Hydrodynamics describes the flow properties of viscoelastic materials. The basic quantities are the density
and velocity
.
Here
is
the density of the material which at time t happens to be at position .
Similarly
is the velocity of the material
which at time t is at position .
A consequence of describing the
flow field this way, is that for example
and
are the densities of two
different amounts of material. If we want to know the change with time of
the density of a given amount of material we shall write
= | |||
= | (5.1) |
A second important concept in hydrodynamics is that of friction . Suppose we have a linear flow field like in Fig. (5.1), where the arrows indicate the velocities. The only nonzero
velocity component is in the y-direction, and it only depends on z. Now
look at a volume element at (x,y,z). It has a velocity v_{y}(z) which is
larger than the velocity
v_{y}(z-dz) of the volume element just below it.
As a result the element at (x,y,z) will be slowed down and the one at
(x,y,z-dz) will be accelerated. Both forces will be proportional to
and to the surface dxdy, with constant of
proportionality .
Similarly the volume element at (x,y,z) will be
accelerated by the one at
(x,y,z+dz). In total our volume element will
feel a force in the y-direction given by
F_{y} | = | ||
= | (5.2) |
F_{y}^{P} | = | P(y)dxdz-P(y+dy)dxdz | |
= | (5.3) |