Hydrodynamics is used to describe the flow of continuous 54 media like gases, many types of fluids and plasmas. It is used in computer modeling to optimize the design of ships, airplanes and sports cars with respect to stability and resistance. The equations are also used for weather prediction, including computer simulations of hurricanes, tornadoes and tsunamis.

The three equations of hydrodynamics are basically conservation laws. They can describe a wide variety of transport phenomena because the equations contain quite a few parameters. These equations were originally obtained through the study of fluids. The system can also be obtained from the Boltzmann equation by analyzing averages of quantities that are conserved in two-particle collisions: mass, momentum and energy.

A very hard problem still facing the mathematical and physical community is to understand in detail the phenomenon of turbulence from first principles, and to construct it as a solution to the Navier–Stokes equations.

The Navier–Stokes equations are based on the conservation of three basic quantities in the underlying particle interactions: mass, momentum and energy. Related to these are three fields: the mass density ρ(r,t), the velocity field u(r,t) and the energy density (per unit mass) ε(r,t). The equations require knowledge of the equations of state of the medium one is studying, one relating the pressure to density and temperature, P = P(ρ,T), and one relating the energy density to density and temperature,ε - ε (r,T).

For dilute gaseous systems these fields are defined as averages over the velocities using the Boltzmann distribution function, as given in the side bar on page 57. The defifining expressions for the fields depend only on r and t. One may then use the Boltzmann equation to derive the set of coupled equations given above, usually called the Navier-Stokes equations.

It is a very general set of equations, and in principle they also describe aerodynamics, though the parameters will be very different. In these equations a number of approximations have been made by introducing several phenomenological parameters which characterize the fluid, a pressure tensor P, the viscosity η, and the thermal conductivity K. We have left out a viscosity-dependent contribution to the third equation for simplicity.