Mathematical and Numerical Analysis of Some FSI Problems - Fluid-Structure Interaction and Biomedical Applications - page 17

O f ;

div u D 0; in (1.31)

p D p in ; on in ; (1.32)

p D p out ; on out ; (1.33)

u n D 0; on 0 : (1.34)

Note that we neglect the fluid domain variations and the equations are set
in the reference configuration O f . Concerning the structure part we consider a
thin clamped elastic structure but whose displacement is only vertical and whose
transverse component satisfied, for instance

s e@ tt Ǜ 2 @ xx LJ 2 @ xx @ t D p; on .0;L/: (1.35)

Moreover since the fluid is inviscid the kinematic condition at the interface writes

†;

u n D @ t ; on (1.36)

Note that here all geometrical and convective nonlinearities have been omitted.
We will assume that all the quantities are regular enough to justify the following
derivations. Due to the incompressibility constraint ( 1.31 ), by taking the divergence
of the fluid equation ( 1.30 ) we obtain that the pressure satisfies

O f ;

p D 0; in (1.37)

@p
@n D f @ t u n; on

†

(1.38)

@p
@n D 0; on 0 ;

(1.39)

together with Dirichlet boundary conditions ( 1.32 ), ( 1.33 )on in and out .
Due to ( 1.36 ), the boundary condition ( 1.38 ) writes

@p
@n D f @ tt on

†:

(1.40)

Consequently we can rewrite the pressure load applied by the fluid on the
structure as p D q f M .@ tt / where q satisfies:

O f ;

q D 0; in (1.41)

@q
@n D 0; on 0 [ †;

(1.42)

q D p in ; on in ; (1.43)

q D p out ; on out ; (1.44)