Fluid Mechanics: Navier Stokes 
NavierStokes Equations 
The motion of a nonturbulent, Newtonian fluid is governed by the NavierStokes equation: 
The above equation can also be used to model
turbulent flow, where the fluid parameters are interpreted as
timeaveraged values.
The timederivative of the fluid velocity in the NavierStokes 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. 
NavierStokes Background 
On the most basic level, laminar (or timeaveraged turbulent) fluid behavior is described by a set of fundamental equations. These equations are: 
The NavierStokes 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 
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