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

2. The variation of with p is very small, i.e. such that in product terms of velocity
and density the latter can be assumed constant.
The ®rst assumption will be relaxed, as we shall see later, allowing some thermal
coupling via the dependence of the ¯uid properties on temperature. In such cases
we shall introduce the coupling iteratively. Here the problem of density-induced
currents or temperature-dependent viscosity (Chapter 5) will be typical.
If the assumptions introduced above are used we can still allow for small compressibility,
noting that density changes are, as a consequence of elastic deformability,
related to pressure changes. Thus we can write
where K is the elastic bulk modulus. This can be written as
or
p
with c ˆ K=
d ˆ K dp …1:27a†
d ˆ 1
dp …1:27b†
c2 @ 1
ˆ
@t c2 @p
@t
being the acoustic wave velocity.
…1:27c†
Equations (1.24) and (1.25) can now be rewritten omitting the energy transport
(and condensing the general form) as
@uj @ 1 @p 1 @
‡ …u
@t @x
jui†‡ ÿ
i @xj @xi 1
c2 @p
@t ‡ @ui ˆ 0 …1:28a†
@xi ji ÿ f j ˆ 0 …1:28b†
With j ˆ 1; 2; 3 this represents a system of four equations in which the variables are
u j and p.
Written in terms of cartesian coordinates we have, in place of Eq. (1.28a),
1
c 2
@p @u
‡
@t @x
@v @w
‡ ‡ ˆ 0 …1:29a†
@y @z
where the ®rst term is dropped for complete incompressibility …c ˆ1†and
@u @
‡
@t @x …u2 †‡ @ @ @p
…uv†‡ …uw†‡1
@y @z @x
with similar forms for y and z. In both forms
1 @
ÿ
@x xx ‡ @
@y xy ‡ @
@z xz ÿ fx ˆ 0 …1:29b†
1 ij ˆ
@ui ‡
@xj @uj @xi where ˆ = is the kinematic viscosity.
Incompressible (or nearly incompressible) ¯ows 11
2 @uk ÿ ij
3 @xk