What Are the Navier-Stokes Equations?

The Navier-Stokes equations govern the motion of fluids and can be seen
as Newton's second law of motion for fluids. In the case of a compressible
Newtonian fluid, this yields: velocity in 3 dimensions and pressure:

velocity ux(x,y,z,t)  partial derivative notation  dux/dt
velocity uy(x,y,z,t)  partial derivative notation  duy/dt
velocity uz(x,y,z,t)  partial derivative notation  duz/dt
pressure p(x,y,z,t)  partial derivative notation  dp/dx

at each  x,y,z,t equation is:

du/dt + ux*du/dx + uy*du/dy + uz*/du/dz =
Fx + Fy + Fz - 1/rho(dp/dx +_ dp/dy + dp/dz) +
mu/rho*(d^2u/dx^2 + d^2u/dy^2 + d^2u/dz^2)

where u is the fluid velocity, p is the fluid pressure,
rho is the fluid density, and mu is the fluid dynamic viscosity. 
The different terms correspond to the inertial forces = 
external forces F, pressure forces p, and viscous forces mu/rho,
applied to the fluid cell. The Navier-Stokes equations were derived by 
Navier, Poisson, Saint-Venant, and Stokes between 1827 and 1845.

These equations are always solved together with the continuity equation:

rho * (ux/dx + uy/dy + uz/dz) = 0
 
The Navier-Stokes equations represent the conservation of momentum, 
while the continuity equation represents the conservation of mass. 

These general equations can be separated into three equations based
on velocities ux(x,y,z,t), uy(x,y,z,t), uz(x,y,z,t)  in three directions,
and rho(x,y,z,t) density at each point in compressible flow,
rho is a constant for in-compressible flow,
and p(x,y,x,t) pressure at each point in flow.

d ux/dt + ux * d ux/dx + uy * d ux/dy + uz * d ux/dz  =
Fx - 1/rho * dp/dx + mu/rho *(d^2u/dx^2 + d^2u/dy^2 + d^2u/dz^2)

d uy/dt + ux * d uy/dx + uy * d uy/dy + uz * d uy/dz  =
Fy - 1/rho * dp/dy + mu/rho *(d^2uy/dx^2 + d^2uy/dy^2 + d^2uy/dz^2)

d uz/dt + ux * d uz/dx + uy * d uz/dy + uz * d uz/dz  =
Fz - 1/rho * dp/dz + mu/rho *(d^2uz/dx^2 + d^2uz/dy^2 + d^2uz/dz^2)

d rho/dt + d rho ux/dx + d rho uy/dy + d rho uz/dy = 0

In Metric Units:
velocity u in meters per second,
fluid density rho in killograms per cubic meter,
pressure p in newtons per square meter,
distance x, y, z in meters,
time t in seconds,
fluid dynamic viscosity mu in newton seconds per square meter,
force F is F/m in newtons per killogram.