170
(Ill. 22)
where the pressure at the surface Pa is a given function.
The sigma equations can be finally written as:
(Ill.23)
Jb
-
Jb 1 J (- Jb) -
-+U.Vb+V3---- A- -/(&=0
dt
dx3 h 2 dx3 dx3
(Ill. 24)
Jt' -
~ + V(hii) = 0
(lII.25) (lII.26)
u=ii+u'
(Ill.27)(Ill.28)
Ill.2.3 Weak formulation
We define the following spaces:
Vo" = { ¢ E L2 (Q)2/ div¢ E L2(Q) , curl¢ E L2(Q), ~ E L2(Q) ;¢.n=O on n
H=L2(Q/ XL2(Q) x L2(Qx)
v = V 0" x Hi(Q) X Hi(Qx)
If (.,.) indifferently represents the inner product into L2(Qx), L2(Q), and into L2(Q/,
applying the integration by parts formula and the boundary conditions, the weak formulation
associated with the equations (III. 12), (III. 13), (III. 19) can be written as :
For ( uo,bo, SO)E H given, find (u,b, Q such as :
+ (~ c::, ~¢ )+ Ji( divu,div¢) + Ji( curlu,curl¢)
H ax3 aX3