In
physics, the
Navier–Stokes equations, named after Claude-Louis Navier and George Gabriel Stokes, describe the motion of fluid substances. These equations arise from applying
Newton's second law to
fluid motion, together with the assumption that the stress in the fluid is the sum of a diffusing viscous term (proportional to the gradient of velocity) and a pressure term - hence describing
viscous flow.
The equations are useful because they describe the physics of many
things of academic and economic interest. They may be used to
model the
weather,
ocean currents, water
flow in a pipe and air flow around a
wing.
The Navier–Stokes equations in their full and simplified forms help
with the design of aircraft and cars, the study of blood flow, the
design of power stations, the analysis of pollution, and many other
things. Coupled with Maxwell's equations they can be used to model and study
magnetohydrodynamics.
| Navier-Stokes Equations |
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The motion of a non-turbulent,
Newtonian fluid is governed by the Navier-Stokes equation:
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The above equation can also be used to model turbulent flow, where the fluid parameters are interpreted
as time-averaged values.
The time-derivative of the fluid velocity in the Navier-Stokes equation is the material derivative,
defined as:
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The material derivative is distinct from a normal derivative because it includes a convection term,
a very important term in fluid mechanics.
This unique derivative will be denoted by a "dot" placed above the variable it operates on.
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| Navier-Stokes Background |
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On the most basic level, laminar (or time-averaged turbulent) fluid behavior is described by a set of
fundamental equations. These equations are:
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The Navier-Stokes equation is obtained by combining the fluid kinematics and constitutive relation into
the fluid equation of motion, and eliminating the parameters D and T. These terms are defined below:
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| Quantity | Symbol | Object | Units |
| fluid stress | T | 2nd order tensor | N/m2 |
| strain rate | D | 2nd order tensor | 1/s |
| unity tensor | I | 2nd order tensor | 1 |
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