Navier-Stokes Equations Paradoxes
ALEXANDR KOZACHOK
Kiev
UKRAINE
[email protected]
http: // www.continuum-paradoxes.narod.ru
Abstract:-In the Navier-Stokes equations, as well as in theory of elasticity, it is possible to eliminate Pressure. For this
purpose it is necessary to take advantage of analogies of an output of the equations of classical theory of elasticity and
hydromechanics. In that case second coefficient (there is hidden in expression for pressure) and the well known
problem of "second viscosity" disappears. Thus the traditional defining relation between pressure and density appears
redundant. In that case the equations of ideally viscous, compressible fluid under the shape coincide with the
equations of theory of elasticity. The defining relation between pressure and density can be used only for model of a
visco-elastic fluid. However, such model has serious limitations.
Key words: -second viscosity, bulk viscosity, Navier-Stokes equations, elastic pressure.
1. Introduction
p  

div u  2uz  uz .
z 3 z
For transition from the equations (1) to the teared
shape (1,a) it is necessary to mean p  p and accept
2
    . This rather an essential singularity is
3
bound to such conjecture. At an output (1) some
nonconventional relationship is accepted in attention
[4, p. 383; 5, p. 71]. This relationship actually implies
presence of two components of pressures: p and
1
p    div u . Therefore
3
(2)
xx   yy  zz  3 p   div u ,   3  2 .
Classical
dynamic
equation
of
viscous
compressible fluid (the equation of the Navier-Stokes)
and in educational [1, p. 174; 2, p. 156], and in the
scientific [3, p. 44, etc.] literature are usually recorded
in the compact vector shape
Fz 
F  grad p  (  )grad(div u )  2u  u , (1)
where F -vector of unit mass force, which sometimes
[3, etc.] skip; p -pressure, which mathematical
interpretation frequently [1, 3, etc.] do not uncover; u
- velocity vector; u - speed-up vector;  - density;
 2 - Laplacian;  - viscosity (in a nomenclature of
others
authors-coefficient of viscosity);  coefficient or in a nomenclature [3, p. 45] second
viscosity coefficient.
Some authors, for example, [4, p. 387] and [6, p.
66], execute a detailed output and record dynamic
equation of compressible fluid in scalar shape
(accessible to the analysis).
In this paper an object in view of the critical
analysis at an output of dynamic equation of
compressible fluid. Therefore we shall record these
equations in the scalar shape by analogy with [4].
However, at an entry it is used others a label. Besides
instead of kinematics viscosity  , as well as in [1], we
shall execute substitution     . In that case the
equations, actually borrowed from [4], becomes
p  
Fx  
div u  2ux  ux ,
x 3 x
p  
(1,a)
Fy  
div u  2u y  u y ,
y 3 y
In the equations (1) magnitude p should be implied
as a component of an complete pressure p . Any of
the quoted authors for some reason does not mention
it. And only in [5, p. 72], it is specified necessity of
refinement of physical sense p .
At an output (1,a) by authors [4] too it is used (2).
Then accept-   0 . As a result the second component
of pressure p is skipped. Therefore a relationship
between components of direct stresses and pressure
(as a whole) becomes [4, p. 384-385]
 xx   yy   zz  3 p  3 p.
(2,a)
For closure of system and (1), and (1,a) the continuity
equation [4, p. 387] is used

(3)
 div(u )  0 ,
t
and also, on analogies to ideal fluid, some (usually
linear) relationship between pressure and density is
used too [3, p. 45]
1
(4)
p  () p  () .
Different way, under condition of (2,a), states an
output of the Navier-Stokes equations the author [6, p.
65-67]. In [6] the final aspect of the equations, in more
teared shape in comparison with (1,a), are recorded
also. The author [6] too uses closing relationship by
analogy with (4). In the same teared shape the NavierStokes equations in [7, p. 803] are recorded. At an
entry of these equations a relationship (2) are used.
Under conditions of   0 , p  p the author [7]
actually comes to (2,a). The author [7] terms

magnitude
as coefficient of the second (bulk)
3
viscosity. As a closing relationship, as well as in [6],
the linear dependence of type (4) is accepted.
Defining relation for direct stresses of viscous
compressible fluid record as well as in theory of
elasticity [6, p. 64-65; 8, p. 144]. They look like a
linear dependence between components of deviators of
stresses and velocities of strains
1
ii  cр  2(ii  ср ), ср  ( xx   yy   zz )
3
1
.
(5)
ср  ( xx   yy   zz ) .
3
3. Problem Solution
At an output of the classical equations both in
theory of elasticity, and in hydromechanics the same
unclosed equations of motion (in stresses) are used.
Thus, defining relation represent linear dependences
between deviators of stresses and strains (an elastic
medium) or of velocities strains (ideally viscous fluid).
These analogies are well-known. However, the final
equations of hydromechanics essentially differ from
the equations of theory of elasticity. For viscous
compressible fluid it is for some reason used also an
additional closing relationship between density  and
pressure p . And p goes into equations of motion
explicitly.
3.1. Closed-loop system of
equations of ideally viscous fluid
dynamic
In view of stated in items 1 and 2 reasons, the
equation (1,a) it is necessary to convert. As well as in
theory of elasticity, pressure p it is possible to
eliminate. In that case gained equations
2
(6)
F  (  )grad(div u )  2u  u ,  
1  2
are recorded by analogy to equations of classical
theory of elasticity after substitution in the left-hand
part of migrations vectors by velocities vectors. It is
necessary to consider, however, possible objection in
occasion of application of expression for
coefficient  . Experimental definition of Poisson
ratio analog  , going into this expression, in lowviscous fluids is inconveniently. In some cases  it is
possible to define only by indirect methods. Therefore
in such cases, obviously, it is necessary to take
advantage of expression for a bulk viscosity. This
expression can be gained by analogy to expression for
a bulk coefficient of elasticity
п
2(1  )
2(1  )
, (7)
ср 
о  о 

3(1  2)
3(1  2) 3(1  2)
where  п - direct viscosity.
As a result of simple transformations from (7), we
shall gain
2. Problem Formulation
Relationships (5) mean the following. The totals of
normal components of tensors in (5) do not depend on
orientation of coordinates axes [4, p. 378; 9, p. 37].
Hence, pressure p  ср in equations of motion (1,a)
we can express through components of direct stresses
according to (2,a). At execution of the same
substitution in (5) we shall gain non-uniform algebraic
system from three equations with three unknown
magnitudes  xx ,  yy ,  zz . After the solution of this
system we, probably, should gain defining relation
both for  ii , and for  ср in the form of functional
connections ii (ii ) . However, addition of three first
equations of system (5) leads to identity 0  0. Trying
to solve system with use Cramer rule [10, p. 15]
appears unsuccessful too. All radicals are gained in the
form of indefiniteness 0 0 . Was possibly, the
following point of view was generated: for closure of
system (1,a), apart from a continuity equation, the
additional relationship is required [2, p. 156-157; 6,
стр.67; 7, p. 804, 811; 9, p. 45-46, etc.]. This
relationship is usually stated in the form of (4). We
shall try to prove the following: the Navier-Stokes
equations for ideally viscous compressible fluid, i.e.
(1,a), together with a continuity equation, can be
converted to a closed-loop system without an
additional relationship.
3о  2
.
(8)
2(3о  )
For poorly compressible fluids, as well as in
classical theory of elasticity, it is possible to accept
  const . Then the continuity equation can be
disregarded. It will allow to simplify searching of the
solution. Instead of four traditional equations (driving
and continuity) it is possible to use (quite correctly)
only three of the same type equations of motion
r
r
r
r&
m
.
(6,a)
r F + (mо + )grad(div u&) + mС 2u&= r u&
3

2
After the solution (6,a) it is possible to consider
a case  о   , div u  0 . It will allow to gain
solution for absolutely incompressible liquid.
ii  2(ii 
In view of (12), relationships (9) and (10) become
3
ii   p  2(ii 
ср ) ,
1  2
2(1  )
 p  ср   p 
ср   p  o о , (13)
1  2
2(1  )
,
о 
3(1  2)
where  о - analog of bulk viscosity, according to (7).
For shearing stresses, as well as in [4, p. 384],
obviously, it is necessary to accept defining relation in
the form
(14)
ij  ij  ij ij  0 .
3.2. Closed-loop dynamic equation of
viscous, compressible, elastic fluids
Dynamic equations of ideally viscous compressible
fluid (6,a) (hence and (1,a)) do not consider elastic
properties at comprehensive
squeezing of real
mediums (for example, gases). Therefore for an output
of equations of motion in view of elastic properties of
a fluid it is possible to take advantage with approach
according to [4, p. 379; 5, p. 71]. This approach,
however, has not been realized by authors [4,5]. As a
result the physical sense of coefficient  ( in[5]) has
remained obscure. To authors [5, p. 73] only managed
to be displayed, that the second viscosity  is positive
magnitude. However, in this motive there are also
other authoritative judgements [11, etc.].
Let's guess, as well as authors [4,5], that
components of direct stresses viscous, compressible,
elastic fluid it is possible to present in the form of the
total of components of two tensors: a focus globe
tensor and symmetrical tensor of the second rank.
Therefore
ii   p  ii , (i  1, 2,3 или x, y, z ) ,
(9)
where
p – elastic component;  ii - viscous
component.
Let's guess also, as well as for the Voight model
that ii  ii  ii . In the further on detailed
calculations we shall try to track reasonableness of an
output of the equations (1) of the elastic, compressible
fluids.
Under the conventional conjecture the elastic
pressure p does not depend on a direction. Therefore,
p can be considered as some hypothetical,
comprehensive pressure. At lack of the viscosity p is
pressure in perfect fluid. If to combine all three
equations (9), we shall gain
 xx   yy   zz  3 p   xx   yy   zz 
(10)
 3( p  p )  3 p  3ср .
Let's take advantage by expression of the bulk
viscosity  о in (13). Then the coefficient  can be
expressed through  and  о according to (8). As a
result, the dynamic equations of viscous,
compressible, elastic fluids are become
r
r
r
r&
m
. (15)
r F - grad %
p + (mо + )grad(div u&) + mС 2u&= r u&
3
r
In view of expression mo grad(div u&) = - p , the
equation (15) can be recorded in the form of (1,a),
meaning, that p  p  p .
The equations (15) formally coincide with the
equations in [5, p. 73]. They differ from (1) only a
possibility of definition of physical sense for
coefficient at the third term, i.e.
    о 

2
2
   о    о     .
3
3
3
(16)
At definition of bulk viscosity for elastic,
compressible fluids from common stresses  ср it is
necessary to subtract an elastic component. Therefore
о 
ср
о

ср  p
о

p p
,
о
(17)
where ср  ср  p , о  o  o .
Now by analogy with (5) we shall record traditional
relationships between components of deviators of
stresses and velocities of strains for a viscous
component
ii  cр  2(ii  ср ), ср   p .
(11)
Then, by analogy of defining relation of
theory, we shall record expressions
and ср   p
3
2(1  )
ср )  p  ср 
ср . (12)
1  2
1  2
For p , without taking into account of mentioned
conflict, probably, it is possible to record
expression so:
ср  Eo  o
p   Eo o   o 
,
(18)
o
where Eo - analog of a bulk coefficient of elasticity for
any nonconventional, elastic model with shear
modulus equal to null.
Experimental definition of the module Eo is
hampered, as
elasticity
for  ii
3
 o for elastic, compressible fluids. However, it is
necessary to consider the following. The requirement
o  0 and a defining relation (4) are means inevitable
transition to traditional perfect fluid. And it is really,
according to last relationship (13), we should to accept
a requirement   0 also.
p ср
,
(19)

o о
i.e. it is necessary to eliminate a viscous component
 ср from common stresses  ср . Therefore, taking into
Eo  
consideration, that
dV  dV0
dm
dm
, dV 
о 
, dV0 
,
dV

0
let's record the first relationship (18) in the form
p  Eo
  0 Eo

(  0 ) .
0
0
(20)
4. Conclusion
For an output of the classical equations of theory of
elasticity and fluid dynamics the same unclosed
equations of motion in stresses are used. Absolutely
equal linear relationships between deviators of stresses
and strains or velocities of strains are applied to their
closure. Therefore, apparently, the final equations
should coincide under the shape. In both cases together
with a continuity equation they should give a closedloop system. However, hydrodynamical equations for
some reason are considered unclosed. They contain in
the explicit shape the additional unknown magnitude pressure p . Therefore for closure of dynamic equation
of ideally viscous fluid the additional relationship
p  () is used. The mentioned difference, as it has
appeared, is bound with following habit. Substitution
of defining relation in equations of motion really leads
to emersion of magnitude p in the final equations. To
eliminate p with direct methods for some reason it is
not possible. Possibly, for this reason in theory of
elasticity the method of superposition of loading is
used. This method allows to eliminate pressure and to
determine the latent constant. In fluid dynamics with
this constant the known problem of "second viscosity"
is linked ". Therefore has appeared false necessity of
an additional closing relationship. Dynamic equation
of ideally viscous, compressible fluid for one and half
centuries have remained in the "not completed" state.
Thus, the conventional point of view, (for example,
[6,7,9, etc.]), about necessity of an additional defining
relation for closure of dynamic equation of ideally
viscous, compressible fluid (Navier-Stokes equations)
- actually it is erroneous. The equations of motion in
stresses, a continuity equation and defining relation
between deviators of stress tensors and velocities of
strains (5) represent a closed-loop system. In case of
badly compressible fluids (   0  const ) the
continuity equation (3), as well as in classical theory
of elasticity, can be skipped. Therefore instead of four
equations it is possible to use, quite correctly, only
three of the same type equations of motion (9). After
the solution of such system the passage to the limit at
о  , div u  0 will lead to solution for
incompressible liquid. Such transformation of motion
equations of incompressible liquid allows to use
numerous effects of classical theory of elasticity for
consideration of inertialess fluxions.
(21)
According to [7, p. 128; 9, p. 47], a sound velocity in
the strained mediums а  p  . Therefore
magnitude а  Eo o with an adequate accuracy it
is possible to assume as the sound velocity. After
derivation (21) this effect is gained. In that case
  0
p
.
(22)
 a 2
o
o
In view of (22), expression for the bulk viscosity
 o becomes
Eo  
o 
ср  a 2 (  0 )
.
(23)
o
Thus, dynamic equation of elastic compressible
fluids, according to Voight model, with zero elastic
shearing stresses, at stationary values  and  o ,
finally become
r
r
r
r&
m
r F - grad(a 2 (r - r 0 ) + (mо + ) div u&) + mС 2u&= r u&
(24)
3
Together with continuity equation the equations
(24) form a closed-loop system. In such aspect these
equations actually imply from resulted in [5, p. 73], if
second coefficient of viscosity  in [5] to add sense of
bulk viscosity  o . Authors [5], apparently, did not
know it. Otherwise they would not admit erroneous
transition to the equations for incompressible liquid.
r
They have ignored the term z div u& under next
r
condition:
div u&= 0 . However, under such
requirement, a bulk viscosity is    .
For badly compressible fluids is   0 . Let,
besides, grad(r - r 0 ) is small value in comparison
with other terms of the equations (24). Then the first
term in brackets and a continuity equation it is possible
to skip and transfer to the equations of ideally viscous,
compressible fluid (9,a).
2
From (16) implies, that coefficients (    ) in
3
(1), really, it is possible to interpret as a bulk viscosity
4
consider cases of one-dimensional and twodimensional loadings. Therefore a problem of
build-up of more correct equations of viscous, elastic
compressible fluids, especially for the analysis of its
behavior at considerable shift strains, is topical.
Pressure p in the equations (1) should be fathomed
as an elastic component of complete pressure p , from
which the viscous component p is eliminated. Thus
2
the requirement     0 together with a
3
requirement p  () means inevitable transition to
2
perfect fluid. In this case expression     o
3
represents a bulk viscosity, and consequently it is
necessary to accept   0 (hence,   0 ). Otherwise,
if   0 , it is necessary to refuse a requirement
p  () and to accept p  p . Under such
requirement a pressure component p caused by
viscous forces too, but not by elastic forces. However
such procedure means transition to ideal viscous fluid
according to (6,a). It is necessary to note, that for ideal
viscous fluid, when transition from (1) to (2,a) is valid,
2
the requirement     0 in the equations (1) is
3
2
observed always. Therefore, magnitude   
3
cannot be considered as a bulk viscosity.
Hydrodynamical equations in the form of (24) is
admissible for the exposition of dynamics only any
nonconventional visco-elastic fluid (truncated Voight
model). This is possible at small shear strains, when
elastic shearing stresses are close to zero. Therefore
because of such limitations this model must be revised.
Thus, often used [7, p. 802, etc.] conjecture
  0 ,  o  0 is erroneous. These magnitudes, as
well as modules G and Eo in theory of elasticity, are
linked among themselves by a linear dependence of
type (16). Besides for incompressible liquid, for
example, o   . Therefore numerous researches of
dynamics of viscous gas, especially in university
courses [7, p. 801-892], requires radical revising.
Researches with use subzero second viscosity [11,
etc.], obviously, require revising also. In view of
enough argued demonstrations in [1, p. 257; 5, p. 73]
magnitude  o in the equations (1) at independent , 
can be only positive. Such inevitable corollary implies
from the third formula (13), if   0.
In summary it is necessary to mark the following.
Most explicitly the problem of physical sense of
second viscosity and pressure p is discussed in the
monograph [12, p. 18, 48, 70.] In view of presence of
assemblage of judgements and hydrodynamic
paradoxes in this motive, this problem is referred to
discharge of the open problems. Whether the author
has solved completely this problem? Most likely, that
has not solved. There is, for example, a conflict with
the conjecture that p does not depend on a direction.
At such conjecture, for example, it is impossible to
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