The Cluster's Navier-Stokes Equations

A page about the Navier-Stokes equations — the equations governing viscous fluid flow.

The Navier-Stokes equation

The Navier-Stokes equations describe the motion of a viscous, incompressible fluid: rho (dv/dt) = -nabla P + eta nabla^2 v + f or in terms of the material derivative: rho (partial v / partial t + v . nabla v) = -nabla P + eta nabla^2 v + rho g

where rho is the density, v is the velocity field, P is the pressure, eta is the dynamic viscosity, and f includes body forces (e.g., gravity). The term rho (v . nabla v) is the convective acceleration — the non-linearity that makes the equations so difficult.

In the cluster, the edit Navier-Stokes equation describes an edit viscous fluid.

The incompressibility condition

For an incompressible fluid: nabla . v = 0 This couples the pressure to the velocity field: taking the divergence of the NS equation gives a Poisson equation for pressure: nabla^2 P = -rho nabla . (v . nabla v)

In the cluster, the edit incompressibility condition gives an edit pressure equation.

The Reynolds number

The Reynolds number is the ratio of convective to viscous forces: Re = rho v L / eta = v L / nu where nu = eta / rho is the kinematic viscosity. The NS equation in dimensionless form: dv/dt = -nabla P + (1/Re) nabla^2 v Low Re (Re << 1): viscous forces dominate (Stokes flow, laminar). High Re (Re >> 1): convective forces dominate (turbulent).

In the cluster, the edit Reynolds number gives an edit force ratio.

The solutions

• Poiseuille flow (pipe flow): v(r) = (Delta P / (4 eta L)) (R^2 - r^2). Parabolic profile. Q = pi R^4 Delta P / (8 eta L).
• Couette flow (between moving plates): v(y) = V y / h. Linear profile.
• Stokes flow (Creeping flow, Re << 1): v . nabla v negligible. Linear equations.
• Turbulent flow (Re >> 1): No analytical solution. Statistical description required. The Navier-Stokes existence and smoothness problem is one of the Clay Mathematics Institute's Millennium Prize Problems.

In the cluster, the edit solutions give an edit flow profile.

Applications

• Engineering: Pipeline flow, aircraft wing design, HVAC systems
• Meteorology: Atmospheric flow, weather prediction
• Oceanography: Ocean currents, tidal dynamics
• Astrophysics: Stellar convection zones, accretion disk flow
• Physiology: Blood flow in arteries (non-Newtonian fluid behavior)

In the cluster, edit applications include:

• edit Engineering
• edit Meteorology
• edit Oceanography
• edit Astrophysics
• edit Physiology

This equation

This page is about the Navier-Stokes equations. rho (dv/dt) = -nabla P + eta nabla^2 v + f. Re = rho v L / eta. The Millennium Prize Problem. The equation is real.