Editing Euler equations (fluid dynamics) (section)

=== Steady flow in material 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 steady [[momentum]] Euler equation:{{sfn|Fay|1994|pp=150-152}}
<math display="block">
\boldsymbol{u}\cdot\nabla \boldsymbol{u} = - \frac{1}{\rho} \nabla p,
</math>

where  <math>\mathbf u</math>, <math>p</math> and <math>\rho</math> denote the [[flow velocity]], the [[pressure]] and the [[density]], respectively.

Let <math>\left\{ \mathbf e_s, \mathbf e_n, \mathbf e_b \right\}</math> 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-hand side of the above equation, the [[convective derivative]] of velocity, can be described as follows:
<math display="block">
  \boldsymbol{u}\cdot\nabla \boldsymbol{u}= u\frac{\partial}{\partial s}(u\boldsymbol{e}_s) = u\frac{\partial u}{\partial s}\boldsymbol{e}_s + \frac{u^2}{R}\boldsymbol{e}_n,</math>
where 
<math display="block">\begin{align}
   \boldsymbol{u} &= u \boldsymbol{e}_s,\\
   \frac{\partial}{\partial s} &\equiv \boldsymbol{e}_s \cdot \nabla,\\
   \frac{\partial\boldsymbol{e}_s}{\partial s} &= \frac{1}{R}\boldsymbol{e}_n,
\end{align}</math>
and <math>R</math> 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 display="block">\begin{align}
  \displaystyle u\frac{\partial u}{\partial s} &= -\frac{1}{\rho}\frac{\partial p}{\partial s},\\
  \displaystyle                  {u^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{align}</math>

For [[barotropic]] flow <math>(\rho = \rho(p))</math>, [[Bernoulli's equation]] is derived from the first equation:
<math display="block">\frac{\partial}{\partial s}\left(\frac{u^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 <math>r</math> be the distance from the center of curvature of the streamline, then the second equation is written as follows:
<math display="block">
\frac{\partial p}{\partial r} = \rho \frac{u^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.{{sfn|Babinsky|2003}} Japanese fluid-dynamicists call the relationship the "Streamline curvature theorem".{{sfn|Imai|1973|p=}}

This "theorem" explains clearly why there are such low pressures in the centre of [[vortex|vortices]],{{sfn|Babinsky|2003}} which consist of concentric circles of streamlines.
This also is a way to intuitively explain why airfoils generate [[lift (force)|lift forces]].{{sfn|Babinsky|2003}}