Durst F. Fluid Mechanics: An Introduction to the Theory of Fluid Flows

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64 3 Physical Basics

= τ

+ τ

−

nm¯u[U

− l) − U

+ l)]. (3.32)

The velocities U

−l)andU

+ l) can be expressed by means of Taylor

series expansions:

− l)=U

) −



∂U

∂x



l + ··· (3.33)

+ l)=U



∂U

∂x



l + ··· (3.34)

Thus one obtains:

mn¯u



−2

∂U

∂x



, (3.35)

i.e.

= −

mn¯ul



∂U

∂x



= −µ



∂U

∂x



. (3.36)

Hence the derivation based on the molecualar theory of ideal gases re-

sulted in a molecular transport of momentum. The resultant force per unit

area τ

turns out to be proportional to the local velocity gradient. The

proportionality factor is the dynamic viscosity of the considered ﬂuid.

For an ideal gas, this reads:

µ =

mn¯ul. (3.37)

If one takes into account the following relationship:

¯u =

8kT

πm

; l =

√

πn

, (3.38)

where d is the molecular diameter, one obtains:

µ =

3π

3/2

√

mkT

. (3.39)

This relationship tells us that, for an ideal gas, µ ∼

√

T , i.e. the viscosity

increases with increasing temperature. Furthermore, the viscosity of an ideal

gas increases with increasing molecular mass, µ ∼

√

m,whereastheviscosity

decreases with increasing molecular size, µ ∼ (1/d

The above considerations were carried out to serve as an introduction to

the derivations of continuum mechanics properties of ﬂuids, using average

molecular properties. Only one transport direction was taken into account

and only the x

momentum term was included in the considerations. The

complete term τ

is derived in Chap. 5 for Newtonian media, hence com-

pleting the considerations of the momentum transport in ideal gases. It is

3.3 Transport Processes in Newtonian Fluids 65

shown that the τ

term comprises essentially three terms, which can all be

described physically with the help of considerations of the momentum trans-

port by ideal gases. A general approach for momentum-transport processes

in ﬂuids is possible, i.e. of ﬂuids diﬀerent to an ideal gas, but this approach

is not undertaken within the framework of this book.

3.3.4 Molecular Transport of Heat and Mass in Gases

When the temperature in a system is not constant spatially, this system is

thermally not homogeneous and heat will be transferred from areas of higher

temperature to areas of lower temperature. For the one-dimensional problem

shown in Fig. 3.10, a heat ﬂux ˙q

Q/(∆t∆x

∆x

), i.e. per unit area and

unit time, will take place, which is proportional to the temperature gradient

existing at position x

= 0; this is known from experiments and is usually

referred to as Fourier’s law of heat conduction

˙q

=˙q

= −λ

∂T

∂x

. (3.40)

The proportionality constant λ is designated as the thermal conductivity of

the ﬂuid or the thermal conductivity coeﬃcient of the ﬂuid.

In this section, considerations will be summarized that are suitable for

understanding the physical causes of heat conductivity from the point of view

of the molecular theory of ideal gases. Hence the derivations are again given

for the model of an ideal gas. Considerations of this kind are most suitable

because the molecular motion, as a basis for heat transport considerations,

can be presented in a simple way.

If one considers a plane A at x

= constant in an ideal gas, where a

temperature gradient exists that can be stated as the derivative of T (x

the heat conduction through plane A can be explained in such a way that

the molecules in both directions traverse plane A and carry the “thermal

energy” with them. When (∂T/∂x

) > 0, the molecules which move through

the plane from the top to the bottom have a higher mean energy than the

molecules which traverse plane A in the opposite direction. The heat ﬂow

Fig. 3.10 Heat transport through a

plane, caused by molecules

Plane A = hetched area

(

)

(

)

{

}

66 3 Physical Basics

through plane A can thus be explained as the diﬀerence between the energy

transports which stem from the upward and downward molecular motions.

The following equations can be given to express this.

The energy ﬂow in the positive x

direction can be given as:

˙q

n¯ue(x

− l), (3.41)

where n is the number of molecules per unit volume, ¯u is the mean molecular

velocity and

n¯u is the number of molecules that traverse the considered

unit area per unit time in the positive x

direction. These molecules had,

on average, the last contact with other molecules in a plane that has the

distance of the mean free path l of the molecules. Concerning the “energy

content” of the molecular ﬂow through the area A = 1, it can be said that

the molecules had an energy at the position (x

−l), bring the same as that

of all the molecules of the ideal gas, i.e. the energy ne(x

− l).

In an analogous way, for the energy ﬂow through the plane A = 1 constant

in the negative x

direction [3.2]:

˙q

−

= −

n¯ue(x

+ l). (3.42)

The net heat ﬂow results from the diﬀerence of the molecular-dependent

energy transports in the positive and negative x

directions:

˙q

=˙q

+˙q

−

n¯u[e(x

− l) − e(x

+ l)]. (3.43)

By means of Taylor series expansion, one obtains:

e(x

− l)=e(x

) −



∂e

∂x



l +



∂

∂x



−··· (3.44)

e(x

+ l)=e(x



∂e

∂x



l +



∂

∂x



+ ··· (3.45)

For the diﬀerence e(x

−l) − e(x

+ l), one obtains in a ﬁrst approximation,

taking only ﬁrst-order derivatives into account:

e(x

− l) − e(x

+ l)=−2l



∂e

∂x



+ ··· (3.46)

and thus for the heat ﬂow:

˙q

= −

n¯ul



∂e

∂x



= −

n¯ul



∂e

∂T



∂T

∂x



. (3.47)

From the derivations given in Sect. 3.3.2 for the “heat energy” of molecules

of an ideal gas:

e = e

∂e

∂T

k = c

. (3.48)

3.3 Transport Processes in Newtonian Fluids 67

The Boltzmann constant k is understood to be a measure of the heat capacity

of a molecule. When one considers again

¯u =

8kT

πm

; l =

√

πn

(3.49)

one obtains

λ =





. (3.50)

Analogous to the viscosity, the heat conductivity increases with increas-

ing temperature, λ ∼

√

T , and decreases with increasing molecular size,

λ ∼ (1/d

). However, it also decreases with increasing molecular mass,

λ ∼ (1/

√

m).

Similarly to the considerations regarding the heat conductivity, where spa-

tially diﬀerent temperatures lead to temperature equilibrium by diﬀusion

processes, spatially varying concentrations of certain molecule types cause

concentration equalization processes that are to be understood analogously

andinordertofollowupsuchprocesses,andstudythemexperimentally,

gaseous radioactive particles could be used as tracers.

In equilibrium, these marked particles are distributed evenly over the avail-

able volume. However, if the concentration of the marked parts is position

dependent where the entire number of the particles per unit volume is con-

stant, this represents a non-equilibrium state which will try to equalize the

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

˙m

= −D

∂c

∂x

(3.51)

can be employed, which is known as Fick’s law, where ˙m

is the mass ﬂow

per unit time and unit area that lies perpendicular to the direction x

, i.e.

the mass ﬂow rate through a plane x

= constant, D is the diﬀusion constant

and c is the concentration of the marked substance. The minus sign expresses

that the particles move from a position of higher concentration to a position

of lower concentration.

Analogously to the molecular theory of heat conduction, the considerations

below lead to the derivation of Fick’s diﬀusion equation and to a relationship

which states the diﬀusion coeﬃcients in terms of molecular properties. It

holds again that the mass ﬂow of the molecules through a plane x

= constant

can be expressed as the diﬀerence in the mass ﬂow of the molecules in the

positive and negative x

directions (Fig. 3.11). In the positive x

direction,

the area ∆x

∆x

of the considered plane is passed by all the molecules, whose

distance from the plane is not larger than ¯u∆t, i.e.

∆x

¯u∆tc(x

68 3 Physical Basics

Fig. 3.11 Transport of marked mole-

cules through a plane perpendicular to

the x

direction

If one considers the particle ﬂow per unit time and area [3.2], the following

relationship results:

˙m

¯uc(x

− l). (3.52)

The molecule concentration that exists at a distance l from the considered

plane is represented by c(x

− l).

Accordingly, for the mass ﬂow rate in the negative x

direction, one can

write:

˙m

−

= −

¯uc(x

+ l). (3.53)

Through Taylor expansions of both c terms in (3.52) and (3.53), one obtains:

c(x

− l)=c(x

) −



∂c

∂x



l + ··· (3.54)

c(x

+ l)=c(x



∂c

∂x



l + ··· (3.55)

Hence this yields for the desired quantity ˙m

˙m

=˙m

+˙m

−

(3.56)

˙m = −

¯ul

∂c

∂x

. (3.57)

A comparison with the diﬀusion equation (3.51) shows that the diﬀusion

coeﬃcient D is determined by molecular transport considerations as:

D =

¯ul. (3.58)

If, on the other hand, one sets:

¯u =

8kT

πm

; l =

√

πn

(3.59)

3.4 Viscosity of Fluids 69

then one obtains

D =

. (3.60)

From this equation, we can infer that the diﬀusion increases constantly with

increasing temperature, D ∼

√

T , and decreases with increasing molecular

mass, D ∼ 1/

√

m. There exists a decrease with the molecular size, D ∼

(1/d

), and also with increasing density of the gas ρ = nm, D ∼ (1/ρ).

3.4 Viscosity of Fluids

The molecular momentum transport in Newtonian ﬂuid ﬂow is given by

= −µ



∂U

∂x

∂U

∂x



− µ



∂U

∂x

. (3.61)

The material property µ is called the dynamic viscosity of a Newtonian

medium, or the shear viscosity of the ﬂuid. The coeﬃcient µ



is deﬁned as

the expansion viscosity coeﬃcient and can be formulated for a Newtonian

medium as follows:



= −

µ (3.62)

with the same physical units for µ and µ



, i.e. [µ]=[µ



]. The dynamic

shear viscosity is a thermodynamic property of a ﬂuid and thus depends

on temperature and pressure. For a Newtonian medium, µ is independent of



∂U

∂x

∂U

∂x



, i.e. τ

is linearly dependent on the velocity gradients

occurring in a ﬂow, or is connected with local ﬂuid-element deformations.

When the τ

= f(S

), relationship (3.61) is non-linear, one speaks of ﬂuids

with non-Newtonian ﬂuid viscosities. Figure 3.12 sketches some of these pos-

sible non-Newtonian ﬂuid properties with pseudo-plastic behavior, i.e. with

increasing shear rate the ﬂuid tends to have lower viscosity. Dilatant ﬂuids,

on the other hand, show an increase in viscosity with an increase in the rate

of deformation and one therefore deﬁnes them as “shear thickening ﬂuids”.

In the diagram, a Bingham ﬂuid also is shown that is characterized by a base

value of τ

before the ﬂuid moves. This book concentrates on the treatment

of Newtonian ﬂuids rather than on ﬂuids with properties of non-Newtonian

media. The non-Newtonian ﬂuid properties indicated in Fig. 3.12 are pre-

sented to point out the existence of more complex ﬂuid properties occuring

in nature and in industry.

The dynamic viscosity of a Newtonian ﬂuid depends indirectly on the

molecular interactions and can therefore be regarded as a thermodynamic

property that varies with temperature and pressure. A complete theory of

this viscosity as a transport property in gases and liquids is still under devel-

opment and the present state can be consulted in the book by Hirschfelder

70 3 Physical Basics

Investigations are carried out

for constant deformation rate

Decrease with

time = thixotropic

No time dependence

Increase

with

time = rheopectic

Plastic

Bingham

Pseudo-

plastic

Dilatant

Newtonian

Deformation rate

Molecular momentum

loss per unit area

Fig. 3.12 Properties of Newtonian and non-Newtonian ﬂuids

et al. [3.5]. For an entire class of ﬂuids, the function µ[T,P]canberepre-

sented in a way that was presented by Kienan [3.6] and which makes use of

a normalized expression such that all values are normalized with the viscos-

ity at the critical point of the ﬂuid. In this way the following expression is

obtained:

= f









(3.63)

which is illustrated in Fig. 3.13, showing that the viscosity of gases increases

with pressure. The viscosity of liquids decreases with temperature. For gases,

there is a very weak dependence of viscosity on pressure and this is generally

neglected in gas-dynamic considerations. In short, Fig. 3.13 indicate that:

• The viscosity of liquids decreases rapidly with temperature.

• The viscosity of gases increases with temperature at moderate pressure

values.

• The viscosity of all liquids increases more or less with pressure.

• The pressure dependence of viscosity of gases is negligible.

The above normalized property

permits the conclusion that for most

ﬂuids the critical pressure is higher than 10 atm and hence conditions for low

density are fulﬁlled very well under atmospheric pressure.

The theory of the physical properties of gases under pressure conditions

P<P

is well established and has been advanced further on the basis of

theories by Maxwell (1831–1879), as indicated by Hirschfelder et al. [3.5] and

Present [3.7]. All of these theories are based on considerations yielding infor-

mation given in Sect. 3.3. In this section, it is explained that the measured

dynamic viscosity of a gas results from the statistical average of molecule-

dependent momentum transport of the motion of ﬂuids. In the case of gases,

3.4 Viscosity of Fluids 71

0,4 0,6 0,8 1,0 2,0 5,0 8,0 10,0

10,0

8,0

5,0

2,0

1,0

0,8

0,5

0,2

0,5

= 0,2

Critical

point

Two phase

region

Low density limit

Density of gas

•

Behavior of liquids

Increase of viscosity

with pressure

Viscosity increase

for gases

(

)

Fig. 3.13 Normalized viscosity as a function of pressure and temperature. Normali-

zation is achieved with the viscosity at the critical point of the ﬂuid

the dynamic viscosity can be given as follows:

µ =

ρlc, (3.64)

where ρ is the density of the ﬂuid, l the mean free path length of the molecular

motion, and c the speed of sound in a gas. For gases under normal pressure

conditions ρl ≈ constant. However, more precise considerations show that ρl

increases slightly with temperature because of the so-called collision integral

Ω

. According to Chapman and Cowling [3.8]:

µ =

210

−3

√

Ω

, (3.65)

where M is the molecular weight of the gas, T the absolute temperature, and

σ the collision cross-section of the molecules. For Ω

= 1 the molecules inter-

act only in the form of collisions. When more complex molecular interactions

exist, Ω

has to be calculated according to following equation:

Ω

≈ 1.147





−0.145



+0.5



−2

. (3.66)

72 3 Physical Basics

Tab l e 3. 1 Stockmayer collision integral values for determining the viscosity of gases

∗

= T/T

Ω

(3.66)

0.3 2.840 2.928

1.0 1.593 1.591

3.0 1.039 1.060

10.0 0.8244 0.8305

30.0 0.7010 0.7015

100.0 0.5887 0.5884

400.0 0.4811 0.4811

Tab l e 3.2 Values for the calculation of the dynamic viscosity of gases according to

the Sutherland equation (3.68)

Gas T

n Error Temperature S Temperature

(K) (mPa s

−1

) (%) range (K) range for

(K) ±2% error (K)

Air 887.65 0.01716 0.666 ±4 745.65–4548.15 521.90 648.15–4548.15

Ar 887.65 0.02125 0.72 ±3 723.15–3648.15 598.15 548.15–3648.15

887.65 0.01370 0.79 ±5 744.4–4098.15 773.15 700.65–4098.15

CO 887.65 0.01657 0.71 ±2 790.65–3648.15 579.40 565.65–3648.15

887.65 0.01663 0.67 ±3 773.15–3648.15 513.15 498.15–3648.15

887.65 0.01919 0.69 ±2 790.65–4773.15 585.65 691.90–4773.15

887.65 0.008411 0.68 ±2 453.15–2748.15 490.65 778.15–2748.15

Water 1210.65 0.01703 1.04 ±3 903.15–3648.15 2210.65 1085.65–3648.15

vapor

The values determined from the Stockmayer potential, e.g. see Bird et al.

[3.1], are compared in Table 3.1 with the values from the above approximation

relationship and they show good agreement.

For routine calculations, the following equation can be used:

≈





, (3.67)

where µ

and T

are corresponding reference values that were obtained from

measurements or calculations on experimental ﬁndings.

In general, the value of n is found to be around 0.7. More precise values

of n are given in Table 3.2 for diﬀerent gases.

More extensive considerations were made by Sutherland [3.9], and these

were based on an intermolecular potential for forces possessing an attractive

part. The resulting Sutherland equation is

≈





+ S

T + S

, (3.68)

where S represents an eﬀective temperature, the so-called “Sutherland con-

stant”. It is also given in Table 3.2 for diﬀerent gases.

3.5 Balance Considerations and Conservation Laws 73

3.5 Balance Considerations and Conservation Laws

Before we conduct detailed considerations on ﬂuid mechanics processes, some

remarks need to be made on the acquisition of information in ﬂuid mechan-

ics, especially on the knowledge gained in analytical ﬂuid mechanics, which

is treated extensively in this book. Starting from conservation laws, analyt-

ical ﬂuid mechanics employs deductive methods to solve various unsolved

problems, i.e. to gain knowledge on existing ﬂow problems. For analytical

solutions, one makes use of derived relations that are based on balance con-

siderations, as known from other ﬁelds of natural and engineering science and

also from everyday observations. In many domains of daily life one acquires,

starting from intuitive knowledge on the existence of conservation laws, use-

ful information from balance considerations which one conducts over deﬁned

ﬁelds, domains, periods, etc. The way in which the changes of considered

quantities of concern take place in detail is often not of interest. Rather, the

“initial and end states of the considered quantities” need to be known. The

changes within the considered domain are due to “inﬂows and outﬂows of the

considered quantity”. Relationships can be established between the changes

within the considered ﬁelds, domains and periods knowing the “inﬂows and

outﬂows”. Considerations of the ﬁnancial circumstances of a person or an

institution are, for example, conducted by establishing balances of income

and expenditure to obtain information on the development of the ﬁnancial

situation. Many more examples of this kind could be cited in order to make

clear the importance of balances for obtaining information in daily life.

In ﬂuid mechanics, we ﬁnd balances of quantities such as mass, momentum,

and energy in almost all ﬁelds of natural and engineering sciences. With these

balances, basic equations are set up with the aid of existing conservation laws

whose solution lead, in the presence of initial and boundary conditions, to the

desired information on quantities. In order to obtain 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 for which balances

are formulated has to be deﬁned precisely to guarantee the unambiguity of

the derived basic laws. A relationship that was derived by considerations over

a certain domain are, in general, not applicable when domain modiﬁcations

have taken place and these are not included in the relationship.

Fluid mechanics is a subject based on the basic laws of mechanics and

thermodynamics and, moreover, uses state equations in the derivations in

order to establish relationships between thermodynamic state changes of a

ﬂuid. These state properties vary in the course of time or in space. However,

the changes of state take place in accordance with the corresponding state

equations while still observing the conservation laws. For the derivation of

the basic equations of ﬂuid mechanics, the basic laws of physics are: