, streaklines, and pathlines|streamline]] as the [[coordinate system]] for describing the [[momentum]] part of the Euler equations:<ref name="Fay">
{{cite book
|author=James A. Fay
|title=Introduction to Fluid Mechanics
|publisher=MIT Press
|year=June 1994
|isbn=0-262-06165-1
}} see "4.5 Euler's Equation in Streamline Coordinates" pp.150-pp.152 (http://books.google.com/books?id=XGVpue4954wC&pg=150)
</ref>
:<math>
\frac{\mathrm{D} \boldsymbol{v}}{\mathrm{D}t} = -\frac{1}{\rho}\nabla p,
</math>
where  '''''v''''', ''p'' and ''ρ'' denote the [[velocity]], the [[pressure]] and the [[density]], respectively.

Let {'''''e'''''<sub>s</sub>, '''''e'''''<sub>n</sub>, '''''e'''''<sub>b</sub> } be a Frenet–Serret [[orthonormal basis]] which consists of a tangential unit vector, a normal unit vector, and a binormal unit vector to the streamline, respectively.
Since a streamline is a curve that is tangent to the velocity vector of the flow, the left-handed side of the above equation, the [[substantial derivative]] of velocity, can be described as follows:
:<math>\begin{align}
\frac{\mathrm{D} \boldsymbol{v}}{\mathrm{D}t} 
&= \boldsymbol{v}\cdot\nabla \boldsymbol{v} \\
&= v\frac{\partial}{\partial s}(v\boldsymbol{e}_s)
     &(\boldsymbol{v} = v \boldsymbol{e}_s ,~ 
      {\partial / \partial s} \equiv \boldsymbol{e}_s\cdot\nabla)\\
&= v\frac{\partial v}{\partial s}\boldsymbol{e}_s 
+ \frac{v^2}{R} \boldsymbol{e}_n &(\because~ \frac{\partial \boldsymbol{e}_s}{\partial s}=\frac{1}{R}\boldsymbol{e}_n),
\end{align}</math>
where ''R'' is the [[radius of curvature (mathematics)|radius of curvature]] of the streamline.

Therefore, the momentum part of the Euler equations for a steady flow is found to have a simple form:
:<math>\begin{cases}
\displaystyle v\frac{\partial v}{\partial s} = -\frac{1}{\rho}\frac{\partial p}{\partial s},\\
\displaystyle {v^2 \over R}                  = -\frac{1}{\rho}\frac{\partial p}{\partial n}    &({\partial / \partial n}\equiv\boldsymbol{e}_n\cdot\nabla),\\
\displaystyle 0                              = -\frac{1}{\rho}\frac{\partial p}{\partial b} &({\partial / \partial b}\equiv\boldsymbol{e}_b\cdot\nabla).
\end{cases} 
</math>

For [[barotropic]] flow ( ''ρ''=''ρ''(''p'') ), [[Bernoulli's equation]] is derived from the first equation:
:<math>
\frac{\partial}{\partial s}\left(\frac{v^2}{2} + \int \frac{\mathrm{d}p}{\rho}\right) =0.
</math> 
The second equation expresses that, in the case the streamline is curved, there should exist a [[pressure gradient]] normal to the streamline because the [[centripetal acceleration]] of the [[fluid parcel]] is only generated by the normal pressure gradient.

The third equation expresses that pressure is constant along the binormal axis.

=== Streamline curvature theorem ===
[[File:Streamlines around a NACA 0012.svg|frame|right|
The "Streamline curvature theorem" states that the pressure at the upper surface of an airfoil is lower than the pressure far away and that the pressure at the lower surface is higher than the pressure far away; hence the pressure difference between the upper and lower surfaces of an airfoil generates a lift force.
]]

Let ''r'' be the distance from the center of curvature of the streamline, 
then the second equation is written as follows:
:<math>
\frac{\partial p}{\partial r} = \rho \frac{v^2}{r}~(>0),
</math>
where <math>{\partial / \partial r} = -{\partial /\partial n}.</math>

This equation states:<blockquote> 
''In a steady flow of an [[inviscid]] [[fluid]] without external forces, the [[center of curvature]] of the streamline lies in the direction of decreasing radial pressure.'' </blockquote>

Although this relationship between the pressure field and flow curvature is very useful, it doesn't have a name in the English-language scientific literature.<ref name=Babinsky>{{citation 
  | journal=Physics Education
  | first=Holger
  | last=Babinsky
  |date=November 2003
  | url=http://www.iop.org/EJ/article/0031-9120/38/6/001/pe3_6_001.pdf 
  | title=How do wings work?
  }}</ref>
Japanese fluid-dynamicists call the relationship the "Streamline curvature theorem".
<ref name="Imai">{{cite book
|author=今井 功 (IMAI, Isao)
|title=『流体力学(前編)』(Fluid Dynamics 1)
|publisher=裳華房 (Shoukabou)
|date=November 1973
|isbn=4-7853-2314-0
|language=Japanese
}}</ref>

This "theorem" explains clearly why there are such low pressures in the centre of [[vortex|vortices]],<ref name=Babinsky/> which consist of concentric circles of streamlines.
This also is a way to intuitively explain why airfoils generate [[lift (force)|lift forces]].<ref name=Babinsky/>

==See also==
* [[Madelung equations]]

==Notes==
{{reflist}}

==Further reading==
*{{cite book | first=G. K. | last=Batchelor | title=An Introduction to Fluid Dynamics | year=1967 | publisher=Cambridge University Press | isbn=0-521-66396-2 }}
*{{cite book | first=Philip A. | last=Thompson| year=1972 | title=Compressible Fluid Flow | publisher=McGraw-Hill | location=New York | isbn=0-07-064405-5 }}
*{{cite book | first=E.F. | last=Toro | title=Riemann Solvers and Numerical Methods for Fluid Dynamics | publisher=Springer-Verlag | year=1999 | isbn=3-540-65966-8}}

==External links==
* [http://wadeawalker.wordpress.com/2013/02/07/fluid-dynamics-super-awesome-or-super-awesomer/ An introductory explanation of the Euler equations at a conceptual level]

[[Category:Concepts in physics]]
[[Category:Equations of fluid dynamics]]