Navier–Stokes equations

In physics, the Navier–Stokes equations (/nævˈjeɪ stoʊks/) are certain partial differential equations which describe the motion of viscous fluid substances, named after French engineer and physicist Claude-Louis Navier and Anglo-Irish physicist and mathematician George Gabriel Stokes. They were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes).

The Navier–Stokes equations mathematically express conservation of momentum and conservation of mass for Newtonian fluids. They are sometimes accompanied by an equation of state relating pressure, temperature and density.^[1] They arise from applying Isaac Newton's second law to fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term—hence describing viscous flow. The difference between them and the closely related Euler equations is that Navier–Stokes equations take viscosity into account while the Euler equations model only inviscid flow. As a result, the Navier–Stokes are a parabolic equation and therefore have better analytic properties, at the expense of having less mathematical structure (e.g. they are never completely integrable).

The Navier–Stokes equations are useful because they describe the physics of many phenomena of scientific and engineering interest. They may be used to model the weather, ocean currents, water flow in a pipe and air flow around a wing. The Navier–Stokes equations, in their full and simplified forms, help with the design of aircraft and cars, the study of blood flow, the design of power stations, the analysis of pollution, and many other things. Coupled with Maxwell's equations, they can be used to model and study magnetohydrodynamics.

The Navier–Stokes equations are also of great interest in a purely mathematical sense. Despite their wide range of practical uses, it has not yet been proven whether smooth solutions always exist in three dimensions—i.e. they are infinitely differentiable (or even just bounded) at all points in the domain. This is called the Navier–Stokes existence and smoothness problem. The Clay Mathematics Institute has called this one of the seven most important open problems in mathematics and has offered a US$1 million prize for a solution or a counterexample.^[2]^[3]

Flow velocity[]

The solution of the equations is a flow velocity. It is a vector field—to every point in a fluid, at any moment in a time interval, it gives a vector whose direction and magnitude are those of the velocity of the fluid at that point in space and at that moment in time. It is usually studied in three spatial dimensions and one time dimension, although two (spatial) dimensional and steady-state cases are often used as models, and higher-dimensional analogues are studied in both pure and applied mathematics. Once the velocity field is calculated, other quantities of interest such as pressure or temperature may be found using dynamical equations and relations. This is different from what one normally sees in classical mechanics, where solutions are typically trajectories of position of a particle or deflection of a continuum. Studying velocity instead of position makes more sense for a fluid, although for visualization purposes one can compute various trajectories. In particular, the streamlines of a vector field, interpreted as flow velocity, are the paths along which a massless fluid particle would travel. These paths are the integral curves whose derivative at each point is equal to the vector field, and they can represent visually the behavior of the vector field at a point in time.

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

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

by setting the Cauchy stress tensor

σ

to be the sum of a viscosity term

τ

(the deviatoric stress) and a pressure term

- p I

(volumetric stress) we arrive at

Cauchy momentum equation (convective form)

$\rho {\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}=-\nabla p+\nabla \cdot {\boldsymbol {\tau }}+\rho \,\mathbf {g}$

where

$.mw-parser-output .sfrac{white-space:nowrap}.mw-parser-output .sfrac.tion,.mw-parser-output .sfrac .tion{display:inline-block;vertical-align:-0.5em;font-size:85%;text-align:center}.mw-parser-output .sfrac .num,.mw-parser-output .sfrac .den{display:block;line-height:1em;margin:0 0.1em}.mw-parser-output .sfrac .den{border-top:1px solid}.mw-parser-output .sr-only{border:0;clip:rect(0,0,0,0);height:1px;margin:-1px;overflow:hidden;padding:0;position:absolute;width:1px}D/D t$ is the material derivative, defined as $\partial / \partial t + u \cdot \nabla$ ,
$ρ$ is the density,
$u$ is the flow velocity,
$\nabla \cdot$ is the divergence,
$p$ is the pressure,
$t$ is time,
$τ$ is the deviatoric stress tensor, which has order 2,
$g$ 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 conservation of mass we can use the mass continuity equation (or simply continuity equation),

{\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:^[4]

Cauchy momentum equation (conservation form)

${\frac {\partial }{\partial t}}(\rho \,\mathbf {u} )+\nabla \cdot (\rho \,\mathbf {u} \otimes \mathbf {u} )=-\nabla p+\nabla \cdot {\boldsymbol {\tau }}+\rho \,\mathbf {g}$

where $\otimes$ is the outer product:

\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.

Compressible flow[]

Remark: here, the Cauchy stress tensor is denoted $σ$ (instead of $τ$ 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 $\nabla u$ .
the stress is linear in this variable: $σ (\nabla u) = C : (\nabla u)$ , where $C$ 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 $V$ is an isotropic tensor; furthermore, since the stress tensor is symmetric, by Helmholtz decomposition it can be expressed in terms of two scalar Lamé parameters, the bulk viscosity $λ$ and the dynamic viscosity $μ$ , as it is usual in linear elasticity:
Linear stress constitutive equation (expression used for elastic solid)
${\boldsymbol {\sigma }}=\lambda (\nabla \cdot \mathbf {u} )\mathbf {I} +2\mu {\boldsymbol {\varepsilon }}$

where $I$ is the identity tensor, $ε (\nabla u) \equiv 1 / 2 \nabla u + 1 / 2 (\nabla u) T$ is the rate-of-strain tensor and $\nabla \cdot u$ is the rate of expansion of the flow. So this decomposition can be explicitly defined as:
${\boldsymbol {\sigma }}=\lambda (\nabla \cdot \mathbf {u} )\mathbf {I} +\mu \left(\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }\right).$

Since the trace of the rate-of-strain tensor in three dimensions is:

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

The trace of the stress tensor in three dimensions becomes:

\operatorname {tr} ({\boldsymbol {\sigma }})=(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]

{\boldsymbol {\sigma }}=\left(\lambda +{\tfrac {2}{3}}\mu \right)\left(\nabla \cdot \mathbf {u} \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 second viscosity $ζ$ ,

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

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

Linear stress constitutive equation (expression used for fluids)

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

Both second viscosity $ζ$ and dynamic viscosity $μ$ need not be constant – in general, they depend on density, on each other (the viscosity is expressed in pressure), and in compressible flows also on temperature. Any equation that makes explicit one of these transport coefficient in the is called an equation of state.^[7]

The most general of the Navier–Stokes equations become

Navier–Stokes momentum equation (convective form)

$\rho {\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}=\rho \left({\frac {\partial \mathbf {u} }{\partial t}}+\mathbf {u} \cdot \nabla \mathbf {u} \right)=-\nabla p+\nabla \cdot \left\{\mu \left(\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }-{\tfrac {2}{3}}(\nabla \cdot \mathbf {u} )\mathbf {I} \right)+\zeta (\nabla \cdot \mathbf {u} )\mathbf {I} \right\}+\rho \mathbf {g} .$

In most cases, the second viscosity $ζ$ can be assumed to be constant. The effect of the volume viscosity $ζ$ is that the mechanical pressure is not equivalent to the thermodynamic pressure:^[8] Since

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

modified pressure is usually introduced

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

to remove the term corresponding to the second viscosity. This difference is usually neglected, sometimes by explicitly assuming

ζ = 0

, but it could have an impact in sound absorption and attenuation and shock waves.^[9] With this simplification, the Navier–Stokes equations become

Navier–Stokes momentum equation (convective form)

$\rho {\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}=\rho \left({\frac {\partial \mathbf {u} }{\partial t}}+\mathbf {u} \cdot \nabla \mathbf {u} \right)=-\nabla {\bar {p}}+\nabla \cdot \left\{\mu \left(\nabla \mathbf {u} +(\nabla \mathbf {u} )^{\mathrm {T} }-{\tfrac {2}{3}}(\nabla \cdot \mathbf {u} )\mathbf {I} \right)\right\}+\rho \mathbf {g} .$

If the dynamic viscosity $μ$ is also assumed to be constant, the equations can be simplified further. By computing the divergence of the stress tensor, since the divergence of tensor $\nabla u$ is $\nabla 2 u$ and the divergence of tensor $(\nabla u) T$ is $\nabla(\nabla \cdot u)$ , one finally arrives to the compressible (most general) Navier–Stokes momentum equation:^[10]

Navier–Stokes momentum equation (convective form)

$\rho {\frac {\mathrm {D} \mathbf {u} }{\mathrm {D} t}}=\rho \left({\frac {\partial \mathbf {u} }{\partial t}}+\mathbf {u} \cdot \nabla \mathbf {u} \right)=-\nabla {\bar {p}}+\mu \,\nabla ^{2}\mathbf {u} +{\tfrac {1}{3}}\mu \,\nabla (\nabla \cdot \mathbf {u} )+\rho \mathbf {g} .$

where $D / Dt$ is the material derivative. The left-hand side changes in the conservation form of the Navier–Stokes momentum equation:

Navier–Stokes momentum equation (conservation form)

${\frac {\partial }{\partial t}}(\rho \,\mathbf {u} )+\nabla \cdot (\rho \,\mathbf {u} \otimes \mathbf {u} )=-\nabla {\bar {p}}+\mu \,\nabla ^{2}\mathbf {u} +{\tfrac {1}{3}}\mu \,\nabla (\nabla \cdot \mathbf {u} )+\rho \mathbf {g} .$

Bulk viscosity is assumed to be constant, otherwise it should not be taken out of the last derivative. The convective acceleration term can also be written as

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

where the vector

(\nabla \times u) \times u

is known as the Lamb vector.

For the special case of an incompressible flow, the pressure constrains the flow so that the volume of fluid elements is constant: isochoric flow resulting in a solenoidal velocity field with $\nabla \cdot u = 0$ .^[11]

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 $\nabla u$ .
the fluid is assumed to be isotropic, as with gases and simple liquids, and consequently $τ$ is an isotropic tensor; furthermore, since the deviatoric stress tensor can be expressed in terms of the dynamic viscosity $μ$ :
Stokes' stress constitutive equation (expression used for incompressible elastic solids)
${\boldsymbol {\tau }}=2\mu {\boldsymbol {\varepsilon }}$

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

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 is called an equation of state.^[7]

The divergence of the deviatoric stress is given by:

\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}

because

\nabla \cdot u = 0

for an incompressible fluid.

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.^[12] For incompressible (uniform density $ρ 0$ ) flows the following identity holds:

{\frac {1}{\rho _{0}}}\nabla p=\nabla \left({\frac {p}{\rho _{0}}}\right)\equiv \nabla w

where $w$ is the specific (with the sense of per unit mass) thermodynamic work, the internal source term. Then the incompressible Navier–Stokes equations are best visualised by dividing for the density:

Incompressible Navier–Stokes equations (convective form)

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

where

[1]

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[3]

[4]

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