Antontsev S.N., Kazhikhov A.V., Monakhov V.N. Boundary Value Problems in Mechanics of Nonhomogeneous Fluids
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8
Chapter
1
1
au
2
ax,
ax,
=-
(
aL
+
L),
p
=Rpe
+
The vector of mass forces
f
is assumed to
be
the prescribed
function.
Equations
(1.25)
represent rather a complex system of a compound
type: the equations
of
momentum and energy, atp
>
0,
n
>
0
are
parabolic
in
respect to the unknown functions
2
and
the continuity equation may be treated as the first order equation
with respect to the density
p.
Model
(1.25)
began to be used for
the solution of applied problems of gas dynamics as early as in
works of Rayleigh, Prandtl and other authors on an approximate
search for the shock wave zone (see
[97,
Ch.
XI,
paragraph
129J).
In the accurate statement, the problem of shock
for
the case of a
one-dimensional steady-state flow was solved by
R.
Becker
152
.
The interest towards equations
(1.25)
has particularly increased
in the recent decades
in
connection with the possibilities for
solving significant applied problems which became available with
the aid
of
computers (see
[73,
111,
1371).
Issues
of
correctness of the boundary-value problems f r
s
stem
(1.25)
began
to
be studied since
1959,
when
G.
Serrin
p1877
for-
mulated the statements of the basic boundary-value problems and
proved the the rems
of
uniqueness
in
the class of smooth solutions.
Following
[l873
we shall give several types of boundary-value
problems for
11.25).
Let the motion of the medium occur in a
bounded region
Q
of
a space
R3,
the boundary of which
r
is
an
impenetrable hard wall. Then the conditions of adhesion are met
on
r
,
i.e.,
~
Another possibility is that the stress vector $,is prescribed on
the boundary
r
E)
while
ulr
=
0.
(1.26)
++
+
+
3
Pn
I
1
-(p
+
z/3
pdiv u)n
+
2
p(D
")Ir
=
H.
(1.27)
Such boundary stresses appear, for example,
in
the free boundary
problems, the boundary itself being defined with the aid of the
I'
so
called kinematical condition
(1.28)
a<
+
-
t
(U
-V
jc),
=
o
at
r
=
{(x,
t>Ic(x,
t>>=
o
which means that any material particle present on the free
boundary may travel but alongside
it.
Besides, it is essential to prescribe the boundary conditions
characterizing the heat condition of the flow. The most general
type
of
boundary conditions for the temperature is the following:
Models
of
Heterogeneous Media
9
+
Here
k
and
x
outward normal to
I?,
a/an
is
the normal derivative. Equality
(1.29) means
that
the heat flow across the boundary
tional
to
the difference in the temperature
of
the
gas
0
of
the wall
x
Two
more variants are possible
are the
known
functione,
n
is
the unit vector of the
and that
I'
is
propor-
@Ir
=
x
and
when the boundary values
of
the temperature
8
or the heat flow
n
are prescribed.
The
first
existence theorem for the Navier-Stokes equat ons
the compressible viscous fluid was obtained by
G.
Nash
tl8l
p"k
1962.
He proved the existence of the classical solution for the
Cauchy problem, when
52
=
R3,
in
the
small
in time. In
this
case,
the velocity,
density and temperature are prescribed
at
the
initial
moment
of
time
t
=
0
In
works [36, 1701 results
[181]
were generalized by somewhat
dif-
ferent methods.
In
mixed boundary-value problems, local existence
theorems were established by
V.A.
Solonnikov
[127
J
(see also [192
1).
At
present, global existence theorems are obtained
only
in the
case
of
a one-dimensional motion with plane waves, when the solu-
tion depends but on a single space Cartesian coordinate and
on
the
aP
a
-
+
-
(pu)
=
0,
at
ax
(1.33)
p(
-+u-->
+-=
u-+pf,
p=wo
at
ax
ax
ax'
au
au
4
+
lJ
-
)
=
M
-
+
u(
-
I'
-
P
-
1
v
=
-
CL
ax
ax
3
ao
ao
a'@
cvp(
at
ax
ax2
where
u
is
the non-zero velocity component.
Out
of
a number
of
works devoted to the investigation
of
the
solvability
in
general,
it
is
neceasary to point out the paper by
Ya.I.Kane1 [69], where the global correctness of the Cauchy
problem for the particular model
of a barotropic motion
of
gas,
when the pressure depends on the density alone, but doea not
depend on
the
tem era ure. Japanese mathematicians N.Itaya [171,
172
]
and
A.
Tani pl911 considered another simlified model of a
viscous gas,
in which
P
=
const, and obtained the existence
theorems as
a
whole both for the Cauchy problem and for the mixed
boundary-value problem.
It
should also be pointed out that
of
late
A.Matsumura
and T.Nishida [179] proved the solvability
of
the
10
Chapter
1
Cauchy problem
as
a
whole
in
time for the general system (1.25)
provided the initial data (1.32) are close to the state of rest.
In
a series of works
V.V.
Shelukhin [140 to 1431 studied the
issues of existence
of
the periodical, near periodical and simply
time-limited solutions for the model systems of equations of
viscous gas, and in
[I061
an investigation of one-dimensional
axisymmetric flows
is
undertaken.
A.V.
Kazhikov [54 to 56,
59,
601
carried out an investigation of the boundary-value problems for
the complete set of equations (1.33). He established the global
existence theorems
for
all the formulated statements of the
boundary-value problems, whereas for the Cauchy problem without
any additional smallness conditions on the initial data he stu-
dies the behaviour of the solutions at the time increased inde-
finitely. These results are presented in Chapter
11.
3.
Heterogeneous viscous incompressible fluid
Another closed model which
is
studied in Chapter
I11
is
obtained
from the gernral set
of
equations of mechanics
if
the incompressi-
bility condition
aP
3
-
+
(u
.V
>p
=
0.
at
is
taken as the equation of state.
This
assumption
with
a
high
degree of precision
is
justified for Siquids. Then from the con-
tinuity equation
it
follows that div
u
=
0,
and the Navier-Stokes
system at
p
=
const
>
0
is
written in the form
V)U
=
p
A
U
-Vp
+
pf,
-9
3
(1.34)
-1
3
aP
+
-
+
(u
v)p
=
0, div
u
=
0.
at
In this case the energy equation
is
separated from the system and
is
solved after the velocity and density are found. The basic
initial-boundary problem for equations (1.341, in the case when
r
=
an
is
an impenetrable hard wall,
is
formulated as follows:
If
on the boundary
I'
there are sections
of
inflow and outflow of
the liquid, then some boundary values of the density must be pre-
scribed on the
inflow
section
I",
where
(3
.
n)
<
0.
Statements
of this type are
discussed in paragraph 3 of Chapter
111.
-3
If
of
to
p
3
const
>
0
,
the liquid
is
called homogeneous, without
loss
generality, one may assume
P
the equations for the velocity and pressure
au
-f
-t
3
3
3
1,
then system (1.34)
is
reduced
+
(1.36)
-
+
(u vlu
=
IJ.
A
u
-
VP
+
f,
div
u
=
0.
at
Models
of
Heterogeneous Media
11
Mathematical investigation of the correct boundary-value problems
for system
(1.36)
began with the works by J.Lere
Various aspects of the theory of equations
(1.363
are treated
in
detail in the monograph by
O.A.
Ladyzhenskaya
L88J.
Ladyzhenskaya,
as
well as her collegues and disciples V.A.Solonnikov,
K.K.
Golov-
kin
et al. made the most significant contribution to the develop-
ment
of
this theory Results of great importance were also recei-
ved by
E.
Hopf
11681,
K.I.
Babenko
[27]
J.-L.Lions
[96],
R.
Finn
[I581
and by
g
number of other authors [see
[34, 122
to
130,
164,
165, 167, 1861).
The interest towards the model of a heterogeneous liquid
(1.34)
is
due to
its
significance for such sections of applied hydrodynamics
as oceanology and hydrology.
A
number
of
problems important for
practice was solved on
its
basis, many of these problems are
listed in book
L195].
For the
first
time system
(1.34)
as a model
for the description of marhe currents seems to be formulated in
C721
The basic mathematical properties
of
system
(1.34)
at
p
#
const
as compared with the model
of
a homogeneous liquid
(1.36)
consist,
firstly, in the presence
of
the additional nonlinear equation of
the
first
order for
p
.
Secondly, the density
p
enters in the
momentum equations
as
the cofactor at the highest derivative
u
which leads to the system being non-linear in the main part. In
respect
of
ideas, the model
of
heterogeneous viscous fluid
is
studied on the basis
f
the methods developed in the book by
0.A.Ladyzhenskaya
1883.
These investigations are presented, in
the
main,
following the works by A.V.Kazhikhov
L51
to
531.
Para-
graph
3
of Chapter
I11
is
devoted to the new class
of
the boundary-
value problems with one-sided constraints. One
of
the problems
under consideration
is
as
follows. Let the liquid flow across the
region
B
,
the boundary
I?
of which
is
made up of three parts:
the impenetrable wall
ro,
the section of inflow r’and that
of
out-
flow
r2.
On
To
the ve;tocity
is
equal to zero, the tangential ve-
locity components on
I?
are known, as well as the density and the
fluid head
p
i
1/2plu
I
‘.
city components and pressure (or the head) are prescribed.Besides,
the normal velocity component on
I”
and
r2
must satisfy the
conditions
of
the type of the inequalities that specify that
I?’
is
the inlet, while
r2
the outlet, i.e.,
1175, 1761.
+
t
+
On the section r‘the tangential velo-
+-I
++
(u
n)lri
5
0,
(u
n)lrz
2
0.
It
was J.-L.Liona
[96,
1781
who
was
the firet to undertake the
development of the theory
of
one-sided problems for the Navier-
Stokes equations
(1.36);
he considered some instances of one-sided
restrictions, under which the solvability of tkese problems can be
proved, but these examples are of interest only from the theoreti-
cal point of view, because the one-sided conditions are devoid
of
any physical meaning. The statements considered here can be inter-
preted physically and are new even in the case
of
a homogeneous
fluid.
We may add that recently there has been
a
marked increase
in
the number of publications on the problems
of
one-sided condi-
tions for the Navier-Stokes operators, see,
for
example,
[115,
153
to
155,
180, 182, 1841.
12
Chapter
1
4.
Ideal incompressible fluid
Chapter
N
considers the classical model
of
hydrodynamics, that is,
Eulerian equations for an ideal incompressible fluid
-3
-3
(1.37)
These equations represent a particular case of model
(1.341,
when
the viscosity
w
is equal to zero.Regardless of the fact that
from the physical point
of
view the model is rather simplified,
it
was used in hydrodynamics long ago and is still
in
wide use. It is
only natural that the theoretical study of the Eulerian equations
is
being carried out rather intensely for more than
a
hundred
years.
Fundamental results in the theory
of
boundary-value problems for
were received
in
works by N.I.Hhter
t42,
447
and L.Lichtenstein
1
1773.
In the first place, they proved
the local correctness
of
the Cauchy problem, i.e., the problem of
the fluid flow which fills the whole space.
In
this case, in addi-
tion to
(1.37)
at the initial moment
of
time
t
=
0
,
the condi-
tions are prescribed
he
Eul
rian equations
(1.37
(1.38)
0 0
+
+
UI
t=o
=
u
(XI,
PIt=,
=
P
(XI,
-3
whe?
Uo(x)
and
div
uo=
0,
0
<
m
I
considered the case when the liquid fills a vessel with hard impe-
netrable walls.
If
the boundaryp
=
aa
is motionless, then the
condition
of
lack
of
flow
po(x)
are sufficiently smooth functions, and
0
p
(x)
5
M
<
co
.
Secondly, they considered the
-3-3
(u
n>lr
=
0.
should be established as the boundary condition.
These results were preceded by works
of
Cauchy, Dirichlet,Riemann,
Helmholtz, Zhukovskii and a number
of
other mathematicians and
mechanics, in which they considered individual particular classes
of
solutions and problems. Following works by
N.M.
Hunter and
L. Lichtenstein,
given ield were
(1.39)
to
1
Elf,
T.
Kato
L1084
and by
a
number of other au hors (see
114,
133,
151,
166,
169,
185
1901).
Investigations
of
L.V. Ov-
eyannikov and his disciples
[46,
104, 1081
into the free-boundary
problem, which proved to be extremely difficult from the mathema-
tical point
of
view, especially id the non-stationary case, should
be
noted particularly. The theory
of
stationary boundary-value
roblems with free boundaries
is
developed quite thoroughly (see
a
39,
40,
101
1).
At
the same time, the problem
of
the flow
of
an ideal liquid
across the given region
Q
rather interesting for its application,
for a
long
time remained not investigated. Since set
of
equations
Models
of
Heterogeneous Media
13
(1.37)
has real characteristics (the characteristic equation of
system (1.37) has
a
triple real root and two imaginary roots),
it
is
essential to prescribe addi4ionZl boundary conditions
on
the
section of inflow
I?‘,
where
(u
-
n)
<
0.
Physical considerations
prompt that
it
is
essential to prescribe on
r’
vector and the density. Yet, the rigorous proof of the correctness
of this problem turned out to be nontrivial.
For the
first
time
N.E.
Kochin
[SO] considered the problem of the
flow of an ideal fluid across
the given region in the model formu-
lation: on the se3tion
of
inflow
r4
the normal component of the
velocity vector
(u
-
n)
<
0
and
all
the three components of the
velocity vortex
w
=
rot
u
were prescribed. The refinement
of the
proof
of
solvability of this problem in the particular case
when the boundary values of the ort x
3
on
I”
are equal to zero,
as presented by
R.M.
Ukhovskii [1327, also by
V.
Zayachkovski
‘p47],
when the boundary values of the normal curl component
(;
*
z)
on r’equal zero. The two-dimensional problem of flow
with a vortex prescribed on the Fnlet was studied by V.I.Yudovitch
[1471 (see also
[5,
1511
1.
Stationary two-dimensional problems of
flow in the model statements were studied by
G.V.
Alekseyev
[2
to
4i (also see c41)). The basic result of Chapter
IV
consist in the
proof of the unique solvability
in
the
small
in time of the prob-
lem of flow in the physical statement, i.e., when on the inlet
I”
the total velocity vector
and
density are defined, and on the
outle
r2
and n the hard wall
Po
the normal velocity component
(see
t58,
63,
6471.
5.
Models
of
multicomponent and multiphase mixtures
Besides the classical equations of hydrodynamics enumerated above,
when solving many modern problems of mechanics use
is
made of more
involved models which take a more accurate account of the hetero-
geneous nature of the composition of real fluids and gases.
It
must
be mentioned that at present there are a great many models for the
description of multicomponent and multiphase mixtures, and all of
them are rather complex both from the theoretical point of view
and in respect of their application for the solution of specific
problems.
On
these grounds none
of
them became generally accepted.
When compiling a closed set of equations of, say, a multicomponent
mixture use
is
made of the continuity equations for the components
of the mixture
the total velocity
-P
*
-P
--t
’”
-
+
div
(p
u
)
=
hi,
i
=
1,
2,
...,
N,
ii
at
(1.40)
where
p,is
the mean density of the
ui
is
the velocity. In this case hi
f
0
provided the various
components may exchange their
mass
as a result of processes of
mixing, ionization, chemical reactions, etc.
As
far as the equa-
tions of momentum are concerned, two various ways of setting them
up may be singled out. The
first
way
is
that the laws of conserva-
tion of momentum are formulated for each component mixture
i-th component of the mixture,
+
14
Chapter
1
pi
Here
P,
I
at
3
is
the stress tensor for the i-th component,
F,
,
are
-"
the terms considering the sxchange of momentum among various
components. The functions
Fij
portional to the difference in the velocities us
-
uJ
the
coefficients of proportionality being defined differently by
different authors, and
&t
appears that
at
present there
is
no
agreement
in
defining
Fij
as well as
P,
Another method
is
based on writting the equations of conservation
of momentum for the mixture
are usually consi2ered to be pro-
+
-t
p
r2
+
(<
v>u
=
div
P
+
pf,
-1
(1.42)
s
3
where
p
=
C
p
is
the density
of
the mixture,
u
is
the mean
mass
velocity
of
the mixture,
pu
=
2
p,u,.
Velocities ui are defined by means of the average celocity of
u
with the aid of the Fick's law of diffusion (see
[97,
Ch.
VIII])
1=
1
+
N+
1=
1
9
3
(1.43)
where
ci
=
p,/p
is
the mass concentration
of
the
i-th component,
h,
is
the coefficient of diffusion.
Another approach
is
usually applied when describing the
so
called
homogeneous mixtures consisting of well mixed components
in
a
liquid
or
gaseous phase,
as
well as solutions, i.e., in those
cases when the components of the mixture practically interact on
the molecular level.
6.
Model of
a
medium with intrinsic degrees of freedom
Vaen setting up
a
closed set of equations for continuous media with
intrinsic degrees of freedom several different approaches are
available, but there
is
no single generally accepted model.
Here we ehall formulate the simplest version of such
a
model,
proposed in
[95]
for the description of the motion of inhomogene-
oua granular medium, which
is
based on the use of the law of con-
servation of angular momentum in form
(1.8).
We shall omit the
details of deriving the equations, referring the reader to
[95
]
and shall give only their form:
Models
of
Heterogeneous Media
15
-P
au
-,
-,+
-,
++
-
+
(U
V>U
=
f
-Vp
+
V
A
U
+
V[W
x
u],
at
at+
4
-,
aw
-,
+--t
+
+
-
+
(u
v>w
=
m
-
F2(p>W,
div
u
=
0.
(1.44)
Here
u
is
the velocity vector,
w
is
the vector of_angular velo-
city of the particle rotation,
vector of the external
mass
forces,
m
is
the density of the ex-
ternal force moments,
v
=const>
0
is
the viscosity,
q
=const>
0
is
the Magnus coefficient,
F2(p)
is
the scalar function characte-
rizing the particle friction,
it
is
defined experimentally, and
in
the case of dry friction
is
of the form
p is+the pressure,
f
is
the
F2(p>
=
n
+
kp,
(1.45)
where
n
=
const
>
0
is
the constant of shearing cohesion,
k
=
const
>
0
is
the friction coefficient. The pressure p in
this case
is
assumed to be non-negative, therefore, along with the
boundary and initial conditions, the condition p(x, t)
L
0
is
added. Since in the momentum equations the pressure
p
is
defined
up to the additive function of time
t,
it
is
possible to set,
as
an additional condition, the minimal values of
p
on the cross-
-sections
t
=
const, i.e.,
min
P(X,
t>
=
p,(t>
L
0,
t
E
[o,
T]
(1.46)
X€Q
Here
51
is
the region of flow,
[0,
!!!I
is
the period of time on
which the solution
is
based. The boundary and initial conditions
in
the simplest case, when the boundary of the flow region
is
a hard motionless well, are of the
form
(see
15
)
If
on
r
There are sections
of
inflow
r’
and
outflow
r2,
then
on
r’
it
is
necessary to prescribe additionaly the values of
7.
Equations of filtration
of
two immiscible incompressible fluids
The theory
of
filtration studies the motion of fluids and gases in
the porous bodies containing an intercommunicating system of
hollows (pores) through which the motion takes place. This field
of hydrodynamics was formed sufficiently long ago, in the middle
of the last century, but a century later
it
had
its
revival.
It
was connected, in the
first
place, with the application of the
method
of
recovering oil under the drive of water or special
solutions.
It
is
natural that the necessity arose to create mathe-
matical models for the description of
the process of driving one
liquid
by
another,
a8
well as
to
develop the methods
of
solution
of specific problems. In describing the motion of
a fluid or gas
in a porous medium there
is
a number of peculiarities
as
compared
-,
w.
16
Chapter
1
with the classical models
of
hydrodynamics. Firstly, the notion of
porosity of the medium where the motion
is
effected
is
introduced.
The porosity
m
=
m(x)
denotes the part of the volume
of
the medium
taken by the hollows
-
pores. Allowing for the porosity
of
the
medium leads to the continuity e uation
(1.3)
taking, in the theory
of filtration, the form
[33,
1161
a
+
-
(mp)
+
div (pv)
=
0,
at
-+
where
p
is
th2 density,
v
is
the rate of filtration related to
the velozity
,u
formula v
=
mu
.
Another difference
is
due to the fact that, in-
stead of equations
(1.6)
of momentum,
in
the theory of filtration
use
is
made of the following
law
of resistance
of
the porous me-
dium to the liquid flow passing through
it
that was obtained ex-
perimentally, Darcy law of resistance
[116]
of the motion
of
the particles of
the fluid by the
-+
(VP
+
Pd,
(1.49)
KO
-’
v=--
-5
IJ.
where
p
is
thed dynamic viscosity of the liquid,
g
is
the gra-
vitational vector, and
KO
=
const
is
the coefficient of penetra-
bility characterising the filtering properties of the medium.
In the case of an inhomogeneous medium
I(,
depends on the space
coordinates
pressure
P
(in this case the porosity
m
represents also the
function of
p
1,
and in the case of non-linear law
of
resistance
-
on the rate of filtration v
=
I
v
I.If
the medium
is
anisotropic,
then
KO
=
{Kij}is
the symmetrical tensor of filtration (from now
on the tensors
will
be designated with capital letters).
There are several ways of empirical derivation of Darcy equations
(1.49) as the approximations to the law of conservation
of
momentum
(1.6)
[116].
One of these methods, which belongs to N.E.Zhukovskii,
assumes that the pressure gradient, gravitational forces and
another, the
so
called resistance force, which
is
assumed
to
be
proportional to the rate
of
filtration, play the most significant
role in the filtrational motions. Thus, neglecting in the exact
equations of motion
i1.6)
au
-’
P-’
p-=pnZ
-
VP
-
Pg
-
-
mu
x
=
(x,,
x2,
x3)
for a compressible medium
-
on the
-+
‘(0
dt
the inertial terms and forces of viscous friction, yeilds, as a
result, equations (1.49).
It
is
considerably more difficult
to
construct mathematical models
of the process of filtration of two immiscible liquids (for
example, water and
oil)
through a porous medium. Experiments de-
monstrate that in this case each of the liquids selects
its
own
fairly stable ways.
At
decreasing of the saturation
by
the
si
(the volume fraction occupied
i-th component) with one
of
the liquids,
the channels are
Models
of
Heterogeneous Media
17
destroyed, become discontinuous and ultimately there remain but
isolated regions filled with this liquid. This phenomenon
is
cal-
led the residual saturation with oii or water, and the correspond-
ing values of
si
are denoted as
si
>
0.
Yet, in describing this rather involved physical process mathema-
tically, the concept
of
continuum can also be applied.
We
shall
consider a two-component liquid as the aggregate of continua
filling one and the
same
volume
of
incompressible porous space.
For each of the continua, besides the saturations
si
,"e shall
introduce
its
own density p,,the rate of filtration
vi
and
pressure
p
.Then, analogous to
(1.481,
the continuity equations
for each component of the liquid may be written in the form
[33,
1161
i
a
--t
-
(mpisi)
+
div
(p,v,)
=
0,
i
=
1,
2.
at
(1.50)
Considering-the qualitative pattern of multiphaae filtration,
M.
Masket (see
1741)
proposed the following formal generalization
of Darcy law for each of the liquids:
where
is,
as before, the coefficient of filtration of the
porous medium for a homogeneous
liquid (or the symmetrical tensor
for an anisotropic medium),
pi
are the coefficients of dynamic
viscosity, and
Eoi
case
Eoi
must
depend on the saturation
si
since part of the
porous volume
is
occupied with other liquid.
tically do not depend on the pressure, rates of flow and other
parameters of liquid flow, was repeatedly borne out by laboratory
experiments (see
1741
1.
According to the definition, the saturations
limits
0
<
sy
I
si
5
1
-
sy,
i
#
j,
s1
+
s
achieving the values
si
=
sy
the movement of the i-th component
ceases, which
is
ensured by satisfying the conditions
'I;oi(s:)
=
0,
i
=
1,2.
!'Then analysing immiscible multiphase
flows
it
is
also necessary
to consider the effect of the forces acting on the interfaces.
When two immiscible
liquids (I and 11) come
in
contact with each
other and the hard surface
of
the pores the interface
r
sepa-
rating the two liquids approaches the hard wall at the contact
angle
8.
If
0
ie
an acute angle, liquid
I
is
called wetting
(it has a tendency to spread in a greater degree over the given
solid body), and liquid
I1
-
non-wetting. On the boundary
there occurs a shock of phase pressures which
is
known as capil-
lary pressure:
KO
are the relative phase penetrabilities.In this
The fact that
Eoi
are just the functions of
si
and prac-
sL
vary within the
=
1
and, on
2
182
rlr
2