A state-of-the-art survey of some aspects of the aerodynamics of highly rarefied gases

<

m ÄI til m

~

'i5îa'te-o

the-Art Survey of Some Aspects of the

AERODYNAMICS OF HIGHLY RAREFIED GASES by

G. N. Patterson First Interim Report Only

MARCH, 1961 UT IA Review No. 18

ARL 39 ::z:: <:!: 0 r-- V'\ r'\ 0 m::z:: _~"., () _c C -::z:: ::I

°0

o:l~ m 0"., Z cV'\ - j :E ...

>

7'\::z:: - j cO m zO _{o r} -me " . , r-..." -4

of the

AERODYNAMICS OF HIGHLY RAREFIED GASES

MARCH, 1961

by

G. N. Patterson

First Interim Report Only

UTIA Review No. 18 ARL 39

Page 5: below Eq. 6 u2 = ui ui

Page 12: Eq. 24 G(

cp)

=

cr

2

I

cos

cp

I

11 Eq. 25 Sc =

~ïïtS"l.

S

l(orqJlsl",cq~<Q

0 Page 24: Eq. 25 = 0 Page 26: Eq. 1 _JdW

dr.

= 0

-

2

Page 33: Eq. 28 _{Cl (Cl} 2

_{+ -}

C2

₎

e-C 1

3e₁

Page 49: Eq. 3 _{f ~foo}

Page 6Q Eq. 1 = - Df

+

P

:-n-Page 72:

S

_sin

_CQ

_{\ cos}

_q>

I

_d

~

= 1

This state-of ... the-art survey is being made possible by contract number AF 33(616)-6990 issued by the Wright Air Development Division. Wright-Patterson AFB. Ohio.

v

This report presents a survey of some aspects of free-molecule and near-free-free-molecule flow. The basic elements of the kinetic theory of molecular collisions and scattering are presented as a prelimJnary to the derivation of the fundamental Boltzmann equation and the Maxwell transfer equations. The equilibrium solution of the Boltzmann equation is obtained and the transfer properties of a gas in equilibrium at rest and in uniform translational motion are considered in detail.

The special form of the Boltzmann equation for free molecule flow is discussed. The boundary conditions for the case o~ a convex body in free molecule flow are found to involve the distribution function which is closely related to reflection phenomena at the surface of the body. The transfer of mass, momentum and energy at an element of surface is calculated and the results applied to determine the free molecule aerodynamics of a sphere.

The subject of near-free-molecule flow is introduced by a consideration of molecular bearns and the effect of collisions on beam strength. An integral equation form for the Boltzmann equation which facilitates an iterative method of solution is shown to be a suitable

basis for a study of the properties of near-free-molecule flow. However, th is method requires the use of a kinetic model of the Boltzmann equation if the problem is to remain mathematically tractable. The required equations to bè used for investigating the aerodynamics of isolated bodies in near-free-molecule flow are then discussed.

This is the first interim report only. The work of modifying and extending this report is in progr~ss.

TABLE OF CONTENTS

Page

NOTATION ii

lo INTRODUCTION 1

2. PROPERTIES OF MOLECULES AND THEIR MOTION 3

3. THE MOLECULAR DISTRIBUTION FUNCTION 3

4. MOLECULAR SCATTERING 7

5. OTHER PROPERTIES OF MOLECULAR COLLISIONS 13

6. BOLTZMANNiS EQUATION 18

7. MAXWELL'S TRANSFER EQUATIONS . 20

8. DISTRIBUTION FUNCTION FOR A GAS IN MOLECULAR

EQUILIBRIUM 26

9. TRANSFER PROPERTIES OF A GAS IN EQUILIBRIUM

AND AT REST 28

v

10. TRANSFER PROPERTJES OF A GAS IN EQUILIBRIUM_

HAVING TRANSLATIONAL MOT ION 39

Ilo BOLTZMANN EQUATION FOR FREE MOLECULE FLOW 45

12. DISTRIBUTION FUNCTYONS FOR MOLECULES REFLECTED

FR OM ASOLID BOUNDARY 47

13. FREE MOLECULE TRA.'N"SFER AT AN ELEMENT OF

SURFACE 52

14. FREE MOLECULE AERODYNAMICS OF THE SPHERE 55

15. MOLECULAR BEAMS 57

16. BOLTZMANN RELATION AS AN INTEGRAL EQUATION 59

17. A KINETIC MODEL OF THE BOLTZMANN EQUATION 62

18. BASIC EQUATIONS FOR AN ISOLATED BODY IN

NOTATION

(Numbers give the page on which the symbol is defined)

a 33 d 65 _fQ) 46 _noo 48 AJ.. 58 dA 35 F 22 N 59 As 58 dA' 37 g 57 N(c) 58 b 33 dW 7 G 7 _Nb 49 B 13, 55 D 60 Gr 10 N

_w

49 c 5 D ₆₂ I 2).,33 _No 57

-

_{c 30 Dn 61 11} 34 _Nco 49 ~,-aci 5 Do 62 1₂ 34 P 6, 65 cm 30 DQ:l 65 13 34 Pr 50 cm 58 E 6 14 34 Pw 50 B cm·· 67 E. 6 j 56 PCb 50 1J 1 cm 52 E_ijk 6

J

19, 14,31 P 60 r cm _w 48 Er 50

J'

14 Pn 61 cm 48 Ew 50 k 17,26,27 p .. 5

co

.

1J c 43 _Eoo 50 ko 17 Po 62 P cr,cr 30, 36 f 4 k. 1 17 P (Xl 65

Cv

43 _fb 46 _k4 17 Q 20 C 27 fe 27, 30 K 16 Qijkl 6 C 30 _fs 47 1 58 r 7, 62 CD 55 fw 48 m 4 ro 7 C·· 68 fo 62 n 4 R 5 1J Cr 52 f1 62

_-

n 39 Rw 54 C

_w

48 _fn+1 61 _~ 67 Reo 54

s 37 0( 7, 51 _~ 12, 51 s 60 o(i 26, 27

n

9 So 61

~

13, 26, 27,48 ~2 9 sa:> 65 <r12 9 s' 65

e.

55

,

51

a-S 40 1: 7 Sc 7

_1'J

67

-r:

r 50 t 4 _~m 63

T

_w 50 T 5

'7

S 63

1:"

co 50 -T _r 50 9 7, 29

cP

7, 8, 29, 37 T W 50

(jC'.,

34

~r

10 T CO 50

k

31, 49

cp'

37 u 5

À

32

_Y

14 ~ ui 5 Àc 34

tf

15, 39 u·· 68

~t

61

_J)b

67 IJ U 8

!

4 ...!li 67 v 57

!

4

Jl

OO 67 x 4

_~i

4, 5 x 4

_Sn

20

-

-xi 4 Sr 10 y. 1 31

tr

10, 14, 19 Zi 31 ~ 4

(co

53

1. INTRODUCTION

The purpose of this report is to review some aspects of the aerodynamics of highly rarefied gases with special reference to the upper atmosphere. It is an interim statement of a continuing work.

This study is restricted at present to electrically neutral particles. In terms of the upper atmosphere we are interested in the aero-dynamic forces experienced by satellites in orbit at altitudes where elctro-dynamic effects are not yet significant. This is not a severe restriction since most present day satellites have perigee altitudes at which the number of charged particles in unit volume is less than one-thousandth of the num-ber of neutral partic1es (Ref. 1. 1).

In the regime thus defined the two major areas of interest are free-molecule flow and near-free-molecule flow, both of which are considered in this report following a consideration of the fundamental concepts of molecular transfer and its relation to the macroscopie proper-ties of a gas.

Since this project will be continued, comments and sugges-tions from the reader are invited. In partieular, material which the reader feels should be considered would be we1come.

The reader' s attention'is directed to a number of reviews of the dynamics of rarefied gases and the bibliographies which they contain

(e. g. Refs. 1. 2, 1. 3, 1. 4) and to the proceedings of the First International Symposium on Rarefied Gas Dynamies (Ref. 1. 5)

't

PART I

FUNDAMENTAL EQUATIONS OF THE

KINETIC THEORY

ot

2. PROPERTIES OF MOLECULES A.ND THElR MOTION

Our concepts of the molecules composing a gas and their motion are necessarily conditioned by the fact that our objective is to

calculate the macroscopic {observable) characteristics of a flowing gas by means of a .mathematically tractable theory. Only those properties of molecules and their collisions which influence the macroscopic motion need be considered. For many flow problems the molecule represented as a hard elastic sphere free from force except on contact with a similar molecule or as a point center of force has proved to be quite useful.

In this report we will be concerned with neutral particles which move in straight lines at constant speed between collisions, i. e. there are no external forces. The molecules are monatomic, i. e. they have no internal degrees of freedom and are subject to translq.tion only.

An important premise on which our theory will be based is the assumption of molecular chaos introduced originally by Boltzmann. Basically, there will be no correlation between the positions and velocities of different particles unless they are within each other' s fields. This assumption is expected to be reasonable for rarefied gases in which the frequency of collisions is small. It holds for equilibrium motion (Ref. 2. 1) and it can be accepted as true for near-equilibrium flows. This subject is discussed in more detail in Ref. 2.2.

Again, since the gas is rarefied, we can assume that the duration of a collision is much less than the time of transit between encoun-ters. This is equivalent to assuming that the characteristic dimension of a molecule (e. g. diam eter of a spherical molecule) is very small compared with the ave rage distance traversed between collisions (mean free path). Thus only binary collisions will be considered. Furthermore, these binary collisions will be cómplete if the elementary interval of time dt, characteristic ()f the flow process, is large compared with the average duration of a collision but small èompared with the average time between collisions. Such an interval permits us to neglect incomplete collisions and the possible effect of a third molecule in the collisions. These con-ditions also imply an average potential energy which is much less than the mean kinetic energy and justify our consideration of the properties of flow-ing gases from the point of view of kinetic theory.

3. THE MOLECULAR DISTRIBUTION FUNCTION

The basic idea of kinetic theory is that all macroscopic (observable) properties of a gas can be deduced from the basic information about the molecules including the force of interaction through the use of the classical laws of motion. The kinetic theory is essentially superseded by the methods of statistical mechanics when the gas is in equilibrium However, when a gas is not in equilibrium, an extension of the methods

of kinetic theory is the only recourse because of mathematical complexity.

Flowing gases exhibit these non-equilibrium properties and even when the

gas is rarefied (effect of molecular collisions minimized) many problems remain mathematically intractable.

If we know the positions and velocities of all the molecules of a monatomic gas at a particular instant, the description of the gas is complete. Assuming a conservative system and using. the Hamilton canonical equations, the microscopic state of a gas at a subsequent instant can be calculated from the given initial state. This method requires de-tails of the initia 1 state which are never available and in fact provides

much more than the macroscopic information required. Even the method of

Gibbs, which involves the calculation of the probability distribution of a gas at a specified time (t) from that at t

=

0 by means of the Liouville equation (Ref. 2. 2), yields more information than is required. If, how-ever, we consider the probability distribution of a single molecule inde-pendent of the state of all other molecules we arrive at the definition of the molecular distribution function (f) which provides adquate information regarding the macroscopic properties of the gas. Considered in the light of its relation to Gibbs'probability distribution, it is found that f is a smooth point function if this property is prossessed by the Gibbs' function (Ref. 2.2). For a monatomic gas (the molecules have no internal degrees of freedom) the molecular distribution function depends on the rectangular co-ordinates of the

(f '

x)-phase space and on the time. The number of molecules in the physlcalvolume dx around x having velocities within d

f

about

r

may be written in theform f(

f-'

x, t) dx

dr.

Let us now consider the moments of the molecular distri-bution function with respect to the velocity. The zero'th moment is

nIx, t)

=

r

f

(f,

x, t)

~

(1)

Since f( ~, x, t) dx d

J

is the number of partic1es in an element of the phase spi"ce,- then(l)

18

the number density in physical space, the integral being taken over all possible molecular velocities (- te <..

I,

<:

00). If

m is the mass of the molecule, then the density of the gas is given by

f

=mn (2)

The mean value of the i th component of velocity in the ele-ment of physical space dx is obtained by averaging all such components possessed by all the molecules in dx over the total number of molecules in dx. The number of molecules in the element of phase space possessing the velocity

t.

is f(

r.,

Xl·' t) dx d

f

and the sum of all i-components of

, t

-velocity for all molecules in dx is

dx

)!j

f d

t

The total number of molecules in dx is ndx. Hence the mean (macro-scopic or observ~ble) i th component of velo city of the molecules in dx about x is

Ui =

*f

~If~

(3)

We note that the tot al momentum in the element of volume of physical space (dx) is

f

m Si f

ti

and according to (3) may be written mnui or

f

ui per unit volum e about x.

We can refer the velocity of a molecule in dx to the mean

value by writing

-C". =u. +c·

.)1 1 1

(4)

where ~ i is the i th component of rand ci is the i-component of the (unseen) velocity of the random moTIon (c). Substituting for

f

i in (3) and

using (1) we see that

-)C[f~

=0 (5)

The kinetic energy of the molecules in the elem ent of phase space is

t

m ~ i

'S

i f d

s

dx and hence the total kinetic energy in unit volume of physical späCe about x is

tm

r

(Ui

+

0;) (ui

+

0;) f

d!;

where u 2 = ui vi

It wil! be seen that this quantity can be separated into a component of energy of mean (macroscopic or observable) motion and a component which arises from the random (unseen) motion. The latter can be iden-tified with the temperature of the gas. The kinetic energy per unit mass of a monatömic gas is

t

RT per degree of freedom. Since there are three degrees of freedom. then for unit volume of gas about ~.

îf

RT _{= îm}

I

_{ei 0; f}

~

₍₇₎

where R is the gas constant per unit mass of gas.

The quantity m Cj f dx

<!i

is the momentum in the element of phase space directed in the j-direction. If this is multiplied by

Cï.

we obtain the rate at which the j-component of momentum is transferred in the i-direction. This is by definition a component of the stress tensor.

i. e. (referred to unit volum e of physical space about x)

p .. = m

fCï

c

J' f d i (8)

This relation does not all ow for intermolecular forces but may be expected to hold for rarefied gases. The scalar pressure used in the thermodynamic approach to fluid mechanics is defined as

p=1/ 3Pü=1/3m

J

ciC;fdj; (9)

Hence according to (7)

p

=

fRT (10)

The random energy in the element of phase space is

~m ci ci f d x d'[ . The rate at which this quantity is transferred in the j -direction is rm ci Cï Cj f dx

<!..!

The amount of this transfer in unit volume of physical space about x is

î

Ei =!m

f

Ci Cj Cj f

<Û

(11) where Ei is the i th component of the heat flow vector~. This definition also neglects the intermolecular forces which are not considered to make a significant contribution in a rarefied gas.

It will be seen th at the macroscopic quantities which define the observable state of a gas are all expressible in terms of moments of f taken with respect to the component of random ve.locity (Ci). The zero order moment is given by (1) and the first order moment (5) is zero. We define the second, third and fourth moments as follows:

Pij (x, t)

=

m

r

ci Cj f

<!,.i

E"

k (x, t)

=

m ( cic. ck f d

S

1J - ) J .

-Qijk! (x, t) = m ) ci Cj Ck

CR

f

è.1

By appropriate contractions of these tensors we obtain

p ..

=

3p E .. ·

=

E· 11 ' l J J 1 (12) (13) (14) (15)

•

4. MOLECULAR SC AT TERING

We consider a beam of molecules moving with the same speed in parallel paths distributed at random.

A

single molecule held in a fixed position will divert some of the beam molecules when collisions occur. The basic question relevant to scattering phenomena is what

fraction of the molecules crossin,g unit area per.p..eIidicula'r.

ta

the beam in unit tiro e wiU be deflected by the fixed scatterer within a specifiedsolid angle

(dw). Let this fraction be G(

<p,

9 ) d~ where

cp

and 9 are the polar angles relevant to the direction of the beam (dw

=

sinq> dep d 9). Then G is called the scattering coefficient. In gener al G will dep end on the physical characteristics of the incident molecules and the fixed scatterer and on their relative velocity (Ref. 4. 1).

A second important characteristic of molecular scattering is the mutual collision cross section. This is defined as the total

fraction of the molecules crossing unit area perpendicular to the beam in unit time scattered over all angles by the intercepting molecule (Sc). Therefore

Sc =

~

G dfl..l

where the integral is taken over all directions .

(1)

Let us consider the dynamics of a beam molecule of mass mand velocity

S

colliding with a fixed molecule. In terms of the

-I

>

FIG. 1

notation indicated in Fig. 1, the angular momentum of the beam molecule about the position of the fixed molecule is m r 2

ck

and the kinetic energy is -!m

(r

2

+

r 2

eX.

2). If angular momentum is conserved, then

(2)

and conservation of energy requires that

~m(r2

+

r2~ 2)

+

U(r) = ~mg 2 (3)

where U(r) is the mutual potential energy of the two molecules. Then from (2)

•

eX

=

Substituting in (3) for ~ and solving for

r

J we have

• r

=

dr dt Dividing (4) by (5) we have do< dr

=

; r

T (4) (5) (6)

The integral of this is a path which is symmetrical ab out the distance of

nearest approach (ro )' Then the total change in 0< during the encounter is

2

S:

(dO( / d r) dr , i. e. the angle of deflection is

o .

cp

= TT - 2

rr.o

(do<) dr (7)

) ro dr

It will be seen that the deflection of spherically symmetrical

molecules such as those considered here is independent of 9 J i. e. since dUo)=

sin ~ d

Cf

dg, the fraction of molecules crossing unit area perpendicular

to the bearp direction in unit time and scattered between

ç

and

cp

+

d

cp

is

(8)

Now according to Fig. 1, the beam molecules which pass through the

annulus 21TTdT are scattered between

cp

and c()

+

d~ i. e.

cp

=

cp

cr) .

Now we may write d ~ =,

cP'(-r)lq-r-.

Then by definition

?..1rTo.T

,

,

or

G-(CQ)

=

(9)

Let us apply these results to molecules represented as

spheres having the properties outlined in Section 2. Then U

=

0

every-where except at contact between the two molecules dlilring which time the

change in eX is negligible and, therefore, the contribution to the integral

may be neglected: For this case ro = ~ (<Tl +(J2)

=

Cï12 (see Fig. 2

J

,

p.ll)

and

Then

Therefore

Also, since cos

dl>< dr

=

-c:

r

2 _{. ~} - ( 1 - -"'-r ) r 2 ... ~

I

cp

'(~J

I ::

~=l

2. 2 ro

sinCÇ>

=

2 sin

~

cos~

=

2 2

and

Therefore the scattering coefficient for spherical molecules is

r;~ - =

4

4-(10) (11) (12) (13) (14)

For these molecules the mutual collision cross-section has a simple geometrical meaning since

It will be noted that G is independent of

cp

for spherical molecules. The significance of this is that if the incident spherical mole

-cules are uniformly distributed then the fixed spherical molecule will scatter them equally in all directions.

In the above calculations the scattering molecule was held fixed. In actual gas flows the scattering molecule is much mOTe likely to be moving before and after collision. Let us consider the case of relativ«7 ,~,

scattering by allowing the scattering molecule to move at velocities _{I _} ~ , . ~

" before and after collision, respectively. Let the corresponding vë.l.ocities for a beam molecule be

r ,

"f,

I Then the relative velocity before collision is

r

r

=

f

1 -

f

2

~d

after the encounter the relative velocity is

~'r

=

-S'

1 -:.-

f";-

where the magnitudes

1"

r '

~'r

are equal but the ~ctors :tr , Y'r have directions which are different by an angle c:pr . Then the scattering coefficient is

(16)

The calculation of Gr ( ~ r) now proceeds according to the rules of relative motion, the scattering molecule being held fixed. Then our cal-culations above applyand G r (

cp

r) = (J122 the initial and final

velocities of the impinging bearn molecule teing

f

__

and

~'

.«

respectively,

..

~!.'/

and the angle of scatter being <P~ (compare Fig. 1).

Relative scattering is a complex problem. We consider a case which will be found later to be important in the study of hypersonic, low-density flows, i. e. the case in which the scattering molecule is initially at rest but free to move after collision. Then

r

2

=

0, and we have from Fig. 3,

(S'1)1 - ($'2)1

=

! r cos ~r (S'1)2 - ( 5"2)2 = J r sin q)r

FIG. 2

FIG. 3

Since momentum is conserved during the collision, then for identical molecules

(.5"\)1

+

(~'2)1 =

Sr

(18) (S\)2

+

(~'2)2

=

0

From these equations (17) and (18) we find th at

(19)

1

where

Q)

is the angle of

r,

with respect to the direction of

r,

(or Ox.). Hence CÇ>r

=

2

cp

We see also th at

(20) and therefore

(21)

Thus the beam molecule, af ter striking the scattering molecule (initially at rest), moves at speed

I

r

r cos~, at an angle

<:()

=

-!

q?

r to its original direction.

We now consider the scattering coefficient. We have seen that the fraction of molecules crossing unit area perpendicular to the beam

in unit time which are scattered between

ct>

and (Q

+

d

<Ç

is

21r G( ~ ) sin

<p

d ~. We note that G( (Ç) ) is still the coefficient of scatter of the incident beam molecules with respect to their initial motion

(the scatterer being initially fixed) independent of what happens to the scatterer af ter collision. Now we must have

(22)

i. e. the number of molecules scattered in an annulus is independent of the coordinate system. Therefore

G (~)

=

Gr

(CP

r)

sin CQ r sin

cp

For identical spherical molecules,

hence G(Cl')

=

a-2 4 sin

2.qJ

, 2

=

a-

2 cos sin

cp

(23) d

cp

r

=

2

dep

(24)

where q-,'2.

-=

cr- for identical molecules. The scattering coefficient for spherical molecules when the scattering molecule is initially at rest is, therefore, proportional to the eosine of the angle through which the beam molecule is deflected. The mutual collision cross-section is

:tr

5

c

~

'2.

ïï

cr- • )

Cc S <j)

'IV,

<ll

"<jI

= ...

11" (f" ,

o

(25)

In general it will be seen that the scattering coefficient will depend on the relative motion and the angle of scatter . It will not be a function of the individual veloeities through the relative positions of the colliding molecules since, according to the principle of molecular chaos, there is no correlation between the positions and velocities of different molecules. Also the scattering coefficient cannot depend on the position of the characteristic line of nearest approach (ro, Fig. 1) or line of

centers (Fig. 2) since if the velocities and all angles remain unchanged during any rigid body rotation of the problem then, according to the principle of

molecular chaos, the new positions of the molecules are just as probable as the old.

5. OTHER PROPERTIES OF MOLECULAR COLLISIONS

Up to this point the collision between two molecules has been considered with a view to determining the scattering coefficient. We now consider the dynamics of collisions between identical molecules from a more general point of view. If no force or potential energy exists between the colliding molecules when they are far apart, the laws of conservation of momentum and energy may be written

=- "('

+

r;'

-

)

1.. ~ ~

L-r

+1. -:.

_I

r.

_I

+~'

_'), (1) If the initial velocities are given, then the final velocities (six unknowns) may be obtained from the above (four) equations in terms of two para-meters. Let us define a unit vector

!3

having the same direction as the change in.velocity for the first molecule (i.e. the direction of

['-F).

N ow we note tha t

and

(

'[ +

I

T,J.

I (

f+$,) -::

I I

r

,"",~

+1,

+2!'5

" Therefore from the momentum and energy equations

:" r '. r:

_{_ . I} I

-

(4)

If we represent the molecules as perfectly elastic spheres, then

t1

lies along the line of impact joining the two centers and the velocity vectors can be resolved parallel and perpendicular to this line (indicated by the subscripts tand n, respectively). Then

1'-:. 1;

+1;

_t,

::.

f..,

₊

_r~

"-

-

-I - 1

-,

(5)

f

I I

_!/

I ":

!'"

+

r~

_{: 1'"}

_+-

t'

-

_ I

-'

Components normal to the line of impact are unchanged, i. e.

1';,' -::

r"

Combining the first of (1) and (4) we have also I

Jt':

1;

- I (7) Then

~ (r-~)

+ft.

- I (8) and

t,' :

f,.,

+

4

-

-'

...

(9)

These equations become

(10)

where

1' .... :

~

-

r

is the relative velo city before collision.

We note that ~ is a unit vector which can vary over the surface of a

sphere of unit radius and is specified by the two direction parameters. Thus these parameters must be assigned in addition to the initial velocities.

It will be noted that the relations (10) express a linear

transformation. By interchanging the roles of

r ,

f ,

and

'S' ,

r;' ,

-

-we can show that this linear transformation is the inverse of itself.

Physically this means th at the conservation equations describe two possiblee

collisions - a direct collision by which the initial velocities

r

1;

become the final veloeities

_,

1" ,

-

r:'

_, and an inverse encount;r by which

the initial values ., I ,

T,

are-changed to the final velocities

r '

1',

The Jacobian (J) fo; (lO),-is therefore equal to the Jacobian (J') for the

inverse. According to the theory of Jacobians JJ'

=

1. Hence J

=

J'

=

1.

We shall make use of this fact in the derivation of the Boltzmann equation. Intermolecular collisions are characterized by functions which remain invariant under the transformation (10). Thus

~

( 1: ,

~)

:

j (

(

r'

J

r,

J)

.

'

is called a collisional invariant. Since (1) and (10) are equivalent, then every collisional function can be written in the form

The summational invariant is a collisional invariant which may be separated into a sum of functions of

:s

and

l;

i. e.

(13)

This invariant is important in our consideration of the equilibrium form of the distribution function and it will be considered in detail here.

We will show that every continuous sum.mational invariant is a linear combination of Us arguments,

r

+

1;, ().

+

r.~

.

In

determining the form of

..y ,

we can restr1ct oürselves to 'variations in

the variables which leave

S'

and ~'unchanged. Then the following

equation for

tf

-~

di!.

+

dS.

t

(

(i = I, 2, 3) must be solved subject to the conditions

dr·+d~_{1 ::> 1}· =0 1 '( . d r ·

+

~1' d j i =0 1 1 ~ 1 1 (14) -,' (15) (16)

where the summation conventional applies. The procedure now is to eliminate by differentiation all unknown functions except one and then solve the resulting differential equation. From (15) d ! i

1

= -

dl i and

substitution in (16) and (14) gives

and

=

0 · = 0 1 (17) (18)

Using the Lagrange method of undetermined multipliers, we multiply (17) by K and add the result to (18) and obtain

l

àtp -

~tv,

+-

K

(~

- !

I )

J

~

ft

d

f,:

\..

'> ('

t,

,J

~

I == 0

I

(19)

But the d Sirepresent arbitrary variations and hence

: 0

(20) Then, solving for K, we have

K

:= (21)

The three quantities corresponding to i = 1, 2, 3 are therefore equal and we have such equations as

(

~

_

_{:t, ')}

(~ _ d

_af,

_d

j t \ _ (

_f"

_{J -}

_S;

_

~

) (

_rJ

dt;

_{s;a. -}

dcJ1 )

d"f~

,

(22)

Differentiating with respect to

5'

3, we have

(23)

(Recall that

~

,

'f;

are independent and

tv,

is not a function of

Ii).

Now the right hand side is independent of

r.

and differentiation with respect

to this variabie gi yes 2.,

d

'4lfi

dIJ

dr,

In a similar way we can show that

d"1.t..f

=

0 ₍₂₄₎

-=0 (25)

(and higher derivatives with respect to

r. )

t

Hence we conclude that depends only on

r...

t

Returning to (22), we differentiate first with respect to ~

with the result

(recall that we have

(

~

_

\

a~

oe.;'

J.Y:

~~ 1"l.,)ol~~

=

i!~-

;i

,

~

is independent of

f, )

and second with respect to

of:l.

ep·y.;

a

L

4'

::: Qf~ _d1:'"L. I ol or, in general ep·l.jI

,}:;1

_a~ ';:

rTf."

d

s:

1-' :

-tJs.~ I 3 (26) f~ and (27) (28) Now we. have seen that

ft.

=

f(

t;. )

only and (28) can hold only if all three derivatlves are equal to t1ie same constant k. Hence

Therefore and or

d<.JI

I

-

:. A (.

4---R. • d

f.

L ' t' (29) (30) (31) (32)

where the k's are constant. Thus we may write for the first term of the summational invariant

6. BOLTZMANN'S EQUATION

We require an equation from which the distribution function may be obtained. Let us consider a gas in which the molecules move with constant speed and direction until a collision occurs which changes the speed and direction in accordance wi th the conservation laws. We can direct our attention to the molecules of a given class, i. e. all molecules which at time t have speeds between

S

and

!

+

d! and directions in the ranges

cp,

cp

+

d

cp

and Qf , 9

+

d 9. In fact we have in mind those molecules in the gas which form a beam, having the same infinitesimal range of speed and direction. A basic question which involves the distribution function is: how many molecules are lost or gained by this beam due to collisions as we follow it through the gas? It will be seen that this is essentially a scattering problem and may be approached with the help of the theory in Section 4.

According to the definition of the distribution function, the number of molecules per unit volume in the beam is f(

S )

d

r

If

s,..

is the speed of a beam molecule relative to a given scattering molecule, th en the number of beam molecules crossing unit area perpendicular to the beam in unit time is

r

r f(rJ~

r.

We have seen that of these

molecules the nurnber scattered

ini'ö

an element of solid angle dw about the direction

cp

to the beam is

'ft-

G-(~J

(r-)

+( !)~,t1U where

G(ep)

f",)

is the scattering. coefficient. Now the number of scatterers in unit volume (having velocities between ~ and

!!./.

ti S, ) is ~

(!t)

~ Y, . ' There-fore the number of molecules lost to the beam due to scattering by collisions with

f,

molecules in d'-\) per unit volume per unit time is

(1)

The total number of molecules lost from unit volume of the beam in unit time "observed following the motion of the beam" is obtained by integrating over all angles of scatter and over all possible velocities which the

scatterer may have, the result being

(2 )

Although each direct collision between a beam molecule (velocity

f,

r

+

d[ ) and a scattering molecule (velocity

!l)

f; oio

ei

f )

results in the Ïos s of a particle from the beam (

S..:::;,

F' ),

there are also

- - 1

inverse collisions between molecules having initial velocities (

Si)

-r~dr.

- .) -J

r,'}

r,'../.

J

!t' )

which scatter particles into the beam (

!

'~

!,

Section

5). FOllowing the argument for direct collisions above, it is found that the number of molecules scattered into unit 'volume of the beam in unit time "observed following the motion of the beam 11 is

Now the product of the differentials

~!'

and

~~:

represents an element of volume in the six-dimensional (

S·_f.'

)-sPäëe and it is related to the

-'=.1

element of volume in the (

F )

!!

)-space by the relation

where the Jacobian

d

f'

J.-r,

=-

\JI

~

ti

<Ar'

- - - !

ei

(!'J

!!

'j

d(l:Js,)

(4) (5)

is evaluated for the linear transformation (5.10). In other works J

=

1 and the two elements of volume are equal. Then the total number of molecules gained in unit volume of the beam in unit time observed as we follow the beam due to inverse collisions of all kinds (i. e., all possible velocities and directions of scatter ) may be written

(6)

I ,

where

-r

and

f,

are given in terms of

'f ,

J;

byequations (5.10).

-

-

-Now we nöte that the rate of change of the number of mole-cules in unit volume of the beam observed while moving with the beam mole-cules at the velocity

7,

r

+

~

(;

is expressed as

t ) , ' t

l

~

+

-r.

di=' ]

~

r

àt

-

0")(

-(7 )

Equating (7) to the sum of (2) and (6) we obtain Boltzmann's equation

This equation is derived on the basis of certain assumptions i

which were introduced in Section 2. We have assumed point molecules '

and this permits us to express f as a function of x,

S ,

t alone. Further-more the collisions are considered to be complete

;0

that the characteristic time interval dt is much less than the average duration of a collision

but much greater than the ave rage time between collisions. It is also implied that f does not vary appreciably over the distance traversed by a molecule in time dt. Thus f is essentially constant over the effective volume occupied by the molecule. We must rule out, therefore, flows involving bodies of molecular size, but bodies with a characteristic

dimension of the order of

À

would be permissible. Perhaps the most

important assumption is th at of molecular chaos discussed in Section 2.

We have seen that it permits us to consider the scattering coefficient to

be a function of

Cl>

and Sr only. As we shall see later th is fact leads

to the Maxwellian distribution function for a gas in equilibrium (Ref. 2.2). In solving a flow problem we must express the boundary conditions in terms of the distribution function. In external aerodynamics

there are two requirements for f, the first that f tends to an asymptotic

value at infinity ( f .... fca ) and the second that f must meet some

require-ment on the surface of the body such as

(9)

where ~n is the component of molecular velocity normal to the surface.

This condition holds if no resultant accumulation of molecules occurs

on the surface and there is no oblation of the body. If the flow occurs in

a container, then the condition at infinity is replaced by the requirement

that the total number of molecules enclosed remains constant, i. e.

~ ~

f

+

~

=

constant (10)

Under similar physical conditions at the wall and body (if any) the distri-bution function must again meet (9). The interaction of gas molecules with the surface of a solid body is a subject for later consideration (see Section 12).

7. MAXWELL'S TRANSFER EQUATIONS

The transfer of mass, momentum and energy by the mole-cular motion can be obtained from the Boltzmann equation. In developing the latter we were concerned with the loss and ga in of molecules of a given

class due to collisions with all other molecules. Let us now suppose th at

each molecule has a property Q (

"'!)

(eg. mass, momenturn, energy,

etc.). Then the amount of Q lost-from the beam due to scattering by

collisions with SI-molecules in dw per unit volume per unit time is

(analogous to (1) in Section, 6). Similarly the amount of Q gained by the beam due to inverse collisions per unit volume per unit time is

Q •

!

r G(

cp

J

~

r) dw • f(

f

I) f( Sll) d

l'

~

f.1

(2)

-

-

-'

Proceeding now in a way similar to that outlined in Section 6 the rate of change of Q in unit volume observed while moving with the beam at a velocity in the range

r

i, Si

+

d.f i due to collisions with all molecules may be written

If we integrate this equation with respect to d! ,then MaxwellIs transfer equation for the rate of change of the total amOünt of Q in unit volume

becomes

We shall write this equation in the form

Let us now examine the symmetry relations which arise from integrals of the type

I

=

~

Q F (

Sr'

CP)

[f' f' I - f flJ dw d

t

~

(6) By interchanging rand

r

_{1 we have}

(7)

i. e. the only change is Q(

s)

~ Q(

f

1)' Now let us adopt the final velocities

f"'

r.'

as the variables of integration. Then

I

=f

Q'l F(ff j -i'f'l)dW dr' df,' (9)

where the latter is obtained by interchanging the variables. We note that during these transformations the relative velocity and angle of scattering

(i. e. F) remained unchanged. Then since d

-

~ d

sl

=

d

'f'

df/1

-

-(see Section 6), we have

I =

t

~

(Q + Qj - Q' - Q '1) F (fl fl1 - f f 1) d w d

r

d

-

ft

(10)

Now this can be rewritten

I =

t

J

(Q + Q1 - Q' - Q' 1) F f' f' 1 dtc>

~

d

f,

-i

~

(Q+Q1- Q ' _{-Q'1)Fii1 dW} df dS; (11)

If we exchange variables, we have

5

(Q + Q1 - Q'-Q'l) F i' f'l dW d! dl',

=

5

(Q' + Q'l - Q - Q1) F i i_{1 dW} d l d

f;

=- )(Q+Q1-Q'-Q'1)F i f₁ dW d! dE'; (12)

Therefore

I "

~ ~

(Q' + Q' 1 - Q - Q1) F i _{f 1 du.;}

'!J

d', (13)

It will be seen therefore that I

=

0 if Q is a summational invariant.

Maxwell' s transfer equation for the quantity Q may therefore be written

in the form

If Q is the mass of the molecule (m), th en according to

(3. 1)

~SQ

f

di'

=

.:L.

(m n)

Also

Since the collision integral is zero, the transfer equation for molecular mass becOInes

(17)

This is the well-known equation of continuity of continuum fluid mechanics. The molecular momentum is also conserved during collisions and hence for Q = m

Si'

the right hand side of (14) is zero. We have

Also

=

~~m

_Ox'

(u. u· +u· c· +u· c· +c· c.)fdf _{1 J i J J l i J _}

\

d

=

O-:x...

_{(m n Ui uj + Pij)}

j

Then the corresponding transfer equation becomes

~ ~u'u'+P'\=O

:r

1 J i J )

\7)(.'

J

We can wri te th\s in th e form

iA'

'4:.

~p a~t'

+""U'Jt.

((.)~ \J+P~' ~L'

4-

~J' ~

0

I.

dt

\

dt

tdXj \

J I J

d){J

OJlj

Two terms cancel on application of (17) above and (20) becomes

- 0

(18)

(19)

(20)

Finally the molecular kinetic energy involved in a collision

is conserved and again the collision integral is zero. Then with Q =

t

m

r

2,

(22)

Also

Then the energy transfer equation becornes

~

(/Ju2 +

3~)

+

.:L-

...

(fu2u. + 3 P u· + 2 u· p .. +E')

=

0 (24)

dl" (

'd"Xj

J J I IJ J

This can be expanded and cancellations made with the help of (17) and (20). We have finally

te

+-

.!L

(~

r\

+

.!..

1":)."

~\Á,

+

l.

c7

E;'

.J

t

fiX.:

i

J .3 '~'J "ï.""

3

-f T . ():>t..J d "Xl

(25)

The equations can be written in forms which make clear the physical

significance of the terms (Ref. 2.2). It should be noted that the

determina-tion of the components of Pij and Ei in terms of the velocity components and their derivatives requires a knowledge of the distribution function.

PART II

TRANSFER PROPERTIES OF A GAS IN MOLECULAR EQUILIBRIUM

8. DISTRIBUTION FUNCTION FOR A GAS IN MOLECULAR EQUILIBRIUM In developing the Boltzmann equation we considered the gain and loss of molecules from a beam with velocity components between

S

i and ri

+

d ~ i . Let us now consider the equilibrium condition which corresponds to the case in which the loss and gain of molecules by the beam are equal and the collisions have no resultant effect on the number of mole-cules in the beam. Mathematically we seek a solution of the Boltzmann equation which is independent of x and t. Then the partial derivative terms in (6. 8) disappear and we have left an integral equation of the form

Then for any function Q( ~ )

l f

(! )

f( :'1) - f(

r ')

f ( t,1) dW

~

=

0

(1)

) 9l:f

_r

G.(

ct

_J

'!.-

îJ[

.f.

("!)t(Sj) -

f

(1"').f(

(J] "''''

~

t!ti

=

0

(2)

Let us choose Q

=

log f. Then from (7. 10) the left hand side of (2) can be written

J

(log f

+

_{log f l} -log f' - log f'l)

Cr

r

G(CP.

E

rU

[f'f'l - f fJ dw d

3

d!'

=

0

- I

- (3)

where f

=

f(

f),

f 1

=

f (

r, ),

etc. Since

f

r G( ~ ,.

S'

r) is positive and the log function is monatomic increasing, then the integrand must be nega-tive over the range of integration and (2) can be satisfied only if

log f

+

log f 1 - log ft - log ft 1

=

0 (4) In other words log f is a summational invariant and may be written

(5&)

where i, j = 1, 2, 3, (see 5.33) or, introducing the five new constants k, ~ ,

eX

i we can write

We can determine k from the relation (3. 1), that is

Thus (7)

where the number density n is a constant throughout the gas. Also from (3. 3)

(e)

To obtain the constant

f3 '

we use (3.7), i. e.

CO

-t=

nRT =

~

m

k~~~

ei Ci e -

c~

l'

Ifl'

~ 'ra-r~

ol'!J

-~

=

~

m

'1I"J~~3'

3 (35 12 120 (9) which reduces to

(10)

The required distribution function for a gas in equilibrium has the form, therefore,

(11)

This is known as the Maxwellian, Gaussian or normal distribution function and the molecular motion of a gas in equilibrium is sometimes referred to as Maxwellian. It will be noted that this distribution function applies to a gas in whlch the macroscopie quantities (n, ui, T) are the same at all points. The pressure tensor has the form (see 3.8),

cc p .. mn

\}\

_ è'l. dC1 dC2 dC3 (12)

=

ci Cj e iJ _{(21T RT)3/2} -CC> where

ds

=

_{d ! l dS 2 dS} 3

=

dc 1 dC2 dC3' Then P .. = mn

r

(_{2RT)5/2 I1 2 I}

J

=0 ifi;J:j iJ C2lt RT)3/2

L

0

=

m nRT

=

t

R T

=

p if i

=

j (13)

Thus the gas in equilibrium supports normal pressure only and no other stress components appear.

The i th component of the heat flow vector becomes (see

3.11)

= mn

=

0 (14)

(21TRT)3/2

Thus a gas in molecular equilibrium does not support a resultant flow of heat (transfer of kinetic energy)

9. TRANSFER PROPERTIES OF A GAS IN EQUILIBRIUM AND AT REST We have seen that the number of molecules in the element of p~ase space is f dx d ~ • or in other words this expression gives the

n~mber of molecules

Gi

the element of physical space dx about x which

have velocity components between .~ and Si

+

d

r i .

In terms of polar coordinates about the xl-axis for the velocity components. the ele-ment of volume in the velocity space is

where

lP

and 9 are the angles specified in Fig. 4.

,1'3

~---~----~--- ~I) ~

o

FIG. 4

lf the gas is at rest, ui

=

0,

r

i

->

ci and we have

(2)

Then the number of molecules in dx having speeds between c and c

+

dc is obtained by integrating with respect to

cp

and 9, i. e. the required number is

c~ ~ ~

(~1r~\f>-

e -

e'dè:è;'~"';>I«lcl«lC.9: ~è'-e-è-~~d.3.. ~)

o

)0

Therefore the number of molecules per unit volume of physical space

hav-ing speeds between c, c

+

dc is ; ; - C 2e _C2 dC where C2

=

ci ci/ 2 RT or we may express this as 4ïi c2fe dc.

The significance of the quantity 2 RT can be readily

determined. The most probable speed which the molecules in unit volume may have is the value of c corresponding to the maximum value of _

h . 4n 2 -C2_ •

.. .. t e quantlty _ _ C e O\C. We have

rn-A(C2 e-C2 )

=

2 C (1 -C2) e-C2

=

0 (4)

which is true if C

=

O. 1 or 00 where C

=

-1 has no physical significance (0 ~ c (GO). A check of the second derivative shows that a maximum value occurs at C

=

1 or c

=

~2 RT

=

cm where cm is the most probable speed. Then we may write

n -C 2

e (5)

The mean speed of the molecules in unit volume is also used frequently. By definition the mean value of the molecular speed in dx referred to cm is

-_C

₌

1 4n

ndx

_fir

Therefore the mean molecular speed is

2 c = - - cm

RF

(6) 2

-"JTr

(7)

The mean free path or ave rage distance the molecules travel between collisions is an important quantity in the transfer process. Applying (6.1) to the equilibrium gas at rest we see that the number of collisions between the molecules having velocities in the range c. c

+

dc • and those with velocities in the range cl' cl

+

d cl' and scattered in the angle in d W per unit volume per unit time is

(8)

where fe is given by (5) above and cr is the magnitude of the relative velocity vector cr = cl - ~. The number of such collisions for all angles of scatter is

~l

(9)

where Sc is the mutual collision cross section (see Section 4). i. e.

Sc

=

2lT

(lJ"

G (

<p )

sin

CQ

d<Ç (10)

If now we integrate over <all possible velocities for the colliding mole-cules. we obtain twice the total number of collisions per unit volume in unit time. where it will be noted that each collision is counted twice. once as the collision of ac-molecule with a cl-molecule and once with the roles of these particles reversed. Thus the total number of collisions in unit volume per unit time,_ is

î

Sc

~

_{dc ) cr fe(c) fe (c1) dq}

Oi

_{00 co}

H

r

dc

)~r

cr e -C 2 -C 2 1 -QO -00 (12)

If all velocities are referred to cm, then (12) becomes

.!.

r\1.

ScC~

(fr

~è

(Ir

è

e

-è'-è'~01

è

l. 1r3

J

J

J - ),

J

~

_ I

-<::0 _00

(13)

Separation of these integrals can be achieved if we write

C _{r I l 1 1} 2

=

(c 1" - c") (c 1" - c")

=

y" y" _{1 1} (14) Let us introduce the quantity ~ where

(15)

Then expression (13) may be written

,V\2S.C""rrc-.

!è1

([~

. . .

~

'i

T~

- •

ë

t

U:

e - . •

cl

è

_r S

I",~

d

<I

d

9

jl}e

Jz:,rJz,dz

3 (16)

where we note that dc dC1

=

IJl

J

y

ot

z

=.!:.

~

y

8 ~

z.

This expression reduces to

(17)

which is the required total number of collisions in unit volume per unit time. Since each collision terminates two free paths, the tot al number of free paths in unit volume in unit time is 2 •

1..

_Vi

n 2 S _{c c ·} C =

~

n2S C

Now the distance travelled in unit time by ac-molecule is c and, since there are ~(c) dc molecules of this c1ass, the total distance traversed by all c-molecules in unit volume in unit time is c fe dc. Therefore the total distance travelled by all molecules in unit volum e per unit tim e is

Therefore. by definition, the mean free path is

(19)

The above value of the mean free path is the average dis-tance traversed between collisions by all the molecules of the gas in the equilibrium state. In some problems it is convenient to use a mean free path related specifically to a particular veloc ity group or beam of molecules within the gas. Let us consider the collisions experienced by a specimen molecule of the c-group. We can make use of the simple theory of the scattering of molecules by a fixed molecule (see Section 4) since, if the random motion is Maxwellian (equilibrium condition), the number of

~-molecules will be unchanged by collisions. Hence when our specimen molecule is knocked out of the ~-group at the first encounter, another molecule replaces it.

Let us consider a frame of reference in which the specimen c-molecule is held fixed. We require first the number of collisions

between this fixed scatterer and the molecules of the ~-group. The relative velocity of the encounter is obtained from

(20) According to (8) the number of c 1-molecules scattered by the single ~-molecule through all possible angles IDunit time is

(21)

or

n (22)

~

To find the total number bf collisions of the gas with the c-molecule in unit time we must integrate over all possible speeds and directions for the c 1 -molecules. Integrating with respect to

<p

and 9 we have the integrär"""

~

:1i

2 .!.

= 211 (C 2

+

Cl - 2C Cl cos

CQ

)2 sincÇ de(>

Let

u:

:

~

n~~;: ~h: ::e;~;

sin

~ ~~

o

We can write it in the form

I

=

1

(=.

~

b ) (a

+

b cos

t:p

)2 d(a

+

b cos

Cf> )

=

-cp · 0

=

~ ~

l

(a

+

b) 3/2 _(a_b)3/2]

(24)

(25)

where we note that a

+

b cos

cp

must be ~ositive for all values of

<p

and

henc e a

+

b "> 0 J a - b

'>

o.

Sinc e a

=

C

+

C 12 and b

= -

2 C Cl J then

(26) Therefore 2 I t: _1_ [(C

+

C 1)3

~

'C - C

11

3]

=

2 (C

+

~)

if C

>

C 3 C Cl 3C 1

=

2 (C 1

+

C 2 ) if C

<

C 1 3C1

Hence integration with respect to

cp

and 9 yields the result

4 n cm Sc _Cl 2 _(C

₊

C 2 1-_--_-) _e -C 1 2 _{d Cl if C)C1}

Rf"

3C

or

4 n cm Sc _{C (Cl}

₊

C 2 ) e-C· _{dC 1 if C}

_<.

_Cl

fïT

3C1

We now integrate over all possible speeds for the gas

molecules ( 0 ~ Cl <. CO ), i. e. we have

(27 )

The integrals involved here are ~ 11 = C

~tC12

e-C 12 dCl = C l - t Cle-C1 2

+

t

~

e-C1 2 d Cl] o 2

è

0

=

c ( - t c e -C + t ) o e-c,2 dCl ) = C (

r;r

cert

-

C -

i

C e -C 2 ) 4 (30) t 12

=

;C

~

_{c 1}4 e-C\ dCl

=

;C

C

-!

Cl e-C12

-j

C~

e-C12

+!

~

e-Cl-dCl o

c

2 1 -C 2 Cl

= - h e

)

3

Hence integration with respect to Cl gives the result

Sc

= n cm Sc

L

~1r

e-C2

+

(C

+

2~)er;'C

]

which is therefore the total number of collisions experienced by the c-molecule of the beam in unit time.

(31)

(32)

(33)

(34)

(35)

The mean free path for the beam molecules (

Àc )

may therefore be defined as the tot al distance travelled by the beam molecule in unit time divided by the total number of collisions which it experiences in ~it time, i. e.

or

Àc'"

êe

~

tin

Sc.

[11i'

e -

è\

(è -\.

,h.)e,...ç

~

]

This ratio is plotted in Fig. 5. r- - -

-X;i

-.---

~

V

~

/

/

/

o

/.0

FIG. 5

One of the basic transfer quantities which we require for a gas in equilibrium is the number of molecules crossing unit area from one side in unit time in the interior of the gas (Fig. 6). We consider

FIG. 6

an element of area dA in the gas. All ~-molecules which cross

ot

A

in time dt must have occupied the volume dx = ~. n dt dA ::! c cos

cp

dt dA

where n is the unit vector norm al to dA. Then the number of molecules of this specific class crossing dA in time dt is fe dx dc or

fe c cos<p dt dA. c 2 sinq:;> dc;;t) de dc (37) where polar coordinates are used in the velocity space. Integrating (37) over all possible velocities of molecules crossing from one side only

(0 ~

cp

~

-n-/2.. )

we obtain

'2.'1i

(:3 fe dc (o"./". sin

cp

COS

cp

d~

( dg. dtdA = nCrn dt <IA

t

J

\~

2Ji=

(38)

Therefore the number of molecules crossing unit area in unit time from one side only in a gas in molecular equilibrium and macroscopically at rest is nCm/2

Jrr

or (114) nc using (7) above. Since the corresponding

number crossing from the other side is the same, we have no resultant flow of mass across the unit area.

We can obtain expression (37) from another point of view. We have seen that the number of molecules in unit volume having speeds between c and c

+

dc was 41"rc2fe dc. If all directions of motion are equally probable, then the number of these ha ving directions lying in the solid angle dW is 41T c 2 fedc . (d

w

14-rr ). Hence the number of mole-cules having speeds in the range c, c

+

dc and direction within dW which

cross dA in time dt is

Now since dw

=

sin

4>

d

q>

d 9, we see that (39) is identical with (37) and we conclude that our assumption that all directions are equally probable in the interior of a gas in equilibrium at rest is valid.

(39)

We can derive other useful results from expression (37).

If it is integrated with respect to direction only, the result is the number

of molecules having speeds between c and c

+

dc which cross dA in time dt in any direction, i. e.

1T

c3fedc dA dt (or 4n C 3 e-C2 de dA dt). It will be seen that high speeds occur more

fre~ently

among molecules crossing a plane than they do in the gas in general.

If (37) is integrated with respect to c only, the number of molecules crossing dA in time dt having directions within the element of solid angle d IA:> is

1 nCm cos~ dw dA dt or

11'

2

JïT

ne

cosq dWdA dt

Thus the directional distribution of molecules emerging from aplane obeys a cosine law.

The number of molecules passing between two elements of surface in a gas in Maxwellian equilibrium and macroscopically at rest can be found with the help of (40). The number of molecules leaving dA

rJA

\

\- ciA

cas

cP

\

...

_FIG.

₇

in unit time in the direction of dA' is

where I n cm - , dA cos ~

cl

w

lr

2

Vïf

= dAl cos

c:P'

s2

Then (41) may be written

,

_{nC m}

-.

-T

2

J"..

,

cos

<p

cosq/ ----~--- dA dAl s2 (41) (42) (43)

The number of molecules leaving dAl in unit time in the direction of dA is

where n cm dA I cos ~. d Lu '

2Jïr

d~' = dA cos

cp

s2 (44) (45)

On substituting in (44), we obtain the same expression as (43). We con-clude that dA and dAl receive equal numbers of molecules from each other in the same interval of time in a gas at rest with its molecules in Maxwellian motion.