On Some Aspects of Systems in Thermofluiddynamics
MICHAEL LAUSTER
Faculty for Economics
University of the German Armed Forces Munich
Werner-Heisenberg-Weg 39, 85577 Neubiberg
GERMANY
Abstract: This paper deals with the mathematical foundations of systems in thermofluiddynamics. The basic
ideas for the formulation of a consistent theory with a minimum of primitive elements are given. From a single
fundamental relation any information on the physical object in regard is retrieved by differentiation processes.
It is shown how and under what conditions the results of classical mechanics fit into the theory. Furthermore,
restrictions and defects of traditional formulations are shown. The advantages of a system theoretical approach
to thermofluiddynamics are explained.
Key-Words:
system theory; thermodynamics; fluiddynamics; space; time; Gibbs space; energy; nonequilibrium; dissipation
1 Space, Time, and Generic Physical
Quantities
One of the major tasks of sciences is the
compilation of data derived from empirical
experiences, to store them in a standardized way,
and to share them amongst scientists.
In the natural sciences, mathematics has proven to
be the appropriate choice for this task: a sharply
defined vocabulary and a singularly strong grammar
make it a unique language which is spoken
worldwide.1
Referring to physics and the engineering sciences
two options to formulate empirical phenomena have
arisen.
Following the evolution-generated experience of
reality physical objects are placed in a space with
length, breadth, and height.2 This space is
homogeneous and isotropic in each of its
directions.3 Objects move through the threedimensional space and their movement is
characterized by a time co-ordinate flowing
uniformly from past to future.4
The appropriate mathematical model of such a
space is a three-dimensional Euclidian space with
orthogonal axis and equal scales in each direction.
The origin as well as the angular position of the axis
may be set arbitrarily.
As for the time co-ordinate the same holds: origin
and unit length of the scale may be chosen
arbitrarily. A space comprised of three spatial and
one temporal co-ordinate will be called a parameter
space.5
In the parameter space the path of objects taken
during their movement is depicted by a continuous
curved line called trajectory. Provided a (positive)
real number for the mass is attached to the
3
1
2
Nevertheless, it should be noted that in no way a one-toone correspondence between the objects of our
empirical experiences and the mathematical structures
used to describe them may be established. This
principal difference makes the description to a certain
extend ambiguous or even erroneous.
There is strong evidence that our experience being
(only) three-dimensional directly corresponds to the
fact that three is the least number of dimensions needed
for two trajectories not to cross even when they are not
strictly parallel. This might have been useful during our
evolution, e.g. not to interfere with foes or to
successfully use weapons like stones or spears.
4
5
Undisputedly, the description given above is more or
less something like an ideal case and a lot of failings
may happen to our senses while experiencing spatial
and temporal phenomena. This will not be discussed
here but may be looked up in the respective literature,
cf. e.g. [1].
The meaning of time and the arrow of time in the
natural sciences has been discussed widely and this is
not the place to delve into a philosophical dispute. A
brief summary of this discussion and an extensive
compilation of literature on this topic can be found in
[2].
Space and time are the projection parameters used by
life forms to control their biological processes in order
to assure their survival, cf. [3].
trajectory, the mathematical model for a body is
given.6
With the Hamiltonian theory, classical mechanics of
multi-body systems found its final formulation.
Here, a set of 2s-many mutually independent
variables xi and pi, i = 1,2,...s, called the canonical
variables position and momentum for each of the smany particles in regard are used to describe the
dynamics by a set of 2s-many coupled first order
partial differential equations of the form
a conservation law results stating that the total
energy is a conserved quantity. Condition (4) shows
a strong connection between the time t and the total
energy given by H.7
Regarding the definitions from mechanics
dxi
H dpi
H
=
;
=
; i = 1, 2, ..., s
dt
pi
dt
xi
it is obvious that any information on the multi-body
system, i.e. the velocities of the particles as well as
the interacting forces may be derived from the
system describing function H by differentiation.
Taking a close look at the Hamilton mechanism, the
canonical equations of motion may be distinguished
into two kinds of relations: (1)1 is a mere definition
for the kinematic velocity while (1)2 is a theorem
stating how the momentum of a moving body may
be changed. Within the context of Hamiltonian
theory, the definition as well as the theorem hold
unconditionally.
The second option for the construction of
descriptions in the natural sciences came up when
classical mechanics was given its final structure and
has been intensely used when thermodynamics
appeared as a new discipline in physics.8 It is far
more abstract than the first one and does not
directly refer to our sensual experiences.
Physical objects are examined with regard to the
quantities they exchange with other objects of the
same or different kind. The quantities exchanged
may be either matter-like, e.g. like substances or
immaterial like e.g. linear or angular momentum. As
a rule, these quantities should be chosen in such a
way that they do not only belong to one specific
discipline but are constitutive for the phenomena
throughout physics. Quantities fulfilling this
condition like, e.g. the total energy, are called
generic physical quantities or simply generics.
Every generic physical quantity is mathematically
represented by a variable belonging to the set of
functions that may be differentiated at least twice.
The two-fold differentiability allows the
construction of stability criteria.
It goes without saying that only a finite number of
generics may be regarded. The creator of the
(1)
Equations (1) are named the canonical equations of
motion.
The canonical equations combine both options for
the description of empirical phenomena. While the
partial derivatives belong to the phase space, the
total ones stem from the parameter space.
The Hamilton function H represents the total energy
of the particles and t is the time parameter. The
evolution in time of any property of the particles
represented by a variable F may be expressed with
the aid of the Hamiltonian H:
F = Fˆ  x1, x 2 , ..., x s , p1, p2 , ..., ps 

dF
=
dt
 F dxi
F dpi 
+
=
dt

pi dt 
i = 1
i
s
  x
 F H
F H 
+


p

pi xi 
i = 1
i
i
s
=
  x
(2)
The expression on the right side of equation (2) is
identified with the famous Poisson bracket by
definition
dF
=: [F,H]
dt
(3)
Substituting the property F by H itself then with
H,H =
6
dH
=0
dt
(4)
Cf. e.g. [4]. This is the kinematic core of classical
mechanics whose task it is to calculate the trajectory of
any body, provided appropriate initial and boundary
conditions are given. In general, discontinuous motions
of macroscopic bodies are not allowed by hypothesis.
This assures the applicability of the calculus of
analysis.
dx
dt
dp
f :=
dt
v :=
7
8
(5)
Equations (1) contain some more conservation laws:
Provided that H is independent of one of the canonical
variables, e.g. xi, then by (1)2 it is evident that the
momentum pi is a conserved quantity.
Cf. [5]. These milestones in physics are connected to
the names of Sir W.R. Hamilton and J.W. Gibbs.
description has to decide which of the effects are
important for the purpose in regard and which are
not. Once these generics are identified and the
respective variables are defined, the description is
complete by definition. Depending on the resulting
number of generics an abstract space spanned by
these variables is generated which is not accessible
by our common empirical experience. It is called
phase space or Gibbs space. Spatial co-ordinates
and the time forming the parameter space do not
belong to the set of variables of the Gibbs space.
Each variable may attain values which are
represented by real numbers. For macroscopic
systems the use of a continuum for the values of a
variable is required by hypothesis. Thus, a
continuous change of the values of a variable may
be established assuring the applicability of the
calculus.
Setting all variables of a Gibbs space to a respective
value, a single point of this space is referenced. The
vector of values is called a state. The junction (in
the sense of set theory) of all possible states for an
object is called a system.
In mathematical terms a system is a relation  of all
the variables X1, X2, ..., Xn, n  , of the Gibbs
space:
  X1, X2 , ..., Xn   0 .
(6)
 is called Gibbs Fundamental Relation (GFR).
The set of variables (as well as the variables
themselves) leading to a relation homogeneous of
degree one is called extensive.9
The differentiability assures that the first partial
derivatives of a specific extensive variable with
respect to another may be calculated. This delivers
the so-called conjugate intensive variables. The
notion “intensive” here refers to such quantities that
are not additive, i.e. they do not change their values
if two identical systems are combined to a single
new one.10
9
The choice of extensive variables is one of the most
fundamental rules of Gibbs-Falkian dynamics. Once the
basic set of variables is identified, the complete
mathematical structure of the theory may be worked out
rather automatically. Any extensive variable possesses a
number of properties, the simplest of which is
additivity. In this context additivity means that the
values of two identical extensive variables may be
added if the two objects to which the variables belong
are composed to form a single object.
10
Prominent examples are the thermodynamical
temperature or pressure of a system. They do not
Objects change their state by exchanging quantities
with other objects. Such a change of state is called a
process. As a hypothesis, no discontinuous
processes occur for macroscopic objects. Therefore,
the mathematical picture of a process is a
continuous sub-set of the system forming a curved
(one-dimensional) line. Each point of that process
path, i.e. each state attained during the process, may
be assigned a certain value of a time parameter. In
other words: At any given time, the object attains a
certain state; the inverse is not true. Thus, the
process in Gibbs space is the analogue of the
trajectory in parameter space.11
Both options for the description of empirical
phenomena have to be combined to achieve a
picture as complete as possible: while in Gibbs
space the question is answered how processes run,
in parameter space it is expressed where and when
the processes take place.
2 Systems in Thermofluiddynamics
In physics, thermodynamics is a prominent example
for the application of systems theory. Here a holistic
view of the respective phenomena and the acting
objects leads to mathematical descriptions that
differ essentially from those used in other
disciplines of physics. One of the promising new
theories in this branch is the so-called Alternative
Theory of non-equilibrium processes (AT) created
by D. Straub [6]. Its primary purpose is to tackle the
problems
of
irreversible
phenomena
in
thermofluiddynamics. However, in the meanwhile it
has been extended to microscopic as well as to
electromagnetic systems (cf. [6] and [7]). Its
mathematical core has been applied to other
quantitative branches of sciences, e.g. national
economics.
The mathematical basis of the AT is the formalism
first found by Gibbs for thermostatic phenomena.
Here every information on the system in regard may
be obtained from a single function by
differentiation. Falk extended this concept by the
insight that linear and angular momentum have to
be included as members of the set of variables. This
changed Gibbs’ thermostatics to a real dynamics.
change when, e.g. two reservoirs of the same gas with
equal temperature and pressure are coupled.
11
In the language of the North-American Hopi Indians the
word for “time” is “koyaanisquatsu”. An adequate
translation would be “state of non-equilibrium tending
towards equilibrium”.
Additionally, it is supplemented by the important
conclusion of Straub that despite the usual
procedures of thermostatics and classical
thermodynamics the fundamental set of variables
has to be extensive (cf. [5]).
Four primitive, i.e. not reducible elements variable, value, state, and system – in the meaning
described above constitute Gibbs-Falkian dynamics.
The only hypothesis necessary to use this method is
the possibility to define extensive variables
mapping the generics for the respective object.
It is the first step and the major task of the user to
identify those variables that are important for the
intended description and to neglect those that are of
minor interest.
Once all necessary variables are identified,
everything works according to the recipe given by
Gibbs and Falk. A Gibbs Fundamental Relation
exists from which any information is to be retrieved
by differentiation.
To get more specific, we look at an object whose
generics are - besides its total energy E* - linear and
angular momentum, represented by the variables P
and L, body forces and momenta F and M, as well
as the traditional thermodynamic variable entropy S,
volume V, and particle number N. Then the Gibbs
Fundamental
Relation
reads

   , P, L, F, M, S, V, N  0 .
The Gibbs Fundamental Relation may be solved
with respect to any of the variables. Usually, the
total energy E* is chosen to be the dependent
variable. Thus, the so-called Gibbs function results:
 = ˆ  P, L, F, M, S, V, N .
(7)
The total derivative of equation (7), the Gibbs main
equation for the system, delivers the intensive
variables as the first partial derivatives of the total
energy E* with respect to the independent variables:
E*
E*
E*
 dP +
 dL +
 dF +
P
L
F
E*
E*
E*
E*
+
 dM +
dS +
dV +
dN =
M
S
V
N
= v  dP +   d L + r  dF +
dE* =
+   dM + T* dS - p* dV +  * dN
(8)
12
12
The negative sign with p* is a mere convention that
shows the direction of changes of the pressure when the
volume is varied.
This defines the linear and angular velocity, the
position vector, the angular position, the
thermodynamic temperature and pressure as well as
the chemical potential, respectively.
The extensivity of the set of variables leads to a
GFR that is homogeneous of degree one. For
objects from physics the respective factor of
homogeneity is the number of particles N of the
object. As a rule for all relevant processes in
thermofluiddynamics
changes
below
the
elementary-particle level do not occur. This results
in the constancy of the number of baryons and
leptons and therefore in the constancy of the mass m
of the object. Hence, for certain applications m may
be used as the factor of homogeneity instead of N.
For the Gibbs function (7) being homogeneous of
degree one, Euler’s rule for homogeneous functions
holds and the total energy E* may be expressed
explicitly:
E* = v  P +   L + r  F +   M + T* S - p* V + *N
(9)
Calculating the total differential of equation (9) and
comparing it to (8), an important conclusion may be
drawn: from the product rule of the calculus we find
that
P  dv + L  d  + F  d r + M  d  +
+ SdT* - Vdp* + Nd *  0
(10)
has to hold so that (9) and (8) are true without
contradiction.
Equation (10) is called Gibbs-Duhem relation. It
states that there is a relation between the intensive
variables of the system. In other words: linear and
angular velocity do not only depend on linear and
angular position but are influenced by the
thermodynamic variables temperature, pressure, and
chemical potential as well. Thus, the effects of
motion and thermodynamics may not be separated
in general.
3 Matter and Forces
One of the most important facts in GFD is Falk’s
observation that besides the thermodynamic
variables like entropy, volume, and number of
particles, the linear and angular momentum are
The asterisk marks the intensive variables for the
moving system while those for the hypothetical state at
rest will be given without an asterisk.
generic members in the variable list of the GFR.
This turns the method from Gibbs’ thermostatics
formulated in [5] to a real dynamical theory that
includes mechanics with all its kinematic aspects,
thermodynamics, and electrodynamics for real
matter.
The question is still to answer how classical physics
in the form of Hamilton’s mechanics and GibbsFalkian dynamics, especially in the formulation of
the AT, can be made compatible.
The first step will be a two-fold Legendre
transformation that changes the set of variables of
the total energy E*. In the variable list of the
Hamilton function H the generalized momentum
and spatial-co-ordinates appear while forces and
momenta are mere definitions within the theory.
Therefore, we transform E* according to
E := E 
* F, M
= E* - F  r - M  
(11)
and exchange forces and momenta against spatial
and angular co-ordinates.
The energy-like variable E is a residual quantity that
results, when the total energy E* is reduced by the
energy forms of motion. Simultaneously, an aspect
of the parameter space is brought into the system
description via the variables r and  .
The Gibbs main equation (8) then transforms into
dE = d E* - F  r - M    =
= dE* - r  dF - F  dr -   dM - M  d =
= v  dP +   dL - F  dr - M  d  +
+ T* dS - p* dV + * dN
(12)13
The second step is to substitute the increments by
total time derivatives and to set the following
conditions:
E
= v :=
P
E
=  :=
L
dr
dt
d
dt
(15)
then
 d

 d

0 = v   P - F  +    L - M  (16)
 dt

 dt

results.
This equation comprises of two parts. It can only be
true for all times t if both parts are equal to zero
simultaneously because P and L are independent
variables.
Besides the trivial solution where linear and angular
velocity vanish identically, this leads to two famous
theorems
dP
=F
dt
dL
=M
dt
(17)
stating that either of the two conserved quantities
linear and angular momentum can only be changed
by the flux of the respective quantities body force
and momenta flowing over the boundaries of the
body. Theorems (17) are well-known from classical
mechanics, especially in its Hamiltonian
formulation. The classical theory of mechanics is
therefore contained in Gibbs-Falkian Dynamics as a
special case.
However, it should be well noted that according to
the assumptions (13) they only hold for isoenergetic
and isentropic processes without chemical reactions
where the volume remains unchanged. Evidently,
these processes may – if possible at all for the
motion of real matter - only be established under
very intricate conditions.
dE
dS
dV
dN
= 0;
= 0;
= 0;
= 0 . (13)
dt
dt
dt
dt
4 Building Systems in Thermofluiddynamics – Top down vs. Bottom up
then (12) turns into
0 = v
dP
dr
dL
d
- F
+ 
- M
dt
dt
dt
dt
.
(14)
Using the kinematic definitions for the velocities
13
The negative signs with f, m indicate the way how these
variables are used in balance equations: Flows from
inside the body crossing the boundary have negative
values, flows to the inside have positive ones.
One of the major conclusions from the GibbsDuhem relation (10) was the statement that the
effects of motion and the thermodynamical behavior
of an object may not be separated in general. The
relation (10) states the dependence of the velocity
(as well as of the angular velocity) on the
thermodynamic variables and vice versa.
In contrast to this, the methods from classical
physics do not reflect this result. Here, theories are
built in a way that could be described as a “bottomup method”. For every physical effect in regard, e.g.
the motion of the object, an idealized part (in this
case the kinetic energy of the object) is created.
Then, all parts are composed by superposition to
make up the complete system. As a rule no
interference between the subsystems is supposed
and no methods are provided to describe these
interactions.
In fluiddynamics, the total energy for a system
usually comprises of three major parts: the kinetic
energy, the potential energy, and the inner energy of
the fluid. Commonly the Duhem-Hadamard
hypothesis of local equilibrium is applied to be able
to use the values from equilibrium for pressure and
temperature and to assure the validity of the
equation of state.
Thus we have
E = Ekin  Epot + U =
=
1
1
ˆ  T, p,  
mv 2 + d02m  2 + Epot r,   + U
2
2
.
(18)14
The AT, however, uses an alternative approach. The
Gibbs function for the object is created in a holistic
manner and includes all physical effects represented
by the respective generics. Any information on the
system is retrieved from the Gibbs function by
differentiation processes. This could be called a
“top-down approach”: the Gibbs function works
like an umbrella covering all aspects of the theory.
For mathematical convenience artificial subsystems
may be separated from this system describing
function.
There are quite some benefits in following this
approach. One of the major aspects is the
explanation how the total energy may be split up
into parts one of which is independent of the motion
of the object and how the classical definition for the
kinetic energy may be incorporated. Thus, let us
assume the energy to be a sum according to
!
E P, r, L, , S, V, N =
= Ekin P, L, S, V, N + Epot r,   + E0  S, V, N
Additionally, the definition for the kinetic energy
Ekin :=
1
1
m v 2 + d02m  2
2
2
(20)
will be applied.
From (19) and (20) it is clear that
Ekin
Ekin
E
E
=
and
=
P
P
L
L
or
v

+ d02m  
P
P
v

2
 = mv 
+ d0 m 
L
L
v = mv 
and
(21)
hold. The latter are coupled, linear first order partial
differential tensor equations possessing non-trivial
solutions which may easily be calculated:
mv = P + m  S, V, N 
d02m = L + d02m   S, V, N 
(22)
Functions  and  bear the dimensions of linear
and angular velocity, respectively. Therefore, they
are called dissipation velocities. They only depend
on the thermodynamic variables and are
independent of the state of motion. The dissipation
velocities are the price for separating from the
energy a part not depending on linear and angular
momentum. They are interconnected by the relation
  
2
=    .
(23)
Furthermore, equation (22)1 shows that the classical
identity
P = mv
only holds for vacuum conditions where the density
 tends to zero and the dissipation velocities
vanish:
  0

, 

0.
(24)
(19)15
14
The parameter d0 has the dimension of a length and is
characteristic for the rotating system.
15
It should be noted that E0 is not identical to the inner
energy U from equation (18). The difference lies in the
values of the intensive variables taken for a moving
system.
The potential energy is assumed to be conservative, i.e.
it does not depend on the state of motion.
After some lengthy algebraic manipulations where
the total differential of the dissipation velocities is
inserted into the Gibbs main equation, the final
conditions for the separation of the total energy are
found:
dE = m v  dv + d02m   d - m   d - d02m   d- F  dr - M  d +

 

+  - i
- l
dS 
s
s 

(25)


 
-  p - 2i 
- 2l 
 dV +

 
 

 

+  - i
- l
dN


 

with an asterisk apply for the moving system, no
equation of state is known for them.
Therefore the Duhem-Hadamard hypothesis is not
an appropriate tool in fluiddynamics: local
equilibria only exist when motion vanishes.17
However, in the context of the AT this hypothesis is
not needed. The AT provides a full spectrum of
theorems sufficient to set up a mathematically
consistent description free of contradictions and
physical paradoxa.
Hence, we can rewrite equation (19) as
E = Ekin  Epot + Ediss + U
where i and l are the specific linear and angular
momentum,  is the mole fraction. A new energy
form appears in equation (25):
(28)
showing that the theorem (18) from classical
mechanics is only valid for the idealized case of
dissipationless motion but not for real flow
processes.
5 Summary
1
1
Ediss := - m 2 - md02  2
2
2
(26)
contains all dissipative effects that may occur from,
e.g. viscous stresses, chemical reactions, etc. It
reflects the interactions of the artificially created
“subsystems” motion and thermodynamics of the
system. It can only be deduced from a top-down
approach with a system describing function like
equation (7) which covers all relevant aspects of a
physical object. It will not be discovered by a
bottom-up construction used in traditional physics.
Furthermore, equation (25) shows the definitions of
the intensive variables for the hypothetical state at
rest where the linear and angular momentum tend to
zero:


T :=  - i 
- l

s
s


p := p - 2i 
- 2l 





 :=  - i 
- l



(27)
Only the variables defined in equations (27) are
connected by an equation of state, they are valid for
the hypothetical state at rest, i.e. for vanishing
momentum. Their values are measured for
thermodynamical equilibria.16 The variables marked
16
There exists a special case, the so-called kinetic
equilibrium, where a moving system consisting of real
To characterize the differences between the
description of phenomena in the parameter space
constituted by spatial and time co-ordinates and the
method used in Gibbs-Falkian Dynamics the notion
of a generic physical quantity has been introduced.
It is the most primitive element of the theory and
may not be reduced to any other element. For the
Alternative Theory of non-equilibrium phenomena
(AT) it is most important that every generic quantity
is mapped to an extensive variable, i.e. a
mathematical element from the set of at least twice
differentiable functions obeying a list of
requirements from which additivity is the simplest
and most significant.
As an example the mathematical model of a singlephase one-component body-field system in linear
and angular motion was introduced. It could be
shown that classical mechanics in the form of
Hamilton’s theory is contained in the AT as a
special case. The restricting conditions for the
applicability of Hamilton mechanics were derived.
The advantages of the top-down approach from
Gibbs-Falkian dynamics and the Alternative Theory
were demonstrated by showing how the energy of a
moving system may be split up correctly into the
kinetic and the inner energy and how the
matter is controlled in such a way that it behaves like no
dissipation would occur. This can be established only
under very intricate conditions, cf. [6].
17
This theoretical result coincides with the experimental
outcomes, cf. [8].
interactions between these two parts are referenced
by the aid of the so-called dissipation energy. The
classical identity between velocity and the linear
momentum is shown to be valid only for the ideal
case of dissipationless motion. At last the
connection between the intensive variables for the
system in motion and for the hypothetical state at
rest is given. It has been explained that the DuhemHadamard hypothesis is inadequate for a realistic
description of flow phenomena because it only
applies to situations where the momentum vanishes,
i.e. for the hypothetical state at rest. This
theoretically deduced result corresponds well with
the experimental outcomes on motion and
equilibria.
References
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Knowledge, Oxford University Press, New York
1985
[2] Straub, D.: Zeitpfeile in der Natur? Versuch
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Kötter, R. (Edts.): Betrachten-BeobachtenBeschreiben, Beschreibungen in Kultur- und
Naturwissenschaften, Wilhelm Fink Verlag, pp.
105-146
[3] De Broglie, L.: Physik und Mikrophysik,
Claassen, Hamburg 1950.
[4] Falk, G.: Physik * Zahl und Realität – Die
begrifflichen und mathematischen Grundlagen
einer
universellen
quantitativen
Naturbeschreibung: Mathematische Physik und
Thermodynamik, Birkhäuser, Basel 1990
[5] Gibbs, J. W.: On the Equilibrium of
Heterogeneous Substances, in:The Scientific
Papers, Vol. I, Thermodynamics, Dover, New
York 1961
[6] Straub, D.: Alternative Mathematical Theory of
Non-equilibrium Phenomena, Academic Press,
San Diego, London, Boston, New York 1996
[7] Lauster, M.: Statistische Grundlagen einer
allgemeinen
quantitativen
Systemtheorie,
Shaker, Aachen 1998
[8] Neumaier, M., Lauster, M., Lippig, V., Waibel,
R., Straub, D.: The Duhem-Hadamard
Hypothesis
in
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