Editing Navier–Stokes equations (section)

==General continuum equations==
{{Main|Derivation of the Navier–Stokes equations}}
{{See also|Cauchy momentum equation#Conservation form}}

The Navier–Stokes momentum equation can be derived as a particular form of the [[Cauchy momentum equation]], whose general convective form is:
<math display="block"> \frac{\mathrm{D} \mathbf{u}}{\mathrm{D} t} = \frac 1 \rho \nabla \cdot \boldsymbol{\sigma} + \mathbf{f}.</math>
By setting the [[Cauchy stress tensor]] <math display="inline">\boldsymbol{\sigma}</math> to be the sum of a viscosity term <math display="inline">\boldsymbol{\tau}</math> (the [[deviatoric stress]]) and a pressure term <math display="inline">-p \mathbf{I}</math> (volumetric stress), we arrive at:

{{Equation box 1
|indent=:
|title='''Cauchy momentum equation''' ''(convective form)''
|equation=<math>
\rho\frac{\mathrm{D} \mathbf{u}}{\mathrm{D} t} = - \nabla p + \nabla \cdot \boldsymbol \tau + \rho\,\mathbf{a}
</math>
|cellpadding
|border
|border colour = #50C878
|background colour = #DCDCDC
}}

where
* <math display="inline">\frac{\mathrm{D}}{\mathrm{D}t}</math> is the [[material derivative]], defined as <math display="inline">\frac{\partial}{\partial t} + \mathbf{u} \cdot \nabla</math>,
* <math display="inline">\rho</math> is the (mass) density,
* <math display="inline">\mathbf{u}</math> is the flow velocity,
* <math display="inline">\nabla \cdot \,</math> is the [[divergence]],
* <math display="inline">p</math> is the [[pressure]],
* <math display="inline">t</math> is [[time]],
* <math display="inline">\boldsymbol{\tau}</math> is the [[Deviatoric stress|deviatoric stress tensor]], which has order 2,
* <math display="inline">\mathbf{a}</math> represents [[body force#acceleration|body acceleration]]s acting on the continuum, for example [[gravity]], [[Fictitious force|inertial accelerations]], [[Coulomb's law|electrostatic accelerations]], and so on.

In this form, it is apparent that in the assumption of an inviscid fluid – no deviatoric stress – Cauchy equations reduce to the [[Euler equations (fluid dynamics)|Euler equations]].

Assuming [[conservation of mass]], with the known properties of [[divergence]] and [[gradient]] we can use the mass [[continuity equation]], which represents the mass per unit volume of a [[homogenous]] fluid with respect to space and time (i.e., [[material derivative]] <math>\frac{\mathbf{D}}{\mathbf{Dt}}</math>) of any finite volume ('''V''') to represent the change of velocity in fluid media:
<math display="block">
\begin{align}
\frac{\mathbf{D}m}{{\mathbf{Dt}}}&={\iiint\limits_V} \left({\frac{\mathbf{D}\rho}{{\mathbf{Dt}}} + \rho (\nabla \cdot \mathbf{u})}\right)dV \\
\frac{\mathbf{D}\rho}{{\mathbf{Dt}}} + \rho (\nabla \cdot{\mathbf{u}})&=\frac{\partial\rho}{\partial t} + ({\nabla \rho}) \cdot{\mathbf{u}} + {\rho}(\nabla \cdot \mathbf{u})= \frac{\partial\rho}{\partial t} + \nabla\cdot({\rho \mathbf{u}})= 0 
\end{align}</math>where
* <math display="inline">\frac{\mathrm{D}m}{\mathrm{D}t}</math> is the [[material derivative]] of [[mass]] per unit volume ([[density]], <math>\rho</math>),
* <math display="inline">{\iiint \limits_V}(F(x_1, x_2, x_3 ,t))dV</math> is the mathematical operation for the [[Volume integral|integration throughout the volume]] (''V''),
* <math display="inline">\frac{\partial }{\partial t}</math> is the [[partial derivative]] mathematical operator,
* <math display="inline">\nabla \cdot \mathbf{u}\,</math> is the [[divergence]] of the flow velocity (<math>\mathbf{u}</math>), which is a [[scalar field]], [[Del|<sup>Note 1</sup>]]
* <math display="inline">{\nabla \rho} \,</math> is the [[gradient]] of [[density]] (<math>\rho</math>), which is the vector derivative of a [[scalar field]], [[Del|<sup>Note 1</sup>]]
<sup>[[Del|Note 1 - Refer to the mathematical operator del represented by the nabla]] (<math>\nabla</math>) [[Del|symbol.]]</sup>

to arrive at the conservation form of the equations of motion. This is often written:<ref>Batchelor (1967) pp. 137 & 142.</ref>

{{Equation box 1
|indent=:
|title='''Cauchy momentum equation''' ''(conservation form)''
|equation=<math>
\frac {\partial}{\partial t} (\rho\,\mathbf{u})
   + \nabla \cdot (\rho\,\mathbf{u} \otimes \mathbf{u})
 = - \nabla p + \nabla \cdot \boldsymbol \tau + \rho\,\mathbf{a}
</math>
|cellpadding
|border
|border colour = #50C878
|background colour = #DCDCDC
}}

where <math display="inline">\otimes</math> is the [[outer product]] of the flow velocity (<math>\mathbf{u}</math>):<math display="block">\mathbf u \otimes \mathbf u = \mathbf u \mathbf u^{\mathrm T}</math>

The left side of the equation describes acceleration, and may be composed of time-dependent and convective components (also the effects of non-inertial coordinates if present). The right side of the equation is in effect a summation of hydrostatic effects, the divergence of deviatoric stress and body forces (such as gravity).

All non-relativistic balance equations, such as the Navier–Stokes equations, can be derived by beginning with the Cauchy equations and specifying the stress tensor through a [[constitutive relation]]. By expressing the deviatoric (shear) stress tensor in terms of [[viscosity]] and the fluid [[Shear velocity|velocity]] gradient, and assuming constant viscosity, the above Cauchy equations will lead to the Navier–Stokes equations below.

===Convective acceleration===
{{See also|Cauchy momentum equation#Convective acceleration}}
[[Image:ConvectiveAcceleration vectorized.svg|thumb|An example of convection. Though the flow may be steady (time-independent), the fluid decelerates as it moves down the diverging duct (assuming incompressible or subsonic compressible flow), hence there is an acceleration happening over position.]]

A significant feature of the Cauchy equation and consequently all other continuum equations (including Euler and Navier–Stokes) is the presence of convective acceleration: the effect of acceleration of a flow with respect to space. While individual fluid particles indeed experience time-dependent acceleration, the convective acceleration of the flow field is a spatial effect, one example being fluid speeding up in a nozzle.