Navier-Stokes Equations |
The motion of a non-turbulent, Newtonian fluid is governed by the Navier-Stokes equation: |
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: |
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. |
Navier-Stokes Background |
On the most basic level, laminar (or time-averaged turbulent) fluid behavior is described by a set of fundamental equations. These equations are: |
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: |
Quantity | Symbol | Object | Units |
fluid stress | T | 2^{nd} order tensor | N/m^{2} |
strain rate | D | 2^{nd} order tensor | 1/s |
unity tensor | I | 2^{nd} order tensor | 1 |