Navier–Stokes equations


The Navier–Stokes equations are partial differential equations which describe the motion of viscous fluid substances. They were named after French engineer and physicist Claude-Louis Navier and the Irish physicist and mathematician Sir George Gabriel Stokes, Bt. They were developed over several decades of progressively building the theories, from 1822 to 1842–1850.
The Navier–Stokes equations mathematically express momentum balance for Newtonian fluids and make use of conservation of mass. They are sometimes accompanied by an equation of state relating pressure, temperature and density. 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 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.
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 problems. 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., whether they are infinitely differentiable 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.

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 studied in three spatial dimensions and one time dimension, 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:
By setting the Cauchy stress tensor to be the sum of a viscosity term and a pressure term , 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, 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 of any finite volume to represent the change of velocity in fluid media:
where
to arrive at the conservation form of the equations of motion. This is often written:
where is the outer product of the flow velocity :
The left side of the equation describes acceleration, and may be composed of time-dependent and convective components. The right side of the equation is in effect a summation of hydrostatic effects, the divergence of deviatoric stress and body forces.
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 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

A significant feature of the Cauchy equation and consequently all other continuum equations 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 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:
Since the trace of the rate-of-strain tensor in three dimensions is the divergence of the flow:
Given this relation, and since the trace of the identity tensor in three dimensions is three:
the trace of the stress tensor in three dimensions becomes:
So by alternatively decomposing the stress tensor into isotropic and deviatoric parts, as usual in fluid dynamics:
Introducing the bulk viscosity,
we arrive to the linear constitutive equation in the form usually employed in thermal hydraulics:
which can also be arranged in the other usual form:
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:
and the deviatoric stress tensor is still coincident with the shear stress tensor , and it has a compressibility term in addition to the incompressible case, which is proportional to the shear viscosity:
Both bulk viscosity and dynamic viscosity 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.
The most general of the Navier–Stokes equations become
in index notation, the equation can be written as
The corresponding equation in conservation form can be obtained by considering that, given the mass continuity equation, the left side is equivalent to:
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 can be assumed to be constant in which case, the effect of the volume viscosity is that the mechanical pressure is not equivalent to the thermodynamic pressure: as demonstrated below.However, this difference is usually neglected most of the time by explicitly assuming. The assumption of setting is called as the Stokes hypothesis. The validity of Stokes hypothesis can be demonstrated for monoatomic gas both experimentally and from the kinetic theory; for other gases and liquids, Stokes hypothesis is generally incorrect. With the Stokes hypothesis, the Navier–Stokes equations become
If the dynamic and bulk 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 is and the divergence of tensor is, one finally arrives to the compressible Navier–Stokes momentum equation:
where is the material derivative. is the shear kinematic viscosity and is the bulk kinematic viscosity. The left-hand side changes in the conservation form of the Navier–Stokes momentum equation.
By bringing the operator on the flow velocity on the left side, one also has:
The convective acceleration term can also be written as
where the vector 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.