Derivation of the Navier-Stokes equation

The derivation of the Navier-Stokes equations is closely related to [*1], that carries out the derivation in detail.

Main thoughts, description of the flow fields

In three-dimensional movement, the flow field is mainly determined by the velocity vector:



velocity components,


unit vectors,

but also by the pressure p and the temperature T. These 5 variables principally change with time and place:

There are five equations available when determining these five variables:

- Continuity equation (preservation of mass).

- 3 momentum / motion equations (preservation of momentum).

- Energy equation (preservation of energy, 1. theorem of thermodynamics).

For the isotropic Newton fluids that are regarded here, the 5 balance equations contain the following material variables that generally depend on temperature and pressure:

- Isobaric specific heat capacity:  


- Viscosity:

- Heat conductivity:

- Density:

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Hydrodynamics of Floods



*1 Schlichting et al. (1997)


Skript Environmental Hydraulic Simulation:

Derivation of the Navier-Stokes equation

2D hydrodynamic models (full part)

Further Information