Antontsev S.N., Kazhikhov A.V., Monakhov V.N. Boundary Value Problems in Mechanics of Nonhomogeneous Fluids
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188
Chapter
4
o
<
rns
p(x,
t)
I
11
<
a,
(x,
t)
E
Q,
and, therefore, the momentum equations (6.1) can be divided by
and in this case an additional non-linearity, related to
,
remains
only
at
p
:
at
-+fi.p)?+-VP=f*
1
-9
at
P
But, as
was
the case for
a
homogenous fluid, the smoothness
of
the
pressure
is
a unit higher than that
of
the velocity,
80
in
this
sense
p
is
a
lower term.
To
reduce the problem to an operator equation, we shall apply the
same operation as
in
the case
of
a
homogenous fluid, i.e. opera-
tions rot
and
div to equations (6.8)
1
3
au,
auj
--t
P
i,J=l
ax,
ax,
div
(
-
vp>
=
-
C
-
-
+
div
f,
(6.10)
-3- -3
rot
u
=
w,
div
u
=
0,
(6.11)
aP
-
+
(?i
*
v)p
=
0.
(6.12)
at
It
should be noted here that there
is
a
slight difference from the
case
of
a
homogenoue fluid: we can't set
div
2
5
0
since the gra-
dient part
of
this vector can't be referred to
p
Therefore,
(6.10) there has
a
corresponding addent in the right-hand part.
Purther on, the boundary conditions
will
be transformed
in
the
same way
as
in
S
2.
For
this
pur
ose we shall again se
a
curvili-
near System
of
coordinates
g
ad
write equations
(9.8)
in new
variables, followed by considering these equations on
r.As
a
re-
sult, we get the following relations on the boundary
r:
1
aP
z
I
aHi
,
(x,
t>
E
so,
---
-f-+
E
V,
-
-
a93
1
aHi
+
(vf
+
r2)
-
-
,
(x, t)
E
sz.
i=i
'pi
as,
Correctness
of
the
Problem
of
Flow
+
For
the vortex
w
the boundary conditions on
189
l?'
assume the
form
-
v25,
4-
PI,
11ap
11
a
-+
ag,
-
Id2-
-
g52=------
1
-
P
H,
as,
2
H,
asi
at
The conditions for the normal component
of
the velocity
(6.14)
(6.15)
for
the density
P
=
PI(&
t>,
(x,
t>
E
sl,
(6.16)
as well as the unitial data
--t
-3
+
(6.17)
0
t=o
=
wo(x>
5
rot
uo(x>,
plt=o
=
p
(x),
x
E
Q.
remain the same. All the baeic considerations
in
constmcting the
operator
A,
the
fixed
points of which give the solution
of
the
problem, also remain unchanged.
A
new
stage is introduced,related
to findi the dendty from equation
(6.12)
with conditiona
(6.16)
and
(6.18
.
The solution
of
equation
(6.12)
is preserved along the
trajectories
Y(S>
which are described by problem
(3.18).
There-
fore,
for
it valid is the formula
(6.18)
P(X,
tl
=
[Po1(X,
t),
where
pO(x),
x
E
2,
t
=
0,
Po
=(
pl(x,
t),
x
E
I?',
t
E
[0,
T].
The
estimate6 for
p(x,
t>
are the same as those for the vortex
described
ins
4.
Let
us
formulate the requirements for the problem data, at which
an inambiguous solvability
of
the problem
of
flowing through
for
a
non-homogenous fluid is guaranteed:
--t
-+
-9
f
E
C'+"(Q,
uo
E
ClW(S),
po
E
cl+ya>,
2
E
c'+"(s
1,
r
E
c'+~(s~>,
p1
E
C'+?S
>.
(6.19)
Apart
from the suppositions relative
to
the smoothness of the given
functions, the agreement conditions
of
initial and boundary data
are also considered met. Besides, as before,
for
the sake of
aim-
190
Chapter
4
plicity we consider the case when the domain
Q
and the parts of the boundary
r0,
I?’,
r2
do not intercross.
Wit-
hout going into details of the proof, since
all
the consideratione
are quite analogoue to those given above, we shall give the
final
re
mlt
.
Theorem 6.1. When fulfillin conditions (6.19) on a sufficiently
emall period of time
problem (6.1)
-
(6.7).
2.
The case of a multiply-conneoted domain
of
the flow
If
the flow domain
is
multiply-connected, then there
is
an additi-
onal peculiarity related to the restoration
of
the velocity field
by the vortex and divergence [81]. This problem naturally adjoins
the problem of expanding the arbitra
and
a
solenoidal components (see [325. Let us cite the additional
conditions to be introduced into the formulation of the problem of
flowing through, when the domain
Q
is
multiply-connected. These
considerations are given in paper
[
132
3
devoted to the model prob-
lem of flowing through with a given vortex at the in-flow
I”,
and
are used here without any cardinal changes.
So,
let domain
Q
be bounded and
(1
+
1)-connected. Let us denote
through
Li,
i
=
1,
2,
...,
lthe totality of the independent con-
toure of the domain
$2
which are not contracted into a single
point. Let
us
consider the given in
Q
vector-functions Zi(x),
i
=
1,
2,...,
which are the solutions of the problems
is
simply-connected
[d,
T’?
there exists
a
unique solution of
vector field into
a
gradient
-9 -9
rot
ui
=
0,
div ui
=
0,
x
E
Q,
9-f
(6.20)
(ui
n)
=
0,
x
E
r,
-+
-9
Sij
(ui
as)
=
bik,
k
=
1,
2,
Lk
...,
1.
The existence, linear independence and smoothness of ui have been
-
81.6)
for the case
of
a
homogenous liquid
p
9
1.
The initial data of problem (1.1)
-
(1.6) obey the requirements
given
in
9
1. Let
us
show that for the case
of
a
domain of the flow
Q
lem (2.1)
-
(2.31, (2.24)
-
(2.31) with the additional conditions
S((ut
+
(U
V)U
-
f)
ui)dx
=
0,
i
=
1,
2,
...,
1,
t
E
[O,T],
a
roved in
[
321. For simplicity, let
us
consider problem (1
.I
(1
+
1)connected
problem (1.1)
-
(1.6)
is
equivalent to prob-
-9
-9
-f+
-9
(6.21
1
-9
Requirements (6.22) guarantee that the initial velocity
u(x,
0)
coincides with
u
(x)
and conditions (6.21) and Helmholtz equa-
tions (2.1) afford that the momentum equations (1.1) are fulfilled.
-+O
Correctness
of
the
Problem
of
Flow
191
k
-k
When constructing successive approximations
Gk,
p
,
u
),
k
=
I,
2,
...,
additional conditions are also introduced. Namely, the velo-
city vector
I'uk
is
defined from the following problem:
,
div
uk
=
0,
x
E
8,
-9k-1
-'
rot
tk
=
w
When solving
this problem, the vector
Itk
is
found as the
sum:
1
+
C
i=1
(6.24)
-'k
+k
U
(x,
t)
=
uo
(X,
t)
+
A:
(-t)ui(x)
k
with the unknown coefficients
Ai(t)
The vector field
3;
termined from the classical problem
[all:
is
de-
+
-9
rot
z;
=
wkcl,
div
u;
=
0,
x
E
52,
(6.25)
++
+-'
(ui
n>lr
=
y(x,
t),
x
L
r,
j:uE
sui)dx=o,
i
=
I,
2,
.
..
,1.
a
k
The coefficients
of
orthogonality, which are a system
of
Ordinary differential equa-
t
ions
:
%(t)
are defined from the additional conditions
1
dAk
1
,XU,..
d=C
j=i
dt
j=i
k
A.
J
I([u.
x
wk-ll.
Ui)dX
+
aJ
--f+
3
+
f>.
ui)dx,
i
=
I,
2,
...,
1.
(6.26)
Here the matrix of the coefficients at the derivatives
is
as
fol-
lows
-3
a
=
f
(ui
u.)dx,
i,
j
=
1,
2,
...,
1.
ij
5-2
J
and
is
non-degenerate due to the linear dependence
of
the vec-
tors
ui,
i
=
1,
2,
...,
1.
The initial
data
for
Ai(t)
are obtained from other conditions
of
orthogonality in problem (6.23) which ensure the coincidence of
<k
and
uo
at
t
=
0:
-9
k
192
Chapter
4
As far as the matrix
(6.26), (6.27)
is unambiguously solvable, and, hence, uniquely
de-
fines the velocity field
tk.
It
should be noted that the orthogo-
nality conditions are chosen in such a way that tat every step of
the method
of
successive approximations the equations
of
type
(4.5)
are fulfilled
{a.
.}
is non-degenerate, the Cauchy problem
=J
The remaining considerations are exactly the same as are In the
case of a simply-connected flow domain.
Theorem
6.2.
Problem
(1.1)
-
(1.6)
is uniquely solvable if the
flow domain
52
is
multiply-connected.
3.
Other formulations
of
the problem
of
flowing through
Here we are going
to
dwell upon some model formulations
of
the
iroblem
of
flowing through which are studied with the same methods.
1
-
5,
we shall limit ourselves to the case when the
fluid
is homogenous, the flow domain
52
is bounded and simply-connected,
the parts
of
the boundary
Pi,
i
=
0,
I,
2,
do not touch one
an0
t
her.
Let
us
first return to the initial problem
(2.1)
-
(2.31,
(2.24)
-
(2.31).
Attention should be paid to the fact that problem
(3.2)
for pressure, and problems
(3.11,
(3.5)
for velocity and vortex
are interrelated
only
through the boundary conditions
(2.30)
on
I?'.
If in
(2.30)
the function
p
then problem
(3.2)
could
be
solved separately after finding the velocity. This
remark results in considering the model problem arishg from
(2.1)
-
(2.31,
(2.24)
-
(2.311,
when we consider the function
p
known on
l'l
in
this case the problem for the vortex and velocity
gets separated, the boundary conditions
(2.28)
-
(2.30)
denoting
that throughout the boundary
l'
the normal velocity component is
given, and that at the in-flow
I?'
we additionally set the normal
vortex component and the tangential components
of
the vector pro-
duct
[".
x
w].
Thus, on
I?'
four boundary conditions are set and,
generally speaking, such a formulation is overspecified. The con-
dition of compatibility
for
this overspecified oblem can
be
found if we allow
for
the fact that the vector
?if
must
be
a rotor,
and,
hence, it
divergence must equal zero.
As
is seen from proving
Lemma
3.1,
for
this purpose it
is
sufficient to ensure the equa-
lity
div
3
=
0
on
l?'.
These considerations demonstrate that
a
general form
of
the formulation
of
the overspecified problem in
question is to
be
as follows.
On
I"
the normal component
5
(in the curvilinear coordinates
q
)
must have the form
in
were known on
I?'
+
Correctness
of
the
Problem
of
Flow
193
where
g,,
g,-
are two arbitrary functions on
S1
and the tangen-
tial components
x
a]
must be expressed by the relations
-3
ag
1
a+
at
H2
a42
ag,
1
a+
v.,C,
-
v,<,
=
2
-
F,
+
-
-
,
(6.29)
v2<.,
-
V,c,
=
-
-
F,
+
-
-
,
at
HI
a%,,,
where
8
is
an arbitrary function
(J,
E
C
(S'))
given on
S'.
Besides, throughout on
I'
we set
v3
=
6
-
%),
as well
as
the initial data
A
closed system equivalent to
(1.1)
consists of the HeImholz equa-
tion
for
the vortex
and the equation for velocity
+-+
-+
rot
u
=
w,
div
u
=
0.
On the boundary
I?
equations (6.32)
in
their projection onto the
axis which
is
directed along the normal, yield the relation
j-3
-3
-3
a<
--
3
-
(rot
[u
x
w]),
+
v3div
w
=
(rot
F)3.
-3+
at
If
the tangential components
of
the vector product
set
on
is
known. Hence,
if
the boundary
data
on
r'
obey the condition
[u
x
u]
are
+
r'
then the
normal
component of the vortex
(rot
@
x
w]).,.
--f
then the equality
div
W
=
owill
hold, and, hence, in
(.1
as
well.
It
can be easily verified that this condition
is
fulfilled when,
and only when, the boundary data are of type (6.281, (6.29).
Comparin
problem 76.28)
-
(6.33)
solvability
is
one of the stages of pro-
ving the basic theorem
1.1.
(6.29) and (2.30) demonstrates that the proof of the
194
Chapter
4
It
should be also noted that formulation of the boundary conditions
on
I”
in
the form of (6.281, (6.29)
is
a modified formulation of
the model problem studied by
N.E.
Kotchin
[80].
In
a
particular
case, when
3,
S
0
on
r
formulas (6.281, (6.29) are
a
general
type of formulating
an
overspecified problem with the
set
vortex
3
at
the ntrance
I”.
This
particular case was investigated in
papers (547,132
I.
Another variation of formulating the problem
of
flowing thro
with the boundary conditions for the vortex, suggested in [53:
considers the case when set are only the tangential components of
1
-D
w
are given
as
far
as the conditions (6.30).
Studying the boundary problem (6.30)
-
(6.34) results
in
the
fol-
lowing scheme.
At
the
first
stage the velocity field
is
defined by
the given vortex and the normal component on the boundary.From the
condition div
d
=
0
we then
find
the normal component
5
of the
vortex on the part of In-flowing
rl.
For this purpose use
is
made
of
equation (6.32)
in
its
projection onto the axis which
is
considered on
r1
as well as the property div
2
=
0
(this pro-
cedure
is
described
in
S3 when proving Lemma
3.1).
As
a
result,
on
q,
S1
we
get
the
equation
which serves to define
5,
on
I”.
Having found the boundary va-
lues of the vortex at the entrance, we solve the problem for the
Helmholtz equation (6.32) with the initial data and boundary con-
ditions on
I”
the condition divz
=
0
is
automatically ensured
on
I’l
and,
hence, in
Q
as well (see Lemma 3.1).
Thus, the construction
of
the operator
of
the boundary problem
(6.30)
-
(6.34) consists of problems
(3.1)
and (3.51, described
in
93, and the solution
of
equation (6.35)
on S’
with
the
initial
data. In this case there
is
no need
to
set the boundary conditions
on the line of
characteristic for (6.357. Equation (6.35)
is
solved by an ordi-
nary method
of
characteristics,
and
the estimates for
5,
on
P1
are deduced analogously to those
obtained for the vortex in [4&
Let
us
now formulate the final result
as
far
as the considered
problems
is
concerned.
i.e. problem (3.5).
In
this case the fulfilment of
conju ation with
ro
if
any, as this curve
is
+
Theorem 6.4. Let
+
wo,E
C~+~(Q>,
rot
f
E
cl+
a
(($1
,
+
y
E
C’:
cL91+a(S),
g
E
C2+01(S
),
=
(hl,
h2>
E
C1+TS1),O
<
a
<
1,
Correctness
of
the
Problem
of
Flow
195
and the corres ondin agreement conditions be met. Then problems
(6.28)
-
(6.337 and 76.30)
-
(6.34) have unique local solutions
Other formulations of the problem of flowing through are related to
various boundary conditions
at
the exit
I?‘.
is
the problem where the pressure
is
given
at
the exit r‘.This
problem has
a
physical sense, and hence
it
is
important not only
from the theoretical point
of
view. The problem of the correctness
of the boundary problems
fox
the Euler e uations with the pressure
set on the boundary has been studied in 764, 1141. Here
is
one
of
PO
sible formulations: the task
is
to find in the domain
@
=Q
x
x
$0,
ditions (1.2) and the boundary conditions
Of
greatest interest
p
the solution to system (1.1
)
which obeys the
initial con-
--t+
u
n
=
0,
(x,
t)
E
so;
u
=
g(x,
t),
(x,
t)
E
s’;
p
=
n(x,
t),
(6.36)
(x,
t)
E
s2.
1
+
Here
g
is
a given vector on
S
given on
52
To transform the problem to the operator equation, let us make use
of
problem (2.2) for the pressure with the boundary conditions
(2.4), (2.5) on
ro
U
r’
and with the given value
p
=
9
on
r2
as
well as
of
the initial system
of
equations for the momentum conser-
,
gn
<
0,
rl
is
a scalar function
vation
-t
-
+
(u
at
aa
-,
++
v)u
=
f
-v
p,
(6.37)
3
au.
au.
-b
Ap
=
-
C
+
div
f.
i,j=i
ax.
axi
J
The equation div
u
=
0
is
then automatically fulfilled, as
it
re-
sults from
(6.37)
that div
u
-+
*
obeys the linear uniform equation
a
+ + +
-
(div
u)
+
(u
v)(div
u)
=
0.
at
It
follows from the conditions at
t
=
0
and boundary cond1tio:s
(1.4).
(2.5)
that div;
=
0
For
details
of
the proof of this problem solvability see 164,
1141.
7.
UNSOLVED
PROBLEMS
1.
Of
greatest interest
in
the theory of equations for an ideal
liquid
is
the problem of the existence
of
the solutions
in
the
whole with respect to time.
As
has been shown in
[148],
we can
hardly hope that the solutions in the whole do exist in the three-
-dimensional problem (in the class where the uniqueness can be pro-
ved). In the case
of
plane-parallel flows (or axis-symmetrical),
however, the chances
of
proving the global solvability are quite
at
t
=
0
and on
I?’.
Hence,divu
E
0,
(x,
t)
E
‘.1*
196
Chapter
4
real. For
a
homogenous fluid
(p
9
const>
0)
in the problem with
the conditions non-flowing throughu
=
0
it
has been made in
[146, 173, 1941.
As
a step to follow,
it
would be
sound
przctice to
prove
the global theorem of existence
in
the problem of flowing
through with the set velocity vector at the entrance.
In
favour of
this opinion the following considerations can be given. The most
general theorem of uniqueness, which has been proved in
11461
for
two-dimensional flows of
a
homogenous liquid, holds when the velo-
city vector
is
bounded. In the case of
a
plane-parallel motion,the
equation for the velocity vortex proves to be much simpler than
in
the three-dimensional case
:
nl
r
(7.1)
aw
-
+
(u
VlJJ
=cp.
at
Here
w
is
a non-zero component of the vortex, in the Cartesian co-
ordinates
w
=
au,/ax,
-
au,/ax,,
cp
the rotor of the vector of mass forces. Therefore,
if
the function
w
is
bounded
at
t
=
0
and
at the entrance
r1
then there
is
an
estimate with respect to the modulus for
w(x,
t)
in
the domain
y
=
Q
x
(0,
T).Thua,
the basic difficulty lies
in
estimating the
vortex at the entrance
I”
where
w
is
not known beforehand.
In
terms of the scheme presented
in
S2.3,
the boundary values
of
w
on
I?’
are expressed through the tangential derivative of the
pressurep. Not strictly speaking, the problem has a
small
parame-
ter, since for pressure
p
we need only the local estimates
in
the
vicinity of
r‘.
2.
No
global theorems have been established for a single case of a
non-homogenous liquid. Therefore,
it
would be interesting to prove
the theorem of existence in the whole for the Euler equations for
a
non-homogenous liquid
in
the plane-parallel flow with the boun-
dary conditions of non-flowing throughout the boundary
r
.
3.
The next set of questions
is
associated with studying statio-
nary flows of an ideal fluid. Even the two-dimensional problem of
flowing through
in
a
physical formulation has not yet been studied.
In
works
2,
3, 41
theorems have been proved only for the model
problems when
at
the entrance
I”
either the normal component and
the vortex, or the velocity vector are given, but then at the
part
of
out-flowing given
is
the tangential component
of
the velo-
city, which
is
non-zero by
all
means. In all probability, the pro-
cedure under discussion can prove useful when considering station-
ary problems as well. Special attention should be paid to three-
-dimensional stationary problems,
for
which no theorems on their
solvability are available at present.
4. In a technical sense, of interest
is
studying the problem of
flowing through in other functional spaces, i.e. in the Sobolpv’s
spaces. These spaces are more natural as compared with the Hol-
der’s ones, since in this case one can more effectively use the
estimates of the types of energy integrals. The problem of flowing
through
with
the given vortex
at
the entrance haa been studied for
solvability in the Sobolev’s space in [47].
5.
As
has been pointed out in
D
6,
quite difficult problems arise
when considering the liquid flows with the
parts
of
in-
and out-
-flowing touching the solid
wall.
Therefore, we can pose as
is
the non-zero component of
Correctness
of
the Problem
of
Flow
197
problematic the problem
of
studying differential properties
of
the
solutions
of
Euler equations
in
the domains with non-smooth boun-
daries (see [75, 76,
991).
On the other hand,
if
the part
of
in-flowing
PI
and the solid
wall
r0
are conjugated
in
a
amooth way, then the problem with the gi-
vp v5locity vector also
remains
unstudied, as the reguirement
(U
")Ir
5
const
C
ponent on
rl.
This condition for smooth solutions can be fulfilled
if
I"
and
Po
either do not cross or are conjugated
at
a
non-zero
angle. This supposition has been
a
corner-stone of the adopted
scheme
of
proving since
it
makes
it
possible to find the tangen-
tial
components
of
the vortex on
r'.
It
has been also used in
Lem-
mas
2.1,
2.2
when proving the equivalency
of
the
initial
and trana-
formed problems.
6. More complex models
of
an ideal liquid are also worth studying.
For instance, multi-component mixtures with the zero viscosity
within the framework
of
the diffusion model from
!S4
in
Chapter
III,
as well as the models with the internal degrees
of
freedom from $5
in Chapter
111.
7.
There
is
one more set
of
interesting problems, related to study-
ing
the flows of compressible ideal fluids. The Euler equations
for a compressible liquid are as follows:
0
has been imposed on the normal velocity com-
a3
+
-PI
-+
-
+
(u
.o)u
+-
OP
=
f,
at
P
ap
-b
-
+
div
(pu)
=
0,
p.=
p(p).
at
The formulation
of
the problem
of
flowing t-rough for
(7.2)
ia
very difficult, since
it
is
not clear how to pose the boundary
conditions on the part
of
in-flowing
P2.
In the case when the
whole boundaryr ie an impermiable solid
wall,
a
local solvability
of
the problem for equations
(7.2)
with the boundary condition
(<
:)Ir
=
0
has
been proved but recently 186, 1573.