Fluid Mechanics: Navier Stokes

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 2nd order tensor N/m2
strain rate D 2nd order tensor 1/s
unity tensor I 2nd order tensor 1



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