Fluid–Structure Interaction in Hemodynamics: Modeling, Analysis, and Numerical Simulation - Fluid-Structure Interaction and Biomedical Applications - page 95

shell in differential form are given by:

K h @ 2 z
@t 2

C 2 @ r

@ z C 3 @ 2 z

D 2 @ 2 r

@t@ z D 3 @ 3 z @t@ z 2 D f z ;

(2.39)

@ z 2

K h @ 2 r
@t 2

C C 0 r C 1 @ 2 r
@ z 2 @ z C C 4 @ 4 r @t D 1 @ 3 r

C C 2 @ z C D 0 @ r

@ z 4 @t@ z 2

CD 2 @ 2 z

@t@ z C D 4 @ 5 r @t@ z 4 D f r ;

(2.40)

where

h 2
12R 2 /; C 1 D

h 3
6

hE
R 2 .1 2 / .1 C

E
R 2 .1 2 / ;C 2 D

h
R

E
1 2 ;

C 0 D

h 3
12

hE E

C 3 D

1 2 ;

C 4 D

1 2 ;

(2.41)

h 2
12R 2 /; D 1 D

h 3
6

h
R 2 C v .1 C

D v
R 2 ;

hD v
R ;

D 0 D 2 D

h 3
12 C v ;

D 3 D hC v ; D 4 D

and

E v
1 v

E v v
1 v

C v WD ; D v WD :

We can write this problem using the operators L el and L vis as:

K h @ 2

@t 2 C L el C L vis @ @t D f ;

(2.42)

where

0
@

1
A

@ 2 z
@ z 2

@ r
@ z C 3

C 2

L el D ; (2.43)

C 0 r C 1 @ 2 r
@ z 2 @ z C C 4 @ 4 r

C C 2 @ z

@ z 4

and

0 1

D 2 @ 2 r

@t@ z D 3 @ 3 z

@ A

@
@t D

@t@ z 2

L vis : (2.44)

@t D 1 @ 3 r @t@ z 2 C D 2 @ 2 z @t@ z C D 4 @ 5 r

D 0 @ r

@t@ z 4

The typical values of the model parameters for the aorta and iliac arteries are
given in Table 2.2