Biomedical Engineering Reference
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for any 0<y<
R
0
.0/
, 0<t<T and for a given function p
in
D p
in
.y;t/.Onthe
opposite, outflow part of the boundary
out
,(see(
1.9
)) we set
u
2
.t;L;y/D 0;
(1.86)
2.j
D
2
.t;L;y/D 0
.v/j/
@
u
1
f
2
j
u
1
j
@x
p C p
out
(1.87)
for any 0<y<
R
0
.L/
, 0<t<Tand for a given function p
out
D p
out
.t;y/.
Note that we require here that the so-called kinematic pressure (or total pressure) is
prescribed on the inflow and outflow boundary. This implies that the fluxes of kinetic
energy on inflow and outflow boundary will disappear in the weak formulation as
already stated in the introduction. Finally, on the remaining part of the boundary,
0
, we set the flow symmetry condition
.
u
/j/
@
u
1
u
2
.t;x;0/D 0; .j
D
@y
.t;x;0/D 0
(1.88)
for any 0<x<L, 0<t<T. The initial conditions read
u
.0;x;y/D 0 for any 0<x<L;0<y<
R
0
.x/
:
(1.89)
The proof of the main result formulated in Theorem
1.1
will be realized in several
steps:
•
approximation of the solenoidal spaces on a moving domain by the artificial
compressibility approach: "-approximation
•
splitting of the boundary conditions (
1.79
)-(
1.80
) by introducing the semi-
pervious boundary: -approximation
•
transformation of the weak formulation on a time-dependent domain
.t/ to a
fixed reference domain
O
f
D .0;L/ .0;1/ using a
given domain deformation
D
1
. Note that we scale in the y-direction to .0;1/ interval. This step requires
that the domain motion is regular enough (here (or
1
) should be at least C
1
in
space, which was not the case in the previous subsections for weak solutions).
•
limiting process for " ! 0; !1and a fixed point on the geometry domain,
respectively.
Weak Formulation
In this subsection our aim is to present the weak formulation of the problem (
1.73
)-
(
1.89
). Assuming that is enough regular (see below) and taking into account the
results from [
26
] we can define the functional spaces that gives sense to the trace of
velocity from W
1;p
.
.t// and thus to define the weak solution of the problem. We
assume that R
0
2 C
0
.0;L/.
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