In this section we shall present two important equations governing the dynamics of fluid flow. First we obtain the equation describing the conservation of mass. Next we present the equation of motion. In order to do this we have to generalize the concept of viscosity as introduced in the previous section.

Consider a volume element with edges (*dx*,0,0), (0,*dy*,0) and (0,0,*dz*).
If we follow the mass in our volume element along the flow during a short
time ,
the line element (*dx*,0,0) will transform into

(5.4) |

Analogous equations hold for the other line elements. The volume after time will be

= | |||

= | (5.5) |

As a result we have

(5.6) |

or using

This equation describes the conservation of mass .

We shall now generalize the concept of viscosity as it was given in section
1. Consider a surface element of size *dA*, and normal .
Let
be the force exerted by the fluid below the surface
element on the fluid above the surface element. Then we define the stress
tensor *S* by

The total force in

F_{i} |
= | ||

= | (5.9) |

where

(5.10) |

Since

(5.11) |

Writing , because is constant along the flow, and dividing by

(5.12) |

This is the Navier-Stokes equation .

Many fluids may be described by assuming that the stress tensor *S* consists
of a part, which is independent of the velocity, and a part which depends
linearly on the derivatives
.
In
hydrodynamics it is shown that the most general stress tensor having these properties then reads

Here is the shear viscosity , is the bulk viscosity , and

Many flow fields of interest may be described assuming that the fluid is
incompressible , i.e. that the density along the flow is
constant. In that case
,
as follows from Eq. (5.7).
Assuming moreover that the velocities are small, and that the second order
non-linear term
may be neglected we
obtain Stokes equations for incompressible flow

= | (5.14) | ||

= | 0. | (5.15) |

These are the equations that we are going to use in the next two sections.