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${\displaystyle \mathbf {y} _{t}+\mathbf {A} (\mathbf {y} )\mathbf {y} _{x}=0.}$

General continuum equations

The Navier–Stokes momentum equation can be derived as a particular form of the Cauchy momentum equation, whose general convective form is

${\displaystyle {\frac {D\mathbf {u} }{Dt}}={\frac {1}{\rho }}\nabla \cdot {\boldsymbol {\sigma }}+\mathbf {g} }$

By setting the Cauchy stress tensor ${\displaystyle {\boldsymbol {\sigma }}}$ to be the sum of a viscosity term ${\displaystyle {\boldsymbol {\tau }}}$ (the deviatoric stress) and a pressure term ${\displaystyle -p\mathbf {I} }$ (volumetric stress) we arrive at

where

In this form, it is apparent that in the assumption of an inviscid fluid -no deviatoric stress- Cauchy equations reduce to the Euler equations.

Assuming conservation of mass we can use the continuity equation, ${\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla \cdot (\rho \,\mathbf {u} )=0}$ to arrive to the conservation form of the equations of motion. This is often written:[1]

where Template:Math is the outer product:

${\displaystyle \mathbf {u} \otimes \mathbf {v} =\mathbf {u} \mathbf {v} ^{\mathrm {T} }.}$

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 velocity gradient, and assuming constant viscosity, the above Cauchy equations will lead to the Navier–Stokes equations below.

Convective acceleration

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.

1. Batchelor (1967) pp. 137 & 142.