Momentum equation for Inviscid fluid flow (a partial case of Navier–Stokes equation):

(1)	$\begin{array}{l}\displaystyle \frac{\partial {\bf u}}{\partial t} + ({\bf u} \cdot \nabla) {\bf u} = - \frac{1}{\rho} \, \nabla p + {\bf g} +\frac{1}{\rho} \cdot {\bf f}_{\rm cnt}\end{array}$

where

$\begin{array}{l}{\bf u}\end{array}$	fluid velocity
$\begin{array}{l}\rho\end{array}$	fluid density
$\begin{array}{l}\nu\end{array}$	fluid kinematic viscosity
$\begin{array}{l}{\bf g}\end{array}$	resulting specific body force exerted on fluid body
$\begin{array}{l}{\bf f}_{\rm cnt}\end{array}$	volumetric density of all contact forces exerted on fluid body

Approximations

Transient 1D Inviscid fluid flow

  
	(2)
	\rho \left(  \frac{\partial u}{\partial t} + u \frac{\partial u}{\partial l} \right)=
-\frac{\partial p}{\partial l}  + \rho \, g \, \cos \theta + f_{\rm cnt, \, l}
  

Steady-state 1D inviscid fluid flow

  
	(3)
	\frac{d p}{d l} =
-\rho \, u \, \frac{d u}{d l}  + \rho \, g \, \cos \theta + f_{\rm cnt, \, l}
  

Bernoulli equation =
Steady-state 1D inviscid fluid flow of incompressible fluid with no friction

  
	(4)
	\frac{p(l)}{\rho} +  \frac{u^2}{2} -  g \cdot z(l)   = \rm const

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Approximations

See also

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Euler equation

Approximations

See also