Influence
dispersion and cross flux in continuous
mechanics
Evelina V. Prozorova
Mathematics & Mechanics Faculty, St. Peterburg State University
University av. 28 , Peterhof, 198504, Russia
prozorova@niimm.spbu.ru
Abstract
Present investigation develops theory of continuous mechanics for formulation general laws of
equilibrium as laws of equilibrium angular momentum. Besides influence of cross flux on
description of continuous mechanics is discussed. Usually the equilibrium conditions for forces
are postulated in spite of that it is special case interpretation. The new model for solution the
problem of flow rarefied gas near bodies is suggested. Layer closed to surface is order of some
radius of interaction molecules. The
modified Navie-Stokes for remainder can be used and
conjugate conditions at surface can be written without the Knudsen layer. You will be able to count
friction and heat flow to the surface by solution the Boltzmann equation without collision integral in
thin layer and by solution the Navie-Stokes equations with addition new terms.
The ChapmanEnskog distribution function as boundary condition for external boundary of thin layer is used with
macroscopic parameters are determined from the Navie-Stokes equations. The inertial N. A.
Kolmogorov region in boundary layer is analyzed. . For normal shock waves angular momentum
does not work but the self- diffusion give additional diffusion. This effect is investigated for
fluctuations of shock waves for number Mach M≈1
1. Introduction
Earlier studies were focused on equations which includes the influence of angular momentum in an elementary
volume and cross flux of physical values [1] through sides of an elementary volume. Some simple examples were
solve for the boundary layer problem, shock waves for number Mach M≈1 and for some elastic tasks [2-5 ]. Present
investigation attempts theory of continuous mechanics as laws of equilibrium for angular momentum. Usually the
equilibrium conditions for forces are postulated in spite of that it is special case of more general conditions of
equilibrium for angular momentum.
Any movement of an elementary volume of liquid today can be considered as a result of the following motion:
quasisolid motion that is translation with selecting pole, rotating motion around this pole and deformation motion.
This theorem was proved by Gelmholtz H. Prandtl L. formulated conception of hardplastic body as the theory of
ideal plasticity. Usually we do not take into account twist velocity into elementary volume. In our opinion many
phenomenons can be explain by modified theory. Instability of shock waves for Mach number Μ≈1 is well known.
For other M experimental data shows instability of shock waves for plasma and reacting gases with endothermic
reactions. The explanation of these effects we have not at present. For classic theory continuity of medium is
breaking. Some experimental facts tell us about importance of gradients of physical values (density, linear
momentum, energy). For classic theory continuity of medium is breaking. At numerical investigation we have
different values of strain on opposite sides of elementary volume (rectangle network) but usually we get average
value and continuity breaks too. This paper reports on influence angular momentum variation in an elementary
volume and cross flux through the sides of an elementary volume on continuous equations. At present used of
continuous mechanics equations were received, as we see it, from insufficiently full physical laws and, hence,
these laws would be revised. Present investigation attempts to formulate the equilibrium conditions as equilibrium
conditions of angular momentum but usually the laws of equilibrium are postulated as equilibrium of force. In
the classical Newton mechanics we have four conservation laws: masses, liner momentum, energy, angular
momentum. In continuous mechanics we use only three first laws. Last law degenerates in symmetric tensor P.
The kinetic theory does not save the situation. The law of angular momentum does not implement in the Boltzmann
equation. The numerical solution of the Boltzmann equation does not contain this mistake if it was solved without
employment of the conservation laws. In this paper the relationships between the Boltzmann equation and
hydrodynamics are discussed. The modified equations of hydrodynamic follow from the modified Boltzmann
equation [1,2]. The modified laws of conservation were received for the particles without structure. The angular
momentum does not contain the new dimension constants. So the similar solution for classic equations are similar
for modified equations. For elasticity theory equations did not change but were proposed another interpretation and
added the angular momentum equation to the classic equations. Taking into account the angular momentum law
nonsymmetrical stress tensor is received. The method for calculation of nonsymmetrical part is suggested.
Now for consideration of angular momentum the theory of brothers E.,Cossssrat , F. Cosserat and their
modifications are used. In this theory new constants was added. Besides usually convective flows the cross flux
performed. As example for density was take into consideration only the flows of density across the surface of an
elementary volume. Conclusion of the equation with the more full physical idea as classical was done by S.V.
Vallander in 1951 year ( for the cross effects ) [3 ]. The self- diffusion, energy and linear momentum were
considered for the perfect continuous gas in supposition about internal energy E E = cv Т. Here cv - the
coefficient of thermal capacity under constant volume. For great gradients of physical values colliding molecules
are belonging to different distribution functions can have different macroparameters ( density, temperature,
velocity) . The influence variation of the distribution function over times of the mean time  between collisions of
particles was suggested B.V.Alexeev [4]. We suggest that the second term R. Taylor series in collision integral
necessary to take into account. It received after expansion the distribution functions in collision integral over space
coordinates near the point x. Differentiating the equilibrium distribution function of collision integral the
structure of formulas by S.V.Vallander can be received
for interacting molecules with hard potential. In other
case it is necessary to take averaged cross section. In general case ( nonstationary ) the collision integral can be
written in form suggested by B. Aleexeev. Constants are defined by the species of concrete potential. Always in
spite of that concrete time for two molecules we will be exploit average time which is proportional to inverse value
of frequency. For numerical solution can be used the average values of collision integral for concrete velocity.
Classical theory predicts the existence of the second viscosity but usually we assumed that it should be take into
account for molecules with inner degree of free or for dense gas. The modified kinetic theory gives the second
viscosity. The modified Boltzmann and Navier-Stokes equations are needed boundary conditions.
Another
problem for the solving of the Boltzmann equation is the asymptotical methods. It is essential that selecting
the local equilibrium distribution function ƒ0 as the basis in solution of the Boltzmann equation by the ChapmanEnskog method exploits
macroscopic parameters in f0 from the Euler equations [ 5 ]. Macroscopic
parameters are determined the Chapman-Enskog distribution function leads to the Euler equations parameters
and tensor P is symmetric. Formally in that way we have values (density, linear moment, energy) with mistake of
the first order. This fact was noted by Hilbert without further use and correction. So for the equilibrium in
collision integral [ 5,6 ]
3/ 2
 m 
f  t , x,    f 0  t , x,    n 

 2kT 
and for nonequilibrium distribution


2
 m 2
exp 
c  , c 2  c12  c22  c32     u 
2
kT



pij m
q m  mc 2  
f  f0 1 
ci c j  i ci 1 

pkT  5kT  
 2 pkT
has the same macroparameters in f0. The nonequilibrium distribution function is such that integral of it contains
only the integral of equilibrium function f0 and gives containing in f0 macroscopic parameters. The remaining
term gives null. Besides it is essentially for the value of viscosity. So it is necessary to do iteration all values
(density, linear moment, energy). It is possible for equation with angular momentum. In classic thus tensor P is
entering as the value p in the Euler equations and as symmetric tensor in the Navier-Stokes equations.
Consequently the theory value p for the Navie-Stokes equations is not equal to value p for the macroscopic
Euler equations.
The order of the new equations (for the density and for the linear momentum, energy) is more than in classical
case. If we deal with continual medium the external boundary condition for boundary layer can be determined
as the value of rotor velocity or as of value normal velocity. That is for the vertical velocity. For the longitudinal
velocity it is need to put friction. In turbulence layer we need to set a friction too [7,8]. For the rarefied gas the
boundary conditions would be included the value gas flow besides the classical boundary conditions [1,2 ]. At
present the theory of description turbulence does not clear in spite of existence a large number theories.
Usually we consider the following theory for turbulence streams: Direct Numerical Simulation, Large Eddy
Simulation, Reynols-Averaged Navie-Stokes and some others. In the previous papers we investigated numerically
and analytical method the modified Blasius problem, the Falkner-Skan problem, the flow near infinite plate. For
the last case the motionless thin film was detected and the Prandtl formula was received. It has been shown that
longitudinal velocity profile for these problems has fluctuations near upper boundary. For the boundary layer
turbulence is beginning here. Essentially that the tasks about the flow in tubes, flow among two infinite plates are
formulated in our textbook as hydrotechnical tasks (middle change is constant). Infinite force is need to ensure
this force. The problem of the inertial N. A. Kolmogorov region in boundary layer is analyzed as linear velocity
can be only near surface but there the velocity is small; so this region far from surface. Our model tell us about
logarithm profile for boundary layer. Gas-surface interaction plays an essential role in processes connected with
atmosphere re-entry vehicles [5]. It is necessary to know aerodynamic characteristics. These conditions are known
badly for rarefied gas and for turbulence streams. To solute the Boltzmann equation is more difficult than NavierStokes equations. So better to solute Navier-Stokes equations. Usually for the classical case near the surface the
Knudsen layer is considered [9]. This layer has the length of order free path. M. Lunc, J. Luboncki, V.C.Liu ,
R.G. Patterson, W. Bule, F.O.Goodman, H.Y.Wachman, R G. Barantsev, Yu. A.Ryzhev, G.V. Dubrovskii and the
others investigated the interaction molecules with surface near freely-molecular simulations. Another type of
works are investigation of nanostructures, but we do not discuss this problem. The analytical study of weak
shock waves with cross flows
across the surface of an elementary volume has been investigated for gas
without structure [10]. For structure gas with threshold energy of destruction was received possibility for movement
it in reverse direction if Mach number is near the unit. The aim of this paper is to find the reason for appearing the
second viscosity for cross flow at large gradients of the physical values, to investigate the self- diffusion which
gives additional diffusion for normal shock waves where angular momentum does not work but the self- diffusion
give additional diffusion. The convective flow has the same value as diffusion flow for the Mach number M≈1. It
will be received in our work. The boundary conditions for rarefied gas near the surface make more accurate than
early. Boltzmann equation and hydrodynamics are discussed. The modified equations of hydrodynamic follow
from the modified Boltzmann equation [1,2]. The new equations with inclusion of angular momentum and cross
flux have only one stretch parameter- the Reynolds number, Prandtl number. So we need not do another methodics
to receive equation for boundary layer.
2. Equations
Our results can be summarized as follows for all continuous mechanics : in phenomenon theory we have four
equilibrium equations but if we choose equilibrium of force the three equilibrium equations and symmetric tensor
are received; so our interpretation bases on traditional theory. The degree of asymmetric stress tensor we can
received from momentum equation ( in projections ) [11]
 yz  zz    xy  yy  zy 
 
y  xz 



 z
   zy   yz  0 ,
y
z   x
y
z 
 x
 yz  zz
 
x  xz 


x

y
z

   xx  yx  zx 


 z
   zx   xz  0

x

y

z
 

  xy  yy  zy 
  xx  yx  zx 
x




 y
   yx   xy  0
y
z 
y
z 
 x
 x
The equations are classic for gas, fluid and solid. We have another interpretation. The angular momentum does not
contain the new dimension constants and nonstationary contains in density and velocity but not have the
nonstationary term for the angular momentum equation. For gas we received the modified Navier-Stokes
equations from the modified Boltzmann equation
 ui
  ui 
 xi
  0.


t xi xi  xi 
P  X
ui
 
 ui u j  Pij  xi ij   i   0.

t
xi 
xi  m
 3
1  
  RT  u 2  
t  2
2  x j

1 2

3
u j  2 RT  2 u   uk Pkj  q j  





1 2

3
u j  2 RT  2 u   uk Pkj  q j   0




We obtain the equation for angular momentum from the modified Boltzmann equation.



xi
xi x j
r
r
r

 px   p y   p z  x j
Pj  M I
x
y
z
x j
 
We should use this equation for definition of value nonsymmetric tensor in gas. Besides the second viscosity can
be received from collision integral and formulas suggested by S.V. Vallander
 f '
f ' f1 ' f f1  f ' f1 ' t ξ '
 t , x, ξ '  f1 '  t , x, ξ '1  
 x
f '

 f1  t , x, ξ ' ξ '1 1  t , x, ξ '1    f f1 
x

f

 f
t ξ  t , x, ξ   f1  t , x, ξ1   ξ1 1  t , x, ξ1  f  t , x, ξ   
x
 x

O  x 

df (0)
(0)  m  c c  1 c 2  ui 
  i j
t 0  f
ij 
dt
3
 x j
 kT 

1 T  m
ci 
2T xi  kT
 2  m
 c  5  kT


ui 
1 2  

 ci c j  3 c  ij  x xi x 

 j
i

3. Beam
Consider protozoa task about beam. Closer definition of equilibrium will be obtained if angular momentum would
take into attention.
Figure 1 Elementary beam.
dM  Qdx  mdx  Ndv  0,
dQ  qdx  0,
M 'Q  m  Nv'  0, Q 'q  0.
 M ' '  q  m'( Nv' )'.
 M ' '  q  m'(( Nv)' )'.
For this problem the solution for general case and without N’ (for N=Const) is the same. Consequently we can
choose the beam of variable cross-section. For beam on elastic foundation results distinguish. For another problem
the solutions can be different too.
4. Infinite plate
The Blasius problem was considered in [12] by numerical and analytical. Some results for infinite plate will be
formulated here. The equation for this case is
2 



d
du
d  d u
   
y
 0.
dy  dy  dy  dy 2 
The inertial N.A. Kolmogorov region in boundary layer can be decribed by this equation too. It followes
from general equation.
Boundary conditions are
u  0,
du
 w ,
y  0,
dy
u U ,
y  .

Integrating gives

du
d 2u
  y 2  Const   w .
dy
dy
There are t-time, y-coordinate, -density, Pi,j – stress tensor, u-velocity, -viscosity. index "w" is relative to
surface. From boundary condition we have Const  w . w is skin friction. Integral of the equation is
u  C ln y  w /   y  Const.
Possible variant to satisfy boundary conditions is that under the
1/ 2
 w 




we
y  , where   , v*  

v

 w 
have ln = 0. Later on diminution velocity takes place up zero, derivative can be very large but zero velocity
observes between surface and y. So layer of the rest liquid is formed. Thickness of this layer is 10 -3 cm. We have
not reliable measurements there. Probably for laminar layer there is no layer with zero velocity. Near the edge the
gradient of the velocity tends to work. It works near the rebuilding region too. Far from edge friction strives to
zero. It does not follow from the theory for semi-infinite plate that the value of the friction is finite but if we
suggest zero friction in the first integral we can get the Karman formula for the mixture length. Equality w = 0
provides u = 0 as y = 0 and u = U as y   and leads to rebuilding of the flow. The profile of the velocity
becomes more completed than near the edge. The region with w = 0 formulated the inertial layer( N.A.
Kolmogorov). This case relies to logarithm profile for boundary layer. If we construct the Falkner-Skan profile in
form of logarithm profile we will receive similar form of profile. It is interesting that asymptotic friction for halfinfinite plate has not the value for infinite plate. In my opinion we have similar situation for tubes.
5. Shock wave
The classical structure normal shock waves can be found in [13]
u  0 D  a , p  u 2  w  p0  0 D 2  p00 ,
w  
du
,
dx
p
 p 1 2
1
D2
dT
 u 
 RT
 S  uw   h0   h00 , S  k ,
 1  2
0 D
2
dx

Here ρ- density, μ-viscosity, internal energy E E = cv Т, p-pressure, u-velocity.
To investigate the weak shock wave for Mach number M = 1 is very difficult. The front of shock wave is not
stability for this regime in experiments. The cause of instability is not clearly. In my opinion the cause is in the
next. Influence of self- diffusion (only of density) leads to
d u d 1 d 
,

dx
dx  dx
du
d 1 d  dp d 4 du
u  u



0,
dx
dx  dx dx dx 3 dx


d
u2  d  d
cV T   cV T  


dx
2  dx  dx


4  du 
d dT d cV T d 
   

3  dx  dx dx dx  dx
2
 p 1 2
1
D2
dT
 u 
 S  uw   h0   h00 , S  k ,
 1  2
0 D
2
dx
Here ρ- density, μ-viscosity, internal energy E E = cv Т, p-pressure, u-velocity.
0 1 d 
0 dx
1
From R.Becker solution (for η)
0 D0
p
 RT

for M = 1,
In our case for gas without structure we have equation for η [14,15]
d D 0
u 0


 0.
dx
1
1
Consider the influence of the new terms (self-diffusion) on changing ρ,u,p,T. Transfer new terms on the right side
of equation as in the first equation. In shock wave we have growth of density. Hence after integration of x we have
larger zero increment density. So for small growth of density velocity should increased, pressure increased and for
small reduction of density we have opposite situation. For M≈1 velocity can change and shock wave can modify to
wave of compression. So we can tell about unstable shock waves for some special parameters of flows. After
decrease of density in shock wave the process go to opposite side. For pressure we have similar process. If gas is
mixture and it contains the light components we can have tongue of light
component at two side of shock wave.
Conclusions
We discuss the problems that can be appearing to consider the angular momentum variation in an elementary
volume near the surface and influence the cross flows through the sides of an elementary volume for great
gradients of the physical values. The problem of boundary layer is discussed.
The unstable shock wave for M≈1 is investigated.
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