Zienkiewicz O.C., Taylor R.L. Vol. 3. The finite - tiera.ru

6 Introduction and the equations of ¯uid dynamics
In the above,
ij ˆ 2 _" ij ÿ ij _" kk
3
ˆ
@ui ‡
@xj @uj @xi 2 @uk ÿ ij
3 @xk …1:12b†
All of the above relationships are analogous to those of elasticity, as we shall note
again later for incompressible ¯ow. We have also mentioned this in Chapter 12 of
Volume 1 where various stabilization procedures are considered for incompressible
problems.
Non-linearity of some ¯uid ¯ows is observed with a coe cient depending on
strain rates. We shall term such ¯ows `non-newtonian'.
1.2.2 Mass conservation
If is the ¯uid density then the balance of mass ¯ow u i entering and leaving an
in®nitesimal control volume (Fig. 1.1) is equal to the rate of change in density
@ @
‡ … u
@t @x
i†
i
or in traditional cartesian coordinates
x 3 ; (z)
@
@t ‡ rT … u† ˆ0 …1:13a†
@ @ @ @
‡ … u†‡ … v†‡ … w† ˆ0 …1:13b†
@t @x @y @z
dx2 ; (dy)
x2 ; (y)
dx 3 ; (dz)
x 1 ; (x)
dx 1 ; (dx)
Fig. 1.1 Coordinate direction and the in®nitesimal control volume.
1.2.3 Momentum conservation ± or dynamic equilibrium
Now the balance of momentum in the jth direction, this is … u j†u i leaving and entering
a control volume, has to be in equilibrium with the stresses ij and body forces f j