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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 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 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 (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 phenomenon 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 (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.

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

where
 * $\frac{\mathrm{D}}{\mathrm{D}t}$ is the material derivative, defined as $\frac{\partial}{\partial t} + \mathbf{u} \cdot \nabla$ ,
 * $\rho$ is the (mass) density,
 * $\mathbf{u}$ is the flow velocity,
 * $\nabla \cdot \,$ is the divergence,
 * $p$ is the pressure,
 * $t$ is time,
 * $\boldsymbol{\tau}$ is the deviatoric stress tensor, which has order 2,
 * $\mathbf{f}$ 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 conservation of mass, with the known properties of divergence and gradient we can use the mass continuity equation, which represents the mass per unit volume of a homogenous fluid with respect to space and time (i.e., material derivative $$\frac{\mathbf{D}}{\mathbf{Dt}}$$) of any finite volume (V) to represent the change of velocity in fluid media: $$ \begin{align} \frac{\mathbf{D}m}&={\iiint\limits_V}({\frac{\mathbf{D}\rho} + \rho (\nabla \cdot \mathbf{u})})dV \\ \frac{\mathbf{D}\rho} + \rho (\nabla \cdot{\mathbf{u}})&=\frac{\partial\rho}{\partial t} + ({\nabla \rho}) \cdot{\mathbf{u}} + {\rho}(\nabla \cdot \mathbf{u})= \frac{\partial\rho}{\partial t} + \nabla\cdot({\rho \mathbf{u}})= 0 \end{align}$$where Note 1 - Refer to the mathematical operator del represented by the nabla ($\nabla$) symbol.
 * $\frac{\mathrm{D}m}{\mathrm{D}t}$ is the material derivative of mass per unit volume (density, $$\rho$$),
 * ${\iiint \limits_V}(F(x_1, x_2, x_3 ,t))dV$ is the mathematical operation for the integration throughout the volume (V),
 * $\frac{\partial }{\partial t}$ is the partial derivative mathematical operator,
 * $\nabla \cdot \mathbf{u}\,$ is the divergence of the flow velocity ($$\mathbf{u}$$), which is a scalar field, Note 1
 * ${\nabla \rho} \,$ is the gradient of density ($$\rho$$), which is the vector derivative of a scalar field, Note 1

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

where $\otimes$ is the outer product of the flow velocity ($$\mathbf{u}$$):$$\mathbf u \otimes \mathbf u = \mathbf u \mathbf u^{\mathrm T}$$

The left side of the equation describes acceleration, and may be composed of time-dependent and convective components (also the effects of non-inertial coordinates if present). The right side of the equation is in effect a summation of hydrostatic effects, the divergence of deviatoric stress and body forces (such as gravity).

All non-relativistic balance equations, such as the Navier–Stokes equations, can be derived by beginning with the Cauchy equations and specifying the stress tensor through a constitutive relation. By expressing the deviatoric (shear) stress tensor in terms of viscosity and the fluid velocity gradient, and assuming constant viscosity, the above Cauchy equations will lead to the Navier–Stokes equations below.

Convective acceleration

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 $\boldsymbol{\tau}$ 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:

 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 $\nabla \mathbf{u}$, or more simply the rate-of-strain tensor: $\boldsymbol{\varepsilon}\left(\nabla \mathbf{u}\right) \equiv \frac{1}{2}\nabla \mathbf{u} + \frac{1}{2} \left(\nabla \mathbf{u}\right)^T$  the deviatoric stress is linear in this variable: $\boldsymbol{\sigma}(\boldsymbol \varepsilon) = -p \mathbf I + \mathbf{C} : \boldsymbol \varepsilon$, where $p$ is independent on the strain rate tensor, $\mathbf{C}$  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 $\mathbf{C}$ is an isotropic tensor; furthermore, since the deviatoric stress tensor is symmetric, by Helmholtz decomposition it can be expressed in terms of two scalar Lamé parameters, the second viscosity $\lambda$  and the dynamic viscosity $\mu$, as it is usual in linear elasticity:

where $\mathbf{I}$ is the identity tensor, and $\operatorname{tr} (\boldsymbol \varepsilon)$  is the trace of the rate-of-strain tensor. So this decomposition can be explicitly defined as: $$\boldsymbol \sigma = -p \mathbf I + \lambda (\nabla\cdot\mathbf{u}) \mathbf I + \mu \left(\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathrm{T}\right).$$

Since the trace of the rate-of-strain tensor in three dimensions is the divergence (i.e. rate of expansion) of the flow: $$\operatorname{tr} (\boldsymbol \varepsilon) = \nabla\cdot\mathbf{u}.$$

Given this relation, and since the trace of the identity tensor in three dimensions is three: $$\operatorname{tr} (\boldsymbol I) = 3.$$

the trace of the stress tensor in three dimensions becomes: $$\operatorname{tr} (\boldsymbol \sigma ) = -3p + (3 \lambda + 2 \mu )\nabla\cdot\mathbf{u}.$$

So by alternatively decomposing the stress tensor into isotropic and deviatoric parts, as usual in fluid dynamics: $$\boldsymbol \sigma = - \left[ p - \left(\lambda + \tfrac23 \mu\right) \left(\nabla\cdot\mathbf{u}\right) \right] \mathbf I + \mu \left(\nabla\mathbf{u} + \left( \nabla\mathbf{u} \right)^\mathrm{T} - \tfrac23 \left(\nabla\cdot\mathbf{u}\right)\mathbf I\right)$$

Introducing the bulk viscosity $\zeta$ , $$ \zeta \equiv \lambda + \tfrac23 \mu ,$$

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

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: $$p = - \frac 1 3 \operatorname{tr} (\boldsymbol \sigma) + \zeta (\nabla\cdot\mathbf{u})$$

and the deviatoric stress tensor $$\boldsymbol \sigma'$$ is still coincident with the shear stress tensor $$\boldsymbol \tau$$ (i.e. the deviatoric stress in a Newtonian fluid has no normal stress components), and it has a compressibility term in addition to the incompressible case, which is proportional to the shear viscosity:

$$\boldsymbol \sigma' = \boldsymbol \tau = \mu \left[\nabla\mathbf{u} + ( \nabla\mathbf{u} )^\mathrm{T} - \tfrac23 (\nabla\cdot\mathbf{u})\mathbf I\right]$$

Both bulk viscosity $\zeta$ and dynamic viscosity $\mu$  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.

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:

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

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 $\zeta$ can be assumed to be constant in which case, the effect of the volume viscosity $\zeta$  is that the mechanical pressure is not equivalent to the thermodynamic pressure: as demonstrated below. $$\nabla\cdot(\nabla\cdot \mathbf u)\mathbf I=\nabla (\nabla \cdot \mathbf u),$$$$ \bar{p} \equiv p - \zeta \, \nabla \cdot \mathbf{u} ,$$ However, this difference is usually neglected most of the time (that is whenever we are not dealing with processes such as sound absorption and attenuation of shock waves, where second viscosity coefficient becomes important) by explicitly assuming $\zeta = 0$. The assumption of setting $\zeta = 0$ 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 $μ$ and bulk $$\zeta$$ 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 $\nabla \mathbf{u}$ is $\nabla^2 \mathbf{u}$  and the divergence of tensor $\left(\nabla \mathbf{u}\right)^{\mathrm T}$  is $\nabla \left(\nabla \cdot \mathbf{u}\right) $, one finally arrives to the compressible Navier–Stokes momentum equation:

where $\frac{\mathrm{D}}{\mathrm{D}t} $ is the material derivative. $$\nu=\frac \mu \rho$$ is the shear kinematic viscosity and $$\xi=\frac \zeta \rho$$ 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, on also has:

The convective acceleration term can also be written as $$ \mathbf u\cdot\nabla\mathbf u = (\nabla\times\mathbf u)\times\mathbf u + \tfrac12\nabla\mathbf u^2,$$ where the vector $ (\nabla \times \mathbf{u}) \times \mathbf{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 \mathbf{u} = 0$.

Incompressible flow
The incompressible momentum Navier–Stokes equation results from the following assumptions on the Cauchy stress tensor:

 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 $\nabla \mathbf{u}$ . the fluid is assumed to be isotropic, as with gases and simple liquids, and consequently $\boldsymbol{\tau}$ is an isotropic tensor; furthermore, since the deviatoric stress tensor can be expressed in terms of the dynamic viscosity $\mu$ :

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:

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.

The divergence of the deviatoric stress in case of uniform viscosity 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 \mathbf{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. the incompressible Navier–Stokes equations are best visualized by dividing for the density:

where $\nu = \frac{\mu}{\rho}$ is called the kinematic viscosity. By isolating the fluid velocity, one can also state:

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

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

In incompressible flows, the pressure field satisfies the Poisson equation,

 * $$\nabla^2 p = - \rho \frac{\partial u_i}{\partial x_k}\frac{\partial u_k}{\partial x_i} = - \rho \frac{\partial^2 u_iu_k}{\partial x_kx_i},$$

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

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

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

The higher-order term, namely the shear stress divergence $\nabla \cdot \boldsymbol{\tau}$, has simply reduced to the vector Laplacian term $\mu \nabla^2 \mathbf{u}$. This Laplacian term can be interpreted as the difference between the velocity at a point and the mean velocity in a small surrounding volume. This implies that – for a Newtonian fluid – viscosity operates as a diffusion of momentum, in much the same way as the 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: $$ \mathbf g = - \nabla \varphi $$ by defining the hydraulic head: $$h \equiv w + \varphi $$

one can finally condense the whole source in one term, arriving to the incompressible Navier–Stokes equation with conservative external field: $$\frac{\partial \mathbf{u}}{\partial t} + (\mathbf{u} \cdot \nabla) \mathbf{u} - \nu \, \nabla^2 \mathbf{u} = - \nabla h.$$

The incompressible Navier–Stokes equations with uniform density and viscosity and conservative external field is the fundamental equation of hydraulics. The domain for these equations is commonly a 3 or less dimensional Euclidean space, for which an 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, $$\begin{align} \frac{\partial\mathbf{u}}{\partial t} &= \Pi^S\left(-(\mathbf{u}\cdot\nabla)\mathbf{u} + \nu\,\nabla^2\mathbf{u}\right) + \mathbf{f}^S \\ \rho^{-1}\,\nabla p &= \Pi^I\left(-(\mathbf{u}\cdot\nabla)\mathbf{u} + \nu\,\nabla^2\mathbf{u}\right) + \mathbf{f}^I \end{align}$$ where $\Pi^S$ and $\Pi^I$  are solenoidal and irrotational projection operators satisfying $\Pi^S + \Pi^I = 1$, and $\mathbf{f}^S$  and $\mathbf{f}^I$  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: $$\Pi^S\,\mathbf{F}(\mathbf{r}) = \frac{1}{4\pi}\nabla\times\int \frac{\nabla^\prime\times\mathbf{F}(\mathbf{r}')}{|\mathbf{r}-\mathbf{r}'|} \, \mathrm{d} V', \quad \Pi^I = 1-\Pi^S$$ 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, is given by, $$\left(\mathbf{w},\frac{\partial\mathbf{u}}{\partial t}\right) = -\bigl(\mathbf{w}, \left(\mathbf{u}\cdot\nabla\right)\mathbf{u}\bigr) - \nu \left(\nabla\mathbf{w}: \nabla\mathbf{u}\right) + \left(\mathbf{w}, \mathbf{f}^S\right)$$

for divergence-free test functions $\mathbf{w}$ 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.

Strong form
Consider the incompressible Navier–Stokes equations for a Newtonian fluid of constant density $\rho$ in a domain $$ \Omega \subset \mathbb R^d \quad (d=2, 3)$$ with boundary $$ \partial \Omega = \Gamma_D \cup \Gamma_N ,$$ being $\Gamma_D$ and $\Gamma_N$  portions of the boundary where respectively a Dirichlet and a Neumann boundary condition  is applied ($\Gamma_D \cap \Gamma_N = \emptyset$ ): $$ \begin{cases} \rho \dfrac{\partial \mathbf{u}}{\partial t} + \rho (\mathbf{u} \cdot \nabla) \mathbf{u} - \nabla \cdot \boldsymbol{\sigma} (\mathbf{u}, p) = \mathbf{f} & \text{ in } \Omega \times (0, T) \\ \nabla \cdot \mathbf{u} = 0 & \text{ in } \Omega \times (0, T) \\ \mathbf{u} = \mathbf{g} & \text{ on } \Gamma_D \times (0, T) \\ \boldsymbol{\sigma} (\mathbf{u}, p) \hat{\mathbf{n}} = \mathbf{h} & \text{ on } \Gamma_N \times (0, T) \\ \mathbf{u}(0)= \mathbf{u}_0 & \text{ in } \Omega \times \{ 0\} \end{cases} $$ $\mathbf{u}$ is the fluid velocity, $p$  the fluid pressure, $\mathbf{f}$  a given forcing term, $$\hat{\mathbf{n}}$$ the outward directed unit normal vector to $\Gamma_N$, and $\boldsymbol{\sigma}(\mathbf{u}, p)$  the viscous stress tensor defined as: $$\boldsymbol{\sigma} (\mathbf{u}, p) = -p \mathbf{I} + 2 \mu \boldsymbol{\varepsilon}(\mathbf{u}).$$ Let $\mu$ be the dynamic viscosity of the fluid, $\mathbf{I}$  the second-order identity tensor and $\boldsymbol{\varepsilon}(\mathbf{u})$  the strain-rate tensor defined as: $$ \boldsymbol{\varepsilon} (\mathbf{u}) = \frac{1}{2} \left(\left( \nabla \mathbf{u} \right) + \left( \nabla \mathbf{u} \right)^\mathrm{T}\right). $$ The functions $ \mathbf{g} $ and $ \mathbf{h} $ are given Dirichlet and Neumann boundary data, while $ \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 $$ \nabla \cdot \left( \nabla \mathbf{f} \right)^\mathrm{T} = \nabla ( \nabla \cdot \mathbf{f} )$$ and exploiting mass conservation, the divergence of the total stress tensor in the momentum equation can also be expressed as: $$ \begin{align} \nabla \cdot \boldsymbol{\sigma} (\mathbf{u}, p) & = \nabla \cdot \left(-p \mathbf{I} + 2 \mu \boldsymbol{\varepsilon}(\mathbf{u}) \right) \\ & = - \nabla p + 2 \mu \nabla \cdot \boldsymbol{\varepsilon}(\mathbf{u}) \\ & = - \nabla p + 2 \mu \nabla \cdot \left [ \tfrac{1}{2} \left(\left(\nabla \mathbf{u} \right) + \left(\nabla \mathbf{u} \right)^\mathrm{T}\right) \right] \\ & = -\nabla p + \mu \left(\Delta \mathbf{u} + \nabla \cdot \left(\nabla \mathbf{u} \right)^\mathrm{T} \right) \\ & = -\nabla p + \mu \bigl( \Delta \mathbf{u} + \nabla  \underbrace{(\nabla \cdot \mathbf{u})}_{=0} \bigr) = -\nabla p + \mu \, \Delta \mathbf{u}. \end{align} $$ Moreover, note that the Neumann boundary conditions can be rearranged as: $$ \boldsymbol{\sigma}(\mathbf{u}, p) \hat{\mathbf{n}} = \left(-p \mathbf{I} + 2 \mu \boldsymbol{\varepsilon}(\mathbf{u})\right)\hat{\mathbf{n}} = -p \hat{\mathbf{n}} + \mu \frac{\partial \boldsymbol u}{\partial \hat{\mathbf{n}}}. $$

Weak form
In order to find the weak form of the Navier–Stokes equations, firstly, consider the momentum equation $$ \rho \frac{\partial \mathbf{u}}{\partial t} - \mu \Delta \mathbf{u} + \rho (\mathbf{u} \cdot \nabla) \mathbf{u} + \nabla p  = \mathbf{f} $$ multiply it for a test function $ \mathbf{v} $, defined in a suitable space $ V $, and integrate both members with respect to the domain $ \Omega $ : $$ \int \limits_\Omega \rho \frac{\partial \mathbf{u}}{\partial t}\cdot \mathbf{v} - \int \limits_\Omega \mu \Delta \mathbf{u} \cdot \mathbf{v} + \int \limits_\Omega \rho (\mathbf{u} \cdot \nabla) \mathbf{u} \cdot \mathbf{v} + \int \limits_\Omega \nabla p \cdot \mathbf{v} = \int \limits_\Omega \mathbf{f} \cdot \mathbf{v} $$ Counter-integrating by parts the diffusive and the pressure terms and by using the Gauss' theorem: $$\begin{align} -\int \limits_\Omega \mu \Delta \mathbf{u} \cdot \mathbf{v} &= \int_\Omega \mu \nabla \mathbf{u} \cdot \nabla \mathbf{v} - \int \limits_{\partial \Omega} \mu \frac{\partial \mathbf{u}}{\partial \hat{\mathbf{n}}} \cdot \mathbf{v} \\ \int \limits_\Omega \nabla p \cdot \mathbf{v} &= -\int \limits_\Omega p \nabla \cdot \mathbf{v} + \int \limits_{\partial \Omega} p \mathbf{v} \cdot {\hat{\mathbf{n}}} \end{align}$$

Using these relations, one gets: $$ \int \limits_\Omega \rho \dfrac{\partial \mathbf{u}}{\partial t}\cdot \mathbf{v} + \int \limits_\Omega \mu \nabla \mathbf{u} \cdot \nabla \mathbf{v} + \int \limits_\Omega \rho (\mathbf{u} \cdot \nabla) \mathbf{u} \cdot \mathbf{v} - \int \limits_\Omega p \nabla \cdot \mathbf{v} = \int \limits_\Omega \mathbf{f} \cdot \mathbf{v} + \int \limits_{\partial \Omega} \left ( \mu \frac{\partial \mathbf{u}}{\partial \hat{\mathbf{n}}} - p \hat{\mathbf{n}}\right) \cdot \mathbf{v} \quad \forall \mathbf{v} \in V. $$ In the same fashion, the continuity equation is multiplied for a test function $x$ belonging to a space $ Q $ and integrated in the domain $ \Omega $ : $$\int \limits_\Omega q \nabla \cdot \mathbf{u} = 0. \quad \forall q \in Q. $$ The space functions are chosen as follows: $$\begin{align} V = \left[H_0^1(\Omega) \right]^d &= \left\{ \mathbf{v} \in \left[H^1(\Omega)\right]^d: \quad \mathbf{v} = \mathbf{0} \text{ on } \Gamma_D \right\}, \\ Q &= L^2(\Omega) \end{align}$$ Considering that the test function $y = h$ vanishes on the Dirichlet boundary and considering the Neumann condition, the integral on the boundary can be rearranged as: $$ \int \limits_{\partial \Omega} \left ( \mu \frac{\partial \mathbf{u}}{\partial \hat{\mathbf{n}}} - p \hat{\mathbf{n}} \right) \cdot \mathbf{v} = \underbrace{ \int \limits_{\Gamma_D} \left ( \mu \frac{\partial \mathbf{u}}{\partial \hat{\mathbf{n}}} - p \hat{\mathbf{n}} \right) \cdot \mathbf{v} }_{ \mathbf{v} = \mathbf{0} \text{ on } \Gamma_D \ } + \int \limits_{\Gamma_N} \underbrace{ \vphantom{\int \limits_{\Gamma_N} } \left ( \mu \frac{\partial \mathbf{u}}{\partial \hat{\mathbf{n}}} - p \hat{\mathbf{n}} \right) }_{= \mathbf{h} \text{ on } \Gamma_N} \cdot \mathbf{v} = \int \limits_{\Gamma_N} \mathbf{h} \cdot \mathbf{v}. $$ Having this in mind, the weak formulation of the Navier–Stokes equations is expressed as: $$\begin{align} &\text{find } \mathbf{u} \in L^2 \left(\mathbb R^+\; \left[H^1(\Omega)\right]^d\right) \cap C^0\left(\mathbb R^+ \; \left[L^2(\Omega)\right]^d\right) \text{ such that: } \\[5pt] &\quad\begin{cases} \displaystyle \int \limits_{\Omega}\rho \dfrac{\partial \mathbf{u}}{\partial t}\cdot \mathbf{v} + \int \limits_{\Omega} \mu \nabla \mathbf{u} \cdot \nabla \mathbf{v} + \int \limits_{\Omega} \rho (\mathbf{u} \cdot \nabla) \mathbf{u} \cdot \mathbf{v} - \int \limits_{\Omega} p \nabla \cdot \mathbf{v} = \int \limits_{\Omega}\mathbf{f} \cdot \mathbf{v} +  \int \limits_{\Gamma_N} \mathbf{h} \cdot \mathbf{v} \quad \forall \mathbf{v} \in V, \\ \displaystyle \int \limits_{\Omega} q \nabla \cdot \mathbf{u} = 0 \quad \forall q \in Q. \end{cases}\end{align} $$

Discrete velocity
With partitioning of the problem domain and defining basis functions on the partitioned domain, the discrete form of the governing equation is $$\left(\mathbf{w}_i, \frac{\partial\mathbf{u}_j}{\partial t}\right) = -\bigl(\mathbf{w}_i, \left(\mathbf{u}\cdot\nabla\right)\mathbf{u}_j\bigr) - \nu\left(\nabla\mathbf{w}_i: \nabla\mathbf{u}_j\right) + \left(\mathbf{w}_i, \mathbf{f}^S\right).$$

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

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. 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, $$(\mathbf{g}_i, \nabla p) = -\left(\mathbf{g}_i, \left(\mathbf{u}\cdot\nabla\right)\mathbf{u}_j\right) - \nu\left(\nabla\mathbf{g}_i: \nabla\mathbf{u}_j\right) + \left(\mathbf{g}_i, \mathbf{f}^I\right)$$

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 $\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 $K $, and a non-inertial frame of reference $K' $, which is translating with velocity $\mathbf{U}(t) $ and rotating with angular velocity $\Omega(t) $ with respect to the stationary frame. The Navier–Stokes equation observed from the non-inertial frame then becomes

Here $\mathbf{x} $ and $\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 $K' $ with respect to $K $ and the fourth term is due to the angular acceleration of $K' $ with respect to $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
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: $$ \begin{align} \frac{\mathbf{D}m}&={\iiint\limits_V}({\frac{\mathbf{D}\rho} + \rho (\nabla \cdot \mathbf{u})})dV \\ \frac{\mathbf{D}\rho} + \rho (\nabla \cdot{\mathbf{u}})&=\frac{\partial\rho}{\partial t} + ({\nabla \rho}) \cdot{\mathbf{u}} + {\rho}(\nabla \cdot \mathbf{u})= \frac{\partial\rho}{\partial t} + \nabla\cdot({\rho \mathbf{u}})= 0 \end{align}$$ A fluid media for which the density ($$\rho$$) is constant is called incompressible. Therefore, the rate of change of density ($$\rho$$) with respect to time $$(\frac{\partial\rho}{\partial t})$$ and the gradient of density $$(\nabla \rho)$$ are equal to zero $$(0)$$. In this case the general equation of continuity, $$\frac{\partial\rho}{\partial t} + \nabla({\rho \mathbf{u}})= 0$$, reduces to: $$\rho(\nabla{\cdot}{\mathbf{u}}) = 0$$. Furthermore, assuming that density ($$\rho$$) is a non-zero constant $$(\rho \neq 0)$$ means that the right-hand side of the equation $$(0)$$ is divisible by density ($$\rho$$). Therefore, the continuity equation for an incompressible fluid reduces further to: $$(\nabla{\cdot{\mathbf{u}}}) = 0$$ This relationship, $(\nabla{\cdot{\mathbf{u}}}) = 0$,  identifies that the divergence of the flow velocity vector ($$\mathbf{u}$$) is equal to zero $$(0)$$, which means that for an incompressible fluid the flow velocity field is a solenoidal vector field or a divergence-free vector field. Note that this relationship can be expanded upon due to its uniqueness with the vector Laplace operator $$(\nabla ^{2} \mathbf{u} =\nabla (\nabla \cdot \mathbf{u} )-\nabla \times (\nabla \times \mathbf{u} ))$$, and vorticity $$(\vec \omega = \nabla \times \mathbf{u})$$ which is now expressed like so, for an  incompressible fluid:$$\nabla ^{2}\mathbf {u} = - (\nabla \times (\nabla \times \mathbf {u} )) = - (\nabla \times \vec \omega)$$

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 $u_z = 0 $ and no dependence of anything on $z $ ), where the equations reduce to: $$\begin{align} \rho \left(\frac{\partial u_x}{\partial t} + u_x \frac{\partial u_x}{\partial x} + u_y \frac{\partial u_x}{\partial y}\right) &= -\frac{\partial p}{\partial x} + \mu \left(\frac{\partial^2 u_x}{\partial x^2} + \frac{\partial^2 u_x}{\partial y^2}\right) + \rho g_x \\ \rho \left(\frac{\partial u_y}{\partial t} + u_x \frac{\partial u_y}{\partial x} + u_y \frac{\partial u_y}{\partial y}\right) &= -\frac{\partial p}{\partial y} + \mu \left(\frac{\partial^2 u_y}{\partial x^2} + \frac{\partial^2 u_y}{\partial y^2}\right) + \rho g_y. \end{align}$$

Differentiating the first with respect to $y $, the second with respect to $x $ and subtracting the resulting equations will eliminate pressure and any conservative force. For incompressible flow, defining the stream function $\psi $ through $$u_x = \frac{\partial \psi}{\partial y}; \quad u_y = -\frac{\partial \psi}{\partial x}$$ 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: $$\frac{\partial}{\partial t}\left(\nabla^2 \psi\right) + \frac{\partial \psi}{\partial y} \frac{\partial}{\partial x}\left(\nabla^2 \psi\right) - \frac{\partial \psi}{\partial x} \frac{\partial}{\partial y}\left(\nabla^2 \psi\right) = \nu \nabla^4 \psi$$

where $\nabla^4 $ is the 2D biharmonic operator and $\nu $ is the kinematic viscosity, $\nu = \frac{\mu}{p} $ . We can also express this compactly using the Jacobian determinant: $$\frac{\partial}{\partial t}\left(\nabla^2 \psi\right) + \frac{\partial\left(\psi, \nabla^2\psi \right)}{\partial(y,x)} = \nu \nabla^4 \psi.$$

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.

Nonlinearity
The Navier–Stokes equations are nonlinear partial differential equations in the general case and so remain in almost every real situation. 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.

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.

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 direct numerical simulation. Attempts to solve turbulent flow using a laminar solver typically result in a time-unsteady solution, which fails to converge appropriately. To counter this, time-averaged equations such as the Reynolds-averaged Navier–Stokes equations (RANS), supplemented with turbulence models, are used in practical computational fluid dynamics (CFD) applications when modeling turbulent flows. Some models include the Spalart–Allmaras, $B$–$A$, $q$–$k$, and 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
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 continuum (it is infinitely divisible and not composed of particles such as atoms or molecules), and is not moving at relativistic velocities. At very small scales or under extreme conditions, real fluids made out of discrete molecules will produce results different from the continuous fluids modeled by the Navier–Stokes equations. For example, capillarity of internal layers in fluids appears for flow with high gradients. For large Knudsen number of the problem, the Boltzmann equation may be a suitable replacement. Failing that, one may have to resort to molecular dynamics or various hybrid methods.

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 Newtonian fluids where the viscosity model is linear; truly general models for the flow of other kinds of fluids (such as blood) do not exist.

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

The boundary condition is the no slip condition. This problem is easily solved for the flow field: $$u(y) = \frac{y - y^2}{2}.$$

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 $u = 0$ that must satisfy: $$\frac{\mathrm{d}^2 f}{\mathrm{d} z^2} + R f^2 = -1; \quad f(-1) = f(1) = 0.$$

This ordinary differential equation is what is obtained when the Navier–Stokes equations are written and the flow assumptions applied (additionally, the pressure gradient is solved for). The nonlinear term makes this a very difficult problem to solve analytically (a lengthy implicit solution may be found which involves elliptic integrals and roots of cubic polynomials). Issues with the actual existence of solutions arise for $R > 1.41 $ (approximately; this is not √2), the parameter $R $ being the Reynolds number with appropriately chosen scales. This is an example of flow assumptions losing their applicability, and an example of the difficulty in "high" Reynolds number flows.

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 Poiseuille flow, Couette flow and the oscillatory 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. 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.

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 $r $ be a constant radius of the inner coil. One set of solutions is given by: $$\begin{align} \rho(x, y, z) &= \frac{3B}{r^2 + x^2 + y^2 + z^2} \\ p(x, y, z) &= \frac{-A^2B}{\left(r^2 + x^2 + y^2 + z^2\right)^3} \\ \mathbf{u}(x, y, z) &= \frac{A}{\left(r^2 + x^2 + y^2 + z^2\right)^2}\begin{pmatrix} 2(-ry + xz) \\ 2(rx + yz) \\ r^2 - x^2 - y^2 + z^2 \end{pmatrix} \\ g &= 0 \\ \mu &= 0 \end{align}$$

for arbitrary constants $A $ and $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 $\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:

Viscous three-dimensional periodic solutions
Two examples of periodic fully-three-dimensional viscous solutions are described in. These solutions are defined on a three-dimensional torus $$ \mathbb{T}^3 = [0, L]^3 $$ and are characterized by positive and negative helicity respectively. The solution with positive helicity is given by: $$\begin{align} u_x &= \frac{4 \sqrt{2}}{3 \sqrt{3}} \, U_0 \left[\, \sin(k x - \pi/3) \cos(k y + \pi/3) \sin(k z + \pi/2) - \cos(k z - \pi/3) \sin(k x + \pi/3) \sin(k y + \pi/2) \,\right] e^{-3 \nu k^2 t} \\ u_y &= \frac{4 \sqrt{2}}{3 \sqrt{3}} \, U_0 \left[\, \sin(k y - \pi/3) \cos(k z + \pi/3) \sin(k x + \pi/2) - \cos(k x - \pi/3) \sin(k y + \pi/3) \sin(k z + \pi/2) \,\right] e^{-3 \nu k^2 t} \\ u_z &= \frac{4 \sqrt{2}}{3 \sqrt{3}} \, U_0 \left[\, \sin(k z - \pi/3) \cos(k x + \pi/3) \sin(k y + \pi/2) - \cos(k y - \pi/3) \sin(k z + \pi/3) \sin(k x + \pi/2) \,\right] e^{-3 \nu k^2 t} \end{align}$$ where $$k = 2 \pi/L$$ is the wave number and the velocity components are normalized so that the average kinetic energy per unit of mass is $$U_0^2/2$$ at $$ t = 0 $$. The pressure field is obtained from the velocity field as $$ p = p_0 - \rho_0 \| \boldsymbol{u} \|^2/2$$ (where $$p_0$$ and $$\rho_0$$ 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 $$\omega =\sqrt{3} \, k \, \boldsymbol{u}$$. 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 graphs that correspond to the Navier–Stokes equations via a perturbation expansion of the fundamental continuum mechanics. Similar to the Feynman diagrams in quantum field theory, these diagrams are an extension of Keldysh's technique for nonequilibrium processes in fluid dynamics. In other words, these diagrams assign graphs to the (often) turbulent phenomena in turbulent fluids by allowing correlated and interacting fluid particles to obey stochastic processes associated to pseudo-random functions in probability distributions.

Representations in 3D
Note that the formulas in this section make use of the single-line notation for partial derivatives, where, e.g. $\partial_x u$ means the partial derivative of $u$  with respect to $x$, and $\partial_y^2 f_\theta$  means the second-order partial derivative of $f_\theta$  with respect to $y$.

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.

Cartesian coordinates
From the general form of the Navier–Stokes, with the velocity vector expanded as $\mathbf{u} = (u_x, u_y, u_z) $, sometimes respectively named $u$ , $v$ , $w$ , we may write the vector equation explicitly, $$\begin{align} x:\ &\rho \left({\partial_t u_x} + u_x \, {\partial_x u_x} + u_y \, {\partial_y u_x} + u_z \, {\partial_z u_x}\right) \\ &\quad= -\partial_x p + \mu \left({\partial_x^2 u_x} + {\partial_y^2 u_x} + {\partial_z^2 u_x}\right) + \frac{1}{3} \mu \ \partial_x \left( {\partial_x u_x} + {\partial_y u_y} + {\partial_z u_z} \right) + \rho g_x \\ \end{align}$$ $$\begin{align} y:\ &\rho \left({\partial_t u_y} + u_x {\partial_x u_y} + u_y {\partial_y u_y} + u_z {\partial_z u_y}\right) \\ &\quad= -{\partial_y p} + \mu \left({\partial_x^2 u_y} + {\partial_y^2 u_y} + {\partial_z^2 u_y}\right) + \frac{1}{3} \mu \ \partial_y \left( {\partial_x u_x} + {\partial_y u_y} + {\partial_z u_z} \right) + \rho g_y \\ \end{align}$$ $$\begin{align} z:\ &\rho \left({\partial_t u_z} + u_x {\partial_x u_z} + u_y {\partial_y u_z} + u_z {\partial_z u_z}\right) \\ &\quad= -{\partial_z p} + \mu \left({\partial_x^2 u_z} + {\partial_y^2 u_z} + {\partial_z^2 u_z}\right) + \frac{1}{3} \mu \ \partial_z \left( {\partial_x u_x} + {\partial_y u_y} + {\partial_z u_z} \right) + \rho g_z. \end{align}$$

Note that gravity has been accounted for as a body force, and the values of $g_x$, $g_y$ , $g_z$ will depend on the orientation of gravity with respect to the chosen set of coordinates.

The continuity equation reads: $$\partial_t \rho + \partial_x (\rho u_x) + \partial_y (\rho u_y) + \partial_z (\rho u_z) = 0.$$

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

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 nonlinear system of partial differential equations for which solutions are difficult to obtain.

Cylindrical coordinates
A change of variables on the Cartesian equations will yield the following momentum equations for $r$, $\phi$ , and $z$ $$\begin{align} r:\ & \rho \left({\partial_t u_r} + u_r {\partial_r u_r} + \frac{u_\varphi}{r} {\partial_\varphi u_r} + u_z {\partial_z u_r} - \frac{u_\varphi^2}{r}\right) \\ &\quad = -{\partial_r p} \\ &\qquad + \mu \left(\frac{1}{r} \partial_r \left(r {\partial_r u_r}\right) +                      \frac{1}{r^2} {\partial_\varphi^2 u_r} + {\partial_z^2 u_r} - \frac{u_r}{r^2} -                      \frac{2}{r^2} {\partial_\varphi u_\varphi} \right) \\ &\qquad + \frac{1}{3}\mu \partial_r \left( \frac{1}{r} {\partial_r\left(r u_r\right)} + \frac{1}{r} {\partial_\varphi u_\varphi} + {\partial_z u_z} \right) \\ &\qquad + \rho g_r \\[8px] \end{align}$$ $$\begin{align} \varphi:\ & \rho \left({\partial_t u_\varphi} + u_r {\partial_r u_\varphi} + \frac{u_\varphi}{r} {\partial_\varphi u_\varphi} + u_z {\partial_z u_\varphi} + \frac{u_r u_\varphi}{r} \right) \\ &\quad = -\frac{1}{r} {\partial_\varphi p} \\ &\qquad + \mu \left(\frac{1}{r} \  \partial_r \left(r {\partial_r u_\varphi}\right)                    + \frac{1}{r^2} {\partial_\varphi^2 u_{\varphi}}                    + {\partial_z^2 u_{\varphi}} - \frac{u_\varphi}{r^2} + \frac{2}{r^2} {\partial_\varphi u_r}\right) \\ &\qquad + \frac{1}{3}\mu \frac{1}{r} \partial_\varphi \left( \frac{1}{r} {\partial_r\left(r u_r\right)} + \frac{1}{r} {\partial_\varphi u_\varphi} + {\partial_z u_z} \right) \\ &\qquad + \rho g_\varphi \\[8px] \end{align}$$ $$\begin{align} z:\ & \rho \left({\partial_t u_z} + u_r {\partial_r u_z} + \frac{u_\varphi}{r} {\partial_\varphi u_z} + u_z {\partial_z u_z}\right) \\ &\quad = -{\partial_z p} \\ &\qquad + \mu \left(\frac{1}{r} \partial_r \left(r {\partial_r u_z}\right)                    + \frac{1}{r^2} {\partial_\varphi^2 u_z} + {\partial_z^2 u_z}\right) \\ &\qquad + \frac{1}{3}\mu \partial_z \left( \frac{1}{r} {\partial_r \left(r u_r\right)}                                           + \frac{1}{r} {\partial_\varphi u_\varphi} + {\partial_z u_z} \right) \\ &\qquad + \rho g_z. \end{align}$$

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: $${\partial_t\rho} + \frac{1}{r} \partial_r \left(\rho r u_r\right) + \frac{1}{r} {\partial_\varphi \left(\rho u_\varphi\right)} + {\partial_z \left(\rho u_z\right)} = 0. $$

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 ($u_\phi = 0$ ), and the remaining quantities are independent of $\phi$ : $$\begin{align} \rho \left({\partial_t u_r} + u_r {\partial_r u_r} + u_z {\partial_z u_r}\right) &= -{\partial_r p} + \mu \left(\frac{1}{r} \partial_r \left(r {\partial_r u_r}\right) + {\partial_z^2 u_r} - \frac{u_r}{r^2}\right) + \rho g_r \\ \rho \left({\partial_t u_z} + u_r {\partial_r u_z} + u_z {\partial_z u_z}\right) &= -{\partial_z p} + \mu \left(\frac{1}{r} \partial_r \left(r {\partial_r u_z}\right) + {\partial_z^2 u_z}\right) + \rho g_z \\ \frac{1}{r} \partial_r\left(r u_r\right) + {\partial_z u_z} &= 0. \end{align}$$

Spherical coordinates
In spherical coordinates, the $r$, $\phi$ , and $\theta$  momentum equations are (note the convention used: $\theta$  is polar angle, or colatitude, $0 \leq \theta \leq \pi$ ): $$\begin{align} r:\ &\rho \left({\partial_t u_r} + u_r {\partial_r u_r} + \frac{u_\varphi}{r \sin\theta} {\partial_\varphi u_r} + \frac{u_\theta}{r} {\partial_\theta u_r} - \frac{u_\varphi^2 + u_\theta^2}{r}\right) \\ &\quad = -{\partial_r p} \\ &\qquad + \mu \left(\frac{1}{r^2} \partial_r \left(r^2 {\partial_r u_r}\right) + \frac{1}{r^2 \sin^2\theta} {\partial_\varphi^2 u_r} + \frac{1}{r^2 \sin\theta} \partial_\theta \left(\sin\theta {\partial_\theta u_r}\right) - 2\frac{u_r + {\partial_\theta u_\theta} + u_\theta \cot\theta}{r^2} - \frac{2}{r^2 \sin\theta} {\partial_\varphi u_\varphi} \right) \\ &\qquad + \frac{1}{3}\mu \partial_r \left( \frac{1}{r^2} \partial_r\left(r^2 u_r\right) + \frac{1}{r \sin\theta} \partial_\theta \left( u_\theta\sin\theta \right) + \frac{1}{r\sin\theta} {\partial_\varphi u_\varphi} \right) \\ &\qquad + \rho g_r \\[8px] \end{align}$$ $$\begin{align} \varphi:\ &\rho \left({\partial_t u_\varphi} + u_r {\partial_r u_\varphi} + \frac{u_\varphi}{r \sin\theta} {\partial_\varphi u_\varphi} + \frac{u_\theta}{r} {\partial_\theta u_\varphi} + \frac{u_r u_\varphi + u_\varphi u_\theta \cot\theta}{r}\right) \\ &\quad = -\frac{1}{r \sin\theta} {\partial_\varphi p} \\ &\qquad + \mu \left(\frac{1}{r^2} \partial_r \left(r^2 {\partial_r u_\varphi}\right) + \frac{1}{r^2 \sin^2\theta} {\partial_\varphi^2 u_\varphi} + \frac{1}{r^2 \sin\theta} \partial_\theta \left(\sin\theta {\partial_\theta u_\varphi}\right) + \frac{2 \sin\theta {\partial_\varphi u_r} + 2 \cos\theta {\partial_\varphi u_\theta} - u_\varphi}{r^2 \sin^2\theta} \right) \\ &\qquad + \frac{1}{3}\mu\frac{1}{r \sin\theta} \partial_\varphi \left( \frac{1}{r^2} \partial_r \left(r^2 u_r\right) + \frac{1}{r \sin\theta} \partial_\theta \left( u_\theta\sin\theta \right) + \frac{1}{r\sin\theta} {\partial_\varphi u_\varphi} \right) \\ &\qquad + \rho g_\varphi \\[8px] \end{align}$$ $$\begin{align} \theta:\ &\rho \left({\partial_t u_\theta} + u_r {\partial_r u_\theta} + \frac{u_\varphi}{r \sin\theta} {\partial_\varphi u_\theta} + \frac{u_\theta}{r} {\partial_\theta u_\theta} + \frac{u_r u_\theta - u_\varphi^2 \cot\theta}{r}\right) \\ &\quad = -\frac{1}{r} {\partial_\theta p} \\ &\qquad + \mu \left(\frac{1}{r^2} \partial_r \left(r^2 {\partial_r u_\theta}\right) + \frac{1}{r^2 \sin^2\theta} {\partial_\varphi^2 u_\theta} + \frac{1}{r^2 \sin\theta} \partial_\theta \left(\sin\theta {\partial_\theta u_\theta}\right) + \frac{2}{r^2} {\partial_\theta u_r} - \frac{u_\theta + 2 \cos\theta {\partial_\varphi u_\varphi}}{r^2 \sin^2\theta} \right) \\ &\qquad + \frac{1}{3}\mu\frac{1}{r} \partial_\theta \left( \frac{1}{r^2} \partial_r \left(r^2 u_r\right) + \frac{1}{r \sin\theta} \partial_\theta  \left( u_\theta\sin\theta \right) + \frac{1}{r\sin\theta} {\partial_\varphi u_\varphi} \right) \\ &\qquad + \rho g_\theta. \end{align}$$

Mass continuity will read: $${\partial_t \rho} + \frac{1}{r^2} \partial_r \left(\rho r^2 u_r\right) + \frac{1}{r \sin\theta}{\partial_\varphi (\rho u_\varphi)} + \frac{1}{r \sin\theta} \partial_\theta \left(\sin\theta \rho u_\theta\right) = 0.$$

These equations could be (slightly) compacted by, for example, factoring $\frac{1}{r^2}$ 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" by Jos Stam, which elaborates one of the methods proposed in Stam's earlier, more famous paper "Stable Fluids" 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. 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.

General references

 * V. Girault and P. A. Raviart. Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms. Springer Series in Computational Mathematics. Springer-Verlag, 1986.
 * 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
 * V. Girault and P. A. Raviart. Finite Element Methods for Navier–Stokes Equations: Theory and Algorithms. Springer Series in Computational Mathematics. Springer-Verlag, 1986.
 * 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
 * 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
 * Temam, Roger (1984): Navier–Stokes Equations: Theory and Numerical Analysis, ACM Chelsea Publishing, ISBN 978-0-8218-2737-6

title: Navier–Stokes equations


text:
Mathematical analysis

Analysis is the branch of mathematics dealing with continuous functions, limits, and related theories, such as differentiation, integration, measure, infinite sequences, series, and analytic functions.

These theories are usually studied in the context of real and complex numbers and functions. Analysis evolved from calculus, which involves the elementary concepts and techniques of analysis. Analysis may be distinguished from geometry; however, it can be applied to any space of mathematical objects that has a definition of nearness (a topological space) or specific distances between objects (a metric space).

Ancient
Mathematical analysis formally developed in the 17th century during the Scientific Revolution, but many of its ideas can be traced back to earlier mathematicians. Early results in analysis were implicitly present in the early days of ancient Greek mathematics. For instance, an infinite geometric sum is implicit in Zeno's paradox of the dichotomy. (Strictly speaking, the point of the paradox is to deny that the infinite sum exists.) Later, Greek mathematicians such as Eudoxus and Archimedes made more explicit, but informal, use of the concepts of limits and convergence when they used the method of exhaustion to compute the area and volume of regions and solids. The explicit use of infinitesimals appears in Archimedes' The Method of Mechanical Theorems, a work rediscovered in the 20th century. In Asia, the Chinese mathematician Liu Hui used the method of exhaustion in the 3rd century CE to find the area of a circle. From Jain literature, it appears that Hindus were in possession of the formulae for the sum of the arithmetic and geometric series as early as the 4th century BCE. Ācārya Bhadrabāhu uses the sum of a geometric series in his Kalpasūtra in.

Medieval
Zu Chongzhi established a method that would later be called Cavalieri's principle to find the volume of a sphere in the 5th century. In the 12th century, the Indian mathematician Bhāskara II used infinitesimal and used what is now known as Rolle's theorem.

In the 14th century, Madhava of Sangamagrama developed infinite series expansions, now called Taylor series, of functions such as sine, cosine, tangent and arctangent. Alongside his development of Taylor series of trigonometric functions, he also estimated the magnitude of the error terms resulting of truncating these series, and gave a rational approximation of some infinite series. His followers at the Kerala School of Astronomy and Mathematics further expanded his works, up to the 16th century.

Foundations
The modern foundations of mathematical analysis were established in 17th century Europe. This began when Fermat and Descartes developed analytic geometry, which is the precursor to modern calculus. Fermat's method of adequality allowed him to determine the maxima and minima of functions and the tangents of curves. Descartes's publication of La Géométrie in 1637, which introduced the Cartesian coordinate system, is considered to be the establishment of mathematical analysis. It would be a few decades later that Newton and Leibniz independently developed infinitesimal calculus, which grew, with the stimulus of applied work that continued through the 18th century, into analysis topics such as the calculus of variations, ordinary and partial differential equations, Fourier analysis, and generating functions. During this period, calculus techniques were applied to approximate discrete problems by continuous ones.

Modernization
In the 18th century, Euler introduced the notion of a mathematical function. Real analysis began to emerge as an independent subject when Bernard Bolzano introduced the modern definition of continuity in 1816, but Bolzano's work did not become widely known until the 1870s. In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra widely used in earlier work, particularly by Euler. Instead, Cauchy formulated calculus in terms of geometric ideas and infinitesimals. Thus, his definition of continuity required an infinitesimal change in x to correspond to an infinitesimal change in y. He also introduced the concept of the Cauchy sequence, and started the formal theory of complex analysis. Poisson, Liouville, Fourier and others studied partial differential equations and harmonic analysis. The contributions of these mathematicians and others, such as Weierstrass, developed the (ε, δ)-definition of limit approach, thus founding the modern field of mathematical analysis. Around the same time, Riemann introduced his theory of integration, and made significant advances in complex analysis.

Towards the end of the 19th century, mathematicians started worrying that they were assuming the existence of a continuum of real numbers without proof. Dedekind then constructed the real numbers by Dedekind cuts, in which irrational numbers are formally defined, which serve to fill the "gaps" between rational numbers, thereby creating a complete set: the continuum of real numbers, which had already been developed by Simon Stevin in terms of decimal expansions. Around that time, the attempts to refine the theorems of Riemann integration led to the study of the "size" of the set of discontinuities of real functions.

Also, various pathological objects, (such as nowhere continuous functions, continuous but nowhere differentiable functions, and space-filling curves), commonly known as "monsters", began to be investigated. In this context, Jordan developed his theory of measure, Cantor developed what is now called naive set theory, and Baire proved the Baire category theorem. In the early 20th century, calculus was formalized using an axiomatic set theory. Lebesgue greatly improved measure theory, and introduced his own theory of integration, now known as Lebesgue integration, which proved to be a big improvement over Riemann's. Hilbert introduced Hilbert spaces to solve integral equations. The idea of normed vector space was in the air, and in the 1920s Banach created functional analysis.

Metric spaces
In mathematics, a metric space is a set where a notion of distance (called a metric) between elements of the set is defined.

Much of analysis happens in some metric space; the most commonly used are the real line, the complex plane, Euclidean space, other vector spaces, and the integers. Examples of analysis without a metric include measure theory (which describes size rather than distance) and functional analysis (which studies topological vector spaces that need not have any sense of distance).

Formally, a metric space is an ordered pair $$(M,d)$$ where $$M$$ is a set and $$d$$ is a metric on $$M$$, i.e., a function

 * $$d \colon M \times M \rightarrow \mathbb{R}$$

such that for any $$x, y, z \in M$$, the following holds:

 * 1) $$d(x,y) \geq 0$$, with equality if and only if $$x = y$$    (identity of indiscernibles),
 * 2) $$d(x,y) = d(y,x)$$    (symmetry), and
 * 3) $$d(x,z) \le d(x,y) + d(y,z)$$    (triangle inequality).

By taking the third property and letting $$z=x$$, it can be shown that $$d(x,y) \ge 0$$    (non-negative).

Sequences and limits
A sequence is an ordered list. Like a set, it contains members (also called elements, or terms). Unlike a set, order matters, and exactly the same elements can appear multiple times at different positions in the sequence. Most precisely, a sequence can be defined as a function whose domain is a countable totally ordered set, such as the natural numbers.

One of the most important properties of a sequence is convergence. Informally, a sequence converges if it has a limit. Continuing informally, a (singly-infinite) sequence has a limit if it approaches some point x, called the limit, as n becomes very large. That is, for an abstract sequence (an) (with n running from 1 to infinity understood) the distance between an and x approaches 0 as n → ∞, denoted
 * $$\lim_{n\to\infty} a_n = x.$$

Real analysis
Real analysis (traditionally, the "theory of functions of a real variable") is a branch of mathematical analysis dealing with the real numbers and real-valued functions of a real variable. In particular, it deals with the analytic properties of real functions and sequences, including convergence and limits of sequences of real numbers, the calculus of the real numbers, and continuity, smoothness and related properties of real-valued functions.

Complex analysis
Complex analysis (traditionally known as the "theory of functions of a complex variable") is the branch of mathematical analysis that investigates functions of complex numbers. It is useful in many branches of mathematics, including algebraic geometry, number theory, applied mathematics; as well as in physics, including hydrodynamics, thermodynamics, mechanical engineering, electrical engineering, and particularly, quantum field theory.

Complex analysis is particularly concerned with the analytic functions of complex variables (or, more generally, meromorphic functions). Because the separate real and imaginary parts of any analytic function must satisfy Laplace's equation, complex analysis is widely applicable to two-dimensional problems in physics.

Functional analysis
Functional analysis is a branch of mathematical analysis, the core of which is formed by the study of vector spaces endowed with some kind of limit-related structure (e.g. inner product, norm, topology, etc.) and the linear operators acting upon these spaces and respecting these structures in a suitable sense. The historical roots of functional analysis lie in the study of spaces of functions and the formulation of properties of transformations of functions such as the Fourier transform as transformations defining continuous, unitary etc. operators between function spaces. This point of view turned out to be particularly useful for the study of differential and integral equations.

Harmonic analysis
Harmonic analysis is a branch of mathematical analysis concerned with the representation of functions and signals as the superposition of basic waves. This includes the study of the notions of Fourier series and Fourier transforms (Fourier analysis), and of their generalizations. Harmonic analysis has applications in areas as diverse as music theory, number theory, representation theory, signal processing, quantum mechanics, tidal analysis, and neuroscience.

Differential equations
A differential equation is a mathematical equation for an unknown function of one or several variables that relates the values of the function itself and its derivatives of various orders. Differential equations play a prominent role in engineering, physics, economics, biology, and other disciplines.

Differential equations arise in many areas of science and technology, specifically whenever a deterministic relation involving some continuously varying quantities (modeled by functions) and their rates of change in space or time (expressed as derivatives) is known or postulated. This is illustrated in classical mechanics, where the motion of a body is described by its position and velocity as the time value varies. Newton's laws allow one (given the position, velocity, acceleration and various forces acting on the body) to express these variables dynamically as a differential equation for the unknown position of the body as a function of time. In some cases, this differential equation (called an equation of motion) may be solved explicitly.

Measure theory
A measure on a set is a systematic way to assign a number to each suitable subset of that set, intuitively interpreted as its size. In this sense, a measure is a generalization of the concepts of length, area, and volume. A particularly important example is the Lebesgue measure on a Euclidean space, which assigns the conventional length, area, and volume of Euclidean geometry to suitable subsets of the $$n$$-dimensional Euclidean space $$\mathbb{R}^n$$. For instance, the Lebesgue measure of the interval $$\left[0, 1\right]$$ in the real numbers is its length in the everyday sense of the word – specifically, 1.

Technically, a measure is a function that assigns a non-negative real number or +∞ to (certain) subsets of a set $$X$$. It must assign 0 to the empty set and be (countably) additive: the measure of a 'large' subset that can be decomposed into a finite (or countable) number of 'smaller' disjoint subsets, is the sum of the measures of the "smaller" subsets. In general, if one wants to associate a consistent size to each subset of a given set while satisfying the other axioms of a measure, one only finds trivial examples like the counting measure. This problem was resolved by defining measure only on a sub-collection of all subsets; the so-called measurable subsets, which are required to form a $\sigma$-algebra. This means that the empty set, countable unions, countable intersections and complements of measurable subsets are measurable. Non-measurable sets in a Euclidean space, on which the Lebesgue measure cannot be defined consistently, are necessarily complicated in the sense of being badly mixed up with their complement. Indeed, their existence is a non-trivial consequence of the axiom of choice.

Numerical analysis
Numerical analysis is the study of algorithms that use numerical approximation (as opposed to general symbolic manipulations) for the problems of mathematical analysis (as distinguished from discrete mathematics).

Modern numerical analysis does not seek exact answers, because exact answers are often impossible to obtain in practice. Instead, much of numerical analysis is concerned with obtaining approximate solutions while maintaining reasonable bounds on errors.

Numerical analysis naturally finds applications in all fields of engineering and the physical sciences, but in the 21st century, the life sciences and even the arts have adopted elements of scientific computations. Ordinary differential equations appear in celestial mechanics (planets, stars and galaxies); numerical linear algebra is important for data analysis; stochastic differential equations and Markov chains are essential in simulating living cells for medicine and biology.

Vector analysis
Vector analysis is a branch of mathematical analysis dealing with values which have both magnitude and direction. Some examples of vectors include velocity, force, and displacement. Vectors are commonly associated with scalars, values which describe magnitude.

Scalar analysis
Scalar analysis is a branch of mathematical analysis dealing with values related to scale as opposed to direction. Values such as temperature are scalar because they describe the magnitude of a value without regard to direction, force, or displacement that value may or may not have.

Other topics

 * Calculus of variations deals with extremizing functionals, as opposed to ordinary calculus which deals with functions.
 * Harmonic analysis deals with the representation of functions or signals as the superposition of basic waves.
 * Geometric analysis involves the use of geometrical methods in the study of partial differential equations and the application of the theory of partial differential equations to geometry.
 * Clifford analysis, the study of Clifford valued functions that are annihilated by Dirac or Dirac-like operators, termed in general as monogenic or Clifford analytic functions.
 * p-adic analysis, the study of analysis within the context of p-adic numbers, which differs in some interesting and surprising ways from its real and complex counterparts.
 * Non-standard analysis, which investigates the hyperreal numbers and their functions and gives a rigorous treatment of infinitesimals and infinitely large numbers.
 * Computable analysis, the study of which parts of analysis can be carried out in a computable manner.
 * Stochastic calculus – analytical notions developed for stochastic processes.
 * Set-valued analysis – applies ideas from analysis and topology to set-valued functions.
 * Convex analysis, the study of convex sets and functions.
 * Idempotent analysis – analysis in the context of an idempotent semiring, where the lack of an additive inverse is compensated somewhat by the idempotent rule A + A = A.
 * Tropical analysis – analysis of the idempotent semiring called the tropical semiring (or max-plus algebra/min-plus algebra).
 * Constructive analysis, which is built upon a foundation of constructive, rather than classical, logic and set theory.
 * Intuitionistic analysis, which is developed from constructive logic like constructive analysis but also incorporates choice sequences.
 * Paraconsistent analysis, which is built upon a foundation of paraconsistent, rather than classical, logic and set theory.
 * Smooth infinitesimal analysis, which is developed in a smooth topos.

Applications
Techniques from analysis are also found in other areas such as:

Physical sciences
The vast majority of classical mechanics, relativity, and quantum mechanics is based on applied analysis, and differential equations in particular. Examples of important differential equations include Newton's second law, the Schrödinger equation, and the Einstein field equations.

Functional analysis is also a major factor in quantum mechanics.

Signal processing
When processing signals, such as audio, radio waves, light waves, seismic waves, and even images, Fourier analysis can isolate individual components of a compound waveform, concentrating them for easier detection or removal. A large family of signal processing techniques consist of Fourier-transforming a signal, manipulating the Fourier-transformed data in a simple way, and reversing the transformation.

Other areas of mathematics
Techniques from analysis are used in many areas of mathematics, including:
 * Analytic number theory
 * Analytic combinatorics
 * Continuous probability
 * Differential entropy in information theory
 * Differential games
 * Differential geometry, the application of calculus to specific mathematical spaces known as manifolds that possess a complicated internal structure but behave in a simple manner locally.
 * Differentiable manifolds
 * Differential topology
 * Partial differential equations

Famous Textbooks

 * Foundation of Analysis: The Arithmetic of Whole Rational, Irrational and Complex Numbers, by Edmund Landau
 * Introductory Real Analysis, by Andrey Kolmogorov, Sergei Fomin
 * Differential and Integral Calculus (3 volumes), by Grigorii Fichtenholz
 * The Fundamentals of Mathematical Analysis (2 volumes), by Grigorii Fichtenholz
 * A Course Of Mathematical Analysis (2 volumes), by Sergey Nikolsky
 * Mathematical Analysis (2 volumes), by Vladimir Zorich
 * A Course of Higher Mathematics (5 volumes, 6 parts), by Vladimir Smirnov
 * Differential And Integral Calculus, by Nikolai Piskunov
 * A Course of Mathematical Analysis, by Aleksandr Khinchin
 * Mathematical Analysis: A Special Course, by Georgiy Shilov
 * Theory of Functions of a Real Variable (2 volumes), by Isidor Natanson
 * Problems in Mathematical Analysis, by Boris Demidovich
 * Problems and Theorems in Analysis (2 volumes), by George Pólya, Gábor Szegő
 * Mathematical Analysis: A Modern Approach to Advanced Calculus, by Tom Apostol
 * Principles of Mathematical Analysis, by Walter Rudin
 * Real Analysis: Measure Theory, Integration, and Hilbert Spaces, by Elias Stein
 * Complex Analysis: An Introduction to the Theory of Analytic Functions of One Complex Variable, by Lars Ahlfors
 * Complex Analysis, by Elias Stein
 * Functional Analysis: Introduction to Further Topics in Analysis, by Elias Stein
 * Analysis (2 volumes), by Terence Tao
 * Analysis (3 volumes), by Herbert Amann, Joachim Escher
 * Real and Functional Analysis, by Vladimir Bogachev, Oleg Smolyanov
 * Real and Functional Analysis, by Serge Lang

license: Creative Commons - Attribution Share-Alike - https://creativecommons.org/licenses/by-sa/4.0/

title: Mathematical analysis


text:
Isocyanate

In organic chemistry, isocyanate is the functional group with the formula R\sN\dC\dO. Organic compounds that contain an isocyanate group are referred to as isocyanates. An organic compound with two isocyanate groups is known as a diisocyanate. Diisocyanates are manufactured for the production of polyurethanes, a class of polymers.

Isocyanates should not be confused with cyanate esters and isocyanides, very different families of compounds. The cyanate (cyanate ester) functional group (R\sO\sC\tN) is arranged differently from the isocyanate group (R\sN\dC\dO). Isocyanides have the connectivity R\sN\tC, lacking the oxygen of the cyanate groups.

Structure and bonding
In terms of bonding, isocyanates are closely related to carbon dioxide (CO2) and carbodiimides (C(NR)2). The C−N=C=O unit that defines isocyanates is planar, and the N=C=O linkage is nearly linear. In phenyl isocyanate, the C=N and C=O distances are respectively 1.195 and 1.173 Å. The C−N=C angle is 134.9° and the N=C=O angle is 173.1°.

Production
Isocyanates are usually produced from amines by phosgenation, i.e. treating with phosgene:
 * RNH2 + COCl2 -> RNCO + 2 HCl

These reactions proceed via the intermediacy of a carbamoyl chloride (RNHC(O)Cl). Owing to the hazardous nature of phosgene, the production of isocyanates requires special precautions. A laboratory-safe variation masks the phosgene as oxalyl chloride.

Another route to isocyanates entails addition of isocyanic acid to alkenes. Complementarily, alkyl isocyanates form by displacement reactions involving alkyl halides and alkali metal cyanates.

Three rearrangement reactions involving nitrenes give isocyanates:
 * Schmidt reaction, a reaction where a carboxylic acid is treated with ammonia and hydrazoic acid yielding an isocyanate.
 * Curtius rearrangement degradation of an acyl azide to an isocyanate and nitrogen gas.
 * Lossen rearrangement, the conversion of a hydroxamic acid to an isocyanate via the formation of an O-acyl, sulfonyl, or phosphoryl intermediate.

With nucleophiles
Isocyanates are electrophiles, and as such they are reactive toward a variety of nucleophiles including alcohols, amines, and even water having a higher reactivity compared to structurally analogous isothiocyanates.

Upon treatment with an alcohol, an isocyanate forms a urethane linkage:
 * ROH + R'NCO -> ROC(O)N(H)R'

where R and R' are alkyl or aryl groups. If a diisocyanate is treated with a compound containing two or more hydroxyl groups, such as a diol or a polyol, polymer chains are formed, which are known as polyurethanes.

 * Polyurethane synthesis.tif

Isocyanates react with water to form carbon dioxide:
 * RNCO + H2O -> RNH2 + CO2

This reaction is exploited in tandem with the production of polyurethane to give polyurethane foams. The carbon dioxide functions as a blowing agent.

Isocyanates also react with amines to give ureas:
 * R2NH + R'NCO -> R2NC(O)N(H)R'

The addition of an isocyanate to a urea gives a biuret:
 * R2NC(O)N(H)R' + R'\'NCO -> R2NC(O)NR'C(O)NHR'\'

Reaction between a di-isocyanate and a compound containing two or more amine groups produces long polymer chains known as polyureas.

Carbodiimides are produced by the decarboxylation of alkyl and aryl isocyanate using phosphine oxides as a catalyst:
 * C6H11NCO -> (C6H11N)2C + CO2

Cyclization
Isocyanates also can react with themselves. Aliphatic diisocyanates can trimerise to from substituted isocyanuric acid groups. This can be seen in the formation of polyisocyanurate resins (PIR) which are commonly used as rigid thermal insulation. Isocyanates participate in Diels–Alder reactions, functioning as dienophiles.

Rearrangement reactions
Isocyanates are common intermediates in the synthesis of primary amines via hydrolysis:
 * Hofmann rearrangement, a reaction in which a primary amide is treated with a strong oxidizer such as sodium hypobromite  or lead tetraacetate to form an isocyanate intermediate.

Common isocyanates
The global market for diisocyanates in the year 2000 was 4.4 million tonnes, of which 61.3% was methylene diphenyl diisocyanate (MDI), 34.1% was toluene diisocyanate (TDI), 3.4% was the total for hexamethylene diisocyanate (HDI) and isophorone diisocyanate (IPDI), and 1.2% was the total for various others. A monofunctional isocyanate of industrial significance is methyl isocyanate (MIC), which is used in the manufacture of pesticides.

Common applications
MDI is commonly used in the manufacture of rigid foams and surface coating. Polyurethane foam boards are used in construction for insulation. TDI is commonly used in applications where flexible foams are used, such as furniture and bedding. Both MDI and TDI are used in the making of adhesives and sealants due to weather-resistant properties. Isocyanates, both MDI and TDI are widely used in as spraying applications of insulation due to the speed and flexibility of applications. Foams can be sprayed into structures and harden in place or retain some flexibility as required by the application. HDI is commonly utilized in high-performance surface-coating applications, including automotive paints.

Health and safety
The risks of isocyanates was brought to the world's attention with the 1984 Bhopal disaster, which caused the death of nearly 4000 people from the accidental release of methyl isocyanate. In 2008, the same chemical was involved in an explosion at a pesticide manufacturing plant in West Virginia.

LD50s for isocyanates are typically several hundred milligrams per kilogram. Despite this low acute toxicity, an extremely low short-term exposure limit (STEL) of 0.07 mg/m3 is the legal limit for all isocyanates (except methyl isocyanate: 0.02 mg/m3) in the United Kingdom. These limits are set to protect workers from chronic health effects such as occupational asthma, contact dermatitis, or irritation of the respiratory tract.

Since they are used in spraying applications, the properties of their aerosols have attracted attention. In the U.S., OSHA conducted a National Emphasis Program on isocyanates starting in 2013 to make employers and workers more aware of the health risks. Polyurethanes have variable curing times, and the presence of free isocyanates in foams vary accordingly.

Both the US National Toxicology Program (NTP) and International Agency for Research on Cancer (IARC) have evaluated TDI as a potential human carcinogen and Group 2B "possibly carcinogenic to humans". MDI appears to be relatively safer and is unlikely a human carcinogen. The IARC evaluates MDI as Group 3 "not classifiable as to its carcinogenicity in humans".

All major producers of MDI and TDI are members of the International Isocyanate Institute, which promotes the safe handling of MDI and TDI.

Toxicity
Isocyanates can present respiratory hazards as particulates, vapors or aerosols. Autobody shop workers are a very commonly examined population for isocyanate exposure as they are repeatedly exposed when spray painting automobiles and can be exposed when installing truck bed liners. Hypersensitivity pneumonitis has slower onset and features chronic inflammation that can be seen on imaging of the lungs. Occupational asthma is a worrisome outcome of respiratory sensitization to isocyanates as it can be acutely fatal. Diagnosis of occupational asthma is generally performed using pulmonary function testing (PFT) and performed by pulmonology or occupational medicine physicians. Occupational asthma is much like asthma in that it causes episodic shortness of breath and wheezing. Both the dose and duration of exposure to isocyanates can lead to respiratory sensitization. Dermal exposures to isocyanates can sensitize an exposed person to respiratory disease.

Dermal exposures can occur via mixing, spraying coatings or applying and spreading coatings manually. Dermal exposures to isocyanates is known to lead to respiratory sensitization. Even when the right personal protective equipment (PPE) is used, exposures can occur to body areas not completely covered. Isocyanates can also permeate improper PPE, necessitating frequent changes of both disposable gloves and suits if they become over exposed.

Flammability
Methyl isocyanate (MIC) is highly flammable. MDI and TDI are much less flammable. Flammability of materials is a consideration in furniture design. The specific flammability hazard is noted on the safety data sheet (SDS) for specific isocyanates.

Hazard minimization
Industrial science attempts to minimize the hazards of isocyanates through multiple techniques. The EPA has sponsored ongoing research on polyurethane production without isocyanates. Where isocyanates are unavoidable but interchangeable, substituting a less hazardous isocyanate may control hazards. Ventilation and automation can also minimizes worker exposure to the isocyanates used.

If human workers must enter isocyanate-contaminated regions, personal protective equipment (PPE) can reduce their intake. In general, workers wear eye protection and gloves and coveralls to reduce dermal exposure  For some autobody paint and clear-coat spraying applications, a full-face mask is required.

The US Occupational Safety and Health Administration (OSHA) requires frequent training to ensure isocyanate hazards are appropriately minimized. Moreover, OSHA requires standardized isocyanate concentration measurements to avoid violating occupational exposure limits. In the case of MDI, OSHA expects sampling with glass-fiber filters at standard air flow rates, and then liquid chromatography.

Combined industrial hygiene and medical surveillance can significantly reduce occupational asthma incidence. Biological tests exist to identify isocyanate exposure; the US Navy uses regular pulmonary function testing and screening questionnaires.

Emergency management is a complex process of preparation and should be considered in a setting where a release of bulk chemicals may threaten the well-being of the public. In the Bhopal disaster, an uncontrolled MIC release killed thousands, affected hundreds of thousands more, and spurred the development of modern disaster preparation.

Occupational exposure limits
Exposure limits can be expressed as ceiling limits, a maximal value, short-term exposure limits (STEL), a 15-minute exposure limit or an 8-hour time-weighted average limit (TWA). Below is a sampling, not exhaustive, as less common isocyanates also have specific limits within the United States, and in some regions there are limits on total isocyanate, which recognizes some of the uncertainty regarding the safety of mixtures of chemicals as compared to pure chemical exposures. For example, while there is no OEL for HDI, NIOSH has a REL of 5 ppb for an 8-hour TWA and a ceiling limit of 20 ppb, consistent with the recommendations for MDI.

United States
The Occupational Safety and Health Administration (OSHA) is the regulatory body covering worker safety. OSHA puts forth permissible exposure limit (PEL) 20 ppb for MDI and detailed technical guidance on exposure assessment.

The National Institutes of Health (NIOSH) is the agency responsible for providing the research and recommendations regarding workplace safety, while OSHA is more of an enforcement body. NIOSH is responsible for producing the science that can result in recommended exposure limits (REL), which can be lower than the PEL. OSHA is tasked with enforcement and defending the enforceable limits (PELs). In 1992, when OSHA reduced the PEL for TDI to the NIOSH REL, the PEL reduction was challenged in court, and the reduction was reversed.

The Environmental Protection Agency (EPA) is also involved in the regulation of isocyanates with regard to the environment and also non-worker persons that might be exposed.

The American Conference of Governmental Industrial Hygienists (ACGIH) is a non-government organization that publishes guidance known as threshold limit values (TLV) for. The TLV is not an OSHA-enforceable value, unless the PEL is the same.

European Union
The European Chemicals Agency (ECHA) provides regulatory oversight of chemicals used within the European Union. ECHA has been implementing policy aimed at limiting worker exposure through elimination by lower allowable concentrations in products and mandatory worker training, an administrative control. Within the European Union, many nations set their own occupational exposure limits for isocyanates.

International groups
The United Nations, through the World Health Organization (WHO) together with the International Labour Organization (ILO) and United Nations Environment Programme (UNEP), collaborate on the International Programme on Chemical Safety (IPCS) to publish summary documents on chemicals. The IPCS published one such document in 2000 summarizing the status of scientific knowledge on MDI.

The IARC evaluates the hazard data on chemicals and assigns a rating on the risk of carcinogenesis. In the case of TDI, the final evaluation is possibly carcinogenic to humans (Group 2B). For MDI, the final evaluation is not classifiable as to its carcinogenicity to humans (Group 3).

The International Isocyanate Institute is an international industry consortium that seeks promote the safe utilization of isocyanates by promulgating best practices.

title: Isocyanate