Navier–Stokes equations

E5106

continuum mechanics model equations of fluid mechanics partial differential equations

The Navier–Stokes equations are fundamental partial differential equations in fluid mechanics that describe how the velocity field of a fluid evolves under forces like pressure and viscosity.

Observed surface forms (5)

Surface form	Occurrences
Navier–Stokes existence and smoothness problem	2
Navier–Stokes equations (hydrodynamic limit with viscosity)	1
Navier–Stokes equations with zero viscosity	1
Stokes equations	1
linearized Navier–Stokes equations	1

Statements (54)

Predicate	Object
instanceOf	continuum mechanics model ⓘ equations of fluid mechanics ⓘ partial differential equations ⓘ
appliesTo	Newtonian fluids ⓘ gases ⓘ liquids ⓘ
associatedPrize	Millennium Prize Problem ⓘ
assumes	Newtonian constitutive relation for stress ⓘ continuum hypothesis ⓘ
basedOn	conservation of mass ⓘ conservation of momentum ⓘ
centralTo	computational fluid dynamics simulations ⓘ turbulence modeling ⓘ
commonlySolvedBy	numerical methods ⓘ
describes	balance of momentum in a fluid ⓘ effects of external body forces on fluids ⓘ effects of pressure forces in fluids ⓘ effects of viscosity in fluids ⓘ evolution of fluid velocity field ⓘ motion of viscous fluid substances ⓘ
difficulty	analytical solutions are rare ⓘ
field	applied mathematics ⓘ continuum mechanics ⓘ fluid mechanics ⓘ
hasComponent	external force term ⓘ inertial term ⓘ pressure gradient term ⓘ viscous diffusion term ⓘ
includes	compressible Navier–Stokes equations ⓘ incompressible Navier–Stokes equations ⓘ laminar flow regime ⓘ steady-state form ⓘ turbulent flow regime ⓘ unsteady form ⓘ
mathematicalForm	nonlinear partial differential equations ⓘ
namedAfter	Claude-Louis Navier ⓘ George Stokes ⓘ surface form: George Gabriel Stokes
nonlinearitySource	convective acceleration term ⓘ
recognizedBy	Clay Mathematics Institute ⓘ
relatedProblem	Navier–Stokes equations self-linksurface differs ⓘ surface form: Navier–Stokes existence and smoothness problem
relatedTo	Euler equations ⓘ Reynolds number ⓘ Stokes flow ⓘ
requires	boundary conditions ⓘ initial conditions ⓘ
unknownsInclude	pressure field ⓘ velocity field ⓘ
usedIn	aerodynamics ⓘ computational fluid dynamics ⓘ engineering design of fluid systems ⓘ hydrodynamics ⓘ oceanography ⓘ weather prediction ⓘ
yearProposed	19th century ⓘ

Referenced by (11)

Full triples — surface form annotated when it differs from this entity's canonical label.

Stokes flow → governedBy → Navier–Stokes equations ⓘ

this entity surface form: Stokes equations

Stokes flow → governedBy → Navier–Stokes equations ⓘ

this entity surface form: linearized Navier–Stokes equations

Boltzmann equation → hasLimit → Navier–Stokes equations ⓘ

this entity surface form: Navier–Stokes equations (hydrodynamic limit with viscosity)

Millennium Prize Problem → hasProblem → Navier–Stokes equations ⓘ

this entity surface form: Navier–Stokes existence and smoothness problem

Euler equations → isLimitCaseOf → Navier–Stokes equations ⓘ

this entity surface form: Navier–Stokes equations with zero viscosity

Claude-Louis Navier → knownFor → Navier–Stokes equations ⓘ

George Stokes → knownFor → Navier–Stokes equations ⓘ

subject surface form: George Gabriel Stokes

Feynman sprinkler problem → relatedConcept → Navier–Stokes equations ⓘ

Navier–Stokes equations → relatedProblem → Navier–Stokes equations self-linksurface differs ⓘ

this entity surface form: Navier–Stokes existence and smoothness problem

Euler equations → relatedTo → Navier–Stokes equations ⓘ

Newtonian fluids → usedIn → Navier–Stokes equations ⓘ

subject surface form: Newtonian fluid