Navier–Stokes equations

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

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

${\textstyle {\boldsymbol {\sigma }}}$

${\textstyle {\boldsymbol {\tau }}}$

${\textstyle -p\mathbf {I} }$

where

• ${\textstyle {\frac {\mathrm {D} }{\mathrm {D} t}}}$ is the material derivative, defined as ${\textstyle {\frac {\partial }{\partial t}}+\mathbf {u} \cdot \nabla }$,
• ${\textstyle \rho }$ is the (mass) density,
• ${\textstyle \mathbf {u} }$ is the flow velocity,
• ${\textstyle \nabla \cdot \,}$ is the divergence,
• ${\textstyle p}$ is the pressure,
• ${\textstyle t}$ is time,
• ${\textstyle {\boldsymbol {\tau }}}$ is the
  deviatoric stress tensor
  , which has order 2,
• ${\textstyle \mathbf {f} }$ represents body accelerations acting on the continuum, for example gravity, inertial accelerations, 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.

Assuming

${\displaystyle {\frac {\mathbf {D} }{\mathbf {Dt} }}}$) of any finite volume (V) to represent the change of velocity in fluid media:

$${\displaystyle {\begin{aligned}{\frac {\mathbf {D} m}{\mathbf {Dt} }}&={\iiint \limits _{V}}({{\frac {\mathbf {D} \rho }{\mathbf {Dt} }}+\rho (\nabla \cdot \mathbf {u} )})dV\\{\frac {\mathbf {D} \rho }{\mathbf {Dt} }}+\rho (\nabla \cdot {\mathbf {u} })&={\frac {\partial \rho }{\partial t}}+({\nabla \rho })\cdot {\mathbf {u} }+{\rho }(\nabla \cdot \mathbf {u} )={\frac {\partial \rho }{\partial t}}+\nabla \cdot ({\rho \mathbf {u} })=0\end{aligned}}}$$

where

• ${\textstyle {\frac {\mathrm {D} m}{\mathrm {D} t}}}$ is the material derivative of mass per unit volume (density, ${\displaystyle \rho }$),
• ${\textstyle {\iiint \limits _{V}}(F(x_{1},x_{2},x_{3},t))dV}$ is the mathematical operation for the integration throughout the volume (V),
• ${\textstyle {\frac {\partial }{\partial t}}}$ is the partial derivative mathematical operator,
• ${\textstyle \nabla \cdot \mathbf {u} \,}$ is the divergence of the flow velocity (${\displaystyle \mathbf {u} }$), which is a scalar field, ^{Note 1}
• ${\textstyle {\nabla \rho }\,}$ is the gradient of density (${\displaystyle \rho }$), which is the vector derivative of a scalar field, ^{Note 1}

^{Note 1 - Refer to the mathematical operator del represented by the nabla (${\displaystyle \nabla }$) symbol.}

to arrive at the conservation form of the equations of motion. This is often written:^[4]

where ${\textstyle \otimes }$ is the outer product of the flow velocity (${\displaystyle \mathbf {u} }$):

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

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

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.

Compressible flow

Remark: here, the deviatoric stress tensor is denoted ${\textstyle {\boldsymbol {\tau }}}$ as it was in the general continuum equations and in the incompressible flow section.

The compressible momentum Navier–Stokes equation results from the following assumptions on the Cauchy stress tensor:^[5]

• the stress is Galilean invariant: it does not depend directly on the flow velocity, but only on spatial derivatives of the flow velocity. So the stress variable is the tensor gradient ${\textstyle \nabla \mathbf {u} }$, or more simply the rate-of-
  strain tensor
  : ${\textstyle {\boldsymbol {\varepsilon }}\left(\nabla \mathbf {u} \right)\equiv {\frac {1}{2}}\nabla \mathbf {u} +{\frac {1}{2}}\left(\nabla \mathbf {u} \right)^{T}}$
• the deviatoric stress is linear in this variable: ${\textstyle {\boldsymbol {\sigma }}({\boldsymbol {\varepsilon }})=-p\mathbf {I} +\mathbf {C} :{\boldsymbol {\varepsilon }}}$, where ${\textstyle p}$ is independent on the strain rate tensor, ${\textstyle \mathbf {C} }$ is the fourth-order tensor representing the constant of proportionality, called the viscosity or elasticity tensor, and : is the double-dot product.
• the fluid is assumed to be
  isotropic
  , as with gases and simple liquids, and consequently ${\textstyle \mathbf {C} }$ is an isotropic tensor; furthermore, since the deviatoric stress tensor is symmetric, by
  second viscosity
  ${\textstyle \lambda }$ and the
  dynamic viscosity
  ${\textstyle \mu }$, as it is usual in linear elasticity:
  Linear stress constitutive equation (expression similar to the one for elastic solid)

  ${\displaystyle {\boldsymbol {\sigma }}({\boldsymbol {\varepsilon }})=-p\mathbf {I} +\lambda \operatorname {tr} ({\boldsymbol {\varepsilon }})\mathbf {I} +2\mu {\boldsymbol {\varepsilon }}}$

  where ${\textstyle \mathbf {I} }$ is the identity tensor, and ${\textstyle \operatorname {tr} ({\boldsymbol {\varepsilon }})}$ is the trace of the rate-of-strain tensor. So this decomposition can be explicitly defined as:

  $${\displaystyle {\boldsymbol {\sigma }}=-p\mathbf {I} +\lambda (\nabla \cdot \mathbf {u} )\mathbf {I} +\mu \left(\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }\right).}$$

  

Since the trace of the rate-of-strain tensor in three dimensions is the divergence (i.e. rate of expansion) of the flow:

$${\displaystyle \operatorname {tr} ({\boldsymbol {\varepsilon }})=\nabla \cdot \mathbf {u} .}$$

Given this relation, and since the trace of the identity tensor in three dimensions is three:

$${\displaystyle \operatorname {tr} ({\boldsymbol {I}})=3.}$$

the trace of the stress tensor in three dimensions becomes:

$${\displaystyle \operatorname {tr} ({\boldsymbol {\sigma }})=-3p+(3\lambda +2\mu )\nabla \cdot \mathbf {u} .}$$

So by alternatively decomposing the stress tensor into isotropic and deviatoric parts, as usual in fluid dynamics:^[6]

$${\displaystyle {\boldsymbol {\sigma }}=-\left[p-\left(\lambda +{\tfrac {2}{3}}\mu \right)\left(\nabla \cdot \mathbf {u} \right)\right]\mathbf {I} +\mu \left(\nabla \mathbf {u} +\left(\nabla \mathbf {u} \right)^{\mathrm {T} }-{\tfrac {2}{3}}\left(\nabla \cdot \mathbf {u} \right)\mathbf {I} \right)}$$

Introducing the bulk viscosity ${\textstyle \zeta }$,

$${\displaystyle \zeta \equiv \lambda +{\tfrac {2}{3}}\mu ,}$$

we arrive to the linear constitutive equation in the form usually employed in thermal hydraulics:^[5]

which can also be arranged in the other usual form:^[7]

$${\displaystyle {\boldsymbol {\sigma }}=-p\mathbf {I} +\mu \left(\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }\right)+\left(\zeta -{\frac {2}{3}}\mu \right)(\nabla \cdot \mathbf {u} )\mathbf {I} .}$$

Note that in the compressible case the pressure is no more proportional to the isotropic stress term, since there is the additional bulk viscosity term: ${\displaystyle p=-{\frac {1}{3}}\operatorname {tr} ({\boldsymbol {\sigma }})+\zeta (\nabla \cdot \mathbf {u} )}$

and the

${\displaystyle {\boldsymbol {\sigma }}'}$

${\displaystyle {\boldsymbol {\tau }}}$

${\displaystyle {\boldsymbol {\sigma }}'={\boldsymbol {\tau }}=\mu \left[\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }-{\tfrac {2}{3}}(\nabla \cdot \mathbf {u} )\mathbf {I} \right]}$

Both bulk viscosity ${\textstyle \zeta }$ and dynamic viscosity ${\textstyle \mu }$ need not be constant – in general, they depend on two thermodynamics variables if the fluid contains a single chemical species, say for example, pressure and temperature. Any equation that makes explicit one of these transport coefficient in the conservation variables is called an equation of state.^[8]

The most general of the Navier–Stokes equations become

in index notation, the equation can be written as^[9]

The corresponding equation in conservation form can be obtained by considering that, given the mass continuity equation, the left side is equivalent to:

${\displaystyle \rho {\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}={\frac {\partial }{\partial t}}(\rho \mathbf {u} )+\nabla \cdot (\rho \mathbf {u} \otimes \mathbf {u} )}$

To give finally:

Apart from its dependence of pressure and temperature, the second viscosity coefficient also depends on the process, that is to say, the second viscosity coefficient is not just a material property. Example: in the case of a sound wave with a definitive frequency that alternatively compresses and expands a fluid element, the second viscosity coefficient depends on the frequency of the wave. This dependence is called the dispersion. In some cases, the second viscosity ${\textstyle \zeta }$ can be assumed to be constant in which case, the effect of the volume viscosity ${\textstyle \zeta }$ is that the mechanical pressure is not equivalent to the thermodynamic pressure:^[10] as demonstrated below.

$${\displaystyle \nabla \cdot (\nabla \cdot \mathbf {u} )\mathbf {I} =\nabla (\nabla \cdot \mathbf {u} ),}$$

$${\displaystyle {\bar {p}}\equiv p-\zeta \,\nabla \cdot \mathbf {u} ,}$$

${\textstyle \zeta =0}$

${\textstyle \zeta =0}$

If the dynamic μ and bulk ${\displaystyle \zeta }$ viscosities are assumed to be uniform in space, the equations in convective form can be simplified further. By computing the divergence of the stress tensor, since the divergence of tensor ${\textstyle \nabla \mathbf {u} }$ is ${\textstyle \nabla ^{2}\mathbf {u} }$ and the divergence of tensor ${\textstyle \left(\nabla \mathbf {u} \right)^{\mathrm {T} }}$ is ${\textstyle \nabla \left(\nabla \cdot \mathbf {u} \right)}$, one finally arrives to the compressible Navier–Stokes momentum equation:^[14]

where ${\textstyle {\frac {\mathrm {D} }{\mathrm {D} t}}}$ is the material derivative. ${\displaystyle \nu }$ is the shear

${\displaystyle \xi }$

The convective acceleration term can also be written as

$${\displaystyle \mathbf {u} \cdot \nabla \mathbf {u} =(\nabla \times \mathbf {u} )\times \mathbf {u} +{\tfrac {1}{2}}\nabla \mathbf {u} ^{2},}$$

${\textstyle (\nabla \times \mathbf {u} )\times \mathbf {u} }$

For the special case of an

velocity field with ${\textstyle \nabla \cdot \mathbf {u} =0}$.

Incompressible flow

The incompressible momentum Navier–Stokes equation results from the following assumptions on the Cauchy stress tensor:^[5]

• the stress is Galilean invariant: it does not depend directly on the flow velocity, but only on spatial derivatives of the flow velocity. So the stress variable is the tensor gradient ${\textstyle \nabla \mathbf {u} }$.
• the fluid is assumed to be
  isotropic
  , as with gases and simple liquids, and consequently ${\textstyle {\boldsymbol {\tau }}}$ is an isotropic tensor; furthermore, since the deviatoric stress tensor can be expressed in terms of the
  dynamic viscosity
  ${\textstyle \mu }$:
  Stokes' stress constitutive equation (expression used for incompressible elastic solids)

  ${\displaystyle {\boldsymbol {\tau }}=2\mu {\boldsymbol {\varepsilon }}}$

  where

  $${\displaystyle {\boldsymbol {\varepsilon }}={\tfrac {1}{2}}\left(\mathbf {\nabla u} +\mathbf {\nabla u} ^{\mathrm {T} }\right)}$$

  
  is the rate-of-
  strain tensor. So this decomposition can be made explicit as:^[5]
  Stokes's stress constitutive equation (expression used for incompressible viscous fluids)

  ${\displaystyle {\boldsymbol {\tau }}=\mu \left[\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }\right]}$

This is constitutive equation is also called the Newtonian law of viscosity. Dynamic viscosity μ need not be constant – in incompressible flows it can depend on density and on pressure. Any equation that makes explicit one of these transport coefficient in the conservative variables is called an equation of state.^[8]

The divergence of the deviatoric stress in case of uniform viscosity is given by:

$${\displaystyle \nabla \cdot {\boldsymbol {\tau }}=2\mu \nabla \cdot {\boldsymbol {\varepsilon }}=\mu \nabla \cdot \left(\nabla \mathbf {u} +\nabla \mathbf {u} ^{\mathrm {T} }\right)=\mu \,\nabla ^{2}\mathbf {u} }$$

${\textstyle \nabla \cdot \mathbf {u} =0}$

Incompressibility rules out density and pressure waves like sound or shock waves, so this simplification is not useful if these phenomena are of interest. The incompressible flow assumption typically holds well with all fluids at low Mach numbers (say up to about Mach 0.3), such as for modelling air winds at normal temperatures.^[16] the incompressible Navier–Stokes equations are best visualized by dividing for the density:^[17]

where ${\textstyle \nu ={\frac {\mu }{\rho }}}$ is called the

If the density is constant throughout the fluid domain, or, in other words, if all fluid elements have the same density, ${\textstyle \rho }$, then we have

where ${\textstyle p/\rho }$ is called the unit pressure head.

In incompressible flows, the pressure field satisfies the

${\displaystyle \nabla ^{2}p=-\rho {\frac {\partial u_{i}}{\partial x_{k}}}{\frac {\partial u_{k}}{\partial x_{i}}}=-\rho {\frac {\partial ^{2}u_{i}u_{k}}{\partial x_{k}x_{i}}},}$

which is obtained by taking the divergence of the momentum equations.

A laminar flow example

Velocity profile (laminar flow):

$${\displaystyle u_{x}=u(y),\quad u_{y}=0,\quad u_{z}=0}$$

for the x-direction, simplify the Navier–Stokes equation:

$${\displaystyle 0=-{\frac {\mathrm {d} P}{\mathrm {d} x}}+\mu \left({\frac {\mathrm {d} ^{2}u}{\mathrm {d} y^{2}}}\right)}$$

Integrate twice to find the velocity profile with boundary conditions y = h, u = 0, y = −h, u = 0:

$${\displaystyle u={\frac {1}{2\mu }}{\frac {\mathrm {d} P}{\mathrm {d} x}}y^{2}+Ay+B}$$

From this equation, substitute in the two boundary conditions to get two equations:

$${\displaystyle {\begin{aligned}0&={\frac {1}{2\mu }}{\frac {\mathrm {d} P}{\mathrm {d} x}}h^{2}+Ah+B\\0&={\frac {1}{2\mu }}{\frac {\mathrm {d} P}{\mathrm {d} x}}h^{2}-Ah+B\end{aligned}}}$$

Add and solve for B:

$${\displaystyle B=-{\frac {1}{2\mu }}{\frac {\mathrm {d} P}{\mathrm {d} x}}h^{2}}$$

Substitute and solve for A:

$${\displaystyle A=0}$$

Finally this gives the velocity profile:

$${\displaystyle u={\frac {1}{2\mu }}{\frac {\mathrm {d} P}{\mathrm {d} x}}\left(y^{2}-h^{2}\right)}$$

It is well worth observing the meaning of each term (compare to the Cauchy momentum equation):

$${\displaystyle \overbrace {{\vphantom {\frac {}{}}}\underbrace {\frac {\partial \mathbf {u} }{\partial t}} _{\text{Variation}}+\underbrace {{\vphantom {\frac {}{}}}(\mathbf {u} \cdot \nabla )\mathbf {u} } _{\text{Divergence}}} ^{\text{Inertia (per volume)}}=\overbrace {{\vphantom {\frac {\partial }{\partial }}}\underbrace {{\vphantom {\frac {}{}}}-\nabla w} _{\begin{smallmatrix}{\text{Internal}}\\{\text{source}}\end{smallmatrix}}+\underbrace {{\vphantom {\frac {}{}}}\nu \nabla ^{2}\mathbf {u} } _{\text{Diffusion}}} ^{\text{Divergence of stress}}+\underbrace {{\vphantom {\frac {}{}}}\mathbf {g} } _{\begin{smallmatrix}{\text{External}}\\{\text{source}}\end{smallmatrix}}.}$$

The higher-order term, namely the shear stress divergence ${\textstyle \nabla \cdot {\boldsymbol {\tau }}}$, has simply reduced to the

${\textstyle \mu \nabla ^{2}\mathbf {u} }$

heat conduction. In fact neglecting the convection term, incompressible Navier–Stokes equations lead to a vector diffusion equation (namely Stokes equations), but in general the convection term is present, so incompressible Navier–Stokes equations belong to the class of convection–diffusion equations

.

In the usual case of an external field being a

conservative field

:

$${\displaystyle \mathbf {g} =-\nabla \varphi }$$

by defining the hydraulic head:

$${\displaystyle h\equiv w+\varphi }$$

one can finally condense the whole source in one term, arriving to the incompressible Navier–Stokes equation with conservative external field:

$${\displaystyle {\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} -\nu \,\nabla ^{2}\mathbf {u} =-\nabla h.}$$

The incompressible Navier–Stokes equations with uniform density and viscosity and conservative external field is the fundamental equation of

orthogonal coordinate reference frame is usually set to explicit the system of scalar partial differential equations to be solved. In 3-dimensional orthogonal coordinate systems are 3: Cartesian, cylindrical, and spherical. Expressing the Navier–Stokes vector equation in Cartesian coordinates is quite straightforward and not much influenced by the number of dimensions of the euclidean space employed, and this is the case also for the first-order terms (like the variation and convection ones) also in non-cartesian orthogonal coordinate systems. But for the higher order terms (the two coming from the divergence of the deviatoric stress that distinguish Navier–Stokes equations from Euler equations) some tensor calculus

is required for deducing an expression in non-cartesian orthogonal coordinate systems. A special case of the fundamental equation of hydraulics is the

Bernoulli's equation

.

The incompressible Navier–Stokes equation is composite, the sum of two orthogonal equations,

$${\displaystyle {\begin{aligned}{\frac {\partial \mathbf {u} }{\partial t}}&=\Pi ^{S}\left(-(\mathbf {u} \cdot \nabla )\mathbf {u} +\nu \,\nabla ^{2}\mathbf {u} \right)+\mathbf {f} ^{S}\\\rho ^{-1}\,\nabla p&=\Pi ^{I}\left(-(\mathbf {u} \cdot \nabla )\mathbf {u} +\nu \,\nabla ^{2}\mathbf {u} \right)+\mathbf {f} ^{I}\end{aligned}}}$$

where ${\textstyle \Pi ^{S}}$ and ${\textstyle \Pi ^{I}}$ are solenoidal and irrotational projection operators satisfying ${\textstyle \Pi ^{S}+\Pi ^{I}=1}$, and ${\textstyle \mathbf {f} ^{S}}$ and ${\textstyle \mathbf {f} ^{I}}$ are the non-conservative and conservative parts of the body force. This result follows from the Helmholtz theorem (also known as the fundamental theorem of vector calculus). The first equation is a pressureless governing equation for the velocity, while the second equation for the pressure is a functional of the velocity and is related to the pressure Poisson equation.

The explicit functional form of the projection operator in 3D is found from the Helmholtz Theorem:

$${\displaystyle \Pi ^{S}\,\mathbf {F} (\mathbf {r} )={\frac {1}{4\pi }}\nabla \times \int {\frac {\nabla ^{\prime }\times \mathbf {F} (\mathbf {r} ')}{|\mathbf {r} -\mathbf {r} '|}}\,\mathrm {d} V',\quad \Pi ^{I}=1-\Pi ^{S}}$$

with a similar structure in 2D. Thus the governing equation is an integro-differential equation similar to Coulomb and Biot–Savart law, not convenient for numerical computation.

An equivalent weak or variational form of the equation, proved to produce the same velocity solution as the Navier–Stokes equation,^[19] is given by,

$${\displaystyle \left(\mathbf {w} ,{\frac {\partial \mathbf {u} }{\partial t}}\right)=-{\bigl (}\mathbf {w} ,\left(\mathbf {u} \cdot \nabla \right)\mathbf {u} {\bigr )}-\nu \left(\nabla \mathbf {w} :\nabla \mathbf {u} \right)+\left(\mathbf {w} ,\mathbf {f} ^{S}\right)}$$

for divergence-free test functions ${\textstyle \mathbf {w} }$ satisfying appropriate boundary conditions. Here, the projections are accomplished by the orthogonality of the solenoidal and irrotational function spaces. The discrete form of this is eminently suited to finite element computation of divergence-free flow, as we shall see in the next section. There one will be able to address the question "How does one specify pressure-driven (Poiseuille) problems with a pressureless governing equation?".

The absence of pressure forces from the governing velocity equation demonstrates that the equation is not a dynamic one, but rather a kinematic equation where the divergence-free condition serves the role of a conservation equation. This all would seem to refute the frequent statements that the incompressible pressure enforces the divergence-free condition.

Weak form of the incompressible Navier–Stokes equations

Strong form

Consider the incompressible Navier–Stokes equations for a Newtonian fluid of constant density ${\textstyle \rho }$ in a domain

$${\displaystyle \Omega \subset \mathbb {R} ^{d}\quad (d=2,3)}$$

with boundary

$${\displaystyle \partial \Omega =\Gamma _{D}\cup \Gamma _{N},}$$

being ${\textstyle \Gamma _{D}}$ and ${\textstyle \Gamma _{N}}$ portions of the boundary where respectively a Dirichlet and a Neumann boundary condition is applied (${\textstyle \Gamma _{D}\cap \Gamma _{N}=\emptyset }$):^[20]

$${\displaystyle {\begin{cases}\rho {\dfrac {\partial \mathbf {u} }{\partial t}}+\rho (\mathbf {u} \cdot \nabla )\mathbf {u} -\nabla \cdot {\boldsymbol {\sigma }}(\mathbf {u} ,p)=\mathbf {f} &{\text{ in }}\Omega \times (0,T)\\\nabla \cdot \mathbf {u} =0&{\text{ in }}\Omega \times (0,T)\\\mathbf {u} =\mathbf {g} &{\text{ on }}\Gamma _{D}\times (0,T)\\{\boldsymbol {\sigma }}(\mathbf {u} ,p){\hat {\mathbf {n} }}=\mathbf {h} &{\text{ on }}\Gamma _{N}\times (0,T)\\\mathbf {u} (0)=\mathbf {u} _{0}&{\text{ in }}\Omega \times \{0\}\end{cases}}}$$

${\textstyle \mathbf {u} }$ is the fluid velocity, ${\textstyle p}$ the fluid pressure, ${\textstyle \mathbf {f} }$ a given forcing term, ${\displaystyle {\hat {\mathbf {n} }}}$ the outward directed unit normal vector to ${\textstyle \Gamma _{N}}$, and ${\textstyle {\boldsymbol {\sigma }}(\mathbf {u} ,p)}$ the viscous stress tensor defined as:^[20]

$${\displaystyle {\boldsymbol {\sigma }}(\mathbf {u} ,p)=-p\mathbf {I} +2\mu {\boldsymbol {\varepsilon }}(\mathbf {u} ).}$$

Let ${\textstyle \mu }$ be the dynamic viscosity of the fluid, ${\textstyle \mathbf {I} }$ the second-order identity tensor and ${\textstyle {\boldsymbol {\varepsilon }}(\mathbf {u} )}$ the strain-rate tensor defined as:^[20]

$${\displaystyle {\boldsymbol {\varepsilon }}(\mathbf {u} )={\frac {1}{2}}\left(\left(\nabla \mathbf {u} \right)+\left(\nabla \mathbf {u} \right)^{\mathrm {T} }\right).}$$

The functions ${\textstyle \mathbf {g} }$ and ${\textstyle \mathbf {h} }$ are given Dirichlet and Neumann boundary data, while ${\textstyle \mathbf {u} _{0}}$ is the initial condition. The first equation is the momentum balance equation, while the second represents the mass conservation, namely the continuity equation. Assuming constant dynamic viscosity, using the vectorial identity

$${\displaystyle \nabla \cdot \left(\nabla \mathbf {f} \right)^{\mathrm {T} }=\nabla (\nabla \cdot \mathbf {f} )}$$

and exploiting mass conservation, the divergence of the total stress tensor in the momentum equation can also be expressed as:^[20]

$${\displaystyle {\begin{aligned}\nabla \cdot {\boldsymbol {\sigma }}(\mathbf {u} ,p)&=\nabla \cdot \left(-p\mathbf {I} +2\mu {\boldsymbol {\varepsilon }}(\mathbf {u} )\right)\\&=-\nabla p+2\mu \nabla \cdot {\boldsymbol {\varepsilon }}(\mathbf {u} )\\&=-\nabla p+2\mu \nabla \cdot \left[{\tfrac {1}{2}}\left(\left(\nabla \mathbf {u} \right)+\left(\nabla \mathbf {u} \right)^{\mathrm {T} }\right)\right]\\&=-\nabla p+\mu \left(\Delta \mathbf {u} +\nabla \cdot \left(\nabla \mathbf {u} \right)^{\mathrm {T} }\right)\\&=-\nabla p+\mu {\bigl (}\Delta \mathbf {u} +\nabla \underbrace {(\nabla \cdot \mathbf {u} )} _{=0}{\bigr )}=-\nabla p+\mu \,\Delta \mathbf {u} .\end{aligned}}}$$

Moreover, note that the Neumann boundary conditions can be rearranged as:^[20]

$${\displaystyle {\boldsymbol {\sigma }}(\mathbf {u} ,p){\hat {\mathbf {n} }}=\left(-p\mathbf {I} +2\mu {\boldsymbol {\varepsilon }}(\mathbf {u} )\right){\hat {\mathbf {n} }}=-p{\hat {\mathbf {n} }}+\mu {\frac {\partial {\boldsymbol {u}}}{\partial {\hat {\mathbf {n} }}}}.}$$

Weak form

In order to find the weak form of the Navier–Stokes equations, firstly, consider the momentum equation^[20]

$${\displaystyle \rho {\frac {\partial \mathbf {u} }{\partial t}}-\mu \Delta \mathbf {u} +\rho (\mathbf {u} \cdot \nabla )\mathbf {u} +\nabla p=\mathbf {f} }$$

multiply it for a test function ${\textstyle \mathbf {v} }$, defined in a suitable space ${\textstyle V}$, and integrate both members with respect to the domain ${\textstyle \Omega }$:^[20]

$${\displaystyle \int \limits _{\Omega }\rho {\frac {\partial \mathbf {u} }{\partial t}}\cdot \mathbf {v} -\int \limits _{\Omega }\mu \Delta \mathbf {u} \cdot \mathbf {v} +\int \limits _{\Omega }\rho (\mathbf {u} \cdot \nabla )\mathbf {u} \cdot \mathbf {v} +\int \limits _{\Omega }\nabla p\cdot \mathbf {v} =\int \limits _{\Omega }\mathbf {f} \cdot \mathbf {v} }$$

Counter-integrating by parts the diffusive and the pressure terms and by using the Gauss' theorem:^[20]

$${\displaystyle {\begin{aligned}-\int \limits _{\Omega }\mu \Delta \mathbf {u} \cdot \mathbf {v} &=\int _{\Omega }\mu \nabla \mathbf {u} \cdot \nabla \mathbf {v} -\int \limits _{\partial \Omega }\mu {\frac {\partial \mathbf {u} }{\partial {\hat {\mathbf {n} }}}}\cdot \mathbf {v} \\\int \limits _{\Omega }\nabla p\cdot \mathbf {v} &=-\int \limits _{\Omega }p\nabla \cdot \mathbf {v} +\int \limits _{\partial \Omega }p\mathbf {v} \cdot {\hat {\mathbf {n} }}\end{aligned}}}$$

Using these relations, one gets:^[20]

$${\displaystyle \int \limits _{\Omega }\rho {\dfrac {\partial \mathbf {u} }{\partial t}}\cdot \mathbf {v} +\int \limits _{\Omega }\mu \nabla \mathbf {u} \cdot \nabla \mathbf {v} +\int \limits _{\Omega }\rho (\mathbf {u} \cdot \nabla )\mathbf {u} \cdot \mathbf {v} -\int \limits _{\Omega }p\nabla \cdot \mathbf {v} =\int \limits _{\Omega }\mathbf {f} \cdot \mathbf {v} +\int \limits _{\partial \Omega }\left(\mu {\frac {\partial \mathbf {u} }{\partial {\hat {\mathbf {n} }}}}-p{\hat {\mathbf {n} }}\right)\cdot \mathbf {v} \quad \forall \mathbf {v} \in V.}$$

In the same fashion, the continuity equation is multiplied for a test function q belonging to a space ${\textstyle Q}$ and integrated in the domain ${\textstyle \Omega }$:^[20]

$${\displaystyle \int \limits _{\Omega }q\nabla \cdot \mathbf {u} =0.\quad \forall q\in Q.}$$

The space functions are chosen as follows:

$${\displaystyle {\begin{aligned}V=\left[H_{0}^{1}(\Omega )\right]^{d}&=\left\{\mathbf {v} \in \left[H^{1}(\Omega )\right]^{d}:\quad \mathbf {v} =\mathbf {0} {\text{ on }}\Gamma _{D}\right\},\\Q&=L^{2}(\Omega )\end{aligned}}}$$

Considering that the test function v vanishes on the Dirichlet boundary and considering the Neumann condition, the integral on the boundary can be rearranged as:^[20]

$${\displaystyle \int \limits _{\partial \Omega }\left(\mu {\frac {\partial \mathbf {u} }{\partial {\hat {\mathbf {n} }}}}-p{\hat {\mathbf {n} }}\right)\cdot \mathbf {v} =\underbrace {\int \limits _{\Gamma _{D}}\left(\mu {\frac {\partial \mathbf {u} }{\partial {\hat {\mathbf {n} }}}}-p{\hat {\mathbf {n} }}\right)\cdot \mathbf {v} } _{\mathbf {v} =\mathbf {0} {\text{ on }}\Gamma _{D}\ }+\int \limits _{\Gamma _{N}}\underbrace {{\vphantom {\int \limits _{\Gamma _{N}}}}\left(\mu {\frac {\partial \mathbf {u} }{\partial {\hat {\mathbf {n} }}}}-p{\hat {\mathbf {n} }}\right)} _{=\mathbf {h} {\text{ on }}\Gamma _{N}}\cdot \mathbf {v} =\int \limits _{\Gamma _{N}}\mathbf {h} \cdot \mathbf {v} .}$$

Having this in mind, the weak formulation of the Navier–Stokes equations is expressed as:^[20]

$${\displaystyle {\begin{aligned}&{\text{find }}\mathbf {u} \in L^{2}\left(\mathbb {R} ^{+}\;\left[H^{1}(\Omega )\right]^{d}\right)\cap C^{0}\left(\mathbb {R} ^{+}\;\left[L^{2}(\Omega )\right]^{d}\right){\text{ such that: }}\\[5pt]&\quad {\begin{cases}\displaystyle \int \limits _{\Omega }\rho {\dfrac {\partial \mathbf {u} }{\partial t}}\cdot \mathbf {v} +\int \limits _{\Omega }\mu \nabla \mathbf {u} \cdot \nabla \mathbf {v} +\int \limits _{\Omega }\rho (\mathbf {u} \cdot \nabla )\mathbf {u} \cdot \mathbf {v} -\int \limits _{\Omega }p\nabla \cdot \mathbf {v} =\int \limits _{\Omega }\mathbf {f} \cdot \mathbf {v} +\int \limits _{\Gamma _{N}}\mathbf {h} \cdot \mathbf {v} \quad \forall \mathbf {v} \in V,\\\displaystyle \int \limits _{\Omega }q\nabla \cdot \mathbf {u} =0\quad \forall q\in Q.\end{cases}}\end{aligned}}}$$

Discrete velocity

With partitioning of the problem domain and defining basis functions on the partitioned domain, the discrete form of the governing equation is

$${\displaystyle \left(\mathbf {w} _{i},{\frac {\partial \mathbf {u} _{j}}{\partial t}}\right)=-{\bigl (}\mathbf {w} _{i},\left(\mathbf {u} \cdot \nabla \right)\mathbf {u} _{j}{\bigr )}-\nu \left(\nabla \mathbf {w} _{i}:\nabla \mathbf {u} _{j}\right)+\left(\mathbf {w} _{i},\mathbf {f} ^{S}\right).}$$

It is desirable to choose basis functions that reflect the essential feature of incompressible flow – the elements must be divergence-free. While the velocity is the variable of interest, the existence of the stream function or vector potential is necessary by the Helmholtz theorem. Further, to determine fluid flow in the absence of a pressure gradient, one can specify the difference of stream function values across a 2D channel, or the line integral of the tangential component of the vector potential around the channel in 3D, the flow being given by Stokes' theorem. Discussion will be restricted to 2D in the following.

We further restrict discussion to continuous Hermite finite elements which have at least first-derivative degrees-of-freedom. With this, one can draw a large number of candidate triangular and rectangular elements from the plate-bending literature. These elements have derivatives as components of the gradient. In 2D, the gradient and curl of a scalar are clearly orthogonal, given by the expressions,

$${\displaystyle {\begin{aligned}\nabla \varphi &=\left({\frac {\partial \varphi }{\partial x}},\,{\frac {\partial \varphi }{\partial y}}\right)^{\mathrm {T} },\\[5pt]\nabla \times \varphi &=\left({\frac {\partial \varphi }{\partial y}},\,-{\frac {\partial \varphi }{\partial x}}\right)^{\mathrm {T} }.\end{aligned}}}$$

Adopting continuous plate-bending elements, interchanging the derivative degrees-of-freedom and changing the sign of the appropriate one gives many families of stream function elements.

Taking the curl of the scalar stream function elements gives divergence-free velocity elements.^[21][22] The requirement that the stream function elements be continuous assures that the normal component of the velocity is continuous across element interfaces, all that is necessary for vanishing divergence on these interfaces.

Boundary conditions are simple to apply. The stream function is constant on no-flow surfaces, with no-slip velocity conditions on surfaces. Stream function differences across open channels determine the flow. No boundary conditions are necessary on open boundaries, though consistent values may be used with some problems. These are all Dirichlet conditions.

The algebraic equations to be solved are simple to set up, but of course are non-linear, requiring iteration of the linearized equations.

Similar considerations apply to three-dimensions, but extension from 2D is not immediate because of the vector nature of the potential, and there exists no simple relation between the gradient and the curl as was the case in 2D.

Pressure recovery

Recovering pressure from the velocity field is easy. The discrete weak equation for the pressure gradient is,

$${\displaystyle (\mathbf {g} _{i},\nabla p)=-\left(\mathbf {g} _{i},\left(\mathbf {u} \cdot \nabla \right)\mathbf {u} _{j}\right)-\nu \left(\nabla \mathbf {g} _{i}:\nabla \mathbf {u} _{j}\right)+\left(\mathbf {g} _{i},\mathbf {f} ^{I}\right)}$$

where the test/weight functions are irrotational. Any conforming scalar finite element may be used. However, the pressure gradient field may also be of interest. In this case, one can use scalar Hermite elements for the pressure. For the test/weight functions ${\textstyle \mathbf {g} _{i}}$ one would choose the irrotational vector elements obtained from the gradient of the pressure element.

Non-inertial frame of reference

The rotating frame of reference introduces some interesting pseudo-forces into the equations through the material derivative term. Consider a stationary inertial frame of reference ${\textstyle K}$ , and a non-inertial frame of reference ${\textstyle K'}$, which is translating with velocity ${\textstyle \mathbf {U} (t)}$ and rotating with angular velocity ${\textstyle \Omega (t)}$ with respect to the stationary frame. The Navier–Stokes equation observed from the non-inertial frame then becomes

Navier–Stokes momentum equation in non-inertial frame

${\displaystyle \rho \left({\frac {\partial \mathbf {u} }{\partial t}}+(\mathbf {u} \cdot \nabla )\mathbf {u} \right)=-\nabla p+\nabla \cdot \left\{\mu \left[\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }-{\tfrac {2}{3}}(\nabla \cdot \mathbf {u} )\mathbf {I} \right]\right\}+\nabla [\zeta (\nabla \cdot \mathbf {u} )]+\rho \mathbf {f} -\rho \left[2\mathbf {\Omega } \times \mathbf {u} +\mathbf {\Omega } \times (\mathbf {\Omega } \times \mathbf {x} )+{\frac {\mathrm {d} \mathbf {U} }{\mathrm {d} t}}+{\frac {\mathrm {d} \mathbf {\Omega } }{\mathrm {d} t}}\times \mathbf {x} \right].}$

Here ${\textstyle \mathbf {x} }$ and ${\textstyle \mathbf {u} }$ are measured in the non-inertial frame. The first term in the parenthesis represents

Coriolis acceleration, the second term is due to centrifugal acceleration

, the third is due to the linear acceleration of ${\textstyle K'}$ with respect to ${\textstyle K}$ and the fourth term is due to the angular acceleration of ${\textstyle K'}$ with respect to ${\textstyle K}$.

Other equations

The Navier–Stokes equations are strictly a statement of the balance of momentum. To fully describe fluid flow, more information is needed, how much depending on the assumptions made. This additional information may include boundary data (no-slip, capillary surface, etc.), conservation of mass, balance of energy, and/or an equation of state.

Continuity equation for incompressible fluid

Main article: Continuity equation

Regardless of the flow assumptions, a statement of the conservation of mass is generally necessary. This is achieved through the mass continuity equation, as discussed above in the "General continuum equations" within this article, as follows:

$${\displaystyle {\begin{aligned}{\frac {\mathbf {D} m}{\mathbf {Dt} }}&={\iiint \limits _{V}}({{\frac {\mathbf {D} \rho }{\mathbf {Dt} }}+\rho (\nabla \cdot \mathbf {u} )})dV\\{\frac {\mathbf {D} \rho }{\mathbf {Dt} }}+\rho (\nabla \cdot {\mathbf {u} })&={\frac {\partial \rho }{\partial t}}+({\nabla \rho })\cdot {\mathbf {u} }+{\rho }(\nabla \cdot \mathbf {u} )={\frac {\partial \rho }{\partial t}}+\nabla \cdot ({\rho \mathbf {u} })=0\end{aligned}}}$$

A fluid media for which the density (${\displaystyle \rho }$) is constant is called incompressible. Therefore, the rate of change of density (${\displaystyle \rho }$) with respect to time ${\displaystyle ({\frac {\partial \rho }{\partial t}})}$ and the gradient of density ${\displaystyle (\nabla \rho )}$ are equal to zero ${\displaystyle (0)}$. In this case the general equation of continuity, ${\displaystyle {\frac {\partial \rho }{\partial t}}+\nabla ({\rho \mathbf {u} })=0}$, reduces to: ${\displaystyle \rho (\nabla {\cdot }{\mathbf {u} })=0}$. Furthermore, assuming that density (${\displaystyle \rho }$) is a non-zero constant ${\displaystyle (\rho \neq 0)}$ means that the right-hand side of the equation ${\displaystyle (0)}$ is divisible by density (${\displaystyle \rho }$). Therefore, the continuity equation for an incompressible fluid reduces further to:

$${\displaystyle (\nabla {\cdot {\mathbf {u} }})=0}$$

This relationship, ${\textstyle (\nabla {\cdot {\mathbf {u} }})=0}$, identifies that the divergence of the flow velocity vector (${\displaystyle \mathbf {u} }$) is equal to zero ${\displaystyle (0)}$, which means that for an

vector Laplace operator

${\displaystyle (\nabla ^{2}\mathbf {u} =\nabla (\nabla \cdot \mathbf {u} )-\nabla \times (\nabla \times \mathbf {u} ))}$, and vorticity ${\displaystyle ({\vec {\omega }}=\nabla \times \mathbf {u} )}$ which is now expressed like so, for an incompressible fluid:

$${\displaystyle \nabla ^{2}\mathbf {u} =-(\nabla \times (\nabla \times \mathbf {u} ))=-(\nabla \times {\vec {\omega }})}$$

Stream function for incompressible 2D fluid

Taking the curl of the incompressible Navier–Stokes equation results in the elimination of pressure. This is especially easy to see if 2D Cartesian flow is assumed (like in the degenerate 3D case with ${\textstyle u_{z}=0}$ and no dependence of anything on ${\textstyle z}$), where the equations reduce to:

$${\displaystyle {\begin{aligned}\rho \left({\frac {\partial u_{x}}{\partial t}}+u_{x}{\frac {\partial u_{x}}{\partial x}}+u_{y}{\frac {\partial u_{x}}{\partial y}}\right)&=-{\frac {\partial p}{\partial x}}+\mu \left({\frac {\partial ^{2}u_{x}}{\partial x^{2}}}+{\frac {\partial ^{2}u_{x}}{\partial y^{2}}}\right)+\rho g_{x}\\\rho \left({\frac {\partial u_{y}}{\partial t}}+u_{x}{\frac {\partial u_{y}}{\partial x}}+u_{y}{\frac {\partial u_{y}}{\partial y}}\right)&=-{\frac {\partial p}{\partial y}}+\mu \left({\frac {\partial ^{2}u_{y}}{\partial x^{2}}}+{\frac {\partial ^{2}u_{y}}{\partial y^{2}}}\right)+\rho g_{y}.\end{aligned}}}$$

Differentiating the first with respect to ${\textstyle y}$, the second with respect to ${\textstyle x}$ and subtracting the resulting equations will eliminate pressure and any conservative force. For incompressible flow, defining the stream function ${\textstyle \psi }$ through

$${\displaystyle u_{x}={\frac {\partial \psi }{\partial y}};\quad u_{y}=-{\frac {\partial \psi }{\partial x}}}$$

results in mass continuity being unconditionally satisfied (given the stream function is continuous), and then incompressible Newtonian 2D momentum and mass conservation condense into one equation:

$${\displaystyle {\frac {\partial }{\partial t}}\left(\nabla ^{2}\psi \right)+{\frac {\partial \psi }{\partial y}}{\frac {\partial }{\partial x}}\left(\nabla ^{2}\psi \right)-{\frac {\partial \psi }{\partial x}}{\frac {\partial }{\partial y}}\left(\nabla ^{2}\psi \right)=\nu \nabla ^{4}\psi }$$

where ${\textstyle \nabla ^{4}}$ is the 2D

biharmonic operator

and ${\textstyle \nu }$ is the

kinematic viscosity

, ${\textstyle \nu ={\frac {\mu }{p}}}$. We can also express this compactly using the Jacobian determinant:

$${\displaystyle {\frac {\partial }{\partial t}}\left(\nabla ^{2}\psi \right)+{\frac {\partial \left(\psi ,\nabla ^{2}\psi \right)}{\partial (y,x)}}=\nu \nabla ^{4}\psi .}$$

This single equation together with appropriate boundary conditions describes 2D fluid flow, taking only kinematic viscosity as a parameter. Note that the equation for

creeping flow

results when the left side is assumed zero.

In

axisymmetric flow another stream function formulation, called the Stokes stream function, can be used to describe the velocity components of an incompressible flow with one scalar

function.

The incompressible Navier–Stokes equation is a

differential algebraic equation

, having the inconvenient feature that there is no explicit mechanism for advancing the pressure in time. Consequently, much effort has been expended to eliminate the pressure from all or part of the computational process. The stream function formulation eliminates the pressure but only in two dimensions and at the expense of introducing higher derivatives and elimination of the velocity, which is the primary variable of interest.

Properties

Nonlinearity

The Navier–Stokes equations are

partial differential equations in the general case and so remain in almost every real situation.^[23][24] In some cases, such as one-dimensional flow and Stokes flow (or creeping flow), the equations can be simplified to linear equations. The nonlinearity makes most problems difficult or impossible to solve and is the main contributor to the turbulence

that the equations model.

The nonlinearity is due to

convective acceleration, which is an acceleration associated with the change in velocity over position. Hence, any convective flow, whether turbulent or not, will involve nonlinearity. An example of convective but laminar (nonturbulent) flow would be the passage of a viscous fluid (for example, oil) through a small converging nozzle. Such flows, whether exactly solvable or not, can often be thoroughly studied and understood.^[25]

Turbulence

Turbulence is the time-dependent chaotic behaviour seen in many fluid flows. It is generally believed that it is due to the inertia of the fluid as a whole: the culmination of time-dependent and convective acceleration; hence flows where inertial effects are small tend to be laminar (the Reynolds number quantifies how much the flow is affected by inertia). It is believed, though not known with certainty, that the Navier–Stokes equations describe turbulence properly.^[26]

The numerical solution of the Navier–Stokes equations for turbulent flow is extremely difficult, and due to the significantly different mixing-length scales that are involved in turbulent flow, the stable solution of this requires such a fine mesh resolution that the computational time becomes significantly infeasible for calculation or

SST models, which add a variety of additional equations to bring closure to the RANS equations. Large eddy simulation

(LES) can also be used to solve these equations numerically. This approach is computationally more expensive—in time and in computer memory—than RANS, but produces better results because it explicitly resolves the larger turbulent scales.

Applicability

Further information: Discretization of Navier–Stokes equations

Together with supplemental equations (for example, conservation of mass) and well-formulated boundary conditions, the Navier–Stokes equations seem to model fluid motion accurately; even turbulent flows seem (on average) to agree with real world observations.

The Navier–Stokes equations assume that the fluid being studied is a

capillarity of internal layers in fluids appears for flow with high gradients.^[27] For large Knudsen number of the problem, the Boltzmann equation may be a suitable replacement.^[28]

Failing that, one may have to resort to molecular dynamics or various hybrid methods.^[29]

Another limitation is simply the complicated nature of the equations. Time-tested formulations exist for common fluid families, but the application of the Navier–Stokes equations to less common families tends to result in very complicated formulations and often to open research problems. For this reason, these equations are usually written for

linear; truly general models for the flow of other kinds of fluids (such as blood) do not exist.^[30]

Application to specific problems

The Navier–Stokes equations, even when written explicitly for specific fluids, are rather generic in nature and their proper application to specific problems can be very diverse. This is partly because there is an enormous variety of problems that may be modeled, ranging from as simple as the distribution of static pressure to as complicated as multiphase flow driven by surface tension.

Generally, application to specific problems begins with some flow assumptions and initial/boundary condition formulation, this may be followed by scale analysis to further simplify the problem.

Parallel flow

Assume steady, parallel, one-dimensional, non-convective pressure-driven flow between parallel plates, the resulting scaled (dimensionless) boundary value problem is:

$${\displaystyle {\frac {\mathrm {d} ^{2}u}{\mathrm {d} y^{2}}}=-1;\quad u(0)=u(1)=0.}$$

The boundary condition is the

no slip condition

. This problem is easily solved for the flow field:

$${\displaystyle u(y)={\frac {y-y^{2}}{2}}.}$$

From this point onward, more quantities of interest can be easily obtained, such as viscous drag force or net flow rate.

Radial flow

Difficulties may arise when the problem becomes slightly more complicated. A seemingly modest twist on the parallel flow above would be the radial flow between parallel plates; this involves convection and thus non-linearity. The velocity field may be represented by a function f(z) that must satisfy:

$${\displaystyle {\frac {\mathrm {d} ^{2}f}{\mathrm {d} z^{2}}}+Rf^{2}=-1;\quad f(-1)=f(1)=0.}$$

This

roots of cubic polynomials

). Issues with the actual existence of solutions arise for ${\textstyle R>1.41}$ (approximately; this is not √2), the parameter ${\textstyle R}$ being the Reynolds number with appropriately chosen scales.^[31] This is an example of flow assumptions losing their applicability, and an example of the difficulty in "high" Reynolds number flows.^[31]

Convection

A type of natural convection that can be described by the Navier–Stokes equation is the Rayleigh–Bénard convection. It is one of the most commonly studied convection phenomena because of its analytical and experimental accessibility.

Exact solutions of the Navier–Stokes equations

Some exact solutions to the Navier–Stokes equations exist. Examples of degenerate cases—with the non-linear terms in the Navier–Stokes equations equal to zero—are

Stokes boundary layer. But also, more interesting examples, solutions to the full non-linear equations, exist, such as Jeffery–Hamel flow, Von Kármán swirling flow, stagnation point flow, Landau–Squire jet, and Taylor–Green vortex.^[32][33][34]

Note that the existence of these exact solutions does not imply they are stable: turbulence may develop at higher Reynolds numbers.

Under additional assumptions, the component parts can be separated.[35]

A two-dimensional example

For example, in the case of an unbounded planar domain with two-dimensional — incompressible and stationary — flow in

polar coordinates (r,φ), the velocity components (u_r,u_φ) and pressure p are:^[36]

$${\displaystyle {\begin{aligned}u_{r}&={\frac {A}{r}},\\u_{\varphi }&=B\left({\frac {1}{r}}-r^{{\frac {A}{\nu }}+1}\right),\\p&=-{\frac {A^{2}+B^{2}}{2r^{2}}}-{\frac {2B^{2}\nu r^{\frac {A}{\nu }}}{A}}+{\frac {B^{2}r^{\left({\frac {2A}{\nu }}+2\right)}}{{\frac {2A}{\nu }}+2}}\end{aligned}}}$$

where A and B are arbitrary constants. This solution is valid in the domain r ≥ 1 and for A < −2ν.

In Cartesian coordinates, when the viscosity is zero (ν = 0), this is:

$${\displaystyle {\begin{aligned}\mathbf {v} (x,y)&={\frac {1}{x^{2}+y^{2}}}{\begin{pmatrix}Ax+By\\Ay-Bx\end{pmatrix}},\\p(x,y)&=-{\frac {A^{2}+B^{2}}{2\left(x^{2}+y^{2}\right)}}\end{aligned}}}$$

A three-dimensional example

For example, in the case of an unbounded Euclidean domain with three-dimensional — incompressible, stationary and with zero viscosity (ν = 0) — radial flow in

Cartesian coordinates (x,y,z), the velocity vector v and pressure p are:^{[citation needed}

]

$${\displaystyle {\begin{aligned}\mathbf {v} (x,y,z)&={\frac {A}{x^{2}+y^{2}+z^{2}}}{\begin{pmatrix}x\\y\\z\end{pmatrix}},\\p(x,y,z)&=-{\frac {A^{2}}{2\left(x^{2}+y^{2}+z^{2}\right)}}.\end{aligned}}}$$

There is a singularity at x = y = z = 0.

A three-dimensional steady-state vortex solution

A steady-state example with no singularities comes from considering the flow along the lines of a Hopf fibration. Let ${\textstyle r}$ be a constant radius of the inner coil. One set of solutions is given by:^[37]

$${\displaystyle {\begin{aligned}\rho (x,y,z)&={\frac {3B}{r^{2}+x^{2}+y^{2}+z^{2}}}\\p(x,y,z)&={\frac {-A^{2}B}{\left(r^{2}+x^{2}+y^{2}+z^{2}\right)^{3}}}\\\mathbf {u} (x,y,z)&={\frac {A}{\left(r^{2}+x^{2}+y^{2}+z^{2}\right)^{2}}}{\begin{pmatrix}2(-ry+xz)\\2(rx+yz)\\r^{2}-x^{2}-y^{2}+z^{2}\end{pmatrix}}\\g&=0\\\mu &=0\end{aligned}}}$$

for arbitrary constants ${\textstyle A}$ and ${\textstyle B}$. This is a solution in a non-viscous gas (compressible fluid) whose density, velocities and pressure goes to zero far from the origin. (Note this is not a solution to the Clay Millennium problem because that refers to incompressible fluids where ${\textstyle \rho }$ is a constant, and neither does it deal with the uniqueness of the Navier–Stokes equations with respect to any turbulence properties.) It is also worth pointing out that the components of the velocity vector are exactly those from the Pythagorean quadruple parametrization. Other choices of density and pressure are possible with the same velocity field:

Other choices of density and pressure

Another choice of pressure and density with the same velocity vector above is one where the pressure and density fall to zero at the origin and are highest in the central loop at z = 0, x² + y² = r²:

$${\displaystyle {\begin{aligned}\rho (x,y,z)&={\frac {20B\left(x^{2}+y^{2}\right)}{\left(r^{2}+x^{2}+y^{2}+z^{2}\right)^{3}}}\\p(x,y,z)&={\frac {-A^{2}B}{\left(r^{2}+x^{2}+y^{2}+z^{2}\right)^{4}}}+{\frac {-4A^{2}B\left(x^{2}+y^{2}\right)}{\left(r^{2}+x^{2}+y^{2}+z^{2}\right)^{5}}}.\end{aligned}}}$$

In fact in general there are simple solutions for any polynomial function f where the density is:

$${\displaystyle \rho (x,y,z)={\frac {1}{r^{2}+x^{2}+y^{2}+z^{2}}}f\left({\frac {x^{2}+y^{2}}{\left(r^{2}+x^{2}+y^{2}+z^{2}\right)^{2}}}\right).}$$

Viscous three-dimensional periodic solutions

Two examples of periodic fully-three-dimensional viscous solutions are described in.^[38] These solutions are defined on a three-dimensional torus ${\displaystyle \mathbb {T} ^{3}=[0,L]^{3}}$ and are characterized by positive and negative helicity respectively. The solution with positive helicity is given by:

$${\displaystyle {\begin{aligned}u_{x}&={\frac {4{\sqrt {2}}}{3{\sqrt {3}}}}\,U_{0}\left[\,\sin(kx-\pi /3)\cos(ky+\pi /3)\sin(kz+\pi /2)-\cos(kz-\pi /3)\sin(kx+\pi /3)\sin(ky+\pi /2)\,\right]e^{-3\nu k^{2}t}\\u_{y}&={\frac {4{\sqrt {2}}}{3{\sqrt {3}}}}\,U_{0}\left[\,\sin(ky-\pi /3)\cos(kz+\pi /3)\sin(kx+\pi /2)-\cos(kx-\pi /3)\sin(ky+\pi /3)\sin(kz+\pi /2)\,\right]e^{-3\nu k^{2}t}\\u_{z}&={\frac {4{\sqrt {2}}}{3{\sqrt {3}}}}\,U_{0}\left[\,\sin(kz-\pi /3)\cos(kx+\pi /3)\sin(ky+\pi /2)-\cos(ky-\pi /3)\sin(kz+\pi /3)\sin(kx+\pi /2)\,\right]e^{-3\nu k^{2}t}\end{aligned}}}$$

where ${\displaystyle k=2\pi /L}$ is the wave number and the velocity components are normalized so that the average kinetic energy per unit of mass is ${\displaystyle U_{0}^{2}/2}$ at ${\displaystyle t=0}$. The pressure field is obtained from the velocity field as ${\displaystyle p=p_{0}-\rho _{0}\|{\boldsymbol {u}}\|^{2}/2}$ (where ${\displaystyle p_{0}}$ and ${\displaystyle \rho _{0}}$ are reference values for the pressure and density fields respectively). Since both the solutions belong to the class of Beltrami flow, the vorticity field is parallel to the velocity and, for the case with positive helicity, is given by ${\displaystyle \omega ={\sqrt {3}}\,k\,{\boldsymbol {u}}}$. These solutions can be regarded as a generalization in three dimensions of the classic two-dimensional Taylor-Green Taylor–Green vortex.

Wyld diagrams

Wyld diagrams are bookkeeping

pseudo-random functions in probability distributions.^[39]

Representations in 3D

Note that the formulas in this section make use of the single-line notation for partial derivatives, where, e.g. ${\textstyle \partial _{x}u}$ means the partial derivative of ${\textstyle u}$ with respect to ${\textstyle x}$, and ${\textstyle \partial _{y}^{2}f_{\theta }}$ means the second-order partial derivative of ${\textstyle f_{\theta }}$ with respect to ${\textstyle y}$.

A 2022 paper provides a less costly, dynamical and recurrent solution of the Navier-Stokes equation for 3D turbulent fluid flows. On suitably short time scales, the dynamics of turbulence is deterministic.[40]

Cartesian coordinates

From the general form of the Navier–Stokes, with the velocity vector expanded as ${\textstyle \mathbf {u} =(u_{x},u_{y},u_{z})}$, sometimes respectively named ${\textstyle u}$, ${\textstyle v}$, ${\textstyle w}$, we may write the vector equation explicitly,

$${\displaystyle {\begin{aligned}x:\ &\rho \left({\partial _{t}u_{x}}+u_{x}\,{\partial _{x}u_{x}}+u_{y}\,{\partial _{y}u_{x}}+u_{z}\,{\partial _{z}u_{x}}\right)\\&\quad =-\partial _{x}p+\mu \left({\partial _{x}^{2}u_{x}}+{\partial _{y}^{2}u_{x}}+{\partial _{z}^{2}u_{x}}\right)+{\frac {1}{3}}\mu \ \partial _{x}\left({\partial _{x}u_{x}}+{\partial _{y}u_{y}}+{\partial _{z}u_{z}}\right)+\rho g_{x}\\\end{aligned}}}$$

$${\displaystyle {\begin{aligned}y:\ &\rho \left({\partial _{t}u_{y}}+u_{x}{\partial _{x}u_{y}}+u_{y}{\partial _{y}u_{y}}+u_{z}{\partial _{z}u_{y}}\right)\\&\quad =-{\partial _{y}p}+\mu \left({\partial _{x}^{2}u_{y}}+{\partial _{y}^{2}u_{y}}+{\partial _{z}^{2}u_{y}}\right)+{\frac {1}{3}}\mu \ \partial _{y}\left({\partial _{x}u_{x}}+{\partial _{y}u_{y}}+{\partial _{z}u_{z}}\right)+\rho g_{y}\\\end{aligned}}}$$

$${\displaystyle {\begin{aligned}z:\ &\rho \left({\partial _{t}u_{z}}+u_{x}{\partial _{x}u_{z}}+u_{y}{\partial _{y}u_{z}}+u_{z}{\partial _{z}u_{z}}\right)\\&\quad =-{\partial _{z}p}+\mu \left({\partial _{x}^{2}u_{z}}+{\partial _{y}^{2}u_{z}}+{\partial _{z}^{2}u_{z}}\right)+{\frac {1}{3}}\mu \ \partial _{z}\left({\partial _{x}u_{x}}+{\partial _{y}u_{y}}+{\partial _{z}u_{z}}\right)+\rho g_{z}.\end{aligned}}}$$

Note that gravity has been accounted for as a body force, and the values of ${\textstyle g_{x}}$, ${\textstyle g_{y}}$, ${\textstyle g_{z}}$ will depend on the orientation of gravity with respect to the chosen set of coordinates.

The continuity equation reads:

$${\displaystyle \partial _{t}\rho +\partial _{x}(\rho u_{x})+\partial _{y}(\rho u_{y})+\partial _{z}(\rho u_{z})=0.}$$

When the flow is incompressible, ${\textstyle \rho }$ does not change for any fluid particle, and its material derivative vanishes: ${\textstyle {\frac {\mathrm {D} \rho }{\mathrm {D} t}}=0}$. The continuity equation is reduced to:

$${\displaystyle \partial _{x}u_{x}+\partial _{y}u_{y}+\partial _{z}u_{z}=0.}$$

Thus, for the incompressible version of the Navier–Stokes equation the second part of the viscous terms fall away (see Incompressible flow).

This system of four equations comprises the most commonly used and studied form. Though comparatively more compact than other representations, this is still a

partial differential equations

for which solutions are difficult to obtain.

Cylindrical coordinates

A change of variables on the Cartesian equations will yield^[16] the following momentum equations for ${\textstyle r}$, ${\textstyle \phi }$, and ${\textstyle z}$^[41]

$${\displaystyle {\begin{aligned}r:\ &\rho \left({\partial _{t}u_{r}}+u_{r}{\partial _{r}u_{r}}+{\frac {u_{\varphi }}{r}}{\partial _{\varphi }u_{r}}+u_{z}{\partial _{z}u_{r}}-{\frac {u_{\varphi }^{2}}{r}}\right)\\&\quad =-{\partial _{r}p}\\&\qquad +\mu \left({\frac {1}{r}}\partial _{r}\left(r{\partial _{r}u_{r}}\right)+{\frac {1}{r^{2}}}{\partial _{\varphi }^{2}u_{r}}+{\partial _{z}^{2}u_{r}}-{\frac {u_{r}}{r^{2}}}-{\frac {2}{r^{2}}}{\partial _{\varphi }u_{\varphi }}\right)\\&\qquad +{\frac {1}{3}}\mu \partial _{r}\left({\frac {1}{r}}{\partial _{r}\left(ru_{r}\right)}+{\frac {1}{r}}{\partial _{\varphi }u_{\varphi }}+{\partial _{z}u_{z}}\right)\\&\qquad +\rho g_{r}\\[8px]\end{aligned}}}$$

$${\displaystyle {\begin{aligned}\varphi :\ &\rho \left({\partial _{t}u_{\varphi }}+u_{r}{\partial _{r}u_{\varphi }}+{\frac {u_{\varphi }}{r}}{\partial _{\varphi }u_{\varphi }}+u_{z}{\partial _{z}u_{\varphi }}+{\frac {u_{r}u_{\varphi }}{r}}\right)\\&\quad =-{\frac {1}{r}}{\partial _{\varphi }p}\\&\qquad +\mu \left({\frac {1}{r}}\ \partial _{r}\left(r{\partial _{r}u_{\varphi }}\right)+{\frac {1}{r^{2}}}{\partial _{\varphi }^{2}u_{\varphi }}+{\partial _{z}^{2}u_{\varphi }}-{\frac {u_{\varphi }}{r^{2}}}+{\frac {2}{r^{2}}}{\partial _{\varphi }u_{r}}\right)\\&\qquad +{\frac {1}{3}}\mu {\frac {1}{r}}\partial _{\varphi }\left({\frac {1}{r}}{\partial _{r}\left(ru_{r}\right)}+{\frac {1}{r}}{\partial _{\varphi }u_{\varphi }}+{\partial _{z}u_{z}}\right)\\&\qquad +\rho g_{\varphi }\\[8px]\end{aligned}}}$$

$${\displaystyle {\begin{aligned}z:\ &\rho \left({\partial _{t}u_{z}}+u_{r}{\partial _{r}u_{z}}+{\frac {u_{\varphi }}{r}}{\partial _{\varphi }u_{z}}+u_{z}{\partial _{z}u_{z}}\right)\\&\quad =-{\partial _{z}p}\\&\qquad +\mu \left({\frac {1}{r}}\partial _{r}\left(r{\partial _{r}u_{z}}\right)+{\frac {1}{r^{2}}}{\partial _{\varphi }^{2}u_{z}}+{\partial _{z}^{2}u_{z}}\right)\\&\qquad +{\frac {1}{3}}\mu \partial _{z}\left({\frac {1}{r}}{\partial _{r}\left(ru_{r}\right)}+{\frac {1}{r}}{\partial _{\varphi }u_{\varphi }}+{\partial _{z}u_{z}}\right)\\&\qquad +\rho g_{z}.\end{aligned}}}$$

The gravity components will generally not be constants, however for most applications either the coordinates are chosen so that the gravity components are constant or else it is assumed that gravity is counteracted by a pressure field (for example, flow in horizontal pipe is treated normally without gravity and without a vertical pressure gradient). The continuity equation is:

$${\displaystyle {\partial _{t}\rho }+{\frac {1}{r}}\partial _{r}\left(\rho ru_{r}\right)+{\frac {1}{r}}{\partial _{\varphi }\left(\rho u_{\varphi }\right)}+{\partial _{z}\left(\rho u_{z}\right)}=0.}$$

This cylindrical representation of the incompressible Navier–Stokes equations is the second most commonly seen (the first being Cartesian above). Cylindrical coordinates are chosen to take advantage of symmetry, so that a velocity component can disappear. A very common case is axisymmetric flow with the assumption of no tangential velocity (${\textstyle u_{\phi }=0}$), and the remaining quantities are independent of ${\textstyle \phi }$:

$${\displaystyle {\begin{aligned}\rho \left({\partial _{t}u_{r}}+u_{r}{\partial _{r}u_{r}}+u_{z}{\partial _{z}u_{r}}\right)&=-{\partial _{r}p}+\mu \left({\frac {1}{r}}\partial _{r}\left(r{\partial _{r}u_{r}}\right)+{\partial _{z}^{2}u_{r}}-{\frac {u_{r}}{r^{2}}}\right)+\rho g_{r}\\\rho \left({\partial _{t}u_{z}}+u_{r}{\partial _{r}u_{z}}+u_{z}{\partial _{z}u_{z}}\right)&=-{\partial _{z}p}+\mu \left({\frac {1}{r}}\partial _{r}\left(r{\partial _{r}u_{z}}\right)+{\partial _{z}^{2}u_{z}}\right)+\rho g_{z}\\{\frac {1}{r}}\partial _{r}\left(ru_{r}\right)+{\partial _{z}u_{z}}&=0.\end{aligned}}}$$

Spherical coordinates

In

spherical coordinates

, the ${\textstyle r}$, ${\textstyle \phi }$, and ${\textstyle \theta }$ momentum equations are^[16] (note the convention used: ${\textstyle \theta }$ is polar angle, or colatitude,^[42] ${\textstyle 0\leq \theta \leq \pi }$):

$${\displaystyle {\begin{aligned}r:\ &\rho \left({\partial _{t}u_{r}}+u_{r}{\partial _{r}u_{r}}+{\frac {u_{\varphi }}{r\sin \theta }}{\partial _{\varphi }u_{r}}+{\frac {u_{\theta }}{r}}{\partial _{\theta }u_{r}}-{\frac {u_{\varphi }^{2}+u_{\theta }^{2}}{r}}\right)\\&\quad =-{\partial _{r}p}\\&\qquad +\mu \left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}{\partial _{r}u_{r}}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\partial _{\varphi }^{2}u_{r}}+{\frac {1}{r^{2}\sin \theta }}\partial _{\theta }\left(\sin \theta {\partial _{\theta }u_{r}}\right)-2{\frac {u_{r}+{\partial _{\theta }u_{\theta }}+u_{\theta }\cot \theta }{r^{2}}}-{\frac {2}{r^{2}\sin \theta }}{\partial _{\varphi }u_{\varphi }}\right)\\&\qquad +{\frac {1}{3}}\mu \partial _{r}\left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}u_{r}\right)+{\frac {1}{r\sin \theta }}\partial _{\theta }\left(u_{\theta }\sin \theta \right)+{\frac {1}{r\sin \theta }}{\partial _{\varphi }u_{\varphi }}\right)\\&\qquad +\rho g_{r}\\[8px]\end{aligned}}}$$

$${\displaystyle {\begin{aligned}\varphi :\ &\rho \left({\partial _{t}u_{\varphi }}+u_{r}{\partial _{r}u_{\varphi }}+{\frac {u_{\varphi }}{r\sin \theta }}{\partial _{\varphi }u_{\varphi }}+{\frac {u_{\theta }}{r}}{\partial _{\theta }u_{\varphi }}+{\frac {u_{r}u_{\varphi }+u_{\varphi }u_{\theta }\cot \theta }{r}}\right)\\&\quad =-{\frac {1}{r\sin \theta }}{\partial _{\varphi }p}\\&\qquad +\mu \left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}{\partial _{r}u_{\varphi }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\partial _{\varphi }^{2}u_{\varphi }}+{\frac {1}{r^{2}\sin \theta }}\partial _{\theta }\left(\sin \theta {\partial _{\theta }u_{\varphi }}\right)+{\frac {2\sin \theta {\partial _{\varphi }u_{r}}+2\cos \theta {\partial _{\varphi }u_{\theta }}-u_{\varphi }}{r^{2}\sin ^{2}\theta }}\right)\\&\qquad +{\frac {1}{3}}\mu {\frac {1}{r\sin \theta }}\partial _{\varphi }\left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}u_{r}\right)+{\frac {1}{r\sin \theta }}\partial _{\theta }\left(u_{\theta }\sin \theta \right)+{\frac {1}{r\sin \theta }}{\partial _{\varphi }u_{\varphi }}\right)\\&\qquad +\rho g_{\varphi }\\[8px]\end{aligned}}}$$

$${\displaystyle {\begin{aligned}\theta :\ &\rho \left({\partial _{t}u_{\theta }}+u_{r}{\partial _{r}u_{\theta }}+{\frac {u_{\varphi }}{r\sin \theta }}{\partial _{\varphi }u_{\theta }}+{\frac {u_{\theta }}{r}}{\partial _{\theta }u_{\theta }}+{\frac {u_{r}u_{\theta }-u_{\varphi }^{2}\cot \theta }{r}}\right)\\&\quad =-{\frac {1}{r}}{\partial _{\theta }p}\\&\qquad +\mu \left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}{\partial _{r}u_{\theta }}\right)+{\frac {1}{r^{2}\sin ^{2}\theta }}{\partial _{\varphi }^{2}u_{\theta }}+{\frac {1}{r^{2}\sin \theta }}\partial _{\theta }\left(\sin \theta {\partial _{\theta }u_{\theta }}\right)+{\frac {2}{r^{2}}}{\partial _{\theta }u_{r}}-{\frac {u_{\theta }+2\cos \theta {\partial _{\varphi }u_{\varphi }}}{r^{2}\sin ^{2}\theta }}\right)\\&\qquad +{\frac {1}{3}}\mu {\frac {1}{r}}\partial _{\theta }\left({\frac {1}{r^{2}}}\partial _{r}\left(r^{2}u_{r}\right)+{\frac {1}{r\sin \theta }}\partial _{\theta }\left(u_{\theta }\sin \theta \right)+{\frac {1}{r\sin \theta }}{\partial _{\varphi }u_{\varphi }}\right)\\&\qquad +\rho g_{\theta }.\end{aligned}}}$$

Mass continuity will read:

$${\displaystyle {\partial _{t}\rho }+{\frac {1}{r^{2}}}\partial _{r}\left(\rho r^{2}u_{r}\right)+{\frac {1}{r\sin \theta }}{\partial _{\varphi }(\rho u_{\varphi })}+{\frac {1}{r\sin \theta }}\partial _{\theta }\left(\sin \theta \rho u_{\theta }\right)=0.}$$

These equations could be (slightly) compacted by, for example, factoring ${\textstyle {\frac {1}{r^{2}}}}$ from the viscous terms. However, doing so would undesirably alter the structure of the Laplacian and other quantities.

Navier–Stokes equations use in games

The Navier–Stokes equations are used extensively in

video games in order to model a wide variety of natural phenomena. Simulations of small-scale gaseous fluids, such as fire and smoke, are often based on the seminal paper "Real-Time Fluid Dynamics for Games"^[43] by Jos Stam, which elaborates one of the methods proposed in Stam's earlier, more famous paper "Stable Fluids"^[44] from 1999. Stam proposes stable fluid simulation using a Navier–Stokes solution method from 1968, coupled with an unconditionally stable semi-Lagrangian advection

scheme, as first proposed in 1992.

More recent implementations based upon this work run on the game systems graphics processing unit (GPU) as opposed to the central processing unit (CPU) and achieve a much higher degree of performance.^[45][46] Many improvements have been proposed to Stam's original work, which suffers inherently from high numerical dissipation in both velocity and mass.

An introduction to interactive fluid simulation can be found in the 2007 ACM SIGGRAPH course, Fluid Simulation for Computer Animation.^[47]

See also

• Adhémar Jean Claude Barré de Saint-Venant
• Boltzmann equation
• Cauchy momentum equation
• Cauchy stress tensor
• Chapman–Enskog theory
• Churchill–Bernstein equation
• Coandă effect
• Computational fluid dynamics
• Continuum mechanics
• Convection–diffusion equation
• Derivation of the Navier–Stokes equations
• Einstein–Stokes equation
• Euler equations
• Hagen–Poiseuille flow from the Navier–Stokes equations
• Millennium Prize Problems
• Newtonian fluid
• Non-dimensionalization and scaling of the Navier–Stokes equations
• Pressure-correction method
• Primitive equations
• Rayleigh–Bénard convection
• Reynolds transport theorem
• Stokes equations
• Supersonic flow over a flat plate
• Vlasov equation

Citations

1. ISBN 9781119967514
   . The main relationships comprising the NS equations are the basic conservation laws for mass, momentum, and energy. To have a complete equation set we also need an equation of state relating temperature, pressure, and density...
2. ^ "Millennium Prize Problems—Navier–Stokes Equation", claymath.org, Clay Mathematics Institute, March 27, 2017, archived from the original on 2015-12-22, retrieved 2017-04-02
3. ^ Fefferman, Charles L. "Existence and smoothness of the Navier–Stokes equation" (PDF). claymath.org. Clay Mathematics Institute. Archived from the original (PDF) on 2015-04-15. Retrieved 2017-04-02.
4. ^ Batchelor (1967) pp. 137 & 142.
5. ^ ^{a b c d} Batchelor (1967) pp. 142–148.
6. ^ Chorin, Alexandre E.; Marsden, Jerrold E. (1993). A Mathematical Introduction to Fluid Mechanics. p. 33.
7. ^ Bird, Stewart, Lightfoot, Transport Phenomena, 1st ed., 1960, eq. (3.2-11a)
8. ^ ^{a b} Batchelor (1967) p. 165.
9. ^ ^{a b} Landau, Lev Davidovich, and Evgenii Mikhailovich Lifshitz. Fluid mechanics: Landau And Lifshitz: course of theoretical physics, Volume 6. Vol. 6. Elsevier, 2013.
10. ^ Landau & Lifshitz (1987) pp. 44–45, 196
11. ^ White (2006) p. 67.
12. ^ Stokes, G. G. (2007). On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids.
13. ^ Vincenti, W. G., Kruger Jr., C. H. (1975). Introduction to physical gas dynamic. Introduction to physical gas dynamics/Huntington.
14. ^ Batchelor (1967) pp. 147 & 154.
15. ^ Batchelor (1967) p. 75.
16. ^ ^{a b c} See Acheson (1990).
17. ISSN 2227-7390
    .
18. ^ Batchelor (1967) pp. 21 & 147.
19. ^ Temam, Roger (2001), Navier–Stokes Equations, Theory and Numerical Analysis, AMS Chelsea, pp. 107–112
20. ^
    ISBN 978-88-470-5522-3
    .
21. ^ Holdeman, J. T. (2010), "A Hermite finite element method for incompressible fluid flow", Int. J. Numer. Methods Fluids, 64 (4): 376–408,
    S2CID 119882803
22. ^ Holdeman, J. T.; Kim, J. W. (2010), "Computation of incompressible thermal flows using Hermite finite elements", Comput. Meth. Appl. Mech. Eng., 199 (49–52): 3297–3304,
    doi:10.1016/j.cma.2010.06.036
23. ISBN 978-0-07-148781-8
    .
24. ISBN 0-486-66110-5
    .
25. ISBN 0-07-051400-3
    .
26. ^ Encyclopaedia of Physics (2nd Edition), R.G. Lerner, G.L. Trigg, VHC publishers, 1991, ISBN (Verlagsgesellschaft) 3-527-26954-1, ISBN (VHC Inc.) 0-89573-752-3
27. S2CID 55317543
28. ISBN 978-0444503305
29. S2CID 122867563
30. ISBN 9783540583530
31. ^
    Bibcode:1989STIN...9123418S
    .
32. ^ Wang, C. Y. (1991), "Exact solutions of the steady-state Navier–Stokes equations", Annual Review of Fluid Mechanics, 23: 159–177,
    doi:10.1146/annurev.fl.23.010191.001111
33. ^ Landau & Lifshitz (1987) pp. 75–88.
34. ^ Ethier, C. R.; Steinman, D. A. (1994), "Exact fully 3D Navier–Stokes solutions for benchmarking", International Journal for Numerical Methods in Fluids, 19 (5): 369–375,
    doi:10.1002/fld.1650190502
35. ^ "Navier Stokes Equations". www.claudino.webs.com. Archived from the original on 2015-06-19. Retrieved 2023-03-11.
36. ^ Ladyzhenskaya, O. A. (1969), The Mathematical Theory of viscous Incompressible Flow (2nd ed.), p. preface, xi
37. ^ Kamchatno, A. M. (1982), Topological solitons in magnetohydrodynamics (PDF), archived (PDF) from the original on 2016-01-28
38. S2CID 216463266
39. ISBN 978-0-19-923652-7
40. S2CID 251693676
    .
41. ^ de' Michieli Vitturi, Mattia, Navier–Stokes equations in cylindrical coordinates, retrieved 2016-12-26
42. ^ Eric W. Weisstein (2005-10-26), Spherical Coordinates, MathWorld, retrieved 2008-01-22
43. S2CID 9353969, archived from the original
    (PDF) on 2020-08-05
44. ^ Stam, Jos (1999), Stable Fluids (PDF), archived (PDF) from the original on 2019-07-15
45. ^ Harris, Mark J. (2004), "38", GPUGems - Fast Fluid Dynamics Simulation on the GPU
46. ^ Sander, P.; Tatarchuck, N.; Mitchell, J.L. (2007), "9.6", ShaderX5 - Explicit Early-Z Culling for Efficient Fluid Flow Simulation, pp. 553–564
47. ^ Robert Bridson; Matthias Müller-Fischer. "Fluid Simulation for Computer Animation". www.cs.ubc.ca.

General references

• Acheson, D. J. (1990), Elementary Fluid Dynamics, Oxford Applied Mathematics and Computing Science Series,
  ISBN 978-0-19-859679-0
• ISBN 978-0-521-66396-0
• Currie, I. G. (1974), Fundamental Mechanics of Fluids,
  ISBN 978-0-07-015000-3
• V. Girault and P. A. Raviart. Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms. Springer Series in Computational Mathematics. Springer-Verlag, 1986.
• OCLC 15017127
• Polyanin, A. D.; Kutepov, A. M.; Vyazmin, A. V.; Kazenin, D. A. (2002), Hydrodynamics, Mass and Heat Transfer in Chemical Engineering, Taylor & Francis, London,
  ISBN 978-0-415-27237-7
• Rhyming, Inge L. (1991), Dynamique des fluides, Presses polytechniques et universitaires romandes
• Smits, Alexander J. (2014), A Physical Introduction to Fluid Mechanics, Wiley,
  ISBN 0-47-1253499
• Temam, Roger (1984): Navier–Stokes Equations: Theory and Numerical Analysis, ACM Chelsea Publishing,
  ISBN 978-0-8218-2737-6
• ISBN 978-0-07-124493-0

External links

• Simplified derivation of the Navier–Stokes equations
• Three-dimensional unsteady form of the Navier–Stokes equations Glenn Research Center, NASA