Brewster H.D. Fluid Mechanics
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52
Physical Basics
of
Fluid
Here n is the number
of
the molecules per unit volume, u is the molecular
velocity and
(i-
nu
) is the molecular number that traverses the considered
area per unit area and time
in
the positive direction x
2
.
These molecules had,
on average, the last contact with other molecules
in
a plane that has the distance
of
the mean free path
of
the molecules. Concerning the "energy content"
of
the molecular flow through x
2
= const., it can be said that the molecules hold
energy there which at position
(x
2
-
I)
is
owned by the elements
of
an ideal
gas, i.e. the energy
e(x2
-I).
In
an
analogous way, for the energy flow through the plane x
2
= const in
the negative
x
2
-direction, holds:
.- =
-.!.nue(x2
+1)
q2
6
The heat flow results from the difference
of
the molecular-dependent
energy streams, i.e. it holds:
q2
=q!
-qi
=.!.nii[e(x2
+1)-e(x2
+1)]
6
By means
of
the Taylor series expansion one obtains:
e(x2 +1) = e(x2)
+(
Be
)1 +
.!.(B2~Jp
+
...
BX2
2
Bx2
For the difference e(x
2
-l)
- e(x2 + l), one obtains in a first approximation:
e(x2
-1)-e(x2
+1)=-21(
Be
)+
...
BX2
and thus for the heat flow:
q2
=
-i
nu1e
(
:~)
=
-i
nm
(:;
)(:~)
From the derivations for the "heat energy"
of
a molecule holds:
3
Be
3
e=ek
=-kT
-=-k=c
2
BT
2 u
The Boltzmann constant k is understood to be a measure
of
the heat
capacity
of
a molecule. When one considers again:
__
~8kT.
1= 1
u -
1tm
'
J2d
2
nn
one obtains
Physical Basics
of
Fluid
53
However, it also increases with the molecular mass
A.
- (l/--Jm).
Similar to the considerations
of
the heat conductivity, where spatially
different temperatures
lea!!,
to temperature-smoothing processes, spatially
different concentrations
of
a certain particle type cause concentration-
smoothing processes that are to be understood analogously and to which the
term
"diffusion"
is
assigned. In order to follow
up
such processes, some gaseous
radioactive particles could be used as trasers.
In equilibrium these marked particles are distributed evenly over the
available volume. However, when the concentration
of
the marked parts is
position dependent where the entire number
of
the particles per unit volume
is constant, this state represents a non-equilibrium which will try to smooth
concentration in the course
of
time by diffusion. This smoothing is possible
by the temperature-dependent motion
of
the molecules. For the mathematical
description
of
diffusion processes the equation can be employed.
8c
.
-D-
m2
= 8x
2
Here
Til2
is the mass flow per unit time and area that runs parallel to the
direction
x
2
through a plane x
2
= const., D is the diffusion constant and c is
Fig. Transport
of
marked Molecules through a plane
The space-dependent concentration
of
the marked substance. The minus
sign expresses that the particles move from the position
of
higher concentration
to the position
of
lower concentration.
Analog to the molecular-theoretical considerations
of
the heat conduction,
the considerations indicated below lead to a derivation
of
the diffusion equation
and to a relationship which states the diffusion constants in molecular sizes. It
holds again that the flow
of
the particles through a plane x
2
= const can be
54 Physical Basics
of
Fluid
expressed as difference
of
the particle flows in positive or negative x
2
direction.
In the positive
x
2
direction, the area
~1
~3
of
the considered plane is traversed
I
by
th~,particles,
whose distance from the plane is not larger than u[}.t, i.e.
'6
~1
~3
u[}.tc(x
2
)·
by
the particles.
If
one considers the particle flow per unit time and area, it holds,
m!
=
~UC(X2
-I)
The particle concentration that exists in the distance I from the considered
plane is
of
relevance.
Accordingly, for the flow in negative x
2
direction holds:
m2
=
~UC(X2
+
I)
.
With
C(x
2
-l)
=
C(X
2
)-(!:}+'"
c(x2 +
l)
=
C(X
2
)+(
:~}
+ ...
yields for the desired quantity:
.
_.+
.-
m2
-
m2
+m2
. _
-!:..ul
Bc
m - 3
~
A 'comparison with the diffusion equation shows that diffusion constant
D was determined to
be
1
D =
-iii
3
If
one sets
on
the other hand
then one obtains:
u =
~8kT;
1tm
2 1
{kT
D =
3nd
2
VTm
l/..Jm.
On the other hand, there is a decrease with the molecular size,
D-
(l/d
2
),
and also with the density
of
the gas p = nm, D -
(lip).
VISCOSITY
OF
FLUIDS
The moleculare momentum transport in Newtonian fluid flow is given
by:
Physical Basics
of
Fluid 55
[
aU
j
au;]
aUk
't
.. =
-P
-a-+-a
-
POij-;:-
Ij
'X;
'Xj U.A,k
The material property
Jl
in the above presentation
of
'tij is defined as
dynamic shear viscosity
of
a Newtonian medium, or as the shear coeffcient
of
viscosity
of
the fluid. The second coeffcient Jl' is defined as dynamic expan-
sion viscosity
coeffcien~
and can be formulated for a Newtonian medium as
follows:
Ii
2
Jl' =
--p
3
with same physical units for
Jl
and
Jl',
i.e.
[Jl]
= [Jl']' The dynamic shear
viscosity
is
a thermodynamic property
of
a fluid and is· thus dependent on
temperature and pressure. For a Newtonian medium,
Jl
is independent
of
E ij =
1
(aU
j
au;)
"2
ax;
+
ax
j
.,
i.e.
tij
is linear with the velocity gradients occurring
in
a
flow,
or
is connected to local fluid-element deformations.
When the connection t
ij
(E ij) is nonlinear, one speaks
of
non-Newtonian
fluid viscosities.
Some
of
these possible non-Newtonian fluid properties with
pseudo-plastic behaviour, i.e. with increasing shear rate the fluid tends to have
lower viscosities. Dilatant fluids on the other hand show an increase
of
viscosity
with the increase
of
the rate
of
deformation and one defines them therefore as
"shear thickening fluids". A Bingham fluid is shown that
is
characterized by
a basic value
tij"
The treatment
of
Newtonian media rather than the fluids with
properties
of
non-Newtonian media. The non-Newtonian fluid are only
presented to point out that fluids with more complex fluid properties exist in
nature.
.:
= Deformationsrate
1£1
't
= Molekulare Impulsrate
pro
Flacheneinheit
Untersuchungen werden
1£
unter konstanter Deforma-
tionsrate
£ durchgfuhrt
Fig. Properties
of
Newtonian and Non-Newtonian Fluids
The dynamic viscosity
of
a Newtonian fluid depends indirectly on the
molecular interactions and can therefore be regarded as a
thermodynamic
56
Physical Basics
of
Fluid
property that varies with temperature and pressure. A complete theory
of
this
viscosity as a transport property
in
gases and liquids
is
still under development
and it can be looked up in the book
of
Hirschfelder. For an entire class
of
fluids, the function /l[T, P] can be presented
in
a description that was presented
by Keenan, and which makes use
of
a normalised expression such that all
values are normalised with the corresponding quantities at their critical state
and following expression
is
obtained:
1:. _
f[(~)'(~)]
f.lc
-
Tc
Pc
It shows, that the viscosity increases with pressure. The viscosity
of
liquids
decreases with temperature. For gases, there is a very weak dependence
of
viscosity on pressure and it is generally neglected in gas dynamic consideration.
• The viscosity
of
liquids decreases rapidly with the temperature.
• The viscosity
of
gases increases with temperature under moderate
pressure values.
• The viscosity
of
all fluids increases with pressure, independent
of
the aggregate state.
• The pressure dependency
of
gases
is
negligible.
10,0
Flussigkeitsverhalten
8,0-+---~Mcr!----------ir----+--I
Zunahme
der
Viskositat
mitdem
Druck
5,0
-t----~nt_~--__,.-----+---+---I
Zwei-
phasen
gebiet
1 ,0
-I-----+--""IIo--~---I~IiC_-
0,8
-+-----t----I'\-:----:;.0¥1----
0,5
-+----~:!oo.j"o~---+-----_+_--if--l
0,2
......
'-+-+-++-++--I--+-~f--+-+-+-+-++-t
0,4
0,6
0,8 1,0 2,0
5,0
8,0 10,0
Fig. Standardized viscosity
as
a
function
of
the
pressure
and
temperature values standardized
with
the
critical values
Physical Basics
of
Fluid
57
The above mentioned property
is
based on the fact that for the most fluids
the critical pressure
is
higher than 10atm and hence conditions for small density
are fulfilled very well under atmospheric pressures.
The theory
of
physical properties
of
gases under pressure conditions P <
Pc is very well developing and has been developed further until today on the
basis
of
theories by Maxwell (1831 - 1879). All
of
these theories are based
on the considerations. In accordance, the measured dynamic viscosity
of
a
fluid results from the statistical average
of
molecule-dependent momentum
transport
of
the motion
of
fluids. In case
of
gases, the dynamic viscosity reads:
2
~
=
-pic
3
where p is the density
of
the fluid, I the mean free path length
of
the
molecular motion and
c = the sound velocity
ofthe
gas. For gases under normal
pressure conditions
pi
""
const. However, more precise considerations show
that
pi
increases slightly with temperature and that this happens because
of
the so-called collision integral Qs- According to Chapman & Cowling I follows:
21
0-
3
JMT
~
=
cr
2
Q
s
In
this formula Mis the molecular weight
of
the gas, Tthe absolute temper-
ature,
(J
the collision cross section
of
the molecules and Q
s
=1
for the mole-
cules interacting only in connection with the collision. When more complex
molecular interactions exist,
Q
s
has to be calculated according to following
formula:
(
T
)-0.145
( T
)-2.0
Q
s
""
1.147
1'c;
+
1'c;
+0.5
T
*=
TIT
Q Q
s
Gl.
c s
0.32.840 2.928
1.01.593 1.591
3.01.039
1.060
10.00.82440.8305
30.00.70100.7015
100.00.58870.5884
400.00.48110.4811
Table: Data for Stockmayer collision integral
values for determining the viscosity
of
gases
The values determined from the Stockmayer potential, were compared in
Table with the values
of
the above approximation relationship. For routine
calculations, the following equations can be used
58
Physical Basics
0/
Fluid
P
(T)n
Po
~
To
where
~o
and
To
are corresponding reference values that were obtained
from measurements or calculations.
In general the value
of
n is around 0.7 More precise values
of
n are
contained in Table for different gases.
TO
Gas
Air
Ar
CO
2
CO
N2
O
2
H2
Water
vapour
110
n
error temperature
S
[K]
[mPa
*s]
[%]
range [K]
[K]
887.650.01716 0.666
±4
745.65-4548.15 521.90
887.650.02125
0.72
±3
723.15-3648.15 598.15
887.650.01370
0.79
±5
744.4 -4098.15
773.15
887.650.01657
0.71
±2
790.65-3648.15 579.40
887.650.01663 0.67
±3
773.15-3648.15 513.15
887.65
0.01919
0.69
±2
790.65-4773.15
585.65
887.650.008411
0.68
±2
453.15-2748.15 490.65
210.650.01703
1.04
±3
903.15-3648.15 2210.65
Table: Values for the Calculation of
the Dynamic Viscosity
of
Gases
temperature
range/or
±2% error [K]
648.15-4548.15
548.15-3648.15
700.65-4098.15
565.65-3648.15
498.15-3648.15
691.90-4773.15
778.15-2748.15
1085.65-3648.15
More extensive considerations were conducted by Sutherland, which were
based on an intermolecular potential
of
forces with an attractive part. The
resulting Sutherland formula is
3
P
(T)2
To
+S
-~
-
---
Po
To
T+S
In this relation S is an effective temperature, the so-called "Sutherland
constant"
.
BALANCE
CONSIDERATIONS
AND
CONSERVATION
LAWS
Before we conduct detailed considerations on fluid mechanics processes,
some remarks have to be made on the acquisition
of
information in fluid
mechanics, especially on the knowledge in analytical fluid mechanics which
is treated in this manuscript. Starting out from conservation laws, analytical
fluid me-chanics employs deductive methods to solve various unsolved
problems, i.e. to make statements on existing flow problems.
Here one makes use
of
derived relations that are based
on
balance
considerations, as the reader knows them from other fields
of
natural and
engineering science or also from everyday observations. In many domains
of
daily life one acquires, starting from in-tuitive knowledge on the existence
of
conservation laws, useful information from balance considerations which one
Physical Basics
of
Fluid
59
conducts on defined fields, domains, periods, etc. The way by which the
changes in quantities
of
our interest take place in detail is often not
of
interest
rather only the
"initial and end states"
of
the considered quantities are
of
interest. These changes are due to
"~n-flows
and outflows", and relations can
be established between changes within the considered fields, domains and
periods and the
"inflows and outflows". Considerations on the financial
circumstances are for example conducted by establishing balances on the
income and expenditure to obtain
with
these data information
on
the
development
of
the financial situation
of
companies or persons. Many more
examples
of
this kind could be cited that make clear the importance
of
balances
for obtaining information in daily life.
We find balances on quantities like mass, momentum, energy etc. in almost
every field
of
natural and engineering sciences. With these balances, basic
equations are set up with the aid
of
eXIsting conservation laws whose solution
leads, in the presence
of
initial and boundary conditions, to the desired
information on quantities.
In
order to obtain a definite information, the balance
considerations have to be based not only on valid conservation laws
(mass
conservation, energy conservation, momentum conservation etc.) but also on
definite specified domains. The field or the domain, on which balances are set
up, has to be defined precisely to guarantee the unambiguity
of
the derived
basic laws. A relation, that was derived by the consideration
of
a field is, in
general, not applicable when domain modifications have taken place which
were not included
in
the relations.
Fluid
mechanics
is
based
on
the
basic
laws
of
mechanics
and
thermodynamics and moreover uses state equations in the derivations in order
to establish relations between the state
of
a fluid. These state properties vary
in the course
oftime
or in space; however, the changes
of
state take place in
accordance with the corresponding state equations while
observing. the
conservation laws. For the detivation
of
the basic equations
of
fluid mechanics
the following physical basic laws are followed:
• Mass conservation law (continuity equation)
• Momentum conservation law (equation
of
momentum) - energy
conservation law (energy equation)
• Conservation for chemical species
• State equations
The above cited basic laws can now be applied to several
"balance
domains".
The size
ofthe
balance space is not important
in
general and it can include
infinitesimal small balance domains (differential considerations) or finite
volumes (integral considerations). Furthermore, the balance domains can lie
in different coordinate systems and can carry out proper motions themselves
60
Physical Basics
of
Fluid
(Lagrangian and
Euler's
ways
of
consideration). In general, once selected bal-
ance domain is usually maintained, however, this is
not
necessary. Changes
are admissible as long as they are known and thus can be included in the balance
considerations.
Generally,
in
fluid mechanics, only integral considerations are made, i.e.
these balances are set-up over different, favorable domains
of
interest. In the
case
of
differential considerations,
one
finds in general attention
only
on
balances with moving fluid elements (Lagrangian
way
of
consideration) or
space-fixed
elements
(Euler's
way
of
consideration).
Both
have
to
be
distinguished strictly and balances should always be
set
up separately for the
Lagrangian and the Eulerian balance spaces.
Mixed
balances lead to errors in general, however, transformations
of
final equations are possible.
It
is for example usual in fluid mechanics to
transform the balance relations derived for a fluid
element
to space-fixed
coordinate systems and thus to obtain balance relations for constant volumes.
The connections between considerations in moving fluid elements and space-
fixed coordinate systems are presented and the equations required for the
transformation are derived. Particular attention is given here to the physical
understanding
of
the
principal connections,
so
that
advantages
and
disadvantages
of
the
different
ways
of
consideration
become
clear.
The
advantages
of
the "Eulerian form"
of
the basic equations are brought
out
with
view to the imposed boundary conditions for obtaining solutions.
On
the other
hand
the Lagrangian considerations
allow
the transfer
of
physical knowledge
on the mechanics
of
moving bodies to fluid mechanics considera-tions.
When stating the basic equations in Lagrange variables, the following
equations yield for a fluid element
.
d(8m)91
• Mass ConservatIOn :
dt
0
d
•
Newton's
2
nd
Law:
d/(8m)91(U
j
)91]
=
L(M
j
)91
+
(M
j
)91
+
L(Oj)91
d d . 1 d(8V)91
• Energy
Conservation:
dt
(e)91
= dt
(q)91
-
P91
8V
91
dt
+
cj>diss
• State
Equation:
e91
=
!(P
91
,T91)undP
91
= !(!(P91,T91)
The above summarizing presentations make it
clear
that generally in fluid
mechanics
considerations
agree
with
principles
that
usually
treated
in
thermodynamics,
e.g.
the
energy
equation
(1st
fundamental
law
of
thermodynamics) and state equations
of
liquids and gases.
The above equations
can
also be expressed in field variables, such that
Physical Basics
of
Fluid
61
the
below
cited
set
of
differential
equations for
density
p, pressure
P,
temperature
T,
internal energy e and three velocity components Uj (j =
1,2,3)
are obtained:
ap
a(pU;)
• Mass conservation' - + = 0
. at
ax;
•
[
aUj aUj
1 ap
a'tij
Newton's 2
nd
law: p
--+U;--
=----+pg,
at
ax;
ax
j
ax;
[
aUj
au.]
2
aUk
'th 'to
=-p
--+-'
+-po
.--
WI
Ij
at
ax
j 3
Ij
aXk
E
t
·
p[ae
+u.~]=-
aq;
_paUj
-'t
..
aUj
• nergy conserva
IOn:
at '
ax;
ax;
ax
j
Ij
ax;
with
q.
=-t-..
aT
,
ax;
• state equations:e =
f(P,
T)
and P =
f(p,
n
Thus seven differential equations are available, when one inserts
'tij
and
qi
in the previously mentioned equations, for altogether five unknowns. With
this a closed system
of
differential equations which is given above, can be
solved for specified initial and boundary conditions. It is therefore
the
respective initial and boundary conditions that define a given flow problem.
The physical basic laws are identical for all flow problems. However, they
comprise the conservation and state equations as well that are usually treated
in thermodynamics.
THERMODYNAMIC
CONSIDERATIONS
The thermodynamic state equations
of
fluids are often used in supplement
for the solution
of
flow problems. However in the present text only "simple
fluids",
i.e. for homogenous liquids and gases for which the thermodynamic
state can be expressed by a relation between pressure, temperature and density
are considered. The statements are possible for substantial as well as for field
quantities, i.e. it holds
P9\
= f(T9\' P9\)
or
P =
f(T,
p)
Thermodynamic state equations are known to be:
P9{
- =
RT9\
(thermodynamically ideal gases)
p9{
P9\ = const (thermodynamically ideal liquids)
If
one defines with
~
= P 9\' T 9\' P9\'
e9\
and with a = P
~
T,
p,
e,
... , the