Antontsev S.N., Kazhikhov A.V., Monakhov V.N. Boundary Value Problems in Mechanics of Nonhomogeneous Fluids
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X
8
. Stabilization of solutions under infinite time increase
......
86
9
.
Unsolved problems
............................................
96
CHAPTER
3
INITIAL-BOUNDARY VALUE PROBLEMS FOR THE NAVIER-STOKES EQUATIONS
OF NONHOMOGENEOUS VISCOUS INCOMPRESSIBLE FLUID
.................
101
1
.
The existence of generalized solutions
......................
101
1
.
Formulation
of
the first boundary problem
...............
101
2
. Construction of approximate solutions ...................
104
3
.
Lemma on the compactness of approximations
..............
108
2
. Differential properties
of
the generalized solutions
........
113
1
. Density continuity with respect to t in the integral
norms
...............................................
113
2
.
Estimation of higher derivatives of the velocity vector
.
120
3
.
Density continuity by Hdlder
............................
122
4
.
The existence of a classical solution
...................
125
3
.
Unilateral boundary problems
................................
127
1
.
Formulation of the problem
..............................
127
2
. Additional information
..................................
128
3
.
The definition of a generalized solution and a scheme
of proving the theorem on solvability
...............
131
4
. Solvability
of
the regularized problem
..................
135
5
.
Proof of Theorem
3.1
....................................
137
6
.
Other formulations of one-side problems
.................
138
4
. A model of an inhomogeneous fluid with diffusion
............
139
5
. Model
of
a medium with intrinsic degrees of freedom
.........
143
6
.
Problems unsolved
...........................................
147
CHAPTER
4
CORRECTNESS OF THE PROBLEM
OF
FLOW THROUGH AN IDEAL INCOMPRESS-
IBLE LIQUID
....................................................
149
1
. Formulation of the problem and the basic result
.............
149
2
.
Equivalent formulations of the boundary problem
.............
153
3
. The method of successive approximations
.....................
162
4
.
A priori estimates
..........................................
172
1
. Velocity estimates
......................................
172
2
.
Pressure estimates
......................................
174
3
.
Estimates for particle trajectories
.....................
176
4
. Vortex estimates
........................................
181
5
. Proof of the existence and the uniqueness of the solution
...
184
6
. Some generalizations of the basic theorem
...................
187
1
.
The case of non-homogenous fluid
........................
187
2
.
The case of a multiply-connected domain of the flow
.....
190
3
.
Other formulations of the problem of flowing through
....
192
7
.
Unsolved problems
...........................................
195
xi
CHAPTER
5
FILTRATION OF IMMISCIBLE LIQUIDS
...............................
199
1
.
Reduction of equations. formulation of problem
..............
199
1
.
Peculiarities of equations. the
choice of the sought
functions
...........................................
199
2
. Initial boundary value problem
..........................
202
3
.
Outline of conditions which provide summarized filtrat-
ion velocity be independent of saturation
...........
203
2
.
Determination of generalized solutions . Problem regulariz-
ation. maximum and compactness principles
...............
203
1
. Generalized solutions
...................................
203
2
. Regularization
..........................................
206
3
. The maximum principle
...................................
206
4
. Compactness principle for a non-stationary
problem
solution
............................................
209
3
.
The theorem of existence of generalized solutions
...........
212
1
. Galerkin approximations in an auxiliary problem I
.......
213
3
.
A priori estimates independent of h
.....................
216
2
.
Compactness of Galerkin's approximations and a limit
transition over
N
...................................
214
4
. Limiting transition over
E
..............................
218
4
. Determination of the summed velocity of inhomogeneous
liquid filtration
.......................................
220
2
. Continuity of solutions over the variable x
.............
223
1
.
The statement of problem
................................
220
3
.
Solutions belonging to the class
).
q
>2
.........
224
4
.
Estimates of higher derivatives of a solution
...........
226
5
. Smooth solutions on t parameter
.........................
228
summed filtration velocity
..............................
230
1
. Problem design
..........................................
230
2
.
The solution properties for parabolic equations
.........
230
3
. A regular problem of a two-phase filtration
.............
233
4
. Singular problem
(0
<s
Q
1)
.............................
238
1
.
Problem design
..........................................
242
2
. Regular problem
.........................................
242
3
.
Degenerating problem
....................................
246
Wi(
5
. On distribution of phase saturation on a given field of a
6
. On a joint problem
..........................................
242
7
.
Stationary problem
..........................................
246
1
. Existence of generalized solutions ......................
246
2
. Choice
of
other searched functions
......................
248
3
. Regular generalized solutions
...........................
249
4
.
Degenerating case
.......................................
250
5
. Uniqueness
of
regular solutions
.........................
251
8
.
Further smoothness of regular non-stationary problem
solutions
...............................................
252
1
.
Introduction
............................................
252
3
. Continuity
of
regular solutions by HBlder
...............
251
4
.
Further smoothness of solutions
.........................
260
5
. Estimate of regular solutions in W;'l(Q')
...............
262
6
. Stability and uniqueness of solutions
...................
264
I
. Asymptotic behaviour of solutions for t
+
-
.............
261
2
.
Auxiliary evaluations and inequalities
..................
253
xii
9
.
Degenerating non-stationary problem
.........................
267
1
.
Introduction
............................................
267
2
.
Estimates for higher derivatives
........................
268
3
.
Continuity
of
solutions by HBlder
.......................
277
4
.
Uniqueness of solution
..................................
278
5
.
Finite velocity
of
disturbance propagation
..............
281
6
.
The finite time
of
solution stabilization
...............
286
10
.
Approximation solution of two-phase filtration problems
.....
289
1
.
Introduction
............................................
289
2
.
Regular problem
.........................................
289
3
.
Degenerating problem
....................................
293
11
.
Unsolved problems
...........................................
293
REFERENCES
......................................................
297
1
CHAPTER
1
MODELS OF THE DYNAMICS
OF
HETEROGENEOUS mDIA
AND THE
BODY
OF
MATHGMATICS
This chapter gives a brief derivation of various models used in
hydrodynamics, formulates statements of initial-boundary problems
and provides the required subsidiary information from functional
analysis and theory of differential equations.
1.
MODELS OF MECHANICS OF HETEROGENEOUS FLUIDS
1.
General equations of motion
of
continuum
and
laws
of
thermodynamics
The set of dynamic equations describing the motio of continuum
involves the following laws
of
conservation (see
ft97,118]):
1)
the equation
of
conservation of
mass
of continuum,
2)
the
3)
the equations of angular momentum,
4)
the law of conservation
of
energy.
This set of equations does not, as a rule, appear closed, and,
in
order to study on its basis various problems on the motion
of
continuum, it is necessary to supplement this set with the rela-
tion-ships characterising certain properties
of
the given medium.
Let us give
the above laws
of
conservation in integral
and
diffe-
rential forms. Let
Q
be a simply-coyected bounded region
of
three-dimensional Euclidean space
R
,
filled voidlessly with a
continuous medium with a density
p.
Assuming the distribution
of
strength h of the sources
of
mass to be known, the continuum
balance equation
may
be
written as follows:
law
of
conservation of momentum,
the
Here
3
=
(uq,
up¶
23>
ie the velocity vector,
r
is the boundary
of
the region
Q
,
n
is the unit vector of the outward normal to
r,
t
is the time.
Equation
(1.1)
means that the rate
of
variation
with time
of
the amount
of
continuum
in
the volume
Q
is
e
ual to
the sum of the flow of the substance through the boundary jandthe
total strength of the sources
of
mass.
Considering the motion with continuously differentiable characte-
ristics and applying the Gauss-Ostrogradski formula to the first
integral
in
the right side of
(1.11,
in
view
of
the volume
being arbitrary, we obtain a differential equation of the balance
of mass
Q
2
Chapter
1
+
div
(pz)
=
h,
(1.2)
Here
div
is the operator
of
the divergence.
If
x
=
(x,
,
x,
,
x,)
is the Cartesian system of coordinates
in
R',
then
-,
3
aw,
i=i
ax,
diVW=
C
-
In a particular case h
8s
the continuity equation
0
,
equation
(1.2)
is
known
in
mechanics
ap
+
at
-
+
div (pu)
=
0.
(1.3)
Let
us
proceed to deriving the equations
of
conservation
of
momentum. The concept of force plays here an important role.
In
continuum mechanics two basic forma
of
forces are distinguished:
the volume (or
mass)
forces whose effect is distributed over all
the points
of
the volume
surface forces distributed over the surface
r
bounding the region
51
.
The concept
of
stress tensor is closely connected with the
surface forces. Let us take
an
arbitrary point
M
of
the continuum
and construct an infinitely mall tetrahedron MABC, whose
IWB,
MBC, MAC faces are para1131 to the coordinate planes, while the
ABC face with the normal
n
=
(n,,
n,,
n,)
is oriented arbitrarily.
Let-us kenoke the forces acting on the faces
of
the tetrahedron by
P,,P,,
P,,
Pn
51
taken for consideration, and the
respectively. From the equilibrium conditions
follows an equality
3+
-9
+
Pn
=
P,n,
+
P,n,
+
Pp,.
(
The three vectors
Pi,
i
=
1,
2,
3
form a tensor
P,
referred
as
the stress tensor. The vector
of
the surface forces
on
the
surface element
dS
with a normal
2
is defined as the product
-+
+
PndS
=
(P
ii)dS.
-
.4
1
to
Let us denote we vector of the mass forces
as
f.
The particular
form
of
vector
f
under consideration.
As
a rule, these are gravitational forces:
2
is the free fall acceleration.
For
a number
of
problems
it
is essential to take into account, say, the Coriolis forces or
electromagnetic forces. The law of conservation
of
momentum may
be written in the form of
is determined individually for each problem
-t
=
z,
This equation expresses the d'hlembert
'8
principle: the forces
affecting the volume, including the inertial forces, are mutually
equilibrated. In the differential form, in the class of conti-
nuously differentiablz motions, this law is written as
du
-#
dt
P-
=
pf
+
div
P.
Models
of
Heterogeneous Media
3
da+
at
Here
=
-
+
(u
0)
,
v
is
the gradient operator,
a a a
3a
ax,
ax,
ax,
j=1
a
xJ
v=
(
-
9-
,
->,(div
P>i
=
c
-
Pij,
i
=
1,
2,
3.
Let
us
dwell on the
angular
momentum equations. The value
M
=
1
p[;
x
t]dQ
-+
a
is
referred to as the angular mome$um (kinetic momentum) of the
volume
56
of the continuum; here
r
is
the radius-vector of a
point of the -&on
Q
with respect to some motionless centre
0.
The velocity
u
of
an
arbitrary point of the volume may be presen-
ted in the
form
of
the
8um
+-3
-3
u
=
u,
'+
u,
+
where-
u,
is
the velocity
of
the centre of
mass
of
the volume
Q
,
and
U
is
the velocity of the considered point with respect to the
centre of mass. Respectively,
r
=
r,
+
R
.
It
means that the
ki-
netic momentum
of
the volume
Q
with respect to the point
0
is
equal to the
sum
of angular momentum of the material particle
placed
in
the centre of
mass
and
with
the
mass
which
is
equal to
the mass
mof
the volume Qwith respect to the centre of mas8,i.e..
-3+
+
+++
-3-
+-B
M
=
[r,
uo]m
+
1
P[R
U]~Q
=
M,
+
M,.
8
Let
us
consider now an infinitely small volume+
Q.
Then the momen-
tum
MI
may be neglected
in
comparison with
M,.
Indeed, let,
e.g.,
Q
be the sphere of the radius
R
which
is
rotating with the
angular velocity yound_tthe axis passing through the centre of
the sphere
52.
Then
M,
=
Iw
=
m12w
,
where
inertia,
and
1
is
the radius
of
inertia
Q
relative to the axis
of
rotation. Since
1
is
of the order
R
,
and
m
=
1
p
d
Q
=
R3
,
I
=
ml2
=
R5
whereas
M,
is
a
value
of
tha_
order
R3.
Hepe, in
the case
of
the-final angular velocities
w
the moment
M,
is
small against
M,
and we may assume that
++
+-t
Mo
=
M
=
1
p[r
x
u]dQ.
Q
+
+
I
is
the moment
of
+
However,
if
the angular velocity+
value
1~11'
ic~
1
tude. In
this
case, for
a
small
volume
of
Q,
the total angular
momentum can be written
in
the
form
is
sufficiently large
for
the
then
30
and
MI
are of the same order of magni-
4
Chapter
1
where is the density
of
distribution
of
the
so
called intrinsic
or internal angular momentum. Introducing intrinsic angular mornen-
tums for consideration, it is essential to assume that there exist
distributed volumetric and surface pairs-creating these momentums.
Let
us
denote their densities as
<
and Q,
,
respectively.
The
equation
of
kinetic momentum may
be
written in integrated
form
as
r,
a;
1
d6
-D+
-D
dQ
+
I
p
-
d
Q
=
I
p([r
x
f]
+
q>dQ
+
a
"LrX
ZJ
Q
dt
D
It
means that the rate of variation
of
angular momentum
of
the
volume
of
continuum is equal to the sum
of
the moments of volume
and surface forces acting
on
51,
and the moments
of
the distribu-
ted volume and surface pairs generated by material objects that
a,re external
in
relation to
Q
.
Analogous to the stress vectzr
P,
,
th,e density
of
the surface pairs
<n
=
(Q
n)
,
where
Q
is some tensor of rank two. Using the
equality
may be presented as
Q,
=
-t+
3
a
+
3
-,
a+
I
[r
Pnldr
=
C
I
-
[:
x
P,]dQ
=
C
I
[r
x
-
P,]d
Q
+
ax
I
r
,=I
a
ax,
i=1
Q
we shall rewrite equation
(1.7)
in the following way:
-,
du
+
1
d2
Jpb
'(z
-f--divP)
a
P
=I
{G+divQ+
C
3
[
"x-]
a3
P,,}dQ.
Q
i,j=l
ax, ax,
Hence, by virtue
of
(1.61,
we obtain, for continuous motions,
the equality
a%
+
9
=
pq
+
div
Q
+
C
P-
dt
i,j=1
ax,
ax,
(1.8)
Models
of
Heterogeneous
Media
5
which
is
the differential form of the
law
of conservation of
angular momentum. In the simplest case, when%=
3
5
0,
equation
(1.8)
speoializes to the equality
E
0
i,
c'
j=1
p$$lPij=o.
(1.9)
It
is
not difficult to check that these relations mean that the
stress tensor
is
synnnetrical
:
-
pij
-
pji
'
(1.10)
+
Therefore,
in
the classical case, whenn=
q
=
0,
Q
=
0,
the
equation of conservation of kinetic moment
is
reduced to the fol-
lowing consequence: the stress tensor
is
symmetrical. In the
general case, formula
(1.8)
is
a
supplementary differential equa-
tion. This equation
is
used in some models of mechanics
of
continua possessing some intrinsic degrees of freedom, e.g., in
the models of granular media.
As
a rule, we use here the momenta1
equation in the form
of
(1.10).
Paragraph
5
of Chapter
I11
is
an
exception.
Let
us
consider now the energy equation. Let
U
denote the speci-
fic (per unit
mass)
internal energy.
In
mechanics of fluids
and
gases the notion
of
intrinsic energy characterizes mainly the
intrinsic thermal motion of the molecules of matter. The
sum
of
the internal and kinetic energies
is
known
as
the total energy. The law
of
conservation of energy
is
formulated
as
follows: the rate
of
change of the total energy of
the volume
Q
of continuum
is
equal
to
the
sum
of the energy flux
across the boundary
r,
powers of the
mass
and surface forces
applied to the given volume
a
,
and the
amount
of energy supplied
from the outside,
per
unit time,
Here
qn
is
the specific amount
of
energy delivered to the given
point of the medium from the outside. Generally, the quantity
qn
is
expressed according to the Fourier
law
where
8
is
the absolute temperature,
n
1
0
is
the thermal con-
ductivity. After the standard transformations by means of the
Gauss-Ostrogradskii formula and equations
(1.31,
(1.6)
we obtain
the differential form of the law
of
conservation of energy:
p
Q
=
P
:
D
+
div
(n
v
01,
dt
(1.12)
6
where
is
the rate
of
1
2
Dij
=
-
(
Chapter
1
strain tensor,
au,
-
+
au,
),
ax
L
the colon denotes the convolution of
(1.13)
the tensors
P
with
D
,
(1.14)
Thus we have obtained equations (1.31, (1.61, (1.10) and (1.12)
which mathematically describe the laws
of
conservation mentioned
in
the beginning
of
the paragraph. This set
of
equations
is
not
closed, therefore
it
is
necessary to introduce some supplementary
relationships, out
of
which the laws
of
thermodynamics should be
mentioned
in
the
first
place. These laws are formulated, firstly,
in the
form
of
the fundamental thermodynamic identity
Qds
=
dU
+
pd(l/p).
(1.15)
where
8
is
the specific entropy,
p
ie the pressure. The second
law
of thermodynamics says that for any individual (made up
of
one and
the same material particles) thermally isolated volume
its
total
entropy does not decrease. This requirement
is
expressed in the
form
of
conditions stipulating the form of the law
of
the stressed
state of medium. The stressed state
of
the continuum
is
characte-
rized by the tensor
P
,
while the law
of
state
of
stress describes
the dependence
of
the occurrhg stresses on the deformation and
the rates
of
strain. Stokes'
law
of
state, according to which
P
=
(-
p
+
Adivu)I
+
2
pD.
(1.16)
is
generally accepted for liquids and gases.
Here
h
and
p
are the coefficients
of
shear and dynamic viscosity,
respectively, and
I
is
the
unit
tensor. For the continua aatis-
fying Stokes' law (1.16) the second law
of
thermodynamics denotes
that
and most commonly
it
is
assumed that
Finally,
it
is
accepted in thermodynamics that, out
of
the five
parameters
(p,
U,
8,
p
and
s),
only two can be independent,
therefore
in
the general case two supplementary relationshi
s
that
are called the equations of state can be added to equality 81.15).
Equations
of
state may be different for various liquids and gases.
If
the equations
of
state relate the magnitudes that appear in
(1.15) under the differential signs, for example,
p
10,
3h
+
2p
2
0,
(1.17)
A
=
-
2/3
p.
u
=
U(e,
PI,
(1.18)
then from (1.15) two more equations follow
au
22
e=-,
p=p
as
aP
and then we have a closed system.
If
the equation
of
state
is
of the form, say,
(1.19)
Models
of
Heterogeneous Media
7
P
=
P(P,
el,
(1.20)
then
a
supplementary equation
is
required,
for
example,
u
=
u(P,
Q>,
(1.21
1
the requirement being that these two relations must be in agreement
with (1.151, i.e.,
(1.22)
Thus,
a closed model for the descri tion
of
motion
of
liquids and
ases can be formed from equations 71.31, (1.61, (1.121, (1.161,
fl.18). (l.lg), to which the expressions
for
the coefficients
A,
p
and
H
must be added. Generally, these coefficients are believed
to be constant, while
in
the general case they are determined ex-
perimentally
as
the functions
of
independent thermodynamic para-
meters.
Now let
us
dwell in more detail
on
some individual particular
models
of
fluid mechanics and formulate the statements
of
the
basic boundary-value problems for the corresponding sets of
equations.
2.
Model
of
viscous perfect polytropic
gas.
The
first
closed set
of
equations which
will
be examined in
Chapter
11,
is
the Navier-Stokes model
of
a perfect polytropic gas.
In
this model
it
is
assumed that equation
of
state (1.20)
is
of
the
form
p
=
RpQ,
R
=
const
>
0.
(1.23)
Gases that satisfy Clapeyron's e uation (1.23) are called perfect
(ideal). Besides, equation (1.217
will
be taken in the form
U
=
cvo
,
cv
=
const
>
0.
(1.24)
These gases are called polytropic.
Finally, let us assume that the coefficients
of
viscosity and
thermal conductivity are also constant:
p
=
const
>
0,
n
=
const
>
0,
h
=
-
z/3
p.
Then the complete set
of
equations can be written in the following
form:
ap
3
-
+
div
(pu)
=
0,
at
-1
-3
-3
+pp=pAu+-pv(divu)
+pf,
3
(1.25)