Appendix C Useful Theoretical Approaches and Formulas
485
Equating the rate of change of momentum to the forces we obtain:
(C.24)
As the parcel is arbitrary, the relationship must hold for all elements, so we
have:
Du
1
-=F--\lp.
Dt
Pw
(C.25)
These equations are called the Euler equations and together with the mass
conservation equation (C.19) they form the basis for the study of any flow of
an incompressible fluid.
Let us now rewrite material derivative Du/ Dt in a slightly different form:
Du au
au
(1 2)
- = - + u . \lu = - + \l -lui + \l x u x u
Dt
at
.
at
2
'
(C.26)
or:
Du au
(1 2)
-
= -
+ \l -lui + (curlu) x u,
Dt
at
2
(C.27)
where definition (C.ll) of rotation has been used. Substituting Eq. (C.27) into
Eq. (C.25) provides another representation of the Euler equations:
au
(1 2)
1
- + \l -lui + (curlu) x u = F - -\lp.
at
2
Pw
(C.28)
This representation clearly demonstrates that the Euler equations also support
a rotational flow of incompressible fluid. However, they have no mechanism for
vorticity diffusion.
Let us now restrict ourselves to the gravity force only, i. e.:
F = -g.
(C.29)
After substituting Eq. (C.29) into Eq. (C.25), we can present the Euler equations for the water motion, induced by gravity, as:
( au
ou
ou
Ou)
Pw - +u-+v-+wot
ox
oy
oz
( OV
ov
ov
Ov)
Pw -+u-+v-+wat
ox
oy
oz
( OW
ow
ow
ow)
Pw -+u-+v-+wot
ox
oy
oz
op
ox
op
oy
o(p + Pwgz)
oz
(C.30)