Recent trends in the mechanics of highly rarefied gases

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RECENT TRENDS IN THE MECHANICS OF HIGHLY RAREFIED GASES

JANUARY. 1960

BY

G. N. PATTERSON

UTIA REVIEW NO. 16 AFOSR - TN - 59 - 790

JANUARY, 1960

BY

G. N. PATTERSON

UTIA REVIEW NO. 16 AFOSR - TN - 59 - 790

A review of some recent investigations in the flow of highly rarefied gases is presented. The basic nature of the transport process in free molecule flow is deduced from the Boltzmann equation for the molecular velocity distribution function. The present semi-empirical state of our knowledge of the reflection of molecules from the surface of a solid is summarized and some directions for research are indicated. The aerodynamic properties of bodies in highly rarefied flows is considered with emphasis on the long cylinder as a case of special interest. The theory is extrapolated to the limit of very high speed ratios or Mach numbers and the results are compared with those deduced from the Newtonian flow theory. The application of recent studies of rarefied gas flows to the development of instruments for the measurement of the pressure, temperature and density of such flows is reviewed in some detail. The use of free molecule probes for the study of boundary layers and shock waves is outlined. The review is brought to a close with a very brief consideration of some factors involved in the collision-free flow of a plasma.

T ABLE OF CONTENT S

NOTATION ii

1. INTRODUCTION 1

2. MICROSCOPIC PROPERTIES OF A GAS 1

3. VELOCITY DISTRIBUTION FUNCTION IN FREE

MOLECULE FLOW 4

4. REFLECTION OF MOLECULES FROM ASOLID BOUNDARY 5 5. TRANSPORT PROPERTIES OF FREE MOLECULE FLOW

AT A SURFACE ELEMENT 9

6. FREE MOLECULE AERODYNAMICS 11

7. THE HYPERSONIC LIMIT OF FREE MOLECULE FLOW

-NEWTONIAN FLOW 13

8. FREE MOLECULE PROBES 15

9. APPLICATIONS OF FREE MOLECULE PROBES 19

W. COLLISION - FREE PLASMA FLOW 21

REFERENCES 24

A A. 1 B

--

B A .. IJ c C. I c m

c

e e ..

iJ

E ~ E

f'

_o h

J

NOTATION

area or coefficient (Eq. 9) according to context coefficient related to heat flux (Eq. 6)

Stefan-Boltzmann constant or magnitude of B

magnetic !ie ld

coefficient related to viscous action (Eq. 6) molecular speed, or speed of light

compommts of molecular velocity most probable molecular speed

molecular speed referred to cm (Eq. 4) drag coefficient

components of molecular velocity referred to cm lift coefficient

pressure coefficient

emissivity or exponential according to context the tensor

energy of molecule or energy flux according to context

e lectric field

rotational energy of a molecule

general velocity distribution functions Maxwell's velocity distribution function

Maxwell form of velocity distribution function suitable for diatomic gas

Debye length

k Kn I m M n N N_cc p q r r· ₁ R S St t T u Boltzmann constant Knudsen number length of tube mass of molecule Mach number

number of molecules per unit vo lume

number of molecules striking unit area in unit time or the number passing through a tube

number of molecules passing through a tube. '!Y,ithout

s triking the wa lls .

number of molecules which enter a tube and collide with the waUs at least once

pressure

macroscopic speed of particles (Newtoniar:. FlowV" charge on i th partic Ie

heat conducted by molecules heat flow due to radiation radius

Larmor radii

gas constant or flux of radiant energy according to context

speed ra.tio (uj c ) m Stanton number time

temperature

magnitude of molecular ve locity vector

.

.

" ,

,

.

U. 1 w

--

x X. 1 X. 1

e

K

A

components of macroscopic velocity C lausing probabHity function (ReL 6)

vector position coordinate position coordinates

external force vector or the ratio x/I thermal accommodation coefficient

accommodation eoefI cient associated with rotational energy of the molecule

ratio of specific heats

thickness of the shock wave

the ratio ER/kT or angle specifying the position of an orifice probe relative to the leading edge of a flat pla.te

inclination of normal of a surface to direction of macroscopic motion, or polar coordinate

coefficient of heat conduction mean free path

ratio of internal coefficient of heat conduction to the translational value or the ratio

J

lCm coefficient of viscosity

coefficient of bulk viscosity

number of excited internal degrees of freedom of a molecule

partiele velocity vector

components of molecular velocity density

tangential momentum accommodatioI1 coefficient shearing stress

polar coordinate

Note: other symbols of limited use and subscripts are explained in the text

•

'

1. INTRODUCTION

The purpose of this paper is to present a review of some

aspects of recent investigations in the mechanics of highly rarefied gases.

The value of a periodic review of the various branches of aerodynamics was clearly demonstrated by the contribution made to modern aeronautical progress by the six-volume work entitled "Aerodynamic Theory" edited

by William Frederick Durand in whose honour this conference is being

held. Many of us wiU remember the encouragement and guidance received

from this inspiring series of books in our early studies of aerodynamics.

Dr. Durand anticipated the importance of the work when he wrote in the

General Preface: "Such a present taking of stock should also be of value and of interest as furnishing a point of departure from which progress

during coming decades may be measured".

Progress in fluid mechanics is now extremely rapid, and the university research institute has a larger responsibility than that of

making original contributions. The somewhat more unselfed task of schoiarly study, coordination and summarizing of new information for

purely educational purposes is also a necessary function and is in

accord-ance with the Durand tradition.

2. MICROSCOPIe PROPERTIES OF A GAS

A classical study of a gas at some specified time would

show the existence of various particles all in different states of motion as represented by their position and velocity. The motion of each particle is the re sult of encounters with other particles, reflections from walls and acceleration induced by externally applied forces. The basic problem

of molecular fluid mechanics is the determination of the macroscopic

properties of a gas flow from the assumed microscopic motion.

It wiU readily be seen that the microscopie picture of a gas

motion can become exceedingly complex. For example, particle

encount-ers can involve complex molecules, dissociation, ionization and

recom-bination. Furthermore the refl.ection of a particle from a wall is a

com-plex process, although at present it can be adequately characterized by empirical coefficients from an engineering point of view. It is logieal

that a basic understanding of the transport properties of a gas should

begin with the simplified microscopic picture associated with rarefied gas flows. In this paper the term "highly rarefied gas" is intended to imply a gas wUh a microscopic motion free of particle-particle collisions. The first part of the paper considers the effect of reflections in free molecule flow. In the second part the action of electromagnetic forces on charged

particles in collision-free flow is discussed briefly.

In the microscopic picture of a gas at time

t

the basic

( 2 )

space d; at time t which have veiocity components lying within the

range

J, ,

~. -+- dj: . This number is proportional to the mag:,..

nitude of the element of volume in space ( c1X ) and the velocity range (d;'. ). It may be written

fdi;

d1:

where f is the ve ocity distribution function. In general f depends on t, xi'

f

. .

This function obeys the Boltzmann

equat'on t

~

+

f

.

~

+

X.

'Of

=

~f

(

1 )

.... t: , oXi '

01:

which states that the change in the number of partlcles having velocities between ~: and

j:

+

d

f:

"as we follow this group" is due to external

forces (Xi) and particle-partic Ie collisions ( Af), the latter being expressed in integral form.

The significance of the velocity distribution function is

seen when it is realized that it forms the basis for the calculation of the

macroscopie properties of the gas flow. i. e. tOC> 1"00-\-00

n(x;

,i)

=

Jd!1

Jor.

J

f(~i,ri'

t)r

J

=-fJdT

_ ... _ " c -IlO

( 2 )

is the expression for particle density (Ref. 1) in a monatomie gas, and

=

( 3 )

is the relation for the average partic1e (mass) velocity . The velocity of a

molecul'9 can be referred to the macroscopic or mass velocity in dXi by writing

r

.

::

U. -\" cl . Other relations are available for the components of the pressure tensor and the heat flow vector.

The basic characteristic of equilibrium or Maxwellian flow is that the number of molecules having velocities in the range

'1i '

~i + d

tt

is unaltered by molecular collisions "as we follow these mole-cules". The number lost by some collisions is regained by others, and a condition of equilibrium in the microscopie motion exists. In this case the velocity distribution function b.as thc Maxwe lian form (Ref. 2)

( 4 )

Here cm is the most probable speed or the maximum point on the distri-bution curve.

N onisentropic flow involves viscosity effects and heat con-duction and can be regarded as a "slight deviation" from equilibrium or

Maxwellian motion. The velocity distribution function for slightly nonis

en-tropic flow is

'

f

~

j, [ /

+

Al

C

i ( \ -

~

è')

+-

A'd

2',

è

_j

J

( 5 )

where

( 6 )

A.

I

and e .. is given in the list of symbols. Note that A. and A .. result from

heat cà1nduction and viscous action, respectively. 1 IJ

The above ve1.ocity distributions apply strictly to a

mona-tomic gas. The distribution function in Maxwellian form for diatomic

mole-cules may be taken to be (Ref. 3)

j

o

e

-[

I

f

(E)

o

( 7 )

where

E::.

EI!

Ik

T and

(ER)cy

is the rotational energy of a molecule in the

tt

th rotational state. This relation is based on the assumption that f

may be factored into a translational term and a rotational term. Note that we now have a translational temperature Tt and a rotational

tem-perature T R'

A distribution function which applies to both monatomic gases and a class of diatomic gases, for which the exchange of energy

between the translational and internal degrees of freedom is rapid, may be

expressed as follows (ReL 4),

tI

=

1,'

(El)

[I

+

A

(

~~1-

è')

+

Al

tl16-}~~+A(I- ~)(

(

8)

+

A:j

ti

~

]

This functton gives the number of molecules in a 6

+v

dimensional

phase space having positions between xi, xi

+

dXi J velocities in the range

~i

,Ji

-h::lf

i

and internal energy between EI' EI

+

dEI ' Note that

A

-

- T

)Ab ·

oU

OX~ ~

(

9)

where

~h

is the bulk viscosity (Ref. 5). Also

A

is the ratio of the heat

conduction coefficient for the internal energy to that for the random

3. VELOCITY DISTRIBUTION FUNCTION IN FREE MOLECULE FLOW

A study of the transport properties of a highly rarefied gas

can be conveniently made by considering first the case of flows in which no

external force is acting. If the velocity distribution function remains the

same with respect to time. then rOl' a point in the interior of the gas, Eq. 1

reduces to

( 10 )

If this equation is divided

b

y,.{f:f;

,the left ha.nd side becomes the

derivative of f in a direction specified by the three direction cosines of

y

.

Therefore, f is constant along a particle trajectory. Equation 4

describes thc situation in free molecule flow where we are concerned largely

with the effect of the reflection of molecuies from surfaces on the

macro-scopie characteristics of thc flow. Thus according to E q. 10 the

contribu-tion of the molecules having velocity components in the infinitesimal range ~ i )

t

i

+

dr·

to the mac roscopic properties of the flow in xi' xi

+

dXi

can be determined by tracing the trajectory back to a surface or the

un-disturbed gas and using the relevant vaiue of f.

This basic property of free molecule motion can be

illus-trated by considering the flow through a circular tube connecting a large

volume of gas with a vacuum (Fig. 1), the diameter (2 r) of the tube being

sma.ll compared wUh the Inean free path ( À ). The number of molecules

entering the tube wHl be

'Pff?-

'

rrr.l.. according to elementary kinetic theory

(Ref. 2). The basic

tran

~pgrt

probLem is to calculate the number of these

which ultimately emerge fr om the tube exit into the evacuated space. For

a tube of length

t

this number is

l

N

(l)

=

Nc~

(L)

-t

JtJJ

{x.)dNcr

("t-)

(

11) D

Where

Nee

is thc number of molecules which proceed from the inlet to exit without colliding wUh the walls, and _{N cr} is thc total number of

mole-cules which enter the tube and collide with the walls at least once (Ref. 6).

Then U d N_cr is the fraction of these molecules which enter the tube

inlet, strike the waU between x, x

+

dx and which either directly or aftel'

further collisions with the wall emerge finally from the tube exit into the

evacuated space. The integral gives the total number of molecules which strike the wall of the tube one or more times anywhere along its length

and finally reached the exit.

It will be seen that the flow through the tube is calculated

in terms of the contribution from the gas volume directly and the flux of

refLected molecules from the waU. Further consideration of this problem

win be deferred until the characteristics of molecular reflection from

4. REFLECTION OF MOLECULES FROM ASOLID BOUNDARY

The reflection process determines the degree of accommo-dation of the state of the reflecting gas to that of the wall. In the limits either no accommodation occurs (specular reflection) or fuU

accommo-dation results (perfectly diffuse reflection). We expect that observed

reflections will involve only partial. accommodation.

If the wall is perfectly smooth, then "mirror-like" or

specular reflection should occur. The velocity components of the incident

molecules normal to the wall are reversed in direction but unchanged in

magnitude on contact with the surface . The presence of the wall is sensed

by the gas only in terms of normal pressure. Since the tangential

momen-turn and energy of the molecules are unaffected by the coUision, no

adjust-ment of the reflecting gas toward the tangential velocity and temperature of the wall occurs.

If the surface is rough, perfectly diffuse reflection can occur. In this case the refiecting molecules have remained trapped by

the wall for sufficient time for the emerging gas to assume the

tempera-ture of the wan and to have lost any tangential mass motion relative to that

of the surface, that is, complete adjustment of the reflecting gas to the

condition of the waU has occurred. The diffuse nature of this reflection

arises from the fact that the ultimate direction of reflection has no

re-lation to the incident direction, and the probability that a molecule wiU

leave the surface at a particular angle is proportional to the cosine of

the angle with respect to the normal.

Gas-surface interactions have been investigated mainly by

the molecular beam technique (Refs: 7,8, 9.). A stream of mol.ecules is directed on a surface element an.d the flux of scattered molecules is measured at various angles relative to the direction of the incident beam

(Fig. 2). The beam is produced by the thermal effusion of molecules

from a generator through an. orifice. Tests on aircraft materials show

that the eosine law of scattering is valid for the spatial flux distribution

of air molecules (Ref. 8). Furthermore, the reflected molecules

possess a mean energy closely consistent with the thermal condition of

the surface .

We can characterize the actual reflection process in a given case in terms of over-all average deviations from "completely diffuse reflection" for which the scattered flux distribution obeys the

eosine law and the emitted molecules are in Maxwellian equilibrium with the surface. The transport of momentum to and from a surface in directions normal and parallel to the wall can be characterized in the

form of accommodation coefficients. For tangential momentum exchanges

we define

where "C"i and Lr are the actua 1 incident and reflected fluxes of tangential momentum, respectively. N ote that the definition of

Or

has essentially the form

c:sr::

(Tt

-lr)

/(

--rt

-

r::_w) since for perfect1y diffuse reflection.

Lw == 0 . Similarly, for the norma1 momentum exchange we may write

--

( 13 )

where Pi and Pr are the actua1 incident and reflected fluxes of normal momentum, respectively, and Pw is the emitted normal momentum flux for comp1etely diffuse reflection. These momentum accommodation coefficients have the following limiting values:

For complete1y specu1ar reflection,

6N

~

<ST::

0 For complete1y diffuse reflection, ~N #(),..:: I

If the interaction involves a combination of completely diffuse and specular reflection, then 6'N is not independent of G"",. and on1y one of these

eoefficients is needed (Ref. 10). In an actua1 physical case, however, it is expected that ~j' and

<SN

wil! be independent.

The above momentum accommodation coefficients wilt be useful for the calcuLation of aerodynamic forces. They have been defined in terms of ma.croscopie quantities that can be determined experimentally. The mo1ecular beam experiments of Hurbut (Ref. 8) indicate that for

ordinary aircraft materiais, 6"",-

=

1. No experimenta1 determination of ()N appears to be available at present though there is evidence to show that

GN';"

I (Ref. 11).

We de fine the therma1 accommodation coefficient as follows:

0<.=

Ei - E('

E"j - E w

( 14 )

where Ei and Er are the actual incident reflected energy fluxes,

respectively, a.nd Ew is the emitted (reflected) energy flux for completely diffuse reflection. We have the following limiting values:

For comp1etely specular reflection, Ei

=

Er' 0<.

=

0

For complete1y diffuse reflection, Er

=

Ew' rÁ

=

1 The characteristics of the mo1ecu1ar motion of a highly rarefied gas can be iltustrated by considering an experiment for the mea-surement of the thermal accommodation coefficient (Ref. 12). We investi-gate the properties of the motion between two infinitely long, concentric cylinders. When no gas exists between two cylinders, the flow of heat wilt be in accordance with the Stefan-Boltzmann law. If the reflection of

radiation is diffuse and the cylinders are made of the same material, then

the heat flow due to radiation is

Q

=

kA,

(T,4-Ta~)

( 15 )

R

.-L ( I _

AI)

AL-e - - A

where e is the gray body emissiv1ty , A 3. is the area of a cylinder, and

k is Boltzmann's constant (see Ref. 12). The subscripts, 1, 2 refer to the

inner and outer cylinders, respectively. Tests can be run to determine e,

and QR can then be calculated in subsequent tests in which a gas is present.

If now a gas is allowed to fill the space between the two

çylinders such that its mean free path is many times the intervening

dis-tance, then heat wiU be conducted between the surfaces by the molecules.

By extending Knudsenis theory of conduction through highly rarefied gases

(Ref. 13). we obtain

Oe

='

F

(-<) [

( 16 )

where Pi' Ti are the gas pressure and temperature and the function

F (0<. ) contains a constant which depends on molecular weight, the ratio

of specific heats, and the geometry of the apparatus. This equation is

determined from a consideration of the translationa 1 energy transported to

and from each surface by the molecules, the effect of intermolecular

collisions being neglected.

Heat can be introduced into the inner cylinder by an

elec-trical method and drawn off the outer cylinder by a water jacket (Q). Then

Qc

=

Q - QR' and we can find the thermal accommodation coefficient from

Eq. 16. By changing the outer cylinder. the value of 0( for a number of

materials can be found.

Values of 0( obtained by this method are as follows

(Ref. 12):

Type of Surfac e

Machined A luminum E tched Bronze

Polished Bronze Etched Aluminum Etched Cast Iron

Machined Bronze

F lat Lacquer on Bronze Polished A luminum Polished Cast Iron Machined Cast Iron

Value of 0.95 - O. 97 0.93 - 0.95 0.91 - 0.94 0.89 - O. 97 0.89 - 0.96 0.89 - O. JO 0.88 - 0. 89 0.87 - 0.95 0.87-0.93 0.87-0.88

for a given gas and surface. independent of stream velocity or the

tem-peratures of gas and surface. More comprehensive data a;pe needed at

high stream (macroscopie) ve10cities and high temperatures. At present

H appears that we may take 0:..

=

1 and 0( between 0.87 and 0.97 for

I

ordinary aircraft materials.

The above considerations hold strictly for a monatomic gas. It is c lear that degrees of accommodation of other energy modes

of the molecule must occur when polyatomic gases reflect from a surface.

For example, if the molecules possess rotation. then the collision of

such molecules wUh a waB wiU involve accommodation of the rotational

energy and we may define a new energy accommodation coefficient in the

form

(ER), -

(ER)w

( 17 )

where (ER) and (ER) refer to the conditions of actual and complete accommoda1ion of

rot~tional

energy, respectively. Some experimental

evidence exists to show that the accommodation coefficients for

trans-lation and rotation are approximately the same (Ref. 14). The subject of

accommodation of ether energy modes, e. g. vibration. dissociation,

awaits theoretical and experimental investigation.

Some interesting new possibilities exist for the study of molecular collisions with surfaces. It appears that studies of wall

interactions are possible either by the use of a small supersonic nozzle

which replaces the usua.l oven souree (Ref. 15) or by the acceleration of

ionized molecules in an etectric field followed by neutralization (Ref. 16).

Since the latter tends to produce molecular energies above that of interest

in aerodynamics, brief mention wil be made here only of possible

applications of the nozzle me hod.

The nozzle source has been found experimentally to pro-duce higher beam intensities than standard effusive methods (Ref. 17).

Intensities of ab out 3 x 101'1 molecules per square centimeter per second

were achieved which is roughly 200 times that obtainable fr om an oven source. It has also been observed (Ref. 18) that higher energies for

heavier molecules can be obtained from a nozzle by mixing the heavy

gas with a light gas. The mean velocity of the heavy molecules approaches

that for the mixture. Furthermore, the light molecules diffuse outward

leaving a core of high-energy, heavy molecules which have lost most of their random motion in the e~pansion process.

Perhaps the most interesting possibility relevant to the use of the nozzle source is the control of the relative amounts of the

tra.ns1ational, rotational and vibrational energies of the molecules in the beam by adjusting the inlet conditions and area ratio of the nozzle. This

technique may therefore lead to a better understanding of th,e accommo-dation of translational, rotational and vibrational energies during a

surface interaction.

5. TRANSPORT PROPERTIES OF FREE MOLECULE FLOW AT A

!

SURFACE ELEMENT

We shall now consider the transfer of mass. momentum,

and energy to and fr om an element of surface of a body in a free molecule

flow. The fundamental characteristic of free molecule flow is that on the

average molecular collisions are very remote from the body and the

transport of mass. momenturn, and energy to a surface by the incident molecules is independent of the transfer of these quantities away from the surface by the reflected molecules. In other words we can treat the

in-coming and emergent streams of molecules separately (Refs. 19, 20).

It is known that very large mean free paths occur in the

upper atmosphere. It should be noted. however. that although the mean

free path rnay be considerable, the number of molecules in the element of volume still large. For example. when the mean free path in the upper atmosphere is 10 ft. , the number of molecules in a cubic inch is still

about 1013 . The definitions of the macroscopie properties of a gas

(p,

J

.

T, ui) through the velocity distribution are still valide

Let us ca1culate the exchange of mass, momenturn, and

energy in a free molecule flow at an element of surface with a normal

anti-parallel to xl assuming that the molecular velocities are

distri-buted according to Maxwell's law (Eq. 4). The number of incident molecules striking unit area in unit time having velocity components in

the range ~~ ) Ft -rdFt

':5

ft

Eo

elf.

dy ...

d

r"

where fo is expressed in terms

of n

i the number density of incident molecules. Then the total number

of molecules incident on unit area in unit time is

( 18 ) These integrals can be evaluated with the help of Eq. 4. The result is

N,

~

11

iJ ::

l

e-

S,:

~

.

.r;r (I+e'f

SN)

l

( 19 ) where

< _

~ =[~

M

...; - C"", 2. ( 20 )

where u is the magnitude of the mass velocity vector and the quantity S is called the speed ratio. For completely diffuse reflection

a 1-"" -\-1)0

N" '"

J

de,

J

d

C~

f

c,

:I,

de.,

~

-co -"" - 0 ( )

where fo is expressed in terms of nw the number density of reflected molecules for completely diffuse reflection. Assuming that all incident molecules are reflected, then Ni

=

_{N w}.

The molecules which have velocities between ~. and

~i

-t- d

~:

transport momentum

"'('(\!I

normal to the surface. Then the contribution to the normal pressure due to the momentum transported by the incident molecules is toD

~

~ ~ f;~:.J:;~_)

t,2

f.

dr

_J ( 22 )

Similarly the normal pressure due to the diffusely reflecting molcules is

P

N

-=

Y'Y'I Ni

rn-.z

RT

", ( 23 )

where we make use of Eq. 21. Now the total normal pressure acting on the surface is

( 24 )

Therefore ~ 00 ~ 00

p

=

~

J

:'f,_i

d

~"'"

f

J,

[(2-';-N)f>

"N

J~ f}~J

(

25 )

which permits us to calculate the normal force per unit area due to the

incident and reflected molecules.

The incident molecules, having velocities in the range

k~

I

I

:

1"

d~

i

transport to the surface a y-component of tangential momen-turn of amount m ~J. The tangential stress in the y- direction resulting from all incident molecules is

re;

= ":

j

~f~ J:~:_~j;f,;'

dt

(

26 )

From the characteristics of completely diffuse reflection,

rr:

=

0 and the

total tangential stress in the y-direction is

( 27 )

These re lations, applied to an element of area on a body

and integrated over the whole surface, will provide the resultant forces on various aerodynamic shapes in free molecule flow.

Let us now consider the ba lance of energy for the element

( 28 )

where Ei> Er are the incident and reflected fluxes of molecular energy, Ri' Rr are the incident and emitted fluxes of radiant energy, and Q is the heat removed from unit area of dA from inside the body ( a known quantity). In accordance with preceding ana lysis

E;

='

~

:

I~X F~~J~~(rK

r

K)

t.

J

t

(

29 )

For completely diffuse reflection the corresponding expression for Ew is

( 30 )

Then according to the definition of the thermal accommodation coefficient

(Eq. 14)

( 31 ) The above calculations are true for a monatomic gas.

6. FREE MOLECULE AERODYNAM1CS

The above analysis has been applied to the circular cylinder in Refs. 21, 22, 23, For a monatomic gas the coefficient of

tota 1 drag is '1.

CD

=

f

[,;% ;

1.~(S'.~)(I.+I,)}~R]

(32)

where 1

0(S) and Il(S} are modified Bessel functions of order 0 and 1, res-pectively. Thus in free molecule flow the drag coefficient depends on the speed ratio S and the temperature ratio T r/ Ti. The temperature ratio can be obtained from the energy balance equation for the cylinder.

( 33 )

where

and the reflection process is such that only the thermal accommodation coefficient is different from one.

The results of measurements of the drag coefficient of an

insulated circular cylinder at various speed ratios as obtained in a low

density tunnel are shown in Fig. 3 (Ref. 24). The trends suggested by

Eq. 32 are borne out experimentally but noticeable deviations do occur

(see curve for diffuse reflection). In particular the results show na de-pendenee on the Knudsen number. Note that a second theoretical curve

for diffuse reflection has been included in which allowance has been made

for the rotational component of the internal energy characteristic of a

diatomic gas (Hef. 24). It is considered that interference effects are responsible for the values of CD which He above the theoretical curve at

low values of S. On the other ha.nd the experimental values of CD lie

below the theoretical curve for diffuse re!1ection at large S. Figure 3 also shows the variation of CD with S when specular reflection occurs. It is interesting that the mea.sured values of CD 1°18 below this curve also.

A recent detaHed investigation of the drag of a range of

cylinders at a speed ratio of 1. 67 (M=2. 0) over a range of Knudsen

numbers has yielded an asymptotic value of CD for the infinite cylinder which checks closely that given On Ref. 24. The variation of CD with Kn through the transition region into the free molecule regime (Ref. 25)

is indicated in Fig. 40 The experimental value of CD becomes independ-ent of Kn above Kn = 4. The asymptotic value is below the theoretical

values of CD calculated on the assumption of diffuse and specular

reflection. It seems apparent that the discrepancy between the theoretical and experimental results cannot be explained by variations in the reflec-tion process. H should be remembered, however, that CD has been

·extrapolated to an infinOtely long cylinder and that fuU free molecule flow cannot be achieved with respect to this dimension.

Calculations of the aerodynamic properties of various

geometrical shapes have been made by several authors (Refs. 25,26, 27, 28).

In Ref. 28 the effects of diffuse and specular reflection on the drag of a

flat plate, cylinder, sphere, and cone have been considered. It is

interesting that the drag coefficient of an inclined flat plate at low angles

of attack is greater for diffuse reHection whHe at large angles of attach CD is larger for specular reflection (Ref. 28). However, the drag

coefficient for cylinders and spheres is always grea ter if the reflection

is diffuse (Fig. 4).

The optimization of geometrical shapes for minimum drag

in free molecule flow has been considered for nose shapes (Refs. 29, 30).

As suggested by the expression for the drag coefficient of a cylinder (Eq. 32). nose drag depends in general on the speed ratio S and the

temperature ratio T riT i (see notation) . The minimization of the nose drag by the method of variational calculus has produced the result that in free molecule flow the nose tip should be flat. In the hypersonic limit the

nose shape is ogival with a blunt tip and is independent of the temperature

temperature ratio. The nose tip remains blunt but the aft shape is convex or concave, depending on whether the body is hot or cold (Fig. 5). These

calculations assume aU accommodation coefficients to be equal to one. The interesting fact that arises from the energy balance

equation is that the insulated cylinder attains a temperature higher than

the stagnation temperature of the incident gas flow. For an insulated

flat plate aligned in the direction of flow of a monatomic gas, the energy

balance equation reduces to T wiT i

=

1

+

112 S2. The stagnation

tempera-ture for a monatomic gas is T. IT i = 1

+

215 S2. Therefore T w"7 T O'

This effect has been found experimentally to exist for circular cylinders

(Ref. 22). On the other hand the cylinder did not attain the fuU

tempera-ture rise predicted by theory. The authors ascribe this to heat losses due to conduction through the end supports and radiation.

7. THE HYPERSONIC LIMIT OF FREE MOLECULE FLOW - NEWTONIAN FLOW

The limit of free molecule flow as the speed ratio or

Mach number tends to a very large value is now of considerable interest.

On the assumption that the transport processes of the incident and

reflected molecules are independent (free molecule flow). the coefficient

of normal pressure acting on a flat plate inclined at an angle

e

to the direction of the mass flow is given blY (Rei. 31)

l.

[_I

I..L

't

~ ~)

-

5:

Cp

=

SL(\

e

SN

lVir

2.$N

J

t

e

+

(I

+

2.~2.

oot

rrr

{-r;. )

(1+

er}

SN) ( 35 ) N ~ SN

Tl

The corresponding relation for the coefficient of skin friction is

~ ~"J-C

J

=

"Sll"\eC()~e[~SNe

-+

l+e'(fSN ] (36)

In these relations SN:=' S Sin

e

where S is the speed ratio, and it

has been assumed that ~"" 0 ,

=

1.

The resultant force on the flat plate depends on the ratio of the temperatures associated with the reflected and incident molecules

(T riT i) as weU as the speed ratio and angle of incidence. This

tempera-ture ratio can be calculated from the energy balance equation, usuaUy

involving radiation effects, and the accomodation coefficient (Rei. 31).

When the speed ratio becomes very large (hypersonic

condition in free molecule flow),

The first limit (C p) will hold only if T riT i remains finite while S be-comes large. The physical significance of these limits can be readily seen. As the speed ratio becomes very large, the macroscopic (mass) velocity of the molecules beeomes much greater than the most probable speed of the random motion, and the molecular motion assumes effec-tively a simple form in which all molecules are moving in parallel paths

at the same speed. Then the number of molecules striking unit area in unit time is Yl

q,

Sin

e

0 The total normal and tangential momenturns carried by the incident molecules to unit area. of the surface are f~l.Si.,,le;

(f.::

m71

)

and

ff

Sine co:sG respectively where ~ is the resultant particle of velocity. The limiting values given in Eq. 37 imply that the reflection process makes no appreciable contribution to the exchange of norma! momentum, i. e. the restriction on T r/Ti as S increases means a low surface temperature and negligible momentum associated

with the reflecting molecules. Of course» the diffuse nature of the reflection is such that the emergent molecules make no resultant contri-bution to tangential momentum at all values of the speed and temperature ratios.

Thc above limits are determined on the assumption that the kinetic theory of gases ean provide an adequate description of the refleetion process. HovJever, when the incident molecules have very high energies, the collisions wUh the wall may deviate appreciably from

thc elastic type, i. e. they may become "plastic". When high energy molee' .. ".·,:

cules are temporarily trapped by the wall, they may be capable of excit-ing rotations and vibrations of the crystal stoms. The vibrational

ampli .ude of the atoms of an iron crystal may be increased to such an extent that local melting occurs on the wall. In this connection it is interesting to note that for melting of iron an energy of 0. 6 eV per atom is required. On the other hand the kinetic energies of a nitrogen

molecule at speeds of WO and 10, 000

mi

sec. are about O. 001 and 12. 2 eV respectively (Ref. 32). It appears to be possible tha.t the temperature associated with the translational motion of the molecules emerging from the wall wiU be high compared with the temperature of the incident mole-cules as the speed ratio increases (T riT i becomes large as S - 00 ). Therefore, for very hot bodies, the limiting value for Cp will be more

accurate if the ratio (T riT i) SN is retained, i. e.

2

"5i/r;

[1+

,[ïr

{!!- ]

2. 'SN

-r

.

L

( 38 )

It is also possible that thc reflection process will involve dissociation and ionization of the emergent gas molecules. The subject of the reflection of high energy molecules from a wall must be left for further theoretical and experimental investigation.

It is interesting to compare the ümit of free molecule flow as the speed ratio increases to a large value with a flow described by Ne·wton. NeNton assumed an inelastic collision between a gas

molecule and the surface of a body in which the normal component of

molecular velocity is destroyed and the tangential component remains

unchanged. The number of molecules striking unit area of a flat plate

in unit time is Y1~ :5 in

e

and each molecule experiences a change in

normal momentum of 1'nt( Sif'l

e

Therefore the normal pressure on the surface is (Ref. 33)

p:;

~ ~2-

Sin 'l..

e

) Cp ( 39 )

which agrees with Eq. 37.

According to Newton's theory, all the reflected molecules move along the surface with an unchanged tangential velocity <:e cos9 . In

this respect Newtonis theory differs from the hypersonic limit of free molecule flow (see Eq. 36).

In Newton's theory no restrietion is made about the density of the partieles striking a body. If we consider a continuum flow

in which the density is kept constant but the Mach number is allowed to become indefinitely large, then the random motion of the incident

mole-cules becomes insignificant compared with the macroscopie motion and

the latter tends to become that described by Newton. Thus the transport

process involves collisions of gas molecules with the body, and

encount-ers between gas molecules themselves are not important by comparison.

It appears, therefore, that both rarefaction and high speed tend to pro-duce a free molecule transport process.

An interesting verïfication of the limiting expression for

C (Eq. 39) is given in ReL 34. The normal pressure around a

hemis-pfi>ere-cylinder model was investigated in a shock tube which simulated

conditions of hypersonic flight (Fig. 6). Since measurements were made

on Mach waves, the results apply only to the supersonic region of the

flow. In spite of a complex flow pattern involving a detached bow wave,

expansion waves. and a boundary layer, the value of Cp given by Eqs. 37, 39 was verified.

8. FREE MOLECULE PROBES

One of the first applications of modern research in the

flow of highly rarefied gases was the use of free molecule probes to study continuurn, transition and slip flow in low density wind tunnels.

While the çharacteristic dimensions of boundary layers, shock waves,

wakes, etd. are large compared with the mean free path, probes can be constructed of sufficiently small size such that they experience free molecule flow under all operating conditions. The properties of the probe are therefore known in terms of free molecule theory. However,

The work of developing free molecule probes was initiated

at the University of California. Thc free molecule aerodynamics of the

cylinder was carried a stage further to inc lude the case of the more

gener-al velocity distribution function for nonisentropic flows (Refs. 10, 35). Thus the properties of a cylindrical wire are known when the probe is placed in a boundary layer or shock wave. Free molecule pressure

probes were subsequently developed at the Institute of Aerophysics (Refs.

36, 37).

Let us consider the pressure probe in the form of an

orifice in the side of a drcular tube of small diameter (Ref. 36). The

speed ratio, pressure, and temperature of an external gas are S,

Pi

Ti

respectively. Inside thc tube the gas is at rest and the pressure and

temperatur become Pr and T r. Equation 19 gives the number of

mole-cules which enter unit area of the orifice in unit time if the random

velocities of the incident molecuLes are distributed according to Maxwell's

law. The number of molecules emerging from the tube through unit area

of the orifice in unit time is given by Eq. 21 with the subscript r

replac-ing w. EquiHbrium corresponds to thc condition of no resultant flow of .

mass through the orifice. Therefore m Ni :: m Nr, or

b

rf

l. 1.

rr

Tt

-

S CCl5 ~

-

~

-=

e

+

r;-

5

C~e

l'

-t

erf

S

cóS~)

P:

t, ( 40 )

where

e

is the angle between thc now directions and the normal to the orifice (Ref. 36). Now it has been found experimentally that various orientations of the orifice have no appreciable effect on the temperature

inside the tube. Furthermore, if we evaluate Eq. 40 for

e.=o

I f[ a,roJ 7T

we obtain thc useful result lo

~

-hl

2frf

J=;

5

=- ( 41 )

It appears therefore that a free molecule pressure probe can give a

direct measuremen. of speed ratio in terms of the forward (Po) sideways

(

Ps

),

and rearward (

e'

)

facing pressures only. Equation 40 has been

checked experimentally in the UTIA low density wind tunnel (Fig. 7, see

also Ref. 38).

Equation 41 holds strictly when the external molecular

motion is Maxwellian (see Eq. 4). The usefulness of the probe can be

considerably increased by calculating its properties in a free molecule,

nonisentropic flow (see Eq. 5). Thus Eq. 40 becomes

~r~

=(I-r-L A

Scos~ _..!...A.f..S

.

Si1lSGsa~ ~co~G+

Al.

5,/8

~ ~e 5 \ S l.. i.. 2.. :l.

- 5 as

e

.'\

- AIR. "SiflB

cose)e

-t-

r:;r

CS CESB (I-terf S

C~.}r7.J

and Eq. 41 has the form h I

A

_lé-~(,+~)

5 - C" ~

In viscous, heat conducting flows in _{which Ai' A ij are smalt compared with}

1, Eqs. 40, 41 will be sufficiently accurate for the determination of

pres-sure and speed ratio.

The calculation of the properties of a free molecule impact probe involves a knowledge of the flow of a highly rarefied gas through a circular tube. The method for determining free molecule flow

through a tube was introduced in Section 3. Referring to Eq. 11 we see

that _{N cc} and N r must be calculated in accordance with the molecular motion of the gasCat the inlet and exit. and the laws of reflection of mole-cules fr om the internal walls. A consideration of the geometry of the

problem show s that

(x=:

-T-)

( 44 ) )

dX

Furthermore. if the reflection is diffuse, the probability function tU

(X, ]))

is the same as tha t for Knudsen flow (~ :::. 0) and can be obtained fr om

Ref~ 6. There remains therefore the calculation of N cc.

Assuming Maxwellian motion, the number of molecules

which enter in unit time through an element of area dA of the tube inlet with speeds between

1\

and

A. "'"

d

t\

;

("

=-

~

lCm

)directed between

't '

t

+

d

f

and

e,

e

-tde

is

ni~/J.cl'1l

e><p

(_Al.

+

25/\ co.sc:p -

:slo)

A

3

_d

_A

_Si"

_cp

COS

f

d

f

cl

G-

ei

A

(

45 )

The integration of this expression over all possible speeds ( 0 ~

A

~ ()O )

over the outlet area ( o~ ~ ~

lP

l

i

a ~

a

~:t"TT) and over the inlet area will

produce an expres _{sion for N cc} in terms of S and D (Ref. 39). The

theoretical variation of ~~rr./p,

rr:.

with the speed ratio S at D

=

0.04

is shown in Fig. 8. Note that die subscripts 1 and 2 refer to free stream

and gauge volume conditions, respectively. An experimental study of

free molecule impact probes in both a low density tunnel and a whirling arm apparatus appears to confirm the theory over the test range of S (Fig. 8).

Let us now extend our calculation of the properties of a

cylinder in nonisentropic free molecule flow. The method outlined in

Section 6 may be used again with the more general velocity distribution

function (Eq. 5). The results are (Ref. 10)

( 46 )

CL..

-

A,~

CLI) ... -

(A~/2-

s)

CL~

( 47 )

where the partial drag and lift coefficients (C D , . • • •• C_L2) are

functions of the speed ratio S and arise from die deviation from Maxwellian

(isentropic) flow. The relative importance of the drag terms is indicated

Maxwellian motion. We conclude that thc aerodynamic forces on a cylinder

in nonisentropic free molecule flow are affected by the non-Maxwell~an

components of the distribution function. In general the nonisentropic effects

are small except in regions of low speed ratio where forces due to the heat

flux terms (A.) become significant.

1

The energy balance equation for the cylinder must also be

reconsidered using the velocity distribution function for nonisentropic flow

(Eq. 5). The Stanton number is defined as follows,

~

St

=

( 48 )

Where Q is the net heat loss per unit time per unit length, r is the

radius, T w is the cylinder temperature, and

7;w

is the equilibrium

cylinder temperature. Then the heat transfer in nonisentropic flow is

governed by the relation

Si

==

st -t(

~~

)

s(

-t

A"

Stil

+

Ah

~l-~

( 49 )

The partial Stanton numbers corresponding to the nonisentropic terms in

the velocity distribution function are small compared with Sto' The energy

balance is only slightly influenced by the viscous stress terms but, as in

the ca.se of aerodynamic forces, the heat flux contribution can be

appreci-able when the speed ratio is low. It should be noted that the Stanton

num-bers are all functions of the speed ratio S and the thermal accommodation

coefficient 0<.. . The basic fact indicated by the above analysis is that

if we know the speed ratio, the accommodation coefficient, and the velocity

distribution function for a given flow ( A., A_{ij )} then the temperature of

the cylinder (or wire) can be calculated.1

The above results apply to a monatomic gas. For a diatomic gas flowing in Maxwellian equilibrium we may include the effect

of rotation by using the a ppropria te ve loc ity distribution function (E q. 7). The method proceeds as outlined above except that we now consider that

the temperatures associated with translation and rotation are different

and we must use two accommodation coefficients ( a<, o<.R, Section 4). For molecular equilibrium the accommodation coefficient and temperatures

(T, T R) are essentially constant in the integration since variations of

these quantities have no effect on the integrations over the velocity space

and around the cylinder (wire). The energy balance equation now becomes

(Ref. 3).

(

0<

)

0.(5)

=

,

~

-t- 0(1<.

bes)

( 50 )

where the original definition of speed ratio is retained and a, bare

functions of S (Ref. 3).

equation for a diatomic gas. If there is no accommodation of rotational energy, then

o<.y

<><.

=

0, and

4-b

( 51 )

Tt

which is the result for a monatomic gas. This might occur if a very

large number of coUisions must take place before the partition of energy

between the translational and rotational degrees of freedom attains the

equilibrium state. On the other hand, ti the rotational accommodation is as go~d as the translational, then O<ft.

/..<.

=

land the energy balance

equahon becomes

-r-

;:!.-

+

7i...

I",

=

bb

ft

( 52 )

The actual value of 0< R.. is a subject for further investi-gation. This question and the form of the energy balance equation for a

cylinder in the nonisentropic, free molecule flow of a diatomic gas are discussed in Ref. 3.

The above discussion indicates the methods by which

pressure arid temperature might be measured in a flow using the principles

of free molecule motion. The determination of density is another import-ant requirement. When the denisty is sufficiently low, it can be measured using an e ectron beam (Ref. 40). If a beam of electrons is projected across the test region in a low density tunne 1, gaseous fluorescence is

excited along the path and visible light emitted. Visual, photographic and

photometric observatÎons may be made (Fig. 10). The total light output

per unit length of beam can be considered to be proportiona 1 to the density of the gas if suitable spectral lines are used. Tests have shown that the

light output in air is satisfactory for steady flow conditions dow to

-4 -2

pressures of 10 mmo Hg. and for transient flows down to about 10 mmo

Hg. The spatial resolution is good and it is pas'Sible to obtain point

measurements of density without disturbing the flow.

9. APPLICATIONS OF FREE MOLECULE PROBES

The free molecule, orifice probe has been used to survey the pressure field around a flat plate aligned in the direction of

flow (Ref. 41). The tests were done in the UTIA low density tunnel at speed ratios of 0.55 and 0.78 under flow conditions such that the mean

free path was about 1/7 of the plate length. Particu1ar attention was given

to the flow within one mean free path of the leading edge.

The characteristics of the free-molecule, orifice probe

are different from those reviewed in Section 8 above since the plate

influences the pressure measurement. If the orifice is located within one mean free path of the leading edge and the probe diameter is less

1) Some of the molecules enter a forward facing orifice from regions upstream of the plate. the relevant velocity distribution function being that of the undisturbed molecular motion.

2} The remaining molecules enter the orifice upon reflection from the side of the plate facing the probe, having a velocity distribution function consistent with the reflection law s and the surface condition of the plate.

3) These two separate streams of molecules do not interfere and may be treated separately.

Under these assumptions the pressure of the flow at the position of the orifice (subscript 1) is given in terms of the measured gauge volume pressure (subscript 2) as follows,

~rT,-

I

PI

~:re

-

~

where

,X

(s) :::

G

(fan

E.)

==

and

'l-ë

S

-t

s(f

(11-

erf

s)

/

I

H

(S,

fan~)

=fo

( 53 ) ( 54 ) ( 55 )

dA

( 56 ) A comparison of theory and experiment is shown in Fig. H. The agreement is reasonable when it is realized that uniformity of the mass velocity distribution in the test section of a low density tunnel is difficult to attain at subsonic speeds. It may be said that qualitatively at least the existence of free molecule flow around the leading edge of a flat plate has been demonstrated.

An application of the free-molecule, equilibrium-temperature probe to the investigation of the normal shock transition is facilitated by the one dimensionality of the macroscopic flow and the absence of asolid boundary condition. It is particularly interesting since it involves the velocity distribution function in the more general form indicated by Eqs. 5, 6 (for a monatomic gas), corresponding to a flow involving visous stress _{(A ij) and} heat flux (Ai)'.

A study of the internal structure of the shock wave is clearly facilitated by increasing the mean free path so that the shock thickness is several thousand times larger and free molecule probes

ar e pra ct ica bIe (Ref. 3). Devices ca.Ued shock holders were developed

to produce a normal shock wave in the low density tunnel. A shock holder

is a thin walled circular cylinder or cone frustrum with a moveable

sect-ion to vary the area of exit opening. As the exit area is opened up, the

curved detached shock front changes to a stable plane wave. In Ref. 3

the variation of temperature with distance through the shock front was

determined from the measured temperature of a fine wire embedded in

the shock zone and aligned parallel to the wave front. The wire diameter

is such that it is at least half the local mean free path. Under these

con-ditions the flow heats the wire to a temperature which depends on the

local speed ratio, static temperature, and the number of excited degrees

of freedom of the gas molecules provided the cylindrical wire is a perfect

conductor and no radiation or end losses occur.

Free-molecule, equilibrium- temperature probes

involve either a temperature-sensitive resistance wire or a butt-welded

thermocouple as a sensing element. The former responds to an

integrat-ed average temperature over its exposed length, and the latter is sensi-tive only at the junction.

Measurements of temperature variation through the

shock wave obtained by this method are reported in Ref. 3. A

compari-son of these results with the temperature profile calculated fr om the

Navier...,Stokes equations (which are based on the more general velocity

distribution function, see Eq. 5, also ReL 31) shows satisfactory

agree-ment if the so-called "second coefficient of viscosity" or "bulk viscosity"

is included in the theory.

A shock wave thickness can be defined in terms of the

temperature profile. A comparison of shock thicknesses determined

experimentally and theoretically by the method described here is given

in Fig. 12. Experimental results based on the optica1 reflectivity of the

shock front in a shock tube are also included. It may be concluded that

the Navier-Stokes equations are sufficient for the description of the

transition through the shock front in air for upstream Mach numbers up

to 2. The equations must include a bulk viscosity,

f'<h ::

f/'

to account

for the effects of rotational relaxation phenomena which tend to increase

the thickness of the wave.

10. COLLISION - FREE PLASMA FLOW

We have been studying flows which correspond to a

simple form of the Boltzmann equation for the velocity distrihution

function. We now proceed to discuss very briefly the more general

collision-free flows in which external forces exert an influence on the partic les. The significant terms in the Boltzmann equation are ( in

F.

9 -..I-_{I "YY')'} L -

~f

f·

_,

t

- 0

( 57 )

where

f,

(i',

f,

t)

is the velocity distribution function and mi is the

partic Ie mass for the i th partiele species. If

(

-

~

...

)

ft

=

~i

E -t

7-

r

X

B

( 58 )

-

-where E and Bare the macroscopie electric and magnetic fields

con-sistent with Maxwell's equations, ~i is the i th particle charge, and c

is the speed of light, then Eq. 57 holds for a plasma at high temperature.

For a completely (50%) ionized gas, i :: 1 for electrons and i:: 2 for ions ' ... and a velocity distribution funet ion must be determined for each of the

two groups of particles.

The macroscopie properties of the flow are obtained

through basic definitions such as

( 59 )

for the number density of partic1es, and

( 60 )

for the average velocity . Thus for an ionized gas the electric current

density is

( 61 )

The mean free path has played an i~portant role in

previousdiscussions. In a plasma flow the definition of the mean free

path requires revision and other length parameters may be associated

with the partic Ie motion.

When a charged partiele moves through a magnetic field

the component of its motion along the lines of magnetic force remains

unchanged. but that component of the motion which is directed

perpendicu-lar to the lines of .force becomes circular under the influence of the mag-ne tic field. The radius of the circular motion can be calculated by

equating the centripetal force to that produced by the magnetic field

(Lorentz force). Thus the spiral motion is characterized by the (Larmor) radii of gyration

r·

t ( 62 )

~

where

vt

is the component of

g.

perpendicular to the magnetic field.

L

Another important length parameter is the distance

over which a point charge is shielded by alocal increase in the

concen-tration of charges of opposite sign, called the Debye length (h). In an

particle-partic Ie collisions or they can arise from the long- range overall effects of the st rong coupling associated with the electromagnetic interactions. In an ionized gas electromagnetic interactions between particles are long range in character and are usually divided into direct particle-particle encounters and long range collective effects. Thus particle-particle collisions are significant within the Debye length. If e is the electronic charge, the Debye length is given by

h

=

J

~ ~:L

;.

(

63 )

where 'fle is the number of electrons per cc., k is the Boltzmann

constant and T is the absolute temperature.

The mean free path in an ionized gas may be defined as the distance which a particle moves at the thermal velocity during the relaxation time of the gas toward translational equilibrium (Ref. 42). It

may

b~

writte:.,

N

(3

:~

)'-

\0) ( ;\

t,- )

(

64 )

In terms of these length parameters. the region of

plasma flow of interest here occurs when both the electron arid ion Larmor radii are shorter than the mean free path. Under these conditions, the Boltzmann equations for the distribution of electron and ion velocities are both dominated by the magnetic term (see Eqs. 57, 58).

The regime of flow outlined in terms of the above

parameters is the region in which it is expected that fusion reactors will operate. It is also of considerable interest in astrophysics. Apart fr om practical applications, two major lines of research are now proceeding in this regime:

(a) studies of the structure and solutions of the Boltzmann equation (b) determination of the flow equations for this regime. This new

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