linguistic:voice_change
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: D u D t = 1 ρ ∇ ⋅ σ + g. {\displaystyle {\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}={\frac {1}{\rho }}\nabla \cdot {\boldsymbol {\sigma }}+\mathbf {g}.} By setting the Cauchy stress tensor σ {\textstyle {\boldsymbol {\sigma }}} to be the sum of a viscosity term τ {\textstyle {\boldsymbol {\tau }}} (the deviatoric stress) and a pressure term − p I {\textstyle -p\mathbf {I} } (volumetric stress), we arrive at:

ρ D u D t = − ∇ p + ∇ ⋅ τ + ρ f {\displaystyle \rho {\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}=-\nabla p+\nabla \cdot {\boldsymbol {\tau }}+\rho \,\mathbf {f} }

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, 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 D D t {\displaystyle {\frac {\mathbf {D} }{\mathbf {Dt} }}}) of any finite volume (V) to represent the change of velocity in fluid media: D m D t = ∭ V (D ρ D t + ρ (∇ ⋅ u)) d V D ρ D t + ρ (∇ ⋅ u) = ∂ ρ ∂ t + (∇ ρ) ⋅ u + ρ (∇ ⋅ u) = ∂ ρ ∂ t + ∇ ⋅ (ρ u) = 0 {\displaystyle {\begin{aligned}{\frac {\mathbf {D} m}{\mathbf {Dt} }}&={\iiint \limits _{V}}({{\frac {\mathbf {D} \rho }{\mathbf {Dt} }}+\rho (\nabla \cdot \mathbf {u})})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{aligned}}} where

to arrive at the conservation form of the equations of motion. This is often written:

∂ ∂ t (ρ u) + ∇ ⋅ (ρ u ⊗ u) = − ∇ p + ∇ ⋅ τ + ρ f {\displaystyle {\frac {\partial }{\partial t}}(\rho \,\mathbf {u})+\nabla \cdot (\rho \,\mathbf {u} \otimes \mathbf {u})=-\nabla p+\nabla \cdot {\boldsymbol {\tau }}+\rho \,\mathbf {f} }

where ⊗ {\textstyle \otimes } is the outer product of the flow velocity (u {\displaystyle \mathbf {u} }): u ⊗ u = u u T {\displaystyle \mathbf {u} \otimes \mathbf {u} =\mathbf {u} \mathbf {u} ^{\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.