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