{"history":[],"article":{"id":"f30a3bd9-ce14-47fb-86f5-c49d6550dd3a","title":"Navier–Stokes equations","slug":"navier-stokes-equations","content":":::summary\n:::\n\n# \nThe **Navier–Stokes equations** are [partial differential equations](https://bliptext.com/articles/partial-differential-equation) which describe the motion of [viscous fluid](https://bliptext.com/articles/viscous-fluid) substances.\nThey were named after French engineer and physicist [Claude-Louis Navier](https://bliptext.com/articles/claude-louis-navier) and the Irish physicist and mathematician [George Gabriel Stokes](https://bliptext.com/articles/sir-george-stokes-1st-baronet).\nThey were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes).\nThe Navier–Stokes equations mathematically express [momentum](https://bliptext.com/articles/momentum) balance for [Newtonian fluids](https://bliptext.com/articles/newtonian-fluid) and make use of [conservation of mass](https://bliptext.com/articles/conservation-of-mass).\nThey are sometimes accompanied by an [equation of state](https://bliptext.com/articles/equation-of-state) relating [pressure](https://bliptext.com/articles/pressure), [temperature](https://bliptext.com/articles/temperature) and [density](https://bliptext.com/articles/density).\nThey arise from applying [Isaac Newton's second law](https://bliptext.com/articles/newton-s-second-law) to [fluid motion](https://bliptext.com/articles/fluid-dynamics), together with the assumption that the [stress](https://bliptext.com/articles/stress-mechanics) in the fluid is the sum of a [diffusing](https://bliptext.com/articles/diffusion) [viscous](https://bliptext.com/articles/viscosity) term (proportional to the [gradient](https://bliptext.com/articles/gradient) of velocity) and a [pressure](https://bliptext.com/articles/pressure) term—hence describing [viscous](https://bliptext.com/articles/viscosity) flow.\nThe difference between them and the closely related [Euler equations](https://bliptext.com/articles/euler-equations-fluid-dynamics) is that Navier–Stokes equations take [viscosity](https://bliptext.com/articles/viscosity) into account while the [Euler equations](https://bliptext.com/articles/euler-equations-fluid-dynamics) model only [inviscid flow](https://bliptext.com/articles/inviscid-flow).\nAs a result, the Navier–Stokes are a [parabolic equation](https://bliptext.com/articles/parabolic-equation) and therefore have better analytic properties, at the expense of having less mathematical structure (e.g. they are never [completely integrable](https://bliptext.com/articles/completely-integrable)).\nThe Navier–Stokes equations are useful because they describe the physics of many phenomena of [scientific](https://bliptext.com/articles/scientific) and [engineering](https://bliptext.com/articles/engineering) interest.\nThey may be used to [model](https://bliptext.com/articles/model-abstract) the weather, [ocean currents](https://bliptext.com/articles/ocean-current), water [flow in a pipe](https://bliptext.com/articles/flow-conditioning) and air flow around a [wing](https://bliptext.com/articles/airfoil).\nThe Navier–Stokes equations, in their full and simplified forms, help with the design of [aircraft](https://bliptext.com/articles/aircraft-design-process) and cars, the study of [blood flow](https://bliptext.com/articles/hemodynamics), the design of [power stations](https://bliptext.com/articles/power-station), the analysis of [pollution](https://bliptext.com/articles/pollution), and many other problems.\nCoupled with [Maxwell's equations](https://bliptext.com/articles/maxwell-s-equations), they can be used to model and study [magnetohydrodynamics](https://bliptext.com/articles/magnetohydrodynamics).\nThe Navier–Stokes equations are also of great interest in a purely mathematical sense.\nDespite their wide range of practical uses, it has not yet been proven whether smooth solutions always [exist](https://bliptext.com/articles/existence-theorem) in three dimensions—i.e., whether they are infinitely differentiable (or even just bounded) at all points in the [domain](https://bliptext.com/articles/domain-mathematical-analysis).\nThis is called the [Navier–Stokes existence and smoothness](https://bliptext.com/articles/navier-stokes-existence-and-smoothness) problem.\nThe [Clay Mathematics Institute](https://bliptext.com/articles/clay-mathematics-institute) has called this one of the [seven most important open problems in mathematics](https://bliptext.com/articles/millennium-prize-problems) and has offered a [US$](https://bliptext.com/articles/us)1 million prize for a solution or a counterexample.\n\n# Flow velocity\nThe solution of the equations is a [flow velocity](https://bliptext.com/articles/flow-velocity).\nIt is a [vector field](https://bliptext.com/articles/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.\nIt 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.\nOnce the velocity field is calculated, other quantities of interest such as [pressure](https://bliptext.com/articles/pressure) or [temperature](https://bliptext.com/articles/temperature) may be found using dynamical equations and relations.\nThis is different from what one normally sees in [classical mechanics](https://bliptext.com/articles/classical-mechanics), where solutions are typically trajectories of position of a [particle](https://bliptext.com/articles/particle) or deflection of a [continuum](https://bliptext.com/articles/continuum-theory).\nStudying velocity instead of position makes more sense for a fluid, although for visualization purposes one can compute various [trajectories](https://bliptext.com/articles/streamlines-streaklines-and-pathlines).\nIn particular, the [streamlines](https://bliptext.com/articles/streamlines-streaklines-and-pathlines) of a vector field, interpreted as flow velocity, are the paths along which a massless fluid particle would travel.\nThese paths are the [integral curves](https://bliptext.com/articles/integral-curve) 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.\n\n# General continuum equations\nThe Navier–Stokes momentum equation can be derived as a particular form of the [Cauchy momentum equation](https://bliptext.com/articles/cauchy-momentum-equation), whose general convective form is:\nBy setting the [Cauchy stress tensor](https://bliptext.com/articles/cauchy-stress-tensor) to be the sum of a viscosity term (the [deviatoric stress](https://bliptext.com/articles/deviatoric-stress)) and a pressure term (volumetric stress), we arrive at:\nwhere\nIn this form, it is apparent that in the assumption of an inviscid fluid – no deviatoric stress – Cauchy equations reduce to the [Euler equations](https://bliptext.com/articles/euler-equations-fluid-dynamics).\nAssuming [conservation of mass](https://bliptext.com/articles/conservation-of-mass), with the known properties of [divergence](https://bliptext.com/articles/divergence) and [gradient](https://bliptext.com/articles/gradient) we can use the mass continuity equation, which represents the mass per unit volume of a [homogenous](https://bliptext.com/articles/homogenous) fluid with respect to space and time (i.e., [material derivative](https://bliptext.com/articles/material-derivative) ) of any finite volume (**V**) to represent the change of velocity in fluid media:\nwhere\nNote 1 - Refer to the mathematical operator del represented by the nabla (\\nabla) symbol.\nto arrive at the conservation form of the equations of motion.\nThis is often written:\nwhere \\otimes is the [outer product](https://bliptext.com/articles/outer-product) of the flow velocity (\\mathbf{u}):\nThe 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).\nThe right side of the equation is in effect a summation of hydrostatic effects, the divergence of deviatoric stress and body forces (such as gravity).\nAll 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](https://bliptext.com/articles/constitutive-relation).\nBy expressing the deviatoric (shear) stress tensor in terms of [viscosity](https://bliptext.com/articles/viscosity) and the fluid [velocity](https://bliptext.com/articles/shear-velocity) gradient, and assuming constant [viscosity](https://bliptext.com/articles/viscosity), the above Cauchy equations will lead to the Navier–Stokes equations below.\n\n## Convective acceleration\nA 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.\nWhile 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.\n# [An example of convection. Though the flow may be steady (time-independent), the fluid decelerates as it moves down the diverging duct (assuming incompressible or subsonic compressible flow), hence there is an acceleration happening over position. | upload.wikimedia.org/wikipedia/commons/5/57/ConvectiveAcceleration///vectorized.svg]\n\n# Compressible flow\nRemark: here, the deviatoric stress tensor is denoted as it was in the [general continuum equations](https://bliptext.com/articles/) and in the [incompressible flow section](https://bliptext.com/articles/).\nThe compressible momentum Navier–Stokes equation results from the following assumptions on the Cauchy stress tensor:\n

the stress is **[Galilean invariant](https://bliptext.com/articles/galilean-invariance)**: it does not depend directly on the flow velocity, but only on spatial derivatives of the flow velocity.\nSo the stress variable is the tensor gradient, or more simply the [rate-of-strain tensor](https://bliptext.com/articles/strain-rate-tensor):
the deviatoric stress is **linear** in this variable:, 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](https://bliptext.com/articles/elasticity-tensor), and : is the [double-dot product](https://bliptext.com/articles/dyadics).
the fluid is assumed to be [isotropic](https://bliptext.com/articles/isotropic), as with gases and simple liquids, and consequently \\mathbf{C} is an [isotropic](https://bliptext.com/articles/isotropic) tensor; furthermore, since the deviatoric stress tensor is symmetric, by [Helmholtz decomposition](https://bliptext.com/articles/helmholtz-decomposition) it can be expressed in terms of two scalar [Lamé parameters](https://bliptext.com/articles/lam-parameters), the [second viscosity](https://bliptext.com/articles/second-viscosity) \\lambda and the [dynamic viscosity](https://bliptext.com/articles/dynamic-viscosity) \\mu, as it is usual in [linear elasticity](https://bliptext.com/articles/linear-elasticity):\nwhere \\mathbf{I} is the [identity tensor](https://bliptext.com/articles/identity-matrix), and is the [trace](https://bliptext.com/articles/trace-linear-algebra) of the rate-of-strain tensor.\nSo this decomposition can be explicitly defined as:\n

\nSince the [trace](https://bliptext.com/articles/trace-linear-algebra) of the rate-of-strain tensor in three dimensions is the [divergence](https://bliptext.com/articles/divergence) (i.e. rate of expansion) of the flow:\nGiven this relation, and since the trace of the identity tensor in three dimensions is three:\nthe trace of the stress tensor in three dimensions becomes:\nSo by alternatively decomposing the stress tensor into **isotropic** and **deviatoric** parts, as usual in fluid dynamics:\nIntroducing the [bulk viscosity](https://bliptext.com/articles/volume-viscosity) \\zeta,\nwe arrive to the linear [constitutive equation](https://bliptext.com/articles/constitutive-equation) in the form usually employed in [thermal hydraulics](https://bliptext.com/articles/thermal-hydraulics):\nwhich can also be arranged in the other usual form:\nNote that in the compressible case the pressure is no more proportional to the [isotropic stress](https://bliptext.com/articles/hydrostatic-stress) term, since there is the additional bulk viscosity term:\nand the [deviatoric stress tensor](https://bliptext.com/articles/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:\nBoth 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.\nAny equation that makes explicit one of these [transport coefficient](https://bliptext.com/articles/transport-coefficient) in the [conservation variables](https://bliptext.com/articles/conservation-variable) is called an [equation of state](https://bliptext.com/articles/equation-of-state).\nThe most general of the Navier–Stokes equations become\nin index notation, the equation can be written as\nThe corresponding equation in conservation form can be obtained by considering that, given the mass [continuity equation](https://bliptext.com/articles/continuity-equation), the left side is equivalent to:\nTo give finally:\nApart 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.\nExample: 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.\nThis dependence is called the dispersion.\nIn some cases, the [second viscosity](https://bliptext.com/articles/volume-viscosity) \\zeta can be assumed to be constant in which case, the effect of the volume viscosity \\zeta is that the mechanical [pressure](https://bliptext.com/articles/pressure) is not equivalent to the thermodynamic [pressure](https://bliptext.com/articles/pressure): as demonstrated below.\nHowever, 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.\nThe assumption of setting \\zeta = 0 is called as the **Stokes hypothesis**.\nThe 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.\nWith the Stokes hypothesis, the Navier–Stokes equations become\nIf the dynamic μ and bulk \\zeta viscosities are assumed to be uniform in space, the equations in convective form can be simplified further.\nBy 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:\nwhere is the [material derivative](https://bliptext.com/articles/material-derivative).\nis the shear [kinematic viscosity](https://bliptext.com/articles/kinematic-viscosity) and is the bulk [kinematic viscosity](https://bliptext.com/articles/kinematic-viscosity).\nThe left-hand side changes in the conservation form of the Navier–Stokes momentum equation.\nBy bringing the operator on the flow velocity on the left side, on also has:\nThe convective acceleration term can also be written as\nwhere the vector is known as the [Lamb vector](https://bliptext.com/articles/lamb-vector).\nFor the special case of an [incompressible flow](https://bliptext.com/articles/incompressible-flow), the pressure constrains the flow so that the volume of [fluid elements](https://bliptext.com/articles/fluid-element) is constant: [isochoric flow](https://bliptext.com/articles/isochoric-process) resulting in a [solenoidal](https://bliptext.com/articles/solenoidal-vector-field) velocity field with.\n\n# Incompressible flow\nThe incompressible momentum Navier–Stokes equation results from the following assumptions on the Cauchy stress tensor:\n

the stress is **[Galilean invariant](https://bliptext.com/articles/galilean-invariance)**: it does not depend directly on the flow velocity, but only on spatial derivatives of the flow velocity.\nSo the stress variable is the tensor gradient .
the fluid is assumed to be [isotropic](https://bliptext.com/articles/isotropic), as with gases and simple liquids, and consequently is an [isotropic](https://bliptext.com/articles/isotropic) tensor; furthermore, since the deviatoric stress tensor can be expressed in terms of the [dynamic viscosity](https://bliptext.com/articles/dynamic-viscosity) \\mu:\nwhere\nis the rate-of-[strain tensor](https://bliptext.com/articles/strain-tensor).\nSo this decomposition can be made explicit as:\n

\nThis is constitutive equation is also called the **[Newtonian law of viscosity](https://bliptext.com/articles/newtonian-fluid)**.\nDynamic viscosity μ need not be constant – in incompressible flows it can depend on density and on pressure.\nAny equation that makes explicit one of these [transport coefficient](https://bliptext.com/articles/transport-coefficient) in the [conservative variables](https://bliptext.com/articles/conservative-variable) is called an [equation of state](https://bliptext.com/articles/equation-of-state).\nThe divergence of the deviatoric stress in case of uniform viscosity is given by:\nbecause for an incompressible fluid.\nIncompressibility rules out density and pressure waves like sound or [shock waves](https://bliptext.com/articles/shock-wave), so this simplification is not useful if these phenomena are of interest.\nThe incompressible flow assumption typically holds well with all fluids at low [Mach numbers](https://bliptext.com/articles/mach-number) (say up to about Mach 0.3), such as for modelling air winds at normal temperatures.\nthe incompressible Navier–Stokes equations are best visualized by dividing for the density:\nwhere is called the [kinematic viscosity](https://bliptext.com/articles/kinematic-viscosity).\nBy isolating the fluid velocity, one can also state:\nIf the density is constant throughout the fluid domain, or, in other words, if all fluid elements have the same density, \\rho, then we have\nwhere p/\\rho is called the unit [pressure head](https://bliptext.com/articles/pressure-head).\nIn incompressible flows, the pressure field satisfies the [Poisson equation](https://bliptext.com/articles/poisson-equation),\nwhich is obtained by taking the divergence of the momentum equations.\nIt is well worth observing the meaning of each term (compare to the [Cauchy momentum equation](https://bliptext.com/articles/cauchy-momentum-equation)):\nThe higher-order term, namely the [shear stress](https://bliptext.com/articles/shear-stress) divergence, has simply reduced to the [vector Laplacian](https://bliptext.com/articles/vector-laplacian) term.\nThis Laplacian term can be interpreted as the difference between the velocity at a point and the mean velocity in a small surrounding volume.\nThis implies that – for a Newtonian fluid – viscosity operates as a diffusion of momentum, in much the same way as the [heat conduction](https://bliptext.com/articles/heat-conduction).\nIn fact neglecting the convection term, incompressible Navier–[Stokes equations](https://bliptext.com/articles/stokes-flow) lead to a vector [diffusion equation](https://bliptext.com/articles/diffusion-equation) (namely [Stokes equations](https://bliptext.com/articles/stokes-flow)), but in general the convection term is present, so incompressible Navier–[Stokes equations](https://bliptext.com/articles/stokes-flow) belong to the class of convection–[diffusion equation](https://bliptext.com/articles/diffusion-equation)s.\nIn the usual case of an external field being a [conservative field](https://bliptext.com/articles/conservative-field):\nby defining the [hydraulic head](https://bliptext.com/articles/hydraulic-head):\none can finally condense the whole source in one term, arriving to the incompressible Navier–Stokes equation with conservative external field:\nThe incompressible Navier–Stokes equations with uniform density and viscosity and conservative external field is the **fundamental equation of [hydraulics](https://bliptext.com/articles/hydraulics)**.\nThe domain for these equations is commonly a 3 or less dimensional [Euclidean space](https://bliptext.com/articles/euclidean-space), for which an [orthogonal coordinate](https://bliptext.com/articles/orthogonal-coordinate) reference frame is usually set to explicit the system of scalar partial differential equations to be solved.\nIn 3-dimensional orthogonal coordinate systems are 3: [Cartesian](https://bliptext.com/articles/cartesian-coordinate-system), [cylindrical](https://bliptext.com/articles/cylindrical-coordinate-system), and [spherical](https://bliptext.com/articles/spherical-coordinate-system).\nExpressing 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.\nBut 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](https://bliptext.com/articles/tensor-calculus) is required for deducing an expression in non-cartesian orthogonal coordinate systems.\nA special case of the fundamental equation of hydraulics is the [Bernoulli's equation](https://bliptext.com/articles/bernoulli-s-equation).\nThe incompressible Navier–Stokes equation is composite, the sum of two orthogonal equations,\nwhere \\Pi^S and \\Pi^I are solenoidal and [irrotational](https://bliptext.com/articles/conservative-vector-field) projection operators satisfying, and and are the non-conservative and conservative parts of the body force.\nThis result follows from the [Helmholtz theorem](https://bliptext.com/articles/helmholtz-decomposition) (also known as the fundamental theorem of vector calculus).\nThe 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.\nThe explicit functional form of the projection operator in 3D is found from the Helmholtz Theorem:\nwith a similar structure in 2D.\nThus the governing equation is an [integro-differential equation](https://bliptext.com/articles/integro-differential-equation) similar to [Coulomb](https://bliptext.com/articles/coulomb-s-law) and [Biot–Savart law](https://bliptext.com/articles/biot-savart-law), not convenient for numerical computation.\nAn equivalent weak or variational form of the equation, proved to produce the same velocity solution as the Navier–Stokes equation, is given by,\nfor divergence-free test functions \\mathbf{w} satisfying appropriate boundary conditions.\nHere, the projections are accomplished by the orthogonality of the solenoidal and irrotational function spaces.\nThe discrete form of this is eminently suited to finite element computation of divergence-free flow, as we shall see in the next section.\nThere one will be able to address the question \"How does one specify pressure-driven (Poiseuille) problems with a pressureless governing equation?\".\nThe 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.\nThis all would seem to refute the frequent statements that the incompressible pressure enforces the divergence-free condition.\n\n## Weak form of the incompressible Navier–Stokes equations\n\n### Strong form\nConsider the incompressible Navier–Stokes equations for a [Newtonian fluid](https://bliptext.com/articles/newtonian-fluid) of constant density \\rho in a domain\nwith boundary\nbeing \\Gamma_D and \\Gamma_N portions of the boundary where respectively a [Dirichlet](https://bliptext.com/articles/dirichlet-boundary-condition) and a [Neumann boundary condition](https://bliptext.com/articles/neumann-boundary-condition) is applied :\n\\mathbf{u} is the fluid velocity, p the fluid pressure, \\mathbf{f} a given forcing term, the outward directed unit normal vector to \\Gamma_N, and the [viscous stress tensor](https://bliptext.com/articles/viscous-stress-tensor) defined as:\nLet \\mu be the dynamic viscosity of the fluid, \\mathbf{I} the second-order [identity tensor](https://bliptext.com/articles/identity-matrix) and the [strain-rate tensor](https://bliptext.com/articles/strain-rate-tensor) defined as:\nThe functions \\mathbf{g} and \\mathbf{h} are given Dirichlet and Neumann boundary data, while is the [initial condition](https://bliptext.com/articles/initial-condition).\nThe first equation is the momentum balance equation, while the second represents the [mass conservation](https://bliptext.com/articles/conservation-of-mass), namely the [continuity equation](https://bliptext.com/articles/continuity-equation).\nAssuming constant dynamic viscosity, using the vectorial identity\nand exploiting mass conservation, the divergence of the total stress tensor in the momentum equation can also be expressed as:\nMoreover, note that the Neumann boundary conditions can be rearranged as:\n\n### Weak form\nIn order to find the weak form of the Navier–Stokes equations, firstly, consider the momentum equation\nmultiply it for a test function \\mathbf{v}, defined in a suitable space V, and integrate both members with respect to the domain \\Omega:\nCounter-integrating by parts the diffusive and the pressure terms and by using the Gauss' theorem:\nUsing these relations, one gets:\nIn the same fashion, the continuity equation is multiplied for a test function q belonging to a space Q and integrated in the domain \\Omega:\nThe space functions are chosen as follows:\nConsidering that the test function\nv\nvanishes on the Dirichlet boundary and considering the Neumann condition, the integral on the boundary can be rearranged as:\nHaving this in mind, the weak formulation of the Navier–Stokes equations is expressed as:\n\n## Discrete velocity\nWith partitioning of the problem domain and defining [basis functions](https://bliptext.com/articles/basis-function) on the partitioned domain, the discrete form of the governing equation is\nIt is desirable to choose basis functions that reflect the essential feature of incompressible flow – the elements must be divergence-free.\nWhile the velocity is the variable of interest, the existence of the stream function or vector potential is necessary by the Helmholtz theorem.\nFurther, 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](https://bliptext.com/articles/stokes-theorem).\nDiscussion will be restricted to 2D in the following.\nWe further restrict discussion to continuous Hermite finite elements which have at least first-derivative degrees-of-freedom.\nWith this, one can draw a large number of candidate triangular and rectangular elements from the [plate-bending](https://bliptext.com/articles/bending-of-plates) literature.\nThese elements have derivatives as components of the gradient.\nIn 2D, the gradient and curl of a scalar are clearly orthogonal, given by the expressions,\nAdopting 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.\nTaking the curl of the scalar stream function elements gives divergence-free velocity elements.\nThe 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.\nBoundary conditions are simple to apply.\nThe stream function is constant on no-flow surfaces, with no-slip velocity conditions on surfaces.\nStream function differences across open channels determine the flow.\nNo boundary conditions are necessary on open boundaries, though consistent values may be used with some problems.\nThese are all Dirichlet conditions.\nThe algebraic equations to be solved are simple to set up, but of course are [non-linear](https://bliptext.com/articles/), requiring iteration of the linearized equations.\nSimilar 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.\n\n## Pressure recovery\nRecovering pressure from the velocity field is easy.\nThe discrete weak equation for the pressure gradient is,\nwhere the test/weight functions are irrotational.\nAny conforming scalar finite element may be used.\nHowever, the pressure gradient field may also be of interest.\nIn this case, one can use scalar Hermite elements for the pressure.\nFor the test/weight functions one would choose the irrotational vector elements obtained from the gradient of the pressure element.\n\n# Non-inertial frame of reference\nThe rotating frame of reference introduces some interesting pseudo-forces into the equations through the [material derivative](https://bliptext.com/articles/material-derivative) term.\nConsider a stationary inertial frame of reference K, and a non-inertial frame of reference K', which is translating with velocity and rotating with angular velocity \\Omega(t) with respect to the stationary frame.\nThe Navier–Stokes equation observed from the non-inertial frame then becomes\nHere \\mathbf{x} and \\mathbf{u} are measured in the non-inertial frame.\nThe first term in the parenthesis represents [Coriolis acceleration](https://bliptext.com/articles/coriolis-acceleration), the second term is due to [centrifugal acceleration](https://bliptext.com/articles/centrifugal-force), 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.\n\n# Other equations\nThe Navier–Stokes equations are strictly a statement of the balance of momentum.\nTo fully describe fluid flow, more information is needed, how much depending on the assumptions made.\nThis additional information may include boundary data ([no-slip](https://bliptext.com/articles/no-slip-condition), [capillary surface](https://bliptext.com/articles/capillary-surface), etc.), conservation of mass, [balance of energy](https://bliptext.com/articles/first-law-of-thermodynamics-fluid-mechanics), and/or an [equation of state](https://bliptext.com/articles/equation-of-state).\n\n## Continuity equation for incompressible fluid\nRegardless of the flow assumptions, a statement of the [conservation of mass](https://bliptext.com/articles/conservation-of-mass) is generally necessary.\nThis is achieved through the mass [continuity equation](https://bliptext.com/articles/continuity-equation), as discussed above in the \"General continuum equations\" within this article, as follows:\nA fluid media for which the [density](https://bliptext.com/articles/density) (\\rho) is constant is called incompressible.\nTherefore, the rate of change of [density](https://bliptext.com/articles/density) (\\rho) with respect to time and the [gradient](https://bliptext.com/articles/gradient) of [density](https://bliptext.com/articles/density) are equal to zero (0).\nIn this case the general equation of continuity,, reduces to:.\nFurthermore, assuming that [density](https://bliptext.com/articles/[density](https://bliptext.com/articles/density)) (\\rho) is a non-zero constant means that the right-hand side of the equation (0) is divisible by [density](https://bliptext.com/articles/[density](https://bliptext.com/articles/density)) (\\rho).\nTherefore, the continuity equation for an [incompressible fluid](https://bliptext.com/articles/incompressible-flow) reduces further to: This relationship, , identifies that the [divergence](https://bliptext.com/articles/divergence) of the flow velocity [vector](https://bliptext.com/articles/vector-field) (\\mathbf{u}) is equal to zero (0), which means that for an [incompressible fluid](https://bliptext.com/articles/incompressible-flow) the [flow velocity field](https://bliptext.com/articles/flow-velocity) is a solenoidal [vector](https://bliptext.com/articles/vector-field) field or a [divergence](https://bliptext.com/articles/divergence)-free [vector](https://bliptext.com/articles/vector-field) field.\nNote that this relationship can be expanded upon due to its uniqueness with the [vector Laplace operator](https://bliptext.com/articles/vector-laplace-operator), and [vorticity](https://bliptext.com/articles/vorticity) which is now expressed like so, for an [incompressible fluid](https://bliptext.com/articles/incompressible-flow):\n\n# Stream function for incompressible 2D fluid\nTaking the [curl](https://bliptext.com/articles/curl-mathematics) of the incompressible Navier–Stokes equation results in the elimination of pressure.\nThis 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:\nDifferentiating the first with respect to y, the second with respect to x and subtracting the resulting equations will eliminate pressure and any [conservative force](https://bliptext.com/articles/conservative-force).\nFor incompressible flow, defining the [stream function](https://bliptext.com/articles/stream-function) \\psi through\nresults 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:\nwhere \\nabla^4 is the 2D [biharmonic operator](https://bliptext.com/articles/biharmonic-operator) and \\nu is the [kinematic viscosity](https://bliptext.com/articles/kinematic-viscosity),.\nWe can also express this compactly using the [Jacobian determinant](https://bliptext.com/articles/jacobian-matrix-and-determinant):\nThis single equation together with appropriate boundary conditions describes 2D fluid flow, taking only kinematic viscosity as a parameter.\nNote that the equation for [creeping flow](https://bliptext.com/articles/creeping-flow) results when the left side is assumed zero.\nIn [axisymmetric](https://bliptext.com/articles/axisymmetric) flow another stream function formulation, called the [Stokes stream function](https://bliptext.com/articles/stokes-stream-function), can be used to describe the velocity components of an incompressible flow with one [scalar](https://bliptext.com/articles/scalar-mathematics) function.\nThe incompressible Navier–Stokes equation is a [differential algebraic equation](https://bliptext.com/articles/differential-algebraic-equation), having the inconvenient feature that there is no explicit mechanism for advancing the pressure in time.\nConsequently, much effort has been expended to eliminate the pressure from all or part of the computational process.\nThe 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.\n\n# Properties\n\n## Nonlinearity\nThe Navier–Stokes equations are [nonlinear](https://bliptext.com/articles/nonlinearity) [partial differential equations](https://bliptext.com/articles/partial-differential-equations) in the general case and so remain in almost every real situation.\nIn some cases, such as one-dimensional flow and [Stokes flow](https://bliptext.com/articles/stokes-flow) (or creeping flow), the equations can be simplified to linear equations.\nThe nonlinearity makes most problems difficult or impossible to solve and is the main contributor to the [turbulence](https://bliptext.com/articles/turbulence) that the equations model.\nThe nonlinearity is due to [convective](https://bliptext.com/articles/convective) acceleration, which is an acceleration associated with the change in velocity over position.\nHence, any convective flow, whether turbulent or not, will involve nonlinearity.\nAn example of convective but [laminar](https://bliptext.com/articles/laminar-flow) (nonturbulent) flow would be the passage of a viscous fluid (for example, oil) through a small converging [nozzle](https://bliptext.com/articles/nozzle).\nSuch flows, whether exactly solvable or not, can often be thoroughly studied and understood.\n\n## Turbulence\n[Turbulence](https://bliptext.com/articles/turbulence) is the time-dependent [chaotic](https://bliptext.com/articles/chaos-theory) behaviour seen in many fluid flows.\nIt is generally believed that it is due to the [inertia](https://bliptext.com/articles/inertia) of the fluid as a whole: the culmination of time-dependent and convective acceleration; hence flows where [inertia](https://bliptext.com/articles/inertia)l effects are small tend to be laminar (the [Reynolds number](https://bliptext.com/articles/reynolds-number) quantifies how much the flow is affected by [inertia](https://bliptext.com/articles/inertia)).\nIt is believed, though not known with certainty, that the Navier–Stokes equations describe turbulence properly.\nThe 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](https://bliptext.com/articles/direct-numerical-simulation).\nAttempts to solve turbulent flow using a laminar solver typically result in a time-unsteady solution, which fails to converge appropriately.\nTo counter this, time-averaged equations such as the [Reynolds-averaged Navier–Stokes equations](https://bliptext.com/articles/reynolds-averaged-navier-stokes-equations) (RANS), supplemented with turbulence models, are used in practical [computational fluid dynamics](https://bliptext.com/articles/computational-fluid-dynamics) (CFD) applications when modeling turbulent flows.\nSome models include the [Spalart–Allmaras](https://bliptext.com/articles/spalart-allmaras-turbulence-model), [k–ω](https://bliptext.com/articles/k-omega-turbulence-model), [k–ε](https://bliptext.com/articles/turbulence-kinetic-energy), and [SST](https://bliptext.com/articles/sst-menter-s-shear-stress-transport) models, which add a variety of additional equations to bring closure to the RANS equations.\n[Large eddy simulation](https://bliptext.com/articles/large-eddy-simulation) (LES) can also be used to solve these equations numerically.\nThis 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.\n\n## Applicability\nTogether 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.\nThe Navier–Stokes equations assume that the fluid being studied is a [continuum](https://bliptext.com/articles/continuum-mechanics) (it is infinitely divisible and not composed of particles such as atoms or molecules), and is not moving at [relativistic velocities](https://bliptext.com/articles/relativistic-velocity).\nAt 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.\nFor example, [capillarity](https://bliptext.com/articles/capillarity) of internal layers in fluids appears for flow with high gradients.\nFor large [Knudsen number](https://bliptext.com/articles/knudsen-number) of the problem, the [Boltzmann equation](https://bliptext.com/articles/boltzmann-equation) may be a suitable replacement.\nFailing that, one may have to resort to [molecular dynamics](https://bliptext.com/articles/molecular-dynamics) or various hybrid methods.\nAnother limitation is simply the complicated nature of the equations.\nTime-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.\nFor this reason, these equations are usually written for [Newtonian fluids](https://bliptext.com/articles/newtonian-fluid) where the viscosity model is [linear](https://bliptext.com/articles/linear); truly general models for the flow of other kinds of fluids (such as blood) do not exist.\n\n# Application to specific problems\nThe 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.\nThis 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](https://bliptext.com/articles/multiphase-flow) driven by [surface tension](https://bliptext.com/articles/surface-tension).\nGenerally, application to specific problems begins with some flow assumptions and initial/boundary condition formulation, this may be followed by [scale analysis](https://bliptext.com/articles/scale-analysis-mathematics) to further simplify the problem.\n# [Visualization of (a) parallel flow and (b) radial flow | upload.wikimedia.org/wikipedia/commons/d/dc/NSConvection///vectorial.svg]\n\n## Parallel flow\nAssume steady, parallel, one-dimensional, non-convective pressure-driven flow between parallel plates, the resulting scaled (dimensionless) [boundary value problem](https://bliptext.com/articles/boundary-value-problem) is:\nThe boundary condition is the [no slip condition](https://bliptext.com/articles/no-slip-condition).\nThis problem is easily solved for the flow field:\nFrom this point onward, more quantities of interest can be easily obtained, such as viscous drag force or net flow rate.\n\n## Radial flow\nDifficulties may arise when the problem becomes slightly more complicated.\nA seemingly modest twist on the parallel flow above would be the radial flow between parallel plates; this involves convection and thus non-linearity.\nThe velocity field may be represented by a function\nf(z)\nthat must satisfy:\nThis [ordinary differential equation](https://bliptext.com/articles/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).\nThe [nonlinear](https://bliptext.com/articles/nonlinearity) term makes this a very difficult problem to solve analytically (a lengthy [implicit](https://bliptext.com/articles/implicit-function) solution may be found which involves [elliptic integrals](https://bliptext.com/articles/elliptic-integral) and [roots of cubic polynomials](https://bliptext.com/articles/cubic-formula)).\nIssues with the actual existence of solutions arise for R > 1.41 (approximately; this is not [√2](https://bliptext.com/articles/square-root-of-2)), the parameter R being the Reynolds number with appropriately chosen scales.\nThis is an example of flow assumptions losing their applicability, and an example of the difficulty in \"high\" Reynolds number flows.\n\n## Convection\nA type of natural convection that can be described by the Navier–Stokes equation is the [Rayleigh–Bénard convection](https://bliptext.com/articles/rayleigh-b-nard-convection).\nIt is one of the most commonly studied convection phenomena because of its analytical and experimental accessibility.\n\n# Exact solutions of the Navier–Stokes equations\nSome exact solutions to the Navier–Stokes equations exist.\nExamples of degenerate cases—with the non-linear terms in the Navier–Stokes equations equal to zero—are [Poiseuille flow](https://bliptext.com/articles/hagen-poiseuille-equation), [Couette flow](https://bliptext.com/articles/couette-flow) and the oscillatory [Stokes boundary layer](https://bliptext.com/articles/stokes-boundary-layer).\nBut also, more interesting examples, solutions to the full non-linear equations, exist, such as [Jeffery–Hamel flow](https://bliptext.com/articles/jeffery-hamel-flow), [Von Kármán swirling flow](https://bliptext.com/articles/von-k-rm-n-swirling-flow), [stagnation point flow](https://bliptext.com/articles/stagnation-point-flow), [Landau–Squire jet](https://bliptext.com/articles/landau-squire-jet), and [Taylor–Green vortex](https://bliptext.com/articles/taylor-green-vortex).\nNote that the existence of these exact solutions does not imply they are stable: turbulence may develop at higher Reynolds numbers.\nUnder additional assumptions, the component parts can be separated.\n\n## A three-dimensional steady-state vortex solution\nA steady-state example with no singularities comes from considering the flow along the lines of a [Hopf fibration](https://bliptext.com/articles/hopf-fibration).\nLet r be a constant radius of the inner coil.\nOne set of solutions is given by:\nfor 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.\n(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](https://bliptext.com/articles/turbulence) properties.) It is also worth pointing out that the components of the velocity vector are exactly those from the [Pythagorean quadruple](https://bliptext.com/articles/pythagorean-quadruple) parametrization.\nOther choices of density and pressure are possible with the same velocity field:\n# [Wire model of flow lines along a Hopf fibration | upload.wikimedia.org/wikipedia/commons/b/ba/Hopfkeyrings.jpg]\n\n## Viscous three-dimensional periodic solutions\nTwo examples of periodic fully-three-dimensional viscous solutions are described in.\nThese solutions are defined on a three-dimensional [torus](https://bliptext.com/articles/torus) and are characterized by positive and negative [helicity](https://bliptext.com/articles/hydrodynamical-helicity) respectively.\nThe solution with positive helicity is given by:\nwhere 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.\nThe pressure field is obtained from the velocity field as (where p_0 and \\rho_0 are reference values for the pressure and density fields respectively).\nSince both the solutions belong to the class of [Beltrami flow](https://bliptext.com/articles/beltrami-flow), the vorticity field is parallel to the velocity and, for the case with positive helicity, is given by.\nThese solutions can be regarded as a generalization in three dimensions of the classic two-dimensional Taylor-Green [Taylor–Green vortex](https://bliptext.com/articles/taylor-green-vortex).\n\n# Wyld diagrams\n**Wyld diagrams** are bookkeeping [graphs](https://bliptext.com/articles/graph-discrete-mathematics) that correspond to the Navier–Stokes equations via a [perturbation expansion](https://bliptext.com/articles/perturbation-theory) of the fundamental [continuum mechanics](https://bliptext.com/articles/continuum-mechanics).\nSimilar to the [Feynman diagrams](https://bliptext.com/articles/feynman-diagram) in [quantum field theory](https://bliptext.com/articles/quantum-field-theory), these diagrams are an extension of [Keldysh](https://bliptext.com/articles/mstislav-keldysh)'s technique for nonequilibrium processes in fluid dynamics.\nIn other words, these diagrams assign [graphs](https://bliptext.com/articles/graph-theory) to the (often) [turbulent](https://bliptext.com/articles/turbulence) phenomena in [turbulent](https://bliptext.com/articles/turbulence) fluids by allowing [correlated](https://bliptext.com/articles/correlation-function) and interacting fluid particles to obey [stochastic processes](https://bliptext.com/articles/stochastic-processes) associated to [pseudo-random](https://bliptext.com/articles/pseudo-random) [functions](https://bliptext.com/articles/function-mathematics) in [probability distributions](https://bliptext.com/articles/probability-distribution).\n\n# Representations in 3D\nNote that the formulas in this section make use of the single-line notation for partial derivatives, where, e.g. means the partial derivative of u with respect to x, and means the second-order partial derivative of f_\\theta with respect to y.\nA 2022 paper provides a less costly, dynamical and recurrent solution of the Navier-Stokes equation for 3D turbulent fluid flows.\nOn suitably short time scales, the dynamics of turbulence is deterministic.\n\n## Cartesian coordinates\nFrom the general form of the Navier–Stokes, with the velocity vector expanded as, sometimes respectively named u, v, w, we may write the vector equation explicitly,\nNote 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.\nThe continuity equation reads:\nWhen the flow is incompressible, \\rho does not change for any fluid particle, and its [material derivative](https://bliptext.com/articles/material-derivative) vanishes:.\nThe continuity equation is reduced to:\nThus, for the incompressible version of the Navier–Stokes equation the second part of the viscous terms fall away (see [Incompressible flow](https://bliptext.com/articles/incompressible-flow)).\nThis system of four equations comprises the most commonly used and studied form.\nThough comparatively more compact than other representations, this is still a [nonlinear](https://bliptext.com/articles/nonlinearity) system of [partial differential equations](https://bliptext.com/articles/partial-differential-equations) for which solutions are difficult to obtain.\n\n## Cylindrical coordinates\nA change of variables on the Cartesian equations will yield the following momentum equations for r, \\phi, and z\nThe 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).\nThe continuity equation is:\nThis cylindrical representation of the incompressible Navier–Stokes equations is the second most commonly seen (the first being Cartesian above).\nCylindrical coordinates are chosen to take advantage of symmetry, so that a velocity component can disappear.\nA very common case is axisymmetric flow with the assumption of no tangential velocity (u_\\phi = 0), and the remaining quantities are independent of \\phi:\n\n## Spherical coordinates\nIn [spherical coordinates](https://bliptext.com/articles/spherical-coordinates), the r, \\phi, and \\theta momentum equations are (note the convention used: \\theta is polar angle, or [colatitude](https://bliptext.com/articles/colatitude), ):\nMass continuity will read:\nThese equations could be (slightly) compacted by, for example, factoring from the viscous terms.\nHowever, doing so would undesirably alter the structure of the Laplacian and other quantities.\n\n# Navier–Stokes equations use in games\nThe Navier–Stokes equations are used extensively in [video games](https://bliptext.com/articles/video-games) in order to model a wide variety of natural phenomena.\nSimulations 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](https://bliptext.com/articles/jos-stam), which elaborates one of the methods proposed in Stam's earlier, more famous paper \"Stable Fluids\" from 1999.\nStam proposes stable fluid simulation using a Navier–Stokes solution method from 1968, coupled with an unconditionally stable semi-Lagrangian [advection](https://bliptext.com/articles/advection) scheme, as first proposed in 1992.\nMore recent implementations based upon this work run on the game systems [graphics processing unit](https://bliptext.com/articles/graphics-processing-unit) (GPU) as opposed to the [central processing unit](https://bliptext.com/articles/central-processing-unit) (CPU) and achieve a much higher degree of performance.\nMany improvements have been proposed to Stam's original work, which suffers inherently from high numerical dissipation in both velocity and mass.\nAn introduction to interactive fluid simulation can be found in the 2007 [ACM SIGGRAPH](https://bliptext.com/articles/acm-siggraph) course, Fluid Simulation for Computer Animation.\n\n# Citations\n\n# General references","current_revision":null,"created_by":"kACnuQmH5k9Nr5VNVEpVeg7XpvUmMctD","created_at":"2025-01-25T17:45:46.869Z","updated_at":"2025-01-25T17:45:46.869Z","search_vector":"'equat':3 'navier':1 'stoke':2","search_content":" \n \n\n# \nThe **Navier–Stokes equations** are [partial differential equations](https://bliptext.com/articles/partial-differential-equation) which describe the motion of [viscous fluid](https://bliptext.com/articles/viscous-fluid) substances.\nThey were named after French engineer and physicist [Claude-Louis Navier](https://bliptext.com/articles/claude-louis-navier) and the Irish physicist and mathematician [George Gabriel Stokes](https://bliptext.com/articles/sir-george-stokes-1st-baronet).\nThey were developed over several decades of progressively building the theories, from 1822 (Navier) to 1842–1850 (Stokes).\nThe Navier–Stokes equations mathematically express [momentum](https://bliptext.com/articles/momentum) balance for [Newtonian fluids](https://bliptext.com/articles/newtonian-fluid) and make use of [conservation of mass](https://bliptext.com/articles/conservation-of-mass).\nThey are sometimes accompanied by an [equation of state](https://bliptext.com/articles/equation-of-state) relating [pressure](https://bliptext.com/articles/pressure), [temperature](https://bliptext.com/articles/temperature) and [density](https://bliptext.com/articles/density).\nThey arise from applying [Isaac Newton's second law](https://bliptext.com/articles/newton-s-second-law) to [fluid motion](https://bliptext.com/articles/fluid-dynamics), together with the assumption that the [stress](https://bliptext.com/articles/stress-mechanics) in the fluid is the sum of a [diffusing](https://bliptext.com/articles/diffusion) [viscous](https://bliptext.com/articles/viscosity) term (proportional to the [gradient](https://bliptext.com/articles/gradient) of velocity) and a [pressure](https://bliptext.com/articles/pressure) term—hence describing [viscous](https://bliptext.com/articles/viscosity) flow.\nThe difference between them and the closely related [Euler equations](https://bliptext.com/articles/euler-equations-fluid-dynamics) is that Navier–Stokes equations take [viscosity](https://bliptext.com/articles/viscosity) into account while the [Euler equations](https://bliptext.com/articles/euler-equations-fluid-dynamics) model only [inviscid flow](https://bliptext.com/articles/inviscid-flow).\nAs a result, the Navier–Stokes are a [parabolic equation](https://bliptext.com/articles/parabolic-equation) and therefore have better analytic properties, at the expense of having less mathematical structure (e.g. they are never [completely integrable](https://bliptext.com/articles/completely-integrable)).\nThe Navier–Stokes equations are useful because they describe the physics of many phenomena of [scientific](https://bliptext.com/articles/scientific) and [engineering](https://bliptext.com/articles/engineering) interest.\nThey may be used to [model](https://bliptext.com/articles/model-abstract) the weather, [ocean currents](https://bliptext.com/articles/ocean-current), water [flow in a pipe](https://bliptext.com/articles/flow-conditioning) and air flow around a [wing](https://bliptext.com/articles/airfoil).\nThe Navier–Stokes equations, in their full and simplified forms, help with the design of [aircraft](https://bliptext.com/articles/aircraft-design-process) and cars, the study of [blood flow](https://bliptext.com/articles/hemodynamics), the design of [power stations](https://bliptext.com/articles/power-station), the analysis of [pollution](https://bliptext.com/articles/pollution), and many other problems.\nCoupled with [Maxwell's equations](https://bliptext.com/articles/maxwell-s-equations), they can be used to model and study [magnetohydrodynamics](https://bliptext.com/articles/magnetohydrodynamics).\nThe Navier–Stokes equations are also of great interest in a purely mathematical sense.\nDespite their wide range of practical uses, it has not yet been proven whether smooth solutions always [exist](https://bliptext.com/articles/existence-theorem) in three dimensions—i.e., whether they are infinitely differentiable (or even just bounded) at all points in the [domain](https://bliptext.com/articles/domain-mathematical-analysis).\nThis is called the [Navier–Stokes existence and smoothness](https://bliptext.com/articles/navier-stokes-existence-and-smoothness) problem.\nThe [Clay Mathematics Institute](https://bliptext.com/articles/clay-mathematics-institute) has called this one of the [seven most important open problems in mathematics](https://bliptext.com/articles/millennium-prize-problems) and has offered a [US$](https://bliptext.com/articles/us)1 million prize for a solution or a counterexample.\n\n# Flow velocity\nThe solution of the equations is a [flow velocity](https://bliptext.com/articles/flow-velocity).\nIt is a [vector field](https://bliptext.com/articles/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.\nIt 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.\nOnce the velocity field is calculated, other quantities of interest such as [pressure](https://bliptext.com/articles/pressure) or [temperature](https://bliptext.com/articles/temperature) may be found using dynamical equations and relations.\nThis is different from what one normally sees in [classical mechanics](https://bliptext.com/articles/classical-mechanics), where solutions are typically trajectories of position of a [particle](https://bliptext.com/articles/particle) or deflection of a [continuum](https://bliptext.com/articles/continuum-theory).\nStudying velocity instead of position makes more sense for a fluid, although for visualization purposes one can compute various [trajectories](https://bliptext.com/articles/streamlines-streaklines-and-pathlines).\nIn particular, the [streamlines](https://bliptext.com/articles/streamlines-streaklines-and-pathlines) of a vector field, interpreted as flow velocity, are the paths along which a massless fluid particle would travel.\nThese paths are the [integral curves](https://bliptext.com/articles/integral-curve) 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.\n\n# General continuum equations\nThe Navier–Stokes momentum equation can be derived as a particular form of the [Cauchy momentum equation](https://bliptext.com/articles/cauchy-momentum-equation), whose general convective form is:\nBy setting the [Cauchy stress tensor](https://bliptext.com/articles/cauchy-stress-tensor) to be the sum of a viscosity term (the [deviatoric stress](https://bliptext.com/articles/deviatoric-stress)) and a pressure term (volumetric stress), we arrive at:\nwhere\nIn this form, it is apparent that in the assumption of an inviscid fluid – no deviatoric stress – Cauchy equations reduce to the [Euler equations](https://bliptext.com/articles/euler-equations-fluid-dynamics).\nAssuming [conservation of mass](https://bliptext.com/articles/conservation-of-mass), with the known properties of [divergence](https://bliptext.com/articles/divergence) and [gradient](https://bliptext.com/articles/gradient) we can use the mass continuity equation, which represents the mass per unit volume of a [homogenous](https://bliptext.com/articles/homogenous) fluid with respect to space and time (i.e., [material derivative](https://bliptext.com/articles/material-derivative) ) of any finite volume (**V**) to represent the change of velocity in fluid media:\nwhere\nNote 1 - Refer to the mathematical operator del represented by the nabla (\\nabla) symbol.\nto arrive at the conservation form of the equations of motion.\nThis is often written:\nwhere \\otimes is the [outer product](https://bliptext.com/articles/outer-product) of the flow velocity (\\mathbf{u}):\nThe 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).\nThe right side of the equation is in effect a summation of hydrostatic effects, the divergence of deviatoric stress and body forces (such as gravity).\nAll 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](https://bliptext.com/articles/constitutive-relation).\nBy expressing the deviatoric (shear) stress tensor in terms of [viscosity](https://bliptext.com/articles/viscosity) and the fluid [velocity](https://bliptext.com/articles/shear-velocity) gradient, and assuming constant [viscosity](https://bliptext.com/articles/viscosity), the above Cauchy equations will lead to the Navier–Stokes equations below.\n\n## Convective acceleration\nA 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.\nWhile 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.\n# [An example of convection. Though the flow may be steady (time-independent), the fluid decelerates as it moves down the diverging duct (assuming incompressible or subsonic compressible flow), hence there is an acceleration happening over position. | upload.wikimedia.org/wikipedia/commons/5/57/ConvectiveAcceleration///vectorized.svg]\n\n# Compressible flow\nRemark: here, the deviatoric stress tensor is denoted as it was in the [general continuum equations](https://bliptext.com/articles/) and in the [incompressible flow section](https://bliptext.com/articles/).\nThe compressible momentum Navier–Stokes equation results from the following assumptions on the Cauchy stress tensor:\n \n the stress is **[Galilean invariant](https://bliptext.com/articles/galilean-invariance)**: it does not depend directly on the flow velocity, but only on spatial derivatives of the flow velocity.\nSo the stress variable is the tensor gradient, or more simply the [rate-of-strain tensor](https://bliptext.com/articles/strain-rate-tensor): \n the deviatoric stress is **linear** in this variable:, 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](https://bliptext.com/articles/elasticity-tensor), and : is the [double-dot product](https://bliptext.com/articles/dyadics). \n the fluid is assumed to be [isotropic](https://bliptext.com/articles/isotropic), as with gases and simple liquids, and consequently \\mathbf{C} is an [isotropic](https://bliptext.com/articles/isotropic) tensor; furthermore, since the deviatoric stress tensor is symmetric, by [Helmholtz decomposition](https://bliptext.com/articles/helmholtz-decomposition) it can be expressed in terms of two scalar [Lamé parameters](https://bliptext.com/articles/lam-parameters), the [second viscosity](https://bliptext.com/articles/second-viscosity) \\lambda and the [dynamic viscosity](https://bliptext.com/articles/dynamic-viscosity) \\mu, as it is usual in [linear elasticity](https://bliptext.com/articles/linear-elasticity):\nwhere \\mathbf{I} is the [identity tensor](https://bliptext.com/articles/identity-matrix), and is the [trace](https://bliptext.com/articles/trace-linear-algebra) of the rate-of-strain tensor.\nSo this decomposition can be explicitly defined as:\n \n \nSince the [trace](https://bliptext.com/articles/trace-linear-algebra) of the rate-of-strain tensor in three dimensions is the [divergence](https://bliptext.com/articles/divergence) (i.e. rate of expansion) of the flow:\nGiven this relation, and since the trace of the identity tensor in three dimensions is three:\nthe trace of the stress tensor in three dimensions becomes:\nSo by alternatively decomposing the stress tensor into **isotropic** and **deviatoric** parts, as usual in fluid dynamics:\nIntroducing the [bulk viscosity](https://bliptext.com/articles/volume-viscosity) \\zeta,\nwe arrive to the linear [constitutive equation](https://bliptext.com/articles/constitutive-equation) in the form usually employed in [thermal hydraulics](https://bliptext.com/articles/thermal-hydraulics):\nwhich can also be arranged in the other usual form:\nNote that in the compressible case the pressure is no more proportional to the [isotropic stress](https://bliptext.com/articles/hydrostatic-stress) term, since there is the additional bulk viscosity term:\nand the [deviatoric stress tensor](https://bliptext.com/articles/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:\nBoth 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.\nAny equation that makes explicit one of these [transport coefficient](https://bliptext.com/articles/transport-coefficient) in the [conservation variables](https://bliptext.com/articles/conservation-variable) is called an [equation of state](https://bliptext.com/articles/equation-of-state).\nThe most general of the Navier–Stokes equations become\nin index notation, the equation can be written as\nThe corresponding equation in conservation form can be obtained by considering that, given the mass [continuity equation](https://bliptext.com/articles/continuity-equation), the left side is equivalent to:\nTo give finally:\nApart 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.\nExample: 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.\nThis dependence is called the dispersion.\nIn some cases, the [second viscosity](https://bliptext.com/articles/volume-viscosity) \\zeta can be assumed to be constant in which case, the effect of the volume viscosity \\zeta is that the mechanical [pressure](https://bliptext.com/articles/pressure) is not equivalent to the thermodynamic [pressure](https://bliptext.com/articles/pressure): as demonstrated below.\nHowever, 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.\nThe assumption of setting \\zeta = 0 is called as the **Stokes hypothesis**.\nThe 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.\nWith the Stokes hypothesis, the Navier–Stokes equations become\nIf the dynamic μ and bulk \\zeta viscosities are assumed to be uniform in space, the equations in convective form can be simplified further.\nBy 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:\nwhere is the [material derivative](https://bliptext.com/articles/material-derivative).\nis the shear [kinematic viscosity](https://bliptext.com/articles/kinematic-viscosity) and is the bulk [kinematic viscosity](https://bliptext.com/articles/kinematic-viscosity).\nThe left-hand side changes in the conservation form of the Navier–Stokes momentum equation.\nBy bringing the operator on the flow velocity on the left side, on also has:\nThe convective acceleration term can also be written as\nwhere the vector is known as the [Lamb vector](https://bliptext.com/articles/lamb-vector).\nFor the special case of an [incompressible flow](https://bliptext.com/articles/incompressible-flow), the pressure constrains the flow so that the volume of [fluid elements](https://bliptext.com/articles/fluid-element) is constant: [isochoric flow](https://bliptext.com/articles/isochoric-process) resulting in a [solenoidal](https://bliptext.com/articles/solenoidal-vector-field) velocity field with.\n\n# Incompressible flow\nThe incompressible momentum Navier–Stokes equation results from the following assumptions on the Cauchy stress tensor:\n \n the stress is **[Galilean invariant](https://bliptext.com/articles/galilean-invariance)**: it does not depend directly on the flow velocity, but only on spatial derivatives of the flow velocity.\nSo the stress variable is the tensor gradient . \n the fluid is assumed to be [isotropic](https://bliptext.com/articles/isotropic), as with gases and simple liquids, and consequently is an [isotropic](https://bliptext.com/articles/isotropic) tensor; furthermore, since the deviatoric stress tensor can be expressed in terms of the [dynamic viscosity](https://bliptext.com/articles/dynamic-viscosity) \\mu:\nwhere\nis the rate-of-[strain tensor](https://bliptext.com/articles/strain-tensor).\nSo this decomposition can be made explicit as:\n \n \nThis is constitutive equation is also called the **[Newtonian law of viscosity](https://bliptext.com/articles/newtonian-fluid)**.\nDynamic viscosity μ need not be constant – in incompressible flows it can depend on density and on pressure.\nAny equation that makes explicit one of these [transport coefficient](https://bliptext.com/articles/transport-coefficient) in the [conservative variables](https://bliptext.com/articles/conservative-variable) is called an [equation of state](https://bliptext.com/articles/equation-of-state).\nThe divergence of the deviatoric stress in case of uniform viscosity is given by:\nbecause for an incompressible fluid.\nIncompressibility rules out density and pressure waves like sound or [shock waves](https://bliptext.com/articles/shock-wave), so this simplification is not useful if these phenomena are of interest.\nThe incompressible flow assumption typically holds well with all fluids at low [Mach numbers](https://bliptext.com/articles/mach-number) (say up to about Mach 0.3), such as for modelling air winds at normal temperatures.\nthe incompressible Navier–Stokes equations are best visualized by dividing for the density:\nwhere is called the [kinematic viscosity](https://bliptext.com/articles/kinematic-viscosity).\nBy isolating the fluid velocity, one can also state:\nIf the density is constant throughout the fluid domain, or, in other words, if all fluid elements have the same density, \\rho, then we have\nwhere p/\\rho is called the unit [pressure head](https://bliptext.com/articles/pressure-head).\nIn incompressible flows, the pressure field satisfies the [Poisson equation](https://bliptext.com/articles/poisson-equation),\nwhich is obtained by taking the divergence of the momentum equations.\nIt is well worth observing the meaning of each term (compare to the [Cauchy momentum equation](https://bliptext.com/articles/cauchy-momentum-equation)):\nThe higher-order term, namely the [shear stress](https://bliptext.com/articles/shear-stress) divergence, has simply reduced to the [vector Laplacian](https://bliptext.com/articles/vector-laplacian) term.\nThis Laplacian term can be interpreted as the difference between the velocity at a point and the mean velocity in a small surrounding volume.\nThis implies that – for a Newtonian fluid – viscosity operates as a diffusion of momentum, in much the same way as the [heat conduction](https://bliptext.com/articles/heat-conduction).\nIn fact neglecting the convection term, incompressible Navier–[Stokes equations](https://bliptext.com/articles/stokes-flow) lead to a vector [diffusion equation](https://bliptext.com/articles/diffusion-equation) (namely [Stokes equations](https://bliptext.com/articles/stokes-flow)), but in general the convection term is present, so incompressible Navier–[Stokes equations](https://bliptext.com/articles/stokes-flow) belong to the class of convection–[diffusion equation](https://bliptext.com/articles/diffusion-equation)s.\nIn the usual case of an external field being a [conservative field](https://bliptext.com/articles/conservative-field):\nby defining the [hydraulic head](https://bliptext.com/articles/hydraulic-head):\none can finally condense the whole source in one term, arriving to the incompressible Navier–Stokes equation with conservative external field:\nThe incompressible Navier–Stokes equations with uniform density and viscosity and conservative external field is the **fundamental equation of [hydraulics](https://bliptext.com/articles/hydraulics)**.\nThe domain for these equations is commonly a 3 or less dimensional [Euclidean space](https://bliptext.com/articles/euclidean-space), for which an [orthogonal coordinate](https://bliptext.com/articles/orthogonal-coordinate) reference frame is usually set to explicit the system of scalar partial differential equations to be solved.\nIn 3-dimensional orthogonal coordinate systems are 3: [Cartesian](https://bliptext.com/articles/cartesian-coordinate-system), [cylindrical](https://bliptext.com/articles/cylindrical-coordinate-system), and [spherical](https://bliptext.com/articles/spherical-coordinate-system).\nExpressing 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.\nBut 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](https://bliptext.com/articles/tensor-calculus) is required for deducing an expression in non-cartesian orthogonal coordinate systems.\nA special case of the fundamental equation of hydraulics is the [Bernoulli's equation](https://bliptext.com/articles/bernoulli-s-equation).\nThe incompressible Navier–Stokes equation is composite, the sum of two orthogonal equations,\nwhere \\Pi^S and \\Pi^I are solenoidal and [irrotational](https://bliptext.com/articles/conservative-vector-field) projection operators satisfying, and and are the non-conservative and conservative parts of the body force.\nThis result follows from the [Helmholtz theorem](https://bliptext.com/articles/helmholtz-decomposition) (also known as the fundamental theorem of vector calculus).\nThe 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.\nThe explicit functional form of the projection operator in 3D is found from the Helmholtz Theorem:\nwith a similar structure in 2D.\nThus the governing equation is an [integro-differential equation](https://bliptext.com/articles/integro-differential-equation) similar to [Coulomb](https://bliptext.com/articles/coulomb-s-law) and [Biot–Savart law](https://bliptext.com/articles/biot-savart-law), not convenient for numerical computation.\nAn equivalent weak or variational form of the equation, proved to produce the same velocity solution as the Navier–Stokes equation, is given by,\nfor divergence-free test functions \\mathbf{w} satisfying appropriate boundary conditions.\nHere, the projections are accomplished by the orthogonality of the solenoidal and irrotational function spaces.\nThe discrete form of this is eminently suited to finite element computation of divergence-free flow, as we shall see in the next section.\nThere one will be able to address the question \"How does one specify pressure-driven (Poiseuille) problems with a pressureless governing equation?\".\nThe 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.\nThis all would seem to refute the frequent statements that the incompressible pressure enforces the divergence-free condition.\n\n## Weak form of the incompressible Navier–Stokes equations\n\n### Strong form\nConsider the incompressible Navier–Stokes equations for a [Newtonian fluid](https://bliptext.com/articles/newtonian-fluid) of constant density \\rho in a domain\nwith boundary\nbeing \\Gamma_D and \\Gamma_N portions of the boundary where respectively a [Dirichlet](https://bliptext.com/articles/dirichlet-boundary-condition) and a [Neumann boundary condition](https://bliptext.com/articles/neumann-boundary-condition) is applied :\n\\mathbf{u} is the fluid velocity, p the fluid pressure, \\mathbf{f} a given forcing term, the outward directed unit normal vector to \\Gamma_N, and the [viscous stress tensor](https://bliptext.com/articles/viscous-stress-tensor) defined as:\nLet \\mu be the dynamic viscosity of the fluid, \\mathbf{I} the second-order [identity tensor](https://bliptext.com/articles/identity-matrix) and the [strain-rate tensor](https://bliptext.com/articles/strain-rate-tensor) defined as:\nThe functions \\mathbf{g} and \\mathbf{h} are given Dirichlet and Neumann boundary data, while is the [initial condition](https://bliptext.com/articles/initial-condition).\nThe first equation is the momentum balance equation, while the second represents the [mass conservation](https://bliptext.com/articles/conservation-of-mass), namely the [continuity equation](https://bliptext.com/articles/continuity-equation).\nAssuming constant dynamic viscosity, using the vectorial identity\nand exploiting mass conservation, the divergence of the total stress tensor in the momentum equation can also be expressed as:\nMoreover, note that the Neumann boundary conditions can be rearranged as:\n\n### Weak form\nIn order to find the weak form of the Navier–Stokes equations, firstly, consider the momentum equation\nmultiply it for a test function \\mathbf{v}, defined in a suitable space V, and integrate both members with respect to the domain \\Omega:\nCounter-integrating by parts the diffusive and the pressure terms and by using the Gauss' theorem:\nUsing these relations, one gets:\nIn the same fashion, the continuity equation is multiplied for a test function q belonging to a space Q and integrated in the domain \\Omega:\nThe space functions are chosen as follows:\nConsidering that the test function\nv\nvanishes on the Dirichlet boundary and considering the Neumann condition, the integral on the boundary can be rearranged as:\nHaving this in mind, the weak formulation of the Navier–Stokes equations is expressed as:\n\n## Discrete velocity\nWith partitioning of the problem domain and defining [basis functions](https://bliptext.com/articles/basis-function) on the partitioned domain, the discrete form of the governing equation is\nIt is desirable to choose basis functions that reflect the essential feature of incompressible flow – the elements must be divergence-free.\nWhile the velocity is the variable of interest, the existence of the stream function or vector potential is necessary by the Helmholtz theorem.\nFurther, 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](https://bliptext.com/articles/stokes-theorem).\nDiscussion will be restricted to 2D in the following.\nWe further restrict discussion to continuous Hermite finite elements which have at least first-derivative degrees-of-freedom.\nWith this, one can draw a large number of candidate triangular and rectangular elements from the [plate-bending](https://bliptext.com/articles/bending-of-plates) literature.\nThese elements have derivatives as components of the gradient.\nIn 2D, the gradient and curl of a scalar are clearly orthogonal, given by the expressions,\nAdopting 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.\nTaking the curl of the scalar stream function elements gives divergence-free velocity elements.\nThe 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.\nBoundary conditions are simple to apply.\nThe stream function is constant on no-flow surfaces, with no-slip velocity conditions on surfaces.\nStream function differences across open channels determine the flow.\nNo boundary conditions are necessary on open boundaries, though consistent values may be used with some problems.\nThese are all Dirichlet conditions.\nThe algebraic equations to be solved are simple to set up, but of course are [non-linear](https://bliptext.com/articles/), requiring iteration of the linearized equations.\nSimilar 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.\n\n## Pressure recovery\nRecovering pressure from the velocity field is easy.\nThe discrete weak equation for the pressure gradient is,\nwhere the test/weight functions are irrotational.\nAny conforming scalar finite element may be used.\nHowever, the pressure gradient field may also be of interest.\nIn this case, one can use scalar Hermite elements for the pressure.\nFor the test/weight functions one would choose the irrotational vector elements obtained from the gradient of the pressure element.\n\n# Non-inertial frame of reference\nThe rotating frame of reference introduces some interesting pseudo-forces into the equations through the [material derivative](https://bliptext.com/articles/material-derivative) term.\nConsider a stationary inertial frame of reference K, and a non-inertial frame of reference K', which is translating with velocity and rotating with angular velocity \\Omega(t) with respect to the stationary frame.\nThe Navier–Stokes equation observed from the non-inertial frame then becomes\nHere \\mathbf{x} and \\mathbf{u} are measured in the non-inertial frame.\nThe first term in the parenthesis represents [Coriolis acceleration](https://bliptext.com/articles/coriolis-acceleration), the second term is due to [centrifugal acceleration](https://bliptext.com/articles/centrifugal-force), 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.\n\n# Other equations\nThe Navier–Stokes equations are strictly a statement of the balance of momentum.\nTo fully describe fluid flow, more information is needed, how much depending on the assumptions made.\nThis additional information may include boundary data ([no-slip](https://bliptext.com/articles/no-slip-condition), [capillary surface](https://bliptext.com/articles/capillary-surface), etc.), conservation of mass, [balance of energy](https://bliptext.com/articles/first-law-of-thermodynamics-fluid-mechanics), and/or an [equation of state](https://bliptext.com/articles/equation-of-state).\n\n## Continuity equation for incompressible fluid\nRegardless of the flow assumptions, a statement of the [conservation of mass](https://bliptext.com/articles/conservation-of-mass) is generally necessary.\nThis is achieved through the mass [continuity equation](https://bliptext.com/articles/continuity-equation), as discussed above in the \"General continuum equations\" within this article, as follows:\nA fluid media for which the [density](https://bliptext.com/articles/density) (\\rho) is constant is called incompressible.\nTherefore, the rate of change of [density](https://bliptext.com/articles/density) (\\rho) with respect to time and the [gradient](https://bliptext.com/articles/gradient) of [density](https://bliptext.com/articles/density) are equal to zero (0).\nIn this case the general equation of continuity,, reduces to:.\nFurthermore, assuming that [[density](https://bliptext.com/articles/density)](https://bliptext.com/articles/[density](https://bliptext.com/articles/density)) (\\rho) is a non-zero constant means that the right-hand side of the equation (0) is divisible by [[density](https://bliptext.com/articles/density)](https://bliptext.com/articles/[density](https://bliptext.com/articles/density)) (\\rho).\nTherefore, the continuity equation for an [[incompressible fluid](https://bliptext.com/articles/incompressible-flow)](https://bliptext.com/articles/incompressible-flow) reduces further to: This relationship, , identifies that the [divergence](https://bliptext.com/articles/divergence) of the flow velocity [vector](https://bliptext.com/articles/vector-field) (\\mathbf{u}) is equal to zero (0), which means that for an [[incompressible fluid](https://bliptext.com/articles/incompressible-flow)](https://bliptext.com/articles/incompressible-flow) the [flow velocity field](https://bliptext.com/articles/flow-velocity) is a solenoidal [vector](https://bliptext.com/articles/vector-field) field or a [divergence](https://bliptext.com/articles/divergence)-free [vector](https://bliptext.com/articles/vector-field) field.\nNote that this relationship can be expanded upon due to its uniqueness with the [vector Laplace operator](https://bliptext.com/articles/vector-laplace-operator), and [vorticity](https://bliptext.com/articles/vorticity) which is now expressed like so, for an [incompressible fluid](https://bliptext.com/articles/incompressible-flow):\n\n# Stream function for incompressible 2D fluid\nTaking the [curl](https://bliptext.com/articles/curl-mathematics) of the incompressible Navier–Stokes equation results in the elimination of pressure.\nThis 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:\nDifferentiating the first with respect to y, the second with respect to x and subtracting the resulting equations will eliminate pressure and any [conservative force](https://bliptext.com/articles/conservative-force).\nFor incompressible flow, defining the [stream function](https://bliptext.com/articles/stream-function) \\psi through\nresults 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:\nwhere \\nabla^4 is the 2D [biharmonic operator](https://bliptext.com/articles/biharmonic-operator) and \\nu is the [kinematic viscosity](https://bliptext.com/articles/kinematic-viscosity),.\nWe can also express this compactly using the [Jacobian determinant](https://bliptext.com/articles/jacobian-matrix-and-determinant):\nThis single equation together with appropriate boundary conditions describes 2D fluid flow, taking only kinematic viscosity as a parameter.\nNote that the equation for [creeping flow](https://bliptext.com/articles/creeping-flow) results when the left side is assumed zero.\nIn [axisymmetric](https://bliptext.com/articles/axisymmetric) flow another stream function formulation, called the [Stokes stream function](https://bliptext.com/articles/stokes-stream-function), can be used to describe the velocity components of an incompressible flow with one [scalar](https://bliptext.com/articles/scalar-mathematics) function.\nThe incompressible Navier–Stokes equation is a [differential algebraic equation](https://bliptext.com/articles/differential-algebraic-equation), having the inconvenient feature that there is no explicit mechanism for advancing the pressure in time.\nConsequently, much effort has been expended to eliminate the pressure from all or part of the computational process.\nThe 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.\n\n# Properties\n\n## Nonlinearity\nThe Navier–Stokes equations are [nonlinear](https://bliptext.com/articles/nonlinearity) [partial differential equations](https://bliptext.com/articles/partial-differential-equations) in the general case and so remain in almost every real situation.\nIn some cases, such as one-dimensional flow and [Stokes flow](https://bliptext.com/articles/stokes-flow) (or creeping flow), the equations can be simplified to linear equations.\nThe nonlinearity makes most problems difficult or impossible to solve and is the main contributor to the [turbulence](https://bliptext.com/articles/turbulence) that the equations model.\nThe nonlinearity is due to [convective](https://bliptext.com/articles/convective) acceleration, which is an acceleration associated with the change in velocity over position.\nHence, any convective flow, whether turbulent or not, will involve nonlinearity.\nAn example of convective but [laminar](https://bliptext.com/articles/laminar-flow) (nonturbulent) flow would be the passage of a viscous fluid (for example, oil) through a small converging [nozzle](https://bliptext.com/articles/nozzle).\nSuch flows, whether exactly solvable or not, can often be thoroughly studied and understood.\n\n## Turbulence\n[Turbulence](https://bliptext.com/articles/turbulence) is the time-dependent [chaotic](https://bliptext.com/articles/chaos-theory) behaviour seen in many fluid flows.\nIt is generally believed that it is due to the [inertia](https://bliptext.com/articles/inertia) of the fluid as a whole: the culmination of time-dependent and convective acceleration; hence flows where [inertia](https://bliptext.com/articles/inertia)l effects are small tend to be laminar (the [Reynolds number](https://bliptext.com/articles/reynolds-number) quantifies how much the flow is affected by [inertia](https://bliptext.com/articles/inertia)).\nIt is believed, though not known with certainty, that the Navier–Stokes equations describe turbulence properly.\nThe 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](https://bliptext.com/articles/direct-numerical-simulation).\nAttempts to solve turbulent flow using a laminar solver typically result in a time-unsteady solution, which fails to converge appropriately.\nTo counter this, time-averaged equations such as the [Reynolds-averaged Navier–Stokes equations](https://bliptext.com/articles/reynolds-averaged-navier-stokes-equations) (RANS), supplemented with turbulence models, are used in practical [computational fluid dynamics](https://bliptext.com/articles/computational-fluid-dynamics) (CFD) applications when modeling turbulent flows.\nSome models include the [Spalart–Allmaras](https://bliptext.com/articles/spalart-allmaras-turbulence-model), [k–ω](https://bliptext.com/articles/k-omega-turbulence-model), [k–ε](https://bliptext.com/articles/turbulence-kinetic-energy), and [SST](https://bliptext.com/articles/sst-menter-s-shear-stress-transport) models, which add a variety of additional equations to bring closure to the RANS equations.\n[Large eddy simulation](https://bliptext.com/articles/large-eddy-simulation) (LES) can also be used to solve these equations numerically.\nThis 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.\n\n## Applicability\nTogether 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.\nThe Navier–Stokes equations assume that the fluid being studied is a [continuum](https://bliptext.com/articles/continuum-mechanics) (it is infinitely divisible and not composed of particles such as atoms or molecules), and is not moving at [relativistic velocities](https://bliptext.com/articles/relativistic-velocity).\nAt 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.\nFor example, [capillarity](https://bliptext.com/articles/capillarity) of internal layers in fluids appears for flow with high gradients.\nFor large [Knudsen number](https://bliptext.com/articles/knudsen-number) of the problem, the [Boltzmann equation](https://bliptext.com/articles/boltzmann-equation) may be a suitable replacement.\nFailing that, one may have to resort to [molecular dynamics](https://bliptext.com/articles/molecular-dynamics) or various hybrid methods.\nAnother limitation is simply the complicated nature of the equations.\nTime-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.\nFor this reason, these equations are usually written for [Newtonian fluids](https://bliptext.com/articles/newtonian-fluid) where the viscosity model is [linear](https://bliptext.com/articles/linear); truly general models for the flow of other kinds of fluids (such as blood) do not exist.\n\n# Application to specific problems\nThe 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.\nThis 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](https://bliptext.com/articles/multiphase-flow) driven by [surface tension](https://bliptext.com/articles/surface-tension).\nGenerally, application to specific problems begins with some flow assumptions and initial/boundary condition formulation, this may be followed by [scale analysis](https://bliptext.com/articles/scale-analysis-mathematics) to further simplify the problem.\n# [Visualization of (a) parallel flow and (b) radial flow | upload.wikimedia.org/wikipedia/commons/d/dc/NSConvection///vectorial.svg]\n\n## Parallel flow\nAssume steady, parallel, one-dimensional, non-convective pressure-driven flow between parallel plates, the resulting scaled (dimensionless) [boundary value problem](https://bliptext.com/articles/boundary-value-problem) is:\nThe boundary condition is the [no slip condition](https://bliptext.com/articles/no-slip-condition).\nThis problem is easily solved for the flow field:\nFrom this point onward, more quantities of interest can be easily obtained, such as viscous drag force or net flow rate.\n\n## Radial flow\nDifficulties may arise when the problem becomes slightly more complicated.\nA seemingly modest twist on the parallel flow above would be the radial flow between parallel plates; this involves convection and thus non-linearity.\nThe velocity field may be represented by a function\nf(z)\nthat must satisfy:\nThis [ordinary differential equation](https://bliptext.com/articles/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).\nThe [nonlinear](https://bliptext.com/articles/nonlinearity) term makes this a very difficult problem to solve analytically (a lengthy [implicit](https://bliptext.com/articles/implicit-function) solution may be found which involves [elliptic integrals](https://bliptext.com/articles/elliptic-integral) and [roots of cubic polynomials](https://bliptext.com/articles/cubic-formula)).\nIssues with the actual existence of solutions arise for R > 1.41 (approximately; this is not [√2](https://bliptext.com/articles/square-root-of-2)), the parameter R being the Reynolds number with appropriately chosen scales.\nThis is an example of flow assumptions losing their applicability, and an example of the difficulty in \"high\" Reynolds number flows.\n\n## Convection\nA type of natural convection that can be described by the Navier–Stokes equation is the [Rayleigh–Bénard convection](https://bliptext.com/articles/rayleigh-b-nard-convection).\nIt is one of the most commonly studied convection phenomena because of its analytical and experimental accessibility.\n\n# Exact solutions of the Navier–Stokes equations\nSome exact solutions to the Navier–Stokes equations exist.\nExamples of degenerate cases—with the non-linear terms in the Navier–Stokes equations equal to zero—are [Poiseuille flow](https://bliptext.com/articles/hagen-poiseuille-equation), [Couette flow](https://bliptext.com/articles/couette-flow) and the oscillatory [Stokes boundary layer](https://bliptext.com/articles/stokes-boundary-layer).\nBut also, more interesting examples, solutions to the full non-linear equations, exist, such as [Jeffery–Hamel flow](https://bliptext.com/articles/jeffery-hamel-flow), [Von Kármán swirling flow](https://bliptext.com/articles/von-k-rm-n-swirling-flow), [stagnation point flow](https://bliptext.com/articles/stagnation-point-flow), [Landau–Squire jet](https://bliptext.com/articles/landau-squire-jet), and [Taylor–Green vortex](https://bliptext.com/articles/taylor-green-vortex).\nNote that the existence of these exact solutions does not imply they are stable: turbulence may develop at higher Reynolds numbers.\nUnder additional assumptions, the component parts can be separated.\n\n## A three-dimensional steady-state vortex solution\nA steady-state example with no singularities comes from considering the flow along the lines of a [Hopf fibration](https://bliptext.com/articles/hopf-fibration).\nLet r be a constant radius of the inner coil.\nOne set of solutions is given by:\nfor 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.\n(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](https://bliptext.com/articles/turbulence) properties.) It is also worth pointing out that the components of the velocity vector are exactly those from the [Pythagorean quadruple](https://bliptext.com/articles/pythagorean-quadruple) parametrization.\nOther choices of density and pressure are possible with the same velocity field:\n# [Wire model of flow lines along a Hopf fibration | upload.wikimedia.org/wikipedia/commons/b/ba/Hopfkeyrings.jpg]\n\n## Viscous three-dimensional periodic solutions\nTwo examples of periodic fully-three-dimensional viscous solutions are described in.\nThese solutions are defined on a three-dimensional [torus](https://bliptext.com/articles/torus) and are characterized by positive and negative [helicity](https://bliptext.com/articles/hydrodynamical-helicity) respectively.\nThe solution with positive helicity is given by:\nwhere 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.\nThe pressure field is obtained from the velocity field as (where p_0 and \\rho_0 are reference values for the pressure and density fields respectively).\nSince both the solutions belong to the class of [Beltrami flow](https://bliptext.com/articles/beltrami-flow), the vorticity field is parallel to the velocity and, for the case with positive helicity, is given by.\nThese solutions can be regarded as a generalization in three dimensions of the classic two-dimensional Taylor-Green [Taylor–Green vortex](https://bliptext.com/articles/taylor-green-vortex).\n\n# Wyld diagrams\n**Wyld diagrams** are bookkeeping [graphs](https://bliptext.com/articles/graph-discrete-mathematics) that correspond to the Navier–Stokes equations via a [perturbation expansion](https://bliptext.com/articles/perturbation-theory) of the fundamental [continuum mechanics](https://bliptext.com/articles/continuum-mechanics).\nSimilar to the [Feynman diagrams](https://bliptext.com/articles/feynman-diagram) in [quantum field theory](https://bliptext.com/articles/quantum-field-theory), these diagrams are an extension of [Keldysh](https://bliptext.com/articles/mstislav-keldysh)'s technique for nonequilibrium processes in fluid dynamics.\nIn other words, these diagrams assign [graphs](https://bliptext.com/articles/graph-theory) to the (often) [turbulent](https://bliptext.com/articles/turbulence) phenomena in [turbulent](https://bliptext.com/articles/turbulence) fluids by allowing [correlated](https://bliptext.com/articles/correlation-function) and interacting fluid particles to obey [stochastic processes](https://bliptext.com/articles/stochastic-processes) associated to [pseudo-random](https://bliptext.com/articles/pseudo-random) [functions](https://bliptext.com/articles/function-mathematics) in [probability distributions](https://bliptext.com/articles/probability-distribution).\n\n# Representations in 3D\nNote that the formulas in this section make use of the single-line notation for partial derivatives, where, e.g. means the partial derivative of u with respect to x, and means the second-order partial derivative of f_\\theta with respect to y.\nA 2022 paper provides a less costly, dynamical and recurrent solution of the Navier-Stokes equation for 3D turbulent fluid flows.\nOn suitably short time scales, the dynamics of turbulence is deterministic.\n\n## Cartesian coordinates\nFrom the general form of the Navier–Stokes, with the velocity vector expanded as, sometimes respectively named u, v, w, we may write the vector equation explicitly,\nNote 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.\nThe continuity equation reads:\nWhen the flow is incompressible, \\rho does not change for any fluid particle, and its [material derivative](https://bliptext.com/articles/material-derivative) vanishes:.\nThe continuity equation is reduced to:\nThus, for the incompressible version of the Navier–Stokes equation the second part of the viscous terms fall away (see [Incompressible flow](https://bliptext.com/articles/incompressible-flow)).\nThis system of four equations comprises the most commonly used and studied form.\nThough comparatively more compact than other representations, this is still a [nonlinear](https://bliptext.com/articles/nonlinearity) system of [partial differential equations](https://bliptext.com/articles/partial-differential-equations) for which solutions are difficult to obtain.\n\n## Cylindrical coordinates\nA change of variables on the Cartesian equations will yield the following momentum equations for r, \\phi, and z\nThe 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).\nThe continuity equation is:\nThis cylindrical representation of the incompressible Navier–Stokes equations is the second most commonly seen (the first being Cartesian above).\nCylindrical coordinates are chosen to take advantage of symmetry, so that a velocity component can disappear.\nA very common case is axisymmetric flow with the assumption of no tangential velocity (u_\\phi = 0), and the remaining quantities are independent of \\phi:\n\n## Spherical coordinates\nIn [spherical coordinates](https://bliptext.com/articles/spherical-coordinates), the r, \\phi, and \\theta momentum equations are (note the convention used: \\theta is polar angle, or [colatitude](https://bliptext.com/articles/colatitude), ):\nMass continuity will read:\nThese equations could be (slightly) compacted by, for example, factoring from the viscous terms.\nHowever, doing so would undesirably alter the structure of the Laplacian and other quantities.\n\n# Navier–Stokes equations use in games\nThe Navier–Stokes equations are used extensively in [video games](https://bliptext.com/articles/video-games) in order to model a wide variety of natural phenomena.\nSimulations 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](https://bliptext.com/articles/jos-stam), which elaborates one of the methods proposed in Stam's earlier, more famous paper \"Stable Fluids\" from 1999.\nStam proposes stable fluid simulation using a Navier–Stokes solution method from 1968, coupled with an unconditionally stable semi-Lagrangian [advection](https://bliptext.com/articles/advection) scheme, as first proposed in 1992.\nMore recent implementations based upon this work run on the game systems [graphics processing unit](https://bliptext.com/articles/graphics-processing-unit) (GPU) as opposed to the [central processing unit](https://bliptext.com/articles/central-processing-unit) (CPU) and achieve a much higher degree of performance.\nMany improvements have been proposed to Stam's original work, which suffers inherently from high numerical dissipation in both velocity and mass.\nAn introduction to interactive fluid simulation can be found in the 2007 [ACM SIGGRAPH](https://bliptext.com/articles/acm-siggraph) course, Fluid Simulation for Computer Animation.\n\n# Citations\n\n# General references","revisionCount":0},"hasMore":false,"nextCursor":null}

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