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

94 Incompressible laminar ¯ow
and, as we know from the discussions in Volume 1, this provides a natural boundary
condition.
Indeed, at this stage it is not necessary to discuss the application of ®nite elements
to this particular equation, which was considered at length in Volume 1 and for which
many solutions are available. 1 In Fig. 4.1 an example of a typical potential solution is
given.
Of course we must be assured that the potential function exists, and indeed
determine what conditions are necessary for its existence. Here we observe that so
far we have not used in the de®nition of the problem the important momentumconservation
equations (4.8), to which we shall now return. However, we ®rst note
that a single-valued potential function implies that
@ 2
@x j @x i
and hence that, using the de®nition (4.9),
ˆ @2
@x i @x j
…4:12†
! 1 ˆ @u1 ÿ
@x2 @u2 ˆ 0 ! 2 ˆ
@x1 @u2 ÿ
@x3 @u3 ˆ 0 ! 3 ˆ
@x2 @u3 ÿ
@x1 @u1 ˆ 0 …4:13†
@x3 This is a statement of the irrotationality of the ¯ow which we see is implied by the
existence of the potential.
Inserting the de®nition of potential into the ®rst term of Eq. (4.8) and using Eqs
(4.7) and (4.13) we can rewrite this equation as
ÿ @
@x i
@
@t
‡ @
@x i
in which P is the potential of the body forces giving these as
This is alternatively written as
g i ˆÿ @P
@x i
1
2 u ju j ‡ p ‡ P ˆ 0 …4:14†
…4:15†
r ÿ @
‡ H ‡ P ˆ 0 …4:16†
@t
where H is the enthalpy, given as H ˆ 1
2 u iu i ‡ p= .
If isothermal conditions pertain, the speci®c energy is constant and the above
implies that
ÿ @ 1
‡
@t 2 uiui ‡ p ‡ P ˆ constant …4:17†
for the whole domain. This can be taken as a corollary of the existence of the potential
and indeed is a condition for its existence. In steady-state ¯ows it provides the wellknown
Bernoulli equation that allows the pressures to be determined throughout
the whole potential ®eld when the value of the constant is established.