Environmental Engineering Reference
In-Depth Information
to a single equation, in two-dimensional space there are two differential equations
given by system (
11.1
), and in three-dimensional space there are three.
Equations (
11.1
) are derived from the principle of momentum conservation,
where momentum is defined as the product
v. From that point of view the Navier-
Stokes equations are for fluid mechanics what is Newton's Law for classical
mechanics. The first two terms in (
11.1
) represent temporal change and advective
transport of momentum. The
r
r
f-term introduces outer forces as for example
gravity. The connection with pressure is given by the
rp
-expression. The final
term, including the viscosity
as parameter, represents the internal friction within
the fluid. A more general formulation of the Navier-Stokes equations can addition-
ally take the effect of compressibility into account (Guyon et al.
1997
).
Detailed derivations of the Navier-Stokes equations can be found in textbooks
on fluid mechanics; see for example Guyon et al. (
1997
). In the derivation of the
formulation (
11.1
) it is assumed that the internal shear stress within the fluid is
proportional to the change of the velocity in transverse direction. The dynamic
viscosity
is the proportionality factor in that relationship, which can also be traced
back to Newton. Water is a Newtonian fluid for which such a relation is valid, while
different formulations result for non-Newtonian fluids. The change of water viscos-
ity in the temperature range between 0
C and 50
C is depicted in Fig.
11.2
.
x 10
-3
1.8
Hagen
Poiseuille
Gavich e.a.
Pawlowski
Lin e.a.
1.7
1.6
1.5
1.4
1.3
1.2
1.1
1
0.9
0.8
0.7
0.6
0.5
0.4
0
10
20
30
40
50
Temperature [°C]
Fig. 11.2 Change of dynamic viscosity of water; according to different authors
