The Navier–Stokes equations ( ) describe the motion of viscous fluids. This system of partial differential equations was named after Claude-Louis Navier and George Gabriel Stokes, who developed them over a few decades of progressive work, from 1822 (Navier) to 1842–1850 (Stokes). Siméon Denis Poisson independently achieved the same results.
The Navier–Stokes equations mathematically express momentum balance for Newtonian fluids and make use of the conservation of mass. They are sometimes accompanied by an equation of state relating pressure, temperature and density. They arise from applying 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 Navier–Stokes equations generalize the Euler equations which only consider inviscid flow.
The Navier–Stokes equations are of great scientific and engineering interest because they may be used to model a wide variety of scenarios. In their full or simplified forms, they can assist in 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 comprise the fundamentals of magnetohydrodynamics.
The Navier–Stokes equations are also of great interest to pure mathematics. The Navier–Stokes existence and smoothness problem concerns whether they have smooth (meaning infinitely differentiable) or bounded solutions in three-dimensional Euclidean space, as opposed to a breakdown with unbounded solutions. This is one of seven Millennium Prize Problems, notable open mathematics problems for which the Clay Mathematics Institute offered $1 million prizes in 2000 for correct solutions. In September 2026, OpenAI announced a claimed counterexample to the existence and smoothness problem. The announcement was followed by a priority dispute, and the claimed counterexample has yet to be independently verified.
Flow velocity The solution of the equations is a flow velocity. It is a vector field—to every point in a fluid, at any moment in a time interval, it gives a vector whose direction and magnitude are those of the velocity of the fluid at that point in space and at that moment in time. It is studied in three spatial dimensions and one time dimension, and higher-dimensional analogues are studied in both pure and applied mathematics. Once the velocity field is calculated, other quantities of interest, such as pressure or temperature, may be found using dynamical equations and relations. This is different from what one normally sees in classical mechanics, where solutions are typically trajectories of the position of a particle or deflection of a continuum. Studying velocity instead of position makes more sense for a fluid, although for visualization purposes, one can compute various trajectories. In particular, the streamlines of a vector field, interpreted as flow velocity, are the paths along which a massless fluid particle would travel. These paths are the integral curves whose derivative at each point is equal to the vector field, and they can represent visually the behavior of the vector field at a point in time.
General continuum equations
The Navier–Stokes momentum equation can be derived as a particular form of the Cauchy momentum equation, whose general convective form is:
By setting the Cauchy stress tensor to be the sum of a viscosity term (the deviatoric stress) and a pressure term (volumetric stress), we arrive at:
where is the material derivative, defined as , is the (mass) density, is the flow velocity, is the divergence, is the pressure, is time, is the deviatoric stress tensor, which has order 2, 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 ) of any finite volume () to represent the change of velocity in fluid media: where is the material derivative of mass per unit volume (density, ), is the mathematical operation for the integration throughout the volume (), is the partial derivative mathematical operator, is the divergence of the flow velocity (), which is a scalar field,Refer to the mathematical operator del represented by the nabla () symbol. is the gradient of density (), which is the vector derivative of a scalar field, to arrive at the conservation form of the equations of motion. This is often written:
where is the outer product of the flow velocity ():
The left side of the equation describes acceleration, and may be composed of time-dependent and convective components (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 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 , or more simply the rate-of-strain tensor: the deviatoric stress is linear in this variable: , where is independent on the strain rate tensor, 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 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 and the dynamic viscosity , as it is usual in linear elasticity:
where is the identity tensor, and is the trace of the rate-of-strain tensor. So this decomposition can be explicitly defined as:
Since the trace of the rate-of-strain tensor in three dimensions is the divergence (i.e. rate of expansion) of the flow:
Given this relation, and since the trace of the identity tensor in three dimensions is three:
the trace of the stress tensor in three dimensions becomes:
So by alternatively decomposing the stress tensor into isotropic and deviatoric parts, as usual in fluid dynamics:
Introducing the bulk viscosity ,
we arrive at the linear constitutive equation in the form usually employed in thermal hydraulics:
which can also be arranged in the other usual form:Bird, Stewart, Lightfoot, Transport Phenomena, 1st ed., 1960, eq. (3.2-11a).
Note that in the compressible case the pressure is no more proportional to the isotropic stress term, since there is the additional bulk viscosity term:
and the deviatoric stress tensor is still coincident with the shear stress tensor (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:
Both bulk viscosity and dynamic viscosity need not be constant – in general, they depend on two thermodynamics variables if the fluid contains a single chemical species, say for example, pressure and temperature. Any equation that makes explicit one of these transport coefficient in the conservation variables is called an equation of state.
The most general of the Navier–Stokes equations become
in index notation, the equation can be written asLandau, Lev Davidovich, and Evgenii Mikhailovich Lifshitz. Fluid mechanics: Landau And Lifshitz: course of theoretical physics, Volume 6. Vol. 6. Elsevier, 2013.
The corresponding equation in conservation form can be obtained by considering that, given the mass continuity equation, the left side is equivalent to:
to give finally:
Apart from its dependence of pressure and temperature, the second viscosity coefficient also depends on the process, that is to say, the second viscosity coefficient is not just a material property. Example: in the case of a sound wave with a definitive frequency that alternatively compresses and expands a fluid element, the second viscosity coefficient depends on the frequency of the wave. This dependence is called the dispersion. In some cases, the second viscosity can be assumed to be constant in which case, the effect of the volume viscosity is that the mechanical pressure is not equivalent to the thermodynamic pressure: as demonstrated below.
However, this difference is usually neglected most of the time (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 . The assumption of setting is called as the Stokes hypothesis.Stokes, G. G. (1845). On the theories of the internal friction of fluids in motion, and of the equilibrium and motion of elastic solids. The validity of Stokes hypothesis can be demonstrated for monoatomic gas both experimentally and from the kinetic theory; for other gases and liquids, Stokes hypothesis is generally incorrect. With the Stokes hypothesis, the Navier–Stokes equations become
If the dynamic and bulk viscosities are assumed to be uniform in space, the equations in convective form can be simplified further. By computing the divergence of the stress tensor, since the divergence of tensor is and the divergence of tensor is , one finally arrives to the compressible Navier–Stokes momentum equation:
where is the material derivative. is the shear kinematic viscosity and is the bulk kinematic viscosity. The left-hand side changes in the conservation form of the Navier–Stokes momentum equation. By bringing the operator on the flow velocity on the left side, one also has:
The convective acceleration term can also be written as
where the vector is known as the Lamb vector.
For the special case of an incompressible flow, the pressure constrains the flow so that the volume of fluid elements is constant: isochoric flow resulting in a solenoidal velocity field with .
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 .