Navier–Stokes equations

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 curves tangent to the instantaneous velocity field. 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:

D𝐮Dt=1ρ∇⋅𝝈+𝐚.

By setting the Cauchy stress tensor 𝝈 to be the sum of a viscosity term 𝝉 (the deviatoric stress) and a pressure term −p𝐈 (volumetric stress), we arrive at:

where

DDt is the material derivative, defined as ∂∂t+𝐮⋅∇,

ρ is the (mass) density,

𝐮 is the flow velocity,

∇⋅ is the divergence,

p is the pressure,

t is time,

𝝉 is the deviatoric stress tensor, which has order 2,

𝐚 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 𝐃𝐃𝐭) of any finite volume (𝐕) to represent the change of velocity in fluid media:

𝐃m𝐃𝐭=∭V(𝐃ρ𝐃𝐭+ρ(∇⋅𝐮))dV𝐃ρ𝐃𝐭+ρ(∇⋅𝐮)=∂ρ∂t+(∇ρ)⋅𝐮+ρ(∇⋅𝐮)=∂ρ∂t+∇⋅(ρ𝐮)=0where

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 𝝉 as it was in the general continuum equations and in the incompressible flow section.

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

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 ∇𝐮, or more simply the rate-of-strain tensor: 𝜺(∇𝐮)≡12∇𝐮+12(∇𝐮)𝖳

the deviatoric stress is linear in this variable: 𝝈(𝜺)=−p𝐈+𝐂:𝜺, where p is independent on the strain rate tensor, 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 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 λ and the dynamic viscosity μ, as it is usual in linear elasticity:

where 𝐈 is the identity tensor, and tr⁡(𝜺) is the trace of the rate-of-strain tensor. So this decomposition can be explicitly defined as:

𝝈=−p𝐈+λ(∇⋅𝐮)𝐈+μ(∇𝐮+(∇𝐮)𝖳).

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

tr⁡(𝜺)=∇⋅𝐮.

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

tr⁡(𝑰)=3.

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 ∇𝐮.

the fluid is assumed to be isotropic, as with gases and simple liquids, and consequently 𝝉 is an isotropic tensor; furthermore, since the deviatoric stress tensor can be expressed in terms of the dynamic viscosity μ:

where

𝜺=12(𝜵𝐮+𝜵𝐮𝖳)

is the rate-of-strain tensor. So this decomposition can be made explicit as:

This 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:

∇⋅𝝉=2μ∇⋅𝜺=μ∇⋅(∇𝐮+∇𝐮𝖳)=μ∇2𝐮

because ∇⋅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:

Weak form of the incompressible Navier–Stokes equations

Consider the incompressible Navier–Stokes equations for a Newtonian fluid of constant density ρ in a domain

Ω⊂ℝd(d=2,3)

with boundary

∂Ω=ΓD∪ΓN,

being ΓD and ΓN portions of the boundary where respectively a Dirichlet and a Neumann boundary condition is applied (ΓD∩ΓN=∅︀):

{ρ∂𝐮∂t+ρ(𝐮⋅∇)𝐮−∇⋅𝝈(𝐮,p)=𝐟 in Ω×(0,T)∇⋅𝐮=0 in Ω×(0,T)𝐮=𝐠 on ΓD×(0,T)𝝈(𝐮,p)𝐧̂=𝐡 on ΓN×(0,T)𝐮(0)=𝐮0 in Ω×{0}

𝐮 is the fluid velocity, p the fluid pressure, 𝐟 a given forcing term, 𝐧̂ the outward directed unit normal vector to ΓN, and 𝝈(𝐮,p) the viscous stress tensor defined as:

𝝈(𝐮,p)=−p𝐈+2μ𝜺(𝐮).

Let μ be the dynamic viscosity of the fluid, 𝐈 the second-order identity tensor and 𝜺(𝐮) the strain-rate tensor defined as:

𝜺(𝐮)=12((∇𝐮)+(∇𝐮)𝖳).

The functions 𝐠 and 𝐡 are given Dirichlet and Neumann boundary data, while 𝐮0 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

∇⋅(∇𝐟)𝖳=∇(∇⋅𝐟)

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

Discrete velocity

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

(𝐰i,∂𝐮j∂t)=−(𝐰i,(𝐮⋅∇)𝐮j)−ν(∇𝐰i:∇𝐮j)+(𝐰i,𝐟S).

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,

∇φ=(∂φ∂x,∂φ∂y)𝖳,∇×φ=(∂φ∂y,−∂φ∂x)𝖳.

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.

Pressure recovery

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

(𝐠i,∇p)=−(𝐠i,(𝐮⋅∇)𝐮j)−ν(∇𝐠i:∇𝐮j)+(𝐠i,𝐟I)

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 𝐠i 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 𝐔(t) and rotating with angular velocity Ω(t) with respect to the stationary frame. The Navier–Stokes equation observed from the non-inertial frame then becomes

Here 𝐱 and 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:

𝐃m𝐃𝐭=∭V(𝐃ρ𝐃𝐭+ρ(∇⋅𝐮))dV𝐃ρ𝐃𝐭+ρ(∇⋅𝐮)=∂ρ∂t+(∇ρ)⋅𝐮+ρ(∇⋅𝐮)=∂ρ∂t+∇⋅(ρ𝐮)=0

A fluid media for which the density ρ is constant is called incompressible. Therefore, the rate of change of ρ with respect to time ∂ρ∂t and the gradient of density ∇ρ are equal to zero. In this case the general equation of continuity, ∂ρ∂t+∇⋅(ρ𝐮)=0, reduces to: ρ(∇⋅𝐮)=0 Furthermore, assuming that ρ≠0 means that the right-hand side of the equation (zero) is divisible by density ρ. Therefore, the continuity equation for an incompressible fluid reduces further to:(∇⋅𝐮)=0

This relationship, (∇⋅𝐮)=0, identifies that the divergence of the flow velocity vector 𝐮 is equal to zero, 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 ∇2𝐮=∇(∇⋅𝐮)−∇×(∇×𝐮), and vorticity 𝝎=∇×𝐮 which is now expressed like so, for an incompressible fluid:∇2𝐮=−(∇×(∇×𝐮))=−(∇×𝝎)

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 uz=0 and no dependence of anything on z), where the equations reduce to:

ρ(∂ux∂t+ux∂ux∂x+uy∂ux∂y)=−∂p∂x+μ(∂2ux∂x2+∂2ux∂y2)+ρgxρ(∂uy∂t+ux∂uy∂x+uy∂uy∂y)=−∂p∂y+μ(∂2uy∂x2+∂2uy∂y2)+ρgy.

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 ψ through

ux=∂ψ∂y;uy=−∂ψ∂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:

∂∂t(∇2ψ)+∂ψ∂y∂∂x(∇2ψ)−∂ψ∂x∂∂y(∇2ψ)=ν∇4ψ

where ∇4 is the 2D biharmonic operator and ν is the kinematic viscosity, ν=μρ. We can also express this compactly using the Jacobian determinant:

∂∂t(∇2ψ)+∂(ψ,∇2ψ)∂(y,x)=ν∇4ψ.

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.

Properties

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, k–ω, k–ε, 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:

d2udy2=−1;u(0)=u(1)=0.

The boundary condition is the no slip condition. This problem is easily solved for the flow field:

u(y)=y−y22.

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

Radial flow

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

d2fdz2+Rf2=−1;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. Time-dependent self-similar solutions of the three-dimensional non-compressible Navier–Stokes equations in Cartesian coordinate can be given with the help of the Kummer's functions with quadratic arguments. For the compressible Navier–Stokes equations the time-dependent self-similar solutions are however the Whittaker functions again with quadratic arguments when the polytropic equation of state is used as a closing condition. 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:

ρ(x,y,z)=3Br2+x2+y2+z2p(x,y,z)=−A2B(r2+x2+y2+z2)3𝐮(x,y,z)=A(r2+x2+y2+z2)2(2(−ry+xz)2(rx+yz)r2−x2−y2+z2)g=0μ=0

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 ρ 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 𝕋3=ℝ3/Lℤ3 and are characterized by positive and negative helicity respectively.

The solution with positive helicity is given by:

ux=4233U0[sin⁡(kx−π3)cos⁡(ky+π3)sin⁡(kz+π2)−cos⁡(kz−π3)sin⁡(kx+π3)sin⁡(ky+π2)]e−3νk2tuy=4233U0[sin⁡(ky−π3)cos⁡(kz+π3)sin⁡(kx+π2)−cos⁡(kx−π3)sin⁡(ky+π3)sin⁡(kz+π2)]e−3νk2tuz=4233U0[sin⁡(kz−π3)cos⁡(kx+π3)sin⁡(ky+π2)−cos⁡(ky−π3)sin⁡(kz+π3)sin⁡(kx+π2)]e−3νk2t

where k=2π/L is the wave number and the velocity components are normalized so that the average kinetic energy per unit of mass is U02/2 at t=0.

The pressure field is obtained from the velocity field as p=p0−ρ0‖𝒖‖2/2 (where p0 and ρ0 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 ω=3k𝒖.

These solutions can be regarded as a generalization in three dimensions of the classic two-dimensional Taylor–Green vortex.

OpenAI announcement

On 8 September 2026, artificial intelligence company OpenAI announced it had solved the Millennium Prize Problem on the existence and smoothness of the incompressible Navier–Stokes equations in three-dimensional Euclidean space. OpenAI stated that the solution to the problem, a counterexample that refers to statements C and D of the problem statement, was developed by its researchers using as many as 10,000 coordinated agents running an internal frontier model, along with a formalization in the Lean proof assistant. The proposed result has not been verified by external mathematicians or the Clay Mathematics Institute, while OpenAI stated it would not claim the Millennium Prize. The announcement was accompanied by a priority dispute with Levent Alpöge (employed at rival AI company Anthropic) and Tristan Buckmaster, who had derived a set of closely related results on the Euler equations. The method used to generate the proposed result built upon a method developed by Diego Córdoba and Luis Martínez Zoroa in 2023 to prove blowup phenomena in related fluid equations.

On 6 October a paper published on arxiv found discrepancies between OpenAI's Lean proof and natural language proof.

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 Mstislav 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. ∂xu means the partial derivative of u with respect to x, and ∂y2fθ means the second-order partial derivative of fθ 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 𝐮=(ux,uy,uz), sometimes respectively named u, v, w, we may write the vector equation explicitly,

x: ρ(∂tux+ux∂xux+uy∂yux+uz∂zux)=−∂xp+μ(∂x2ux+∂y2ux+∂z2ux)+13μ ∂x(∂xux+∂yuy+∂zuz)+ρgx

y: ρ(∂tuy+ux∂xuy+uy∂yuy+uz∂zuy)=−∂yp+μ(∂x2uy+∂y2uy+∂z2uy)+13μ ∂y(∂xux+∂yuy+∂zuz)+ρgy

z: ρ(∂tuz+ux∂xuz+uy∂yuz+uz∂zuz)=−∂zp+μ(∂x2uz+∂y2uz+∂z2uz)+13μ ∂z(∂xux+∂yuy+∂zuz)+ρgz.

Note that gravity has been accounted for as a body force, and the values of gx, gy, gz will depend on the orientation of gravity with respect to the chosen set of coordinates.

The continuity equation reads:

∂tρ+∂x(ρux)+∂y(ρuy)+∂z(ρuz)=0.

When the flow is incompressible, ρ does not change for any fluid particle, and its material derivative vanishes: DρDt=0. The continuity equation is reduced to:

∂xux+∂yuy+∂zuz=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, ϕ, and z

r: ρ(∂tur+ur∂rur+uφr∂φur+uz∂zur−uφ2r)=−∂rp+μ(1r∂r(r∂rur)+1r2∂φ2ur+∂z2ur−urr2−2r2∂φuφ)+13μ∂r(1r∂r(rur)+1r∂φuφ+∂zuz)+ρgr

φ: ρ(∂tuφ+ur∂ruφ+uφr∂φuφ+uz∂zuφ+uruφr)=−1r∂φp+μ(1r ∂r(r∂ruφ)+1r2∂φ2uφ+∂z2uφ−uφr2+2r2∂φur)+13μ1r∂φ(1r∂r(rur)+1r∂φuφ+∂zuz)+ρgφ

z: ρ(∂tuz+ur∂ruz+uφr∂φuz+uz∂zuz)=−∂zp+μ(1r∂r(r∂ruz)+1r2∂φ2uz+∂z2uz)+13μ∂z(1r∂r(rur)+1r∂φuφ+∂zuz)+ρgz.

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:

∂tρ+1r∂r(ρrur)+1r∂φ(ρuφ)+∂z(ρuz)=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 (uϕ=0), and the remaining quantities are independent of ϕ:

ρ(∂tur+ur∂rur+uz∂zur)=−∂rp+μ(1r∂r(r∂rur)+∂z2ur−urr2)+ρgrρ(∂tuz+ur∂ruz+uz∂zuz)=−∂zp+μ(1r∂r(r∂ruz)+∂z2uz)+ρgz1r∂r(rur)+∂zuz=0.

Spherical coordinates

In spherical coordinates, the r, ϕ, and θ momentum equations are (note the convention used: θ is polar angle, or colatitude, 0≤θ≤π):

r: ρ(∂tur+ur∂rur+uφrsin⁡θ∂φur+uθr∂θur−uφ2+uθ2r)=−∂rp+μ(1r2∂r(r2∂rur)+1r2sin2θ∂φ2ur+1r2sin⁡θ∂θ(sin⁡θ∂θur)−2ur+∂θuθ+uθcot⁡θr2−2r2sin⁡θ∂φuφ)+13μ∂r(1r2∂r(r2ur)+1rsin⁡θ∂θ(uθsin⁡θ)+1rsin⁡θ∂φuφ)+ρgr

φ: ρ(∂tuφ+ur∂ruφ+uφrsin⁡θ∂φuφ+uθr∂θuφ+uruφ+uφuθcot⁡θr)=−1rsin⁡θ∂φp+μ(1r2∂r(r2∂ruφ)+1r2sin2θ∂φ2uφ+1r2sin⁡θ∂θ(sin⁡θ∂θuφ)+2sin⁡θ∂φur+2cos⁡θ∂φuθ−uφr2sin2θ)+13μ1rsin⁡θ∂φ(1r2∂r(r2ur)+1rsin⁡θ∂θ(uθsin⁡θ)+1rsin⁡θ∂φuφ)+ρgφ

θ: ρ(∂tuθ+ur∂ruθ+uφrsin⁡θ∂φuθ+uθr∂θuθ+uruθ−uφ2cot⁡θr)=−1r∂θp+μ(1r2∂r(r2∂ruθ)+1r2sin2θ∂φ2uθ+1r2sin⁡θ∂θ(sin⁡θ∂θuθ)+2r2∂θur−uθ+2cos⁡θ∂φuφr2sin2θ)+13μ1r∂θ(1r2∂r(r2ur)+1rsin⁡θ∂θ(uθsin⁡θ)+1rsin⁡θ∂φuφ)+ρgθ.

Mass continuity will read:

∂tρ+1r2∂r(ρr2ur)+1rsin⁡θ∂φ(ρuφ)+1rsin⁡θ∂θ(sin⁡θρuθ)=0.

These equations could be (slightly) compacted by, for example, factoring 1r2 from the viscous terms. However, doing so would undesirably alter the structure of the Laplacian and other quantities.