liirary, N.W. t'i""

JUN 2 9 1965

^ecUnlcai riote 92o. 3/Jlf

STABILITY OF TWO-PHASE ANNULAR FLOW

IN A VERTICAL PIPE

STEPHEN JARVIS, Jr.

U. S. DEPARTMENT OF COMMERCE

NATIONAL BUREAU OF STANDARDS
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NATIONAL BUREAU OF STANDARDS

technical ^ote 3/4

ISSUED June/, 1965

STABILITY OF TWO-PHASE ANNULAR FLOW

IN A VERTICAL PIPE

Stephen Jarvis, Jr. National Bureau of Standards

Boulder, Colorado

NBS Technical Notes are designed to supplement the Bureau's regular publications program. They provide a means for making available scientific data that are of transient or limited interest. Technical Notes may be

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CONTEIWS

Page

Abstract 1

1. Introduction 1

2. The Model and Basic Equations 3

3- Fundamental Solution System for Small Viscosity 8

k. Singular Point r = 15

5. Critical layer Solutions 18

6. The Eigenvalue Prohlem 21

7. Base Flow 31

8. Numerical Computation 3^

9. Results 38

10. References 4l

11. Glossary 42

Appendix 1 k^

Appendix 2 50

Appendix 3 55

Appendix k 58

Appendix 5 63

Appendix 6 67

Figure 1 71

Figure 2 72

Figure 3 73

Figure h Jh

Figure 5 75

Figure 6 76

Figure 7 77

Figure 8 78

Figure 9 79

Figure 10 80

Table 1 81

Table 2 82

Table 3

- Qh

111
Page

Table k------- 85

Table 5 86

Table 6 87

Table 7 88

IV
stability of Two -Phase Annular Flow in a Vertical Pipe

Stephen Jarvis, Jr.

The theory of hydrodynamic stability for infinitesimal disturbances

is applied to the steady symmetric annular flow of two incompressible

fluids in a vertical pipe . The resulting 12 x 12 complex determinant for

the determination of curves of neutral stability is reduced^ by suitable

approximations^ to an 8 x 8 one, and numerical methods are used to determine

some neutral cui^es for air -water mixtures

.

1. Introduction

In this report, an analysis is made of the stability of two-component

annular flow in vertical tubes . The approach employs the well-known

stability theory of infinitesimal disturbances

Because of its importance in many engineering applications (e.g.,

fluid transport, heat exchangers), two-phase flow in tubes and orifices

has received a great deal of attention in the last decade. But because

the flow is very complex, even in straight vertical pipes, most of the

effort has been devoted to accumulating a large quantity of experimental

Information, and to attempting to correlate data in particular cases with

theories based on empirical relations . Furthermore, most of the work deals

with the apparently simple (though important) case of annular flow without

heat transfer or phase exchange; yet the results to date for this case are theoretically unsatisfactory and of modest practical utility. It is not

the aim of this paper to correct this situation, but merely to apply the

theory of hydrodynamic stability to this simple case in a more general way

than has been used heretofore in this area. The stability theory has a record of significant (though not uniform) success

.

Restricting ourselves to two-component incompressible flow in

vertical pipes (e.g., air and water), the flows resulting experimentally

are readily described in terms of "modes" dependent essentially on the

ratio/L*, of gas flow rate Q , to liquid flow rate Q . As-Aj' increases from

g •<- zero, we have several modes manifesting themselves in the following order:

A. Aerated flow: gas bubbles entrained in an otherwise normal liquid

flow.

B. Piston flow: the gas collects into a train of regularly-shaped

slugs which nearly span the pipe

.
C. Churn flow: a chaotic flow of large gas and liquid masses of

irregular form.

D. Wave entrainment : the liquid lies against the pipe wall^ while

the gas fills the central core^ the interface being marked "by large

waves which nearly span the pipe

.

E. Annular flow: the large waves of D^ damp to form a harely rippled

surface on the liquid film.

F. Entrained flow: liquid droplets are torn from the film surface

and carried along with the gas

.

One is tempted to attribute to annular flow^ E, a low-A^ instability

leading to wave entrainment^ D, and a high->Vinstab illty leading to droplet entrainment, F. There is, however, a conceptual difficulty in studying

the stability of this annular flow via infinitesimal disturbances on an

otherwise steady flow. The gas flow and liquid flows are usually turbulent

in cases of interest, while the small Interfacial ripples are known to be

responsible (probably thru profile drag) for a significant part of the

pressure drop. To what extent the small wavy disturbances admitted in the

stability analysis can qualitatively or quantitatively approximate the

experimentally observed flow can only be determined a posteriori. Some

encouragement is given by the results of the only other stability study

[2] of two -phase flow known to the author, that of Feldman . He considered

the oversimplified model comprising two-dimensional film flow on a flat

wall, with a linear velocity profile in both the film and the semiinfinite gas flow above it . While the critical Reynolds number determined

was very poor, the destabilizing effects of surface tension and gravity

( normal to the wall) were demonstrated. The most Important result, however, was the existence of instability, although for a homogeneous flow with linear

velocity profile (Couette flow), the analysis is known to predict stability

for all Reyiolds numbers (contrary to the physical facts howeverl). The

[3] instability must therefore be associated with the interface rather than

with the inception of turbulence in either fluid. However, Feldman appears

to have erred in his formulation of the interface conditions, which may

cause the failure of his result to match film cooling Reynolds numbers

.
The results obtained here^ when reduced to Feldman's case, give very much

lower critical Reynolds niombers, and agree in an acceptable manner with

[h] film data of Charvonia

r o

1 Squire's result for homogeneous two-dimensional flow, which states

that three-dimensional infinitesimal disturbances are more stable than

two-dimensional ones, is not clearly applicable to Feldman's model, and

is not applicable to the model considered here. (Squire's result appears

to fail for finite disturbances .

)

2. The Model and Basic Equations

The model chosen for this analysis- is that of vertical annular flow

of two incompressible fluids in a circular pipe of radius L. The core fluid is referred to as the gas, the

film fluid is referred to as the liquid

.

Gravity is assumed to act downward, in the

( - z) - direction. An undisturbed flow 7^\

is assumed to exist with cylindrical symmetry having only a vertical

velocity component W (r) . We assume W is continuous, W (l) = 0, and'

V\/(h')<0 for < r < L. This latter assximption simplifies the analysis

by permitting, at most, one critical layer while placing a limitation

on the influence of gravity which is realized in most flows of practical

interest. The pressure drop ^ - ~ T" is assumed constant. In the gas

and liquid, respectively, we have the pressures:

&

P. - - ^^ ^ P..

The constant pressure difference is associated with surface tension at the

interface

. p ^ P - P

=^*

where x^ is the undisturbed interface radius andcr the coefficient of sur- face tension. While the base flow W (r) is assumed to satisfy the NavierStokes equations in each medium, more general flows will be considered;

they are discussed in Section 7- The remainder of the analysis requires
no further specification of the base flov. On this base flow are superimposed small disturbances to the velocities (radial^ azimuthal and axial^

respectively) and the pressure of the form:

e

6 <K^(z - c^f) + irv) ^

(1)

whose amplitudes are taken to be sufficiently small that the linearized

equations of flow suffice for their determination. Only real parts are taken to be physically significant. The longitudinal wave number is

'^tf }^ gives the rotational mode^ and c^^ is the wave speed: thus

C^ = C^^ t i. Cy.-r ' The sign of c^^ determines whether the disturbance

will grow ( c^i- > O ) or decay ( c^x '^ ^ )• When ( o<^ kvi ) are given,

c^ is to be determined from the analysis

[5] From the Navier-Stokes equations in cylindrical coordinates , we

obtain the following ordinary differential equations by linearization:

Am ir IT

-^a cr ^ -; -

(2)

r

o- +

j. m

r

T = /M. - ^* ^

tK

(3)

/\iv+ plVw + Lo^^TT^

e ^^ ^-^ 1 W " f ^ IV

' —^ W — dC^ vv (M

/

I / ^w

/4(r; ^ 10^,

f

(wr^;-- c^j

(5)

ii-
where (a .--t) must be given their respective values in the film and in the

core. At the center of the pipe^ r = 0_, we require the velocity components

to he bounded. At the wall^ r = L, the disturbance components must vanish:

M Cq)^ (.^(qJ^ ^ (c>) bounded (6)

(l)= ir{L) ^ u; (/-)== o . u (7)

We assume the interface to be disturbed in the same way as the base

flow^ and put for the interface

where again / F~^ / < < / is assumed. Since on the interface^ with V^

the complete velocity vector^

we find, using W = W (x^)

l=r = + ..^^^-) - (9)

Now across the interface, u, v, W + w must be continuous for this

viscous flow. The remaining physical conditions are that the total

shearing stress be continuous across the interface, while the total normal

stress jump, A T^^ be given in terms of the mean curvature of the interface

where T' is again the surface tension coefficient. The base flow has

the normal stress

rr o and the shear stress '^^-^ ~ '^ ~ ^^ ^^ (T^

for laminar flow. The complete stress tensor is then
U"

(10)

0-, o ii ^ ^r^r- i- ^-r -h ^-M. t oc^ uj ) H^

where Ll ^ = ^

The unit outward normal to the interface has the components

where j ^ ^ I I *^ i I « I > where

y* ^ o (11)

Thus the force on the interface has the components (neglecting quadratic

terms in the disturbances )

(^ r(^)\ ^^-^ - ^^^^JE^ . (12)

Now a tedious calculation of the mean curvature K^ of the interface gives

9

to the first order in small quantities. (in this calculation^ only the

real part of / is used, leading to Yf^ which is the real part of (l3).)
Noting that on the imdlsturhed interface:

r.

l-r

c..d

3

I I q wli'c4

(IM

= o

for the base flow^ and putting

l^A-l - T.(xJ . r.'C.jF F, 7-0

to first order, we determine the jiomp conditions for the distiirbances

.

jar

r ^ X^ . In the following <^ A - A I

.

(S^M = O J- = i^ H w

rx.j

i^ ^* W - c. 5~W. = <:^ (15)

t cJ cr ^*- / )

^U^'J^ - W M - U^

s. O" rv\ Ui (16)

--M. o<^ ^^ (W,~ O

S(<< vA^y ^ <- '^if '^ f-n + u

^*^

The twelve conditions {6, ^, 15 and 16) are homogeneous in (u, v, w, jt) as are the pair of 6th order systems (2—5), associated with the film and

core flows . This will "be seen in Section 6 to lead to an eigenvalue

problem for the determination of c^ as a function of the other parameters

of the problem. In" fact, we shall be led to require that a complex determinant A - Aj^ + t A vanish:

A, = o

We shall, in effect, set <^j, j-=^ ^ , and eliminate C^^ between these real

equations, obtaining a single relation among all the parameters oc^^^^ ^ etc., of the problem which when satisfied, assures that the solutions so
specified will be neutrally stable, that is, they will neither grow nor

decay. This hypersurface in the space of the parameters will, in general,

separate a region of stable solutions from a region of unstable solutions..

We shall now proceed to the details of determining solutions to the

system (2—5), bearing in mind that C^ will be taken to be real.

3. Fundamental Solution System for Small Viscosity

The customary approach to solving the ccmplicated set of (2—5) is to

assume that the viscosity -m.-^ O . The resulting solutions can be proved

to be rigorously asymptotic in domains devoid of the singularities of the

solutions, i.e., r— and /^-^, where VV(/~'cj== C^ (if such exists), the

so-called "critical layer". We shall require that when f^ exists for real

^*.

Setting x^^ <5, the system (2-5) may be reduced to the system:

T

(17)

A

This system is evidently of 2nd order, and are the so-called "inviscid

equations". They provide only two of the independent set of 6 solutions

we must obtain for (2—5) in order to be able to satisfy general boundary

conditions . Before proceeding to the development of a complementary set

of four solutions, we shall establish certain facts about the solutions

of (17)' The points /"=o and l~^ fZ ^.re regular singular points of (l?) •
It does not appear possible to integrate these equations in closed

form for general values of the parameters rv) and cV^ for the profiles

YV(l^)of interest . In studying the stability of plane boixndary layer flow,

the simpler equation T - ry~7 T - oc^ ir ^ d is obtained which can be integrated in terms of hypergeometric functions for a useful profile. On the

other hand, it turns out that the neutral stability curve is associated

with o^^ L << / (where L'^ is the boundary layer thickness), so that the

solutions may be determined adequately by two terms of an expansion in

3

oC^ . In the present case, we could solve the equation in terms of

(untabulated) hypergeometric fimctions for o^-^ /__ -* O , but Feldman's

results indicate the neutral stability curve involves values cx^ between

the values 0.01 -$' ^^ (/- " ^it) ^ 1^ so that we must be prepared to

have wave numbers of the order of the inverse film thickness^ too large

for this approximation.

When a critical layer occurs in the flow at r'^ /^ , it will be

apparent from Section 5 that certain phases must be adopted to continue

properly asymptotic solutions across the singularity:

O ^ c^^ (r- r^) ^ IT

^-f^

^

A <

^

(18)

where

t ^ -^^ w_4 c.^-'^r^M. c = ^ ^ n- (19)

It is convenient to define here specific solutions of the inviscid

equation which will be used in the numerical work. For this we shall
introduce the dimensionless quantities:

Vff)- ^fc), (20)

and put

The inviscid equations for v,// m/ are then

We define a pair of independent solutions of (22) in the

liquid film "by their values at the (assumed regular) point ? = X :

(23)

Their values at the wall f = I (r = L) will be denoted by ( Tl

^ j J'- -> J*-

/it — <r L> ) which may be complex.

In the gas core, we shall define a unique H = ^ -—^ - by the

requirement (since <^ ^ o )

:

(Any linearly independent solution I '

^ (f) is singular at t - o .) Jxi Section 8, we shall assume a core flow of the form:

Then with

^ ' ^ ^ . p

«= ^ (^^J^)^

""

10
// [ ^j /3 J \n^J satisfies the equations

r

Q- ^l - h;-h; + ^^,w -(-vr^ (25)

for real and Imaginary parts . The initial condition for H is

^ 4-.

' (26)

H.C-) cr- rvo -f' O"" -* o Q(l^n^)

I -

+ a-( s^-' ). . The critical layer is at cr = 9"^ = /2 . Of course, f—/ ( o") = for

o^< p.

We now turn to the derivation of a complementary set of solutions

for ( 2-5). The approach is based on the results of characteristic theory

for inviscid flows, in which jumps in tangential velocity across streamlines are permitted. Then for small viscosity, we expect derivatives across the base flow streamlines (y = constant) to be large. We set:

^M^

LT

UJ

l'^;

,x-<.

'/>

I

^K. "''TT ^

I

A^

-<6 c^ t- ei (t^) . c

(21)

Substituting these in (2-5), and ordering in powers

11
of .-f-^ , the leading terms in each equation are homogeneous algebraic

equations in \"^o "^a "^ 771) • For solutions to exists the determinant

of the coefficients must vanish. This leads to two determinations of

9,M= ^

(28)

The next highest order set of equations leads, for each g, to two independent 1st order differential systems in ( T^o "^ o "^ "77^ ):

5^'^ ^^; . r^ V ^'. ^^ U -''^

with the auxiliary specifications:

Equations 25 and 26 together with the solutions of (17)

enahle us to write the general solution in the gas core:

tv = -

^.

^ ^r ^/ " ^6 ^.

u = ' ^*^

12
IT ^ C^ir -h C, X. s", ^ ^(. "^ (29)

''' L(c, c^c^cj.

The latter term in IT need not be explicitly given. A similar solution can "be written for the liquid film.

Clearly, the solutions (29) are singular at ^~\'^y ^c), "but the

conditions (18) determine their correct continuations.

However, since boundary conditions on the general solution must be

applied atr= (O^ A* jL), it is desirable to derive solutions which are regular at these points without restricting a jiriori the location of the

critical layer. Furthermore, this analysis will determine the continuation

conditions (18)

.

For convenience later -in joining solutions, we shall show how the

solutions (29) behave as f -^ <^ and /"" —^ -^^ .

r-M:v

Writing A (P/ = A ^ ^ and

-^^ } Jit- A

^ I W'

P' " - if" a^v-.J"'

we find as

13
w •—^

' i C ry Ei 4. ; (^ ^;?

Po r-

(30)

V / _ » /

." / /~

o A^^-* ^'P- ;::^/= ' "-

r I ^G "a

r —> X»:

Writing

A -^> ^

+ t (r - A,)

c - A^-'^^r^^',

^..

F. = ^

14
Then as r-^ X^^:

LT u'H''^ ,:)'''"( ^'^'p' + iA_Ei

^ ^r ^, ^ C, ^,

w 3 . <- ^^ —4

P.

'^"•'

f C^ U.^ f C^ t^,

- ^'^^ '^__ / C, E^ p^ _ c C^ E^

I G u

k. Singular Point r = . Regular solutions to the system (2-5) may "be olDtained by introducing the singular - layer coordinate:

and expanding the dependent variables and coefficients in the equations

15
'^2 in powers of -^

^^<j ~

ir

'^? ^

Substituting in (2-^), one can derive a single factorable 4th order

equation inlLtPutting D - ^ '

The remaining variables are determined hy the 2nd order auxiliary system

These equations are readily solved in terms of Bessel functions. In

applying the regularity conditions at r = 0, we must distinguish the

cases m = and m ^ 0. The required solutions are as follows:

16
the latter term depending on ^^.

From the known asymptotic "behavior of the Bess el functions^ one can deduce forms for these solutions as i'3^ ^ o

' —^ '^ • These are matched

with the corresponding asymptotic solutions evaluated as r*—> O (30)^

and relations between the constants 3. (Zy -are found. Elimination of

the O- gives three relations among the C ^^ required for regularity of

the solutions on the axis. Details of this procedure are given in iCppendix 3 where the notation "is defined. We obtain the following:

r

^2 7 ^, F^

C, ^ C

^ o

f

^ ^3 Pc ^ ^ (m = 0) (34)

c. - y^'

^t P^ ^ ^ ^ ^ (°i 7^ 0) (35)

/•* 6?rv,

17
5 . Critical Layer Solutions

We shall now derive a set of solutions to the system (2-5) which

are regular in a neighborhood of the singular point, yet are of the same character as the asymptotic solutions outside such a neighborhood.

We shall assume that in a neighborhood of a fixed (though arbitrary)

point M

^ . p , tfr-rj f M(^'^f)

lA/ < . 0(r-^A- -hro ' (^Cr-n)

{

and confine ourselves to a neighborhood of /^^ in which the higher order

terms are negligible.. If /" = f^ , p =• o ; no restriction will be placed

on the magnitude of p.

Let -'A

Let

( '^\

Ul

u

\ r)

f V(t)

'^

Vfr)

^ "' U (V)

V

^TTCr)

To the highest order in .^ , we obtain the 6th order system

v " - r 1^ - o

u'" - r tx' t !_(. = o

7T'= f'i'U"- CU.)

X, (36)

The first two do not depend on the geometry of the problem. They occur

in stability analyses in general, and lead to the well-known Tietjen's

function and related functions

.

18
Let

r I , 3

i

I--

I ^3

V. - "-f e

Jt ^ 3t '^t

V

c.

C.

-(37)

C

where the contour C,( oo c^ to oo C ) passes over t ~ o and

-c r

JT CC'^^C- "fo £^6-"^) passes under t ^ . We write the

expressions:

r " , 4 i

c

'-'j

(38)

VJlaen /\ ^

^ /r ^

:^ o ^ this set satisfies (36). The invlscid

solutions ( TT If ) are appended as extensions to the approximations to

the inviscid solutions which are obtained from the pole contributions of

(37).

It is readily shown by the method of steepest descents that with the

specification (18)

— < ^

19
we have the asymptotic evaluations

:

//. r"Y^^ ^-^J.^rr-';;

_ 5 -.-?/* /^;iK

e C

/- r

-/- /< ^j^^rr-J^

(39)

^ ^

Specifying /"I = '^^ , ( /^ ^ A" ) in (38)^ and using the asymptotic

forms (39)^ we readily identify the coefficients /\ ^ with the O^^

of (31):

/A,

A.

V

/F^''^t'''x,'j 'H

L o<^ .'«-< /A

(ito)

A.

'A.

A

Thus the solution is determined in the core near the core-film interface

in terms of the L> . . An entirely analogous procedure can he applied in the liquid film.

That is, asymptotic forms analogous to (29) can he joined to critical

layer forms analogous to (38) near the Interface (/"-*X^, ^ ^ ^it)

and near the wall {t--'-^L,f^^L). It turns out, however, that

with the approximation to he made in Section 6 on exponential orders of

magnitude, the same result is obtained when it is assixmed that a critical

20
layer form Is valid throughout the film and the evaluations of C and t

are made locally:

A

t =

The ( ) refers to evaluations in the film. Thus throughout the fllm^ we

take:

^ ^ 'A A A A

u- A,3, . A,r^ . A, M, . A, .,

I ^ A A ^ A A Cuo ^

A

T

where ( ^, , ^ ^ = ( ?^ ( C ), V, ( C ))

.

6. The Eigenvalue Prohlem

The regiHarity conditions (6) have given three linear homogeneous

conditions (3^^ 35) on the C. . Applying the wall conditions (7) to

the film solution (38) leads to three linear homogeneous conditions in,

the >^i : A i ^ AX ^ ^^^ ^ A ^. = ^

21
^

where 7^^ ^ Z^( C (L )).

The six interface conditions (15^ I6) lead to six linear,

homogeneous relations among the {A U , ) • We shall not write them

down explicitly because of their length, but shall refer to them as

(kk) and (45) respectively:

6 linear homogeneous relations in(/vi^^j) / (^^)

derived from interface conditions (15, 16) I . (45)

The twelve linear homogeneous relations (3^ or 35^ ^3^ ^^^ ^5)

/^ suffice to determine non-trivial ( C; /\ ; ) "to within a single scaling

factor if the twelfth order determinant of the coefficient vanishes

.

This determinant is given explicitly in Appendix 2 in the following form:

A ^

(hk) Interface Velocity Conditions

(^5) Interface Shear Conditions

(^3)

Wall Conditions 1

{3h) or (35)

1 Axis Conditions

^.

( t ,^ = 1,...,12)

In the determinant, we have used:

--uH. = n^^ -<-<-

The prime reduction to be made in this determinant is the elimination

of exponentially small terms, i.e., terms down C>(e /for some X

from the leading terms. To do this, we first replace row ^, /\ ^

^ by

(row 9); R^ =f^^ + ^^ Rj . This makes A^,= 0, A^^ = 0,

m 4

fL = "^o. ^ ;^x. ^.. '

22
and does not effect the other elements . Nov we divide columns 1 through h

/\ --\ ^ ^\

\ij, respectively, (^cx, Ick, 2^^^ V^x ^ ^^^ columns 7 through 10

by, respectively, ( /\ ,^ ^ ^ .f j A ^^ ^ A ^ ,o )^ ^o^ 1° ^y ^ /? ^ ^o^ ^

^y A J ,^ . Then the first 6 rows are purely algebraic in y<<. (i.e., algebraic

in ^^ as ^-*^). On the other hand (39):

KL

part of \

for integral (k,n), where y- is algebraic in its arguments. Now the real

^<r^

^^ (^ < -c)

I c^^-^ (t> rJ

is monotonically decreasing with increasing p in each medium. Hence

for some positive <^ independent of --h. associated with L - x and is

exponentially small as .-h. —> C . Qn the other hand.

^^^ ' -A I LI eT ^

/I X

is exponentially large as ^^-< -> O. . Furthermore, ^A, // _

is exponentially large as y-< ~^ ^ , with X related to the core radius x. Then

'7 ^57

a

- '/= A ^

-" 2

23
Clearly the largest terms in the determinant will contain the factors

A -/\ A

"Which is

Then dropping the exponentially small terms ^ the vanishing of the 12 x 12

determinant leads to the vanishing of the 8x8 determinant obtained from

the cofactor of

^,./^..- ^„„- ^.=

in which we also set

O

Now we replace row (3) hy /7~y /"'^o^ iX) + ^o^ (3)/ } row (5)

by|row (5) + -^^row (6) -[^^^^^^J ^°^ (iL and row (6) by /row (6)- ^—

row (5)L in that order^ and use the row and colimm multipliers:

row mult col mult 11 11

1

m

2 m 3

3 m k i

m

^ x./\r -^^ 5 1

6

m

-* 81. 8 m lo-rn 9 m 12 1

We obtain the 8x8 determinant as shown in Appendix 2

.

One can now examine algebraic orders of magnitude in <<. as C°-> <^,

and expand the determinant in the form

2k
One obtains in this way, retaining the top two orders, a bilinear form

A^f ^ L 8, e,

of 11 terms, where the o- are the 3^3 minors of the "inviscid" terms,

(cols. 5^ 6, 12), while the O- are dependent on the viscous elements,

(cols. 1, 3^ ^^ 8^ lO) . The result is still sufficiently complicated to

make further analysis fruitless and require numerical methods for further

evaluation

.

The 8x8 determinant will be put in a final dimensionless form by

setting

r' (wj

(^6)

so that Re is a Reynold's number, °2~ "the Froude number (dependent on sign g*), and Z- a dimensionless surface tension parameter.

We divide column 5 T^y ^^ k~ " ^ ^ ^ column 6 by '-^^^^^ —^ } an<3.

column 12 by ''j^ • We have, then, finally:

o = /I = /A,^i (^7)

25
A

=

-/ o o -I

o -I o

o

o

o

'4.

^ -fv?

^^/ ^63

O

^ ^^7 ^

o

°

'^Pt o / /I

A

A

A

A

Jr

-/r

rr

t*r

i*^

SI 12

4 ^.. ^

I A.. A., o

fL

o A^ A A o

o o

a o

(^)

The values of the n • • follow

.

(The validity of the hasic reduction to an 8 x 8 determinant is

difficult to assess. At some points of the critical curves computed,

-2 .5

the exponential factors dropped were calculated to he down by only e from the factors retained, which is certainly not as much as one would

wish. The effect of this on the location of the critical curves is,

nevertheless, probably not very great, although it would be desirable to

have some results for the complete 12 x 12 case.)

26
o( I- C^ n^o "^

O X

53

J. mL Ir 1. V tL-^

^ sx

A^3 i c<;( ](^ {/-ojJ I ^x - ^ z

•/x

no - /

rj

^xiJ-t ^{/^- ^'jj* '^"^.x'ljdrl

^X ^ct\

?..sx

A ^J v'x t

A1-/- i « %L r 9

A^,= -k-/J m

6?

-f

/\

n,X v^' / X K*x

27
^^

,0 " ^.•^' /^ V^- r, vi, ' ^.v^ ^ ^ '^ ^c^m.X R C^-C)

i \ ^ K K.

^.,0- -'^Mt^

-yr: ]' ( c'A y.'OX

y.a p(

'^.c - '^

A.,r-'^

^j.= -'^^ (^^ ^^'

Ol

/\

7k'

J 6, c< f/- cj

^

'T^

J /2 r/-^^

/-/ f <?< X (/^ ^"x r?^-?

A fr

A '^i*

ra.(/-c;

o^ya^fy-cJA

A + (a

5"i,« F-tL/, xX /

28
A

A ^, ^ ^vi « «<

' X

A 3 s-u (i^ f^y '.o< k'\..0, j,lli- ^=^.

m

7^' __x3

/I

s- /A m (.,-^)-^'.'h[(/^^^-^)^ -^,^^"-x>7'J

'r/- cj

/I

rvi. f X , J^

6^ cX /- C

/\

6. G = - X

o^ X

.^

cxr X -

A L 13

JL.

m . - o? - /- c

-h H.

l- c a

i-o

^^u '- '' c Q..

A

^ X O-cJ

ir 1^ c^.

-' ^-n^.^^^Q ^1- ex: c

SL

29
/ \ m,

^

z ^^ = £ 0^ X (m - // - —^

cx(l-c) m^ 7^

, -~//-e/

m, c< J2^ ^ ^

A>i .f^-J

'/, ^.- ii-^.li^y

K^ *''

J f^'^

30
7. Base Flow

The inviscid disturbance solutions depend on the entire base flow

velocity profile TK ( f )

, while the viscous solutions depend on certain

local properties of "ry . Strictly speaking a stability analysis has

meaning only when applied to an exact steady solution of the Navier-Stokes

equations. (it has been applied with success, however_, to boundary layer

flow when the streamlines are assimied to be parallel to the wall.) The

velocity profile for laminar pipe flow is exactly a function of the radius

only. Feldman's analysis, however, shows instability of a two-phase flow

to disturbances of wave length associated with film thickness while instability in homogeneous Poiseuille flow occurs only for wave lengths

comparable to the radius. Then to some extent, we may be able to distinguish in the thin film flow two types of instability, the one associated

with the gas core, the other with the film-core interface. It may

be possible that the core be turbulent while in a neighborhood of the thin

film the turbulent fluctuations could be of so small a magnitude as to be

contained in a linearized stability theory. Near the film in this case the turbulent core makes itself felt through a radical change in mean

velocity profile and shear stress . Then in a non-rigorous way one might

extend the stability analysis to thin film flows with velocity profiles

of a turbulent type. This extension is not included in this note.

We shall adopt velocity profiles in the gas core and liquid film

as follows

^ff) ^ /< - / f

(1,9)

-hqCf) ^ - Tr/O- rj f XO'd' (50)

The core flow profile is of the correct fonu for . laminar flow

.

A determination of the constants ( K^ f^ IV, A ) must be made.

The requirements ^r^ = - ^ = constant^ lA/- W (i^J only, lead

to the base flow equation

.^(W\ ^ WJ . ('''-pg'J-'', (52)

31
(55)

which has the general solution in non-dimensional form

In the gas core then

while in the liquid film,, to 2nd order in film thickness

:

'W, (fj- A,£^ii-(i' r)) * A,

= A.fO- r) ^ id- rr]

4^ IV/

We have the wall condition

the interface velocity conditions '^ W = >;CaJ = / (57)

and the interface shear condition

^. 'K' M - >^'W (58)

32
to determine the imknown constants . Identifying the resulting forms with

equations 49 through ^1, we find:

7^ = - Tr'O' r) * \('' r)

< A - _ -^. . ^- / ^ ^

V= ^L^— (60)

L / 12.6-xV

"^ I '^ / C/ ^ ^^^6^ ' ^)

>;' = - ^^^

??;' = yr/- ^ AG-.;

33
8 . N-uinerical Computation

It did not appear that anything much could "be done analytically with

this 8x8 complex determinant^ even in the thin film case with linear

velocity profiles^ corresponding to Feldman's study. He, indeed, used

numerical methods also

.

The y^ = limit, the so-called "inviscid limit, " may he obtained

from the determinant, and correspondffto setting (See Appendix 6)

:

o A,, A

A.-<-<.

'

A^^Wr A -/6. A

A.rfi- A

s. «a

O

= O

The same result is readily found by solving the inviscid boundary value

problem from scratch. (Feldman goes to same length to excuse the fact

that his two inviscid limits differ, when in fact they do not, as a simple algebraic manipulation shows'.) While the inviscid solution cannot

give a "critical Reynolds number" for stability, it locates any non-stable

domains in the ( o< XcL ) plane which extend to infinity in Oca_. , and provides the inviscid limit of one of the two functions defined by the

vanishing of the 8x8 complex determinant A;

Searching this limit curve, F(cxr o,D^c)=Oin the approach used, enabled

us to find a zero A ( o<;^ OOj^ c ) = quite easily, even though the

resulting critical Reynolds number (oSJ was not very large . Varying the

neutral wave speed c by small steps, one is able to trace out a neutral

stability curve:

F (c< Y^ c) - O

34
defined by the parameter C . These curves have a nose defining a minimum ( )Pk o< ) for which the flow is unstable. The position o/ this

nose^ the critical 0'^ = Jq, , is then sought for other flows by small

variation of the other parameters

<

m

KYI o m

Z^

All this would require a great deal of computing, and attempts to

simplify A further were made.

First, all the elements A ^

. were expanded in the top two orders in

»cL , and then A itself was expanded, retaining only the top two orders

again. It was found that the order of magnitude of ^^^ , ^

{\Q>Qi) , was

fer too small to justify the asymptotic expansion used, values of C of

(y^ (lO ) occurring in some cases, where C— (^ (lO) would be required

to use asymptotic forms for the /\- (w zi Ny« Since full range numeric 7 Pl ' ' . f=. ical tabulations of 7 and 2_ are sparse \t\\, and with too few

significant figures in the mid-range 2 < |c| < 8, and since complex

routines for ^^^ ( ^ )^ ^d~<. (C ) had recently been written and checked

out , further routines were written (see Appendix 4) to permit at

least four-place evaluation of the required functions on < | C | < <^ . Also, the entire determinant is calculated using single-precision complex

arithmetic with no restriction on the order of "^.j^*

A more satisfactory simplification was the approximations made for

H and for ( 77^^ ^ ^ r u > ~^^l

^ uu )• ^^^ "^^^ inviscid

equations for 7T (^) ^^^3. H (

7-- ) are solved numerically, typical times

for these alone were a minute. The approximating forms used (see Appendix

35
E) are very good for thin films h ^ O . I , and probably adequate to

l^ ^ O. S^ ' Numerical checks have shown that "^^^ is not significantly

affected by using the approximate forms for thin films ^ but no comparisons

were made at the thicker films values because of the long computing times

required. Our primary interest is in the thin film anyway, as being the

normal form in annular flow.

In its final form, in evaluation of the elements A and A took

about 0.003 minutes. A single

^ ( ^7 ) curve for (O.OOl < ^ < O.3)

took 30 minutes at a minimum.

In actual fact, the variations of the nose of the critical curve with

respect to the parameters computed were:

ftl.JL ^'tSo. 1^:^ 5Ar 1-^1 ^ ^^

where

3^ ^^^1' hi = i^iffu ^r

defining, respectively, gravity and surface tension parameters independent

of flow rates, and an interface azimuthal wave number factor, or "wave

pitch". Also, the derivative:

was computed on the nose curve to measure the rate of growth of the disturbance moving off the neutral line in the direction of increasing flU^ < It is found from the requirement:

36
where Oo^ c> c -t i o C

H J- and c- - o on neutral cui^es. This complex equation leads to:

^Yl.^

the partial derivatives "being estimated by taking small independent changes

'—

>

i^ Oa^ and real c . We must expect "o c if the flow is

'/=

unstable for va'lues of U«(Xp exceeding the values on the neutral curve

.

Because of the extensive computing required^ most computing was done

for the air-water case^ taking

m = 'f 7. <^

t^, - ^ i4-, ^

The automated computation process^ curve following by using a Unear-variation predictor-corrector method^ broke down for thick films, and could

not be continued. In the cases shown,, all values appeared numerically

sound; values showing instability have not been included.

To be more precise;, starting from some known point on an ( (^^ , OCL^ ) curve, iVp was stepped by a small increment, then ( {^ c ) readjusted to

lie on curve, using the complex equation

^A- c 'lA r O ^^ r ^o<' " ^^ " -^^^ '' '^F ^ ^ o

to determine ( ^ c^ a )d^ )

, and small variations in ( '<i= ^ c \g^ ) to

compute the partial derivatives . We continue to follow this curve through

decreasing values of "^ until liCi. increases, and call the smallest

value ( OCiup ) . In a similar way, starting from a point on one

( >^^ , Tq_- ) curve, one finds a point on another curve ( '><^ , ocL^ )

37
corresponding to changing some parameter P into p

'^ ^p "by solving

and this cuanre is again followed to a nose ((^1(q,^) , ('^^).<^ )• Breakdown of this procedure appears to occur when the estimates of partial

derivatives fail to give useful predictions . While one would like to take

smaller variation then to compute partial derivatives, one is then limited

"by the four to five place accuracy of the computations

.

^ At each point near the nose of an ( c=c^ , 0<^ ) - curve, variations

?=v

-^ in the position of the critical curve for fixed o( are made for

P L parameters /7/^o*^/'^^(S r ^^ ^ as an estimate of the change to be expected r /"--^

, Much of the

d p

irregularity of these derivatives is doubtless due to the irregularity of

the determination of ( ^^^)^.,^ , due to the discrete stepping of o^^ . 9 . Results

Because of the large number of parameters involved in this stability

analysis and the large computing times involved, it has not been practical

to generate neutral stability curves ( o<'f=. , Ycl ) and critical curves

(( ^iJ2j_^ )^^^ } n ) for the wide variety of materials and flow conditions

common in modem engineering. For this reason, the results given are all

for fluids with density ratio rY\= 8l4.4 and viscosity ratio ry^^= ^I'O,

reasonably good values for an air-water system. The derivatives

I t-j ^^ ; ^rv) / "^^^ "^^ used to estimate the critical curve changes for

fluids with somewhat different ratios M^ and kw , . Tables I through VII contain the computed data on the critical curves

as fimctions of the basic parameters o-^ h ^

fi '^ . The values of

( Yc.-: ) } <^ and ( od^p )^^^ given are those nearest the nose of the

particular neutral stability curve ( '<^ >'<^r) obtained by stepping

through discrete increments, a change of from one to five percent as seemed desirable . This discretization in >(p is the major cause of the

irregularity in the differences in derivatives shown with respect to r) , as was remarked earlier. Data are plotted in figures 1-9

•

38
The tables are arranged in order of increasing surface tension parameter, < CT < .23 (lO) . Within each table, the non-gravity flows are given first, all variables given as fTinctions of film thickness n for

increasing values of wave pitch p. In the last two tables. Tables VI and

VII , for which a gravity flow is given in each case, the P = curve develops complexities at larger values at n which are difficult to trace

with assurance because of the rapid changes . There is most certainly more

than one curve involved, and consequent intersections evidenced by changes

in sign of 'TvS" ' ^^^ linear extrapolation method cannot cope with

these complications . For values of n smaller than those at which these

problems arise, however, there is a change in the sign of ~^f^. , indicating that the axially symmetric wave is not the least stable for

larger values of n . The scattered values of P included beyond this

indicate the locus of -^l F =0, and should give then the lowest

^]^

' ^

critical Reynolds number ( 6ci_, ) in its dependence on both in and p.

Trends are not, in general, easily characterized, and the following

generalizing remarks should be checked in individual cases:

(1) In the absence of gravity, surface tension is destabilizing for very

thin films, stabilizing for thick films'^. For gravity flows, a second

destabilizing trend occurs at the thickest films studied.

^( For the very thick films, n > .V^ , an unaccounted for irregularity in

^^r I olF "With h occurs . ) ^

(2) For very thin films, ^—*^ is positive while .5— is negative.

o I These signs change at approximately the same small value of n , so that

for all larger in , increased viscosity ratio is destabilizing, increased

density ratio is stabilizing.

(3) For very thin films, increasing the downward (- g ) acceleration due to

gravity destabilized the flow, while for non-thin films it stabilizes the

flow. (Again, an unexplained irregularity in z. £ appears for very Q 0-

thick films .

)

{k) Thin films are more stable for P > than for P = 0, while for non- thin films, there are many cases in which the reverse is true.

39
Figure 10 is extracted from Charvonia's paper, and shows a disturlDance

instability line based on apparent droplet entrainment occurring in experimental air-water flows downward in vertical pipes. Charvonia's dimensionless

parameters are thickness parameter 1^ and liquid Reynolds number A/r? :

2 1

'/3 "r^lfit

N

1

^

where &O" is the peripheral liquid flow rate:

for thin films. Using his pipe radius L = 1.25" and /M. = 2 .09 lo"''

( /f^j^J

I

The critical curves of figures 8 and 9 are plotted as curves 8 and 9 on

figure 10. Charvonia's experiments lead to:

a = ^ja—:2-i—p ^ a ^^ oCicJ

cr- = ^^l = o. ;i 3 o oy

corresponding to curve 9- Curve 8 represents intermediate values for cj

and U^ .

ACKCTOWLEDGEMEnW ; The author wishes to thank Miss Garney Hardy and Mr. M. D. Bunch for their

strenuous efforts in programming portions of the numerical scheme.

ll-O
10. References

[l] C C. Lln^ "Theory of Hydrodynamlc Stability^" Cambridge University

Press, (1955).

[2] S. Feldman, "On the Hydrodynamlc Stability of Two Viscous Incompressible

Fluids In Parallel Uniform Shearing Motion/' J. Fluid Mech. 2_, 3^3^

(1957)

.

[3] T. Brooke Benjamin,, "Three -Dimensional Disturbances on an Unstable

F13jn/' J. Fluid Mech. 10, 3, kO\, (1961) .

[k] D. A. Charvonla, "A Study of the Mean Thickness of the Liquid Film

and the Characteristics of the Interfacial Surface in Annular, TwoPhase Flow in a Vertical Pipe, " Purdue University, Lafayette, Indiana,

(May 1959).

[5] H. Schlicting, "Boundary Layer Theory," McGraw-Hill, (1960)

.

[6] H. Holstein, "Uber die Aussere und Innere Reibungsschlcht bel Storungen

Laminarer Stromungen, " Z. Angwe . Math. Mech. 30 ^ 25, (l950)

[7] 0. N. Strand, Private Commiinlcation, (196^).

[8] H. B. Squire, "On the Stability of Three -Dimensional Distribution of

Viscous Fluid between Parallel Walls," Proc . Roy. Soc . London, A,

lh2, (1933).

1^1
11. Glossary

Dimensions are indicated in brackets: [L] lengthy [T] time, [M] mass, and

combinations of them.

Subscripts x and L indicate evaluations at ? = x and f = L, respectively.

Cap, ( ), implies a liquid film quantity.

A(r'^) = L oc' ^ p (W - c^): coefficient in linearized equations (2-h) . [ML T]

A. .: elements of the stability determinant, A.

'J . ^jT) ^= ~ —^ —^— -^^a : quantity in A, see p. 30. [l]

c = c>,+ i C^ : complex wave speed; c = O on neutral curve, [l]

c„ = W c. [LT""^] * x

E: real (A)

F: imaginary (a)

r : Froude number, /

•*- . LU

JiflL

5 = ^^ y'^ I ^ s-^t^Kv-c?* : a gravity number independent of flow

rates, [l] ^ r -2i

cj : gravitational constant . L LT J

/? : film thickness, 1 - x /l. [l]

H: parameter based on inviscid disturbances in the gas core, [l]

*

IV1 ri : nimibers defined by asymptotic expansions of viscous flow near

r = (see Appendix 3)

L: pipe radius . [L]

rvy : azimuthal mode number in wave factor ^

/\ rvij, : viscosity ratio, '^/^ . [l]

/^ Kvij : density ratio, f / P

• [l]

Fo f Fi • exponential factors occurring in viscous flow in core (see

pp. 13-1^).

^= L 0^ X (m^-/)- '^^ - : quantity in A.., see p. 30. [l]

\2.
^^=^/x . [L]

^(f) = 7T'(f).

r : radial coordinate. [L]

c(tj!_ = ^ ^ , pipe Reynolds n\aiii"ber. [l]

Ou, = h lu_. } film Reynolds number, [l]

( Vql ) • critical film Reynolds womber, a minimxim with respect to ocT

, dependent on other parameters, [l]

t. : time coordinate [t]

'^{^)} 1^ (^ ) } '^ {f ) • radial,, azim.uthal and axial velocity coefficients

i. «'• C Z

'* - C^ t *-) + L m V' r -It for disturbance wave factor e . LLT J ¥ (r ): inidisturbed axial velocity profile. [LT ] ?r(r) = w /w^.

x: x^L. [1]

x„: undisturbed interface radius. [L]

y (C ), 2 ( C) : functions describing flow near viscous critical layer

yy^ = c; see (37) •

2 : axial coordinate. LLJ

©< : o^-^L. ^ t OC^ Z* r -It

o<'^: wave number for axial component of wave factor G . LL J

P =

•—

-^ ; interface azimuthal wave rnjmber factor^ or "wave pitch", [l]

-2 -2

Y : axial base flow pressure gradient: [ML T ]

J ,

« : numbers defined by asymptotic expansions of viscous flow near

Z-' = (see Appendix 3)-

A = £ t t. F : complex determinant whose vanishing defines neutral

stability cuirves

.

C : coordinate basic to viscous flow near critical layer 7^"= c.

^3
X-: dynamic viscosity. [ML T ] r= r/L. [1]

Tit")'' pressure coefficient for disturbance wave factor

e . [ML T ]

7/ ( r ): inviscid pressure coefficient.

TT if)' pressure coefficient satisfying TTj ( i° ) —^ f as

JT ( r ) : pressure coefficient satisfying

( 7r^(x), 6?^(x)) = (1,0),

~n^(. ( ^ ) • pressure coefficient satisfying

( rr,{x), o,(x)) = (0,1).

p : density. [ML ]

cr= ^ f : a surface tension nioniber independent of flow rates, [l]

_2

O" "5^ = surface tension. [MT ] cr

» 7 = ^^n aTTT"' : Weber's number, [l]

^^° ( /^ ) : /"^ -component of base flow stress tensor. [ML T ]

^—>- -1 -P

fo i ^ )' ^ ^ -component of base flow stress tensor. [ML T ] f : azimuthal coordinate, in radians

.

kk
Appendix 1

12 X 12 DETERMINANT

Elements of the basic 12 x 12 determinant are given in the following

charts

.

h^
[^-- •^ n r

S

uo/n (.5 )

-=» (6)]

«

7' *<

I v^

(

C w. J ,/^ i^-^^y

<N

<fv^^

I <^\<X ^^£

<t>4

I

*i

1 3'

<^M X

S

X

<rxi <rvi

51V

<-^[^X

I <hsi

Of / CWW/»i / fTl ^/

•r^i

I

nf f^

o

I

<IH4

W

46
o

K

j C/ o/<_< oi in vT u.,77, C-rj M]

o o

>

/ Co/Mm ,n-

^- y''

r> o <rM

<r>si'

-1<{

^iv*

Q Q Q

n<

<M

rCJt-t /H 1 / «-W /f-l| ^-. y7

o o

0-

O

rv

^7
Of

j C e. I <^r\ x/\ 11 <^ r n. 0j-(9)

]

4.

--s.^ •X K .. V ^

'd^ z 'f

X

1

1 3^

\

Q

n f ^ ,il 2 ^ yj

^ _<^ ^^ >C H ft

11 ^ :!:' w ^, < >.

ij--

It 3"«< ? ex. 1^

V

1^

1 ^

^

5 -t- X

f^

r C-0 turn n 7 ^^'i'h ^-yj

K ^ X

><

1 ?.

/^ "^ <? o r^ 1 CJi? ^ ^ n ^

Ih

Of or ,r >-> ^

/

d.

£1

> ^ 'T ti

£ - fV r^ ^ \4> ^3

48
nr

->. ^ o —

o

>. o -

(' -> _ b

v/^ ?

-«

V.

^hs o '7 1 V1

(^ o -

1^ ^

0.

1^ o

1

1 <

—t 1 r^ <^ o- ;- m

h9
Appendix 2

nWERMEDIATE 8x8 DETERMINAm'

Elements of the 8x8 determinant after the exponential reduction and

certain sealing operations have been performed are given in the following

charts. Quantities are still dimensional.

50
w

3

1 i

3^

1 V

3 T 5>f

5>C

O

o

^

X X

fcxflfrNi'*

<rN4

>Kr

Vt'N''

>

51
1"*X

*

«

02^ 5

X

n

O

4-

o

4.

I

ii

>?

i

4- Oars < X

ol 01

X. O

^

o

sj •^

o

0--

52
ry n

3

j^y^-

^K

- >s

«

5

<^

o Q

\3 C<^/< f^ o L^ / 77) frj - ro j

or ^ >

53
<

y

It

4-

. X

-IV**

O

0^^

^

^

V

I

o Q

><

1 ^

'J *

.^-i^

h

s v'

[C.K yy\ n vT UJ ,th (r)-^(C)}

^ ^ Do 0^

3h
Appendix 3

Letting Z ^ "? Ag % we have the asymptotic expansion:

(-2f ,< ^

F ^ ^ )

'-' ^ (f^'4' ' '- ^^^]

. ,^.e-'^-^=^^/,. S^Ji)]

where the C^.jv, ^^s asymptotic series in ^ which vanish as 121-*°'. Then

for some constant i ^ ^ :

by integration by parts involving new series • For 100 = 0, we obtain the asymptotic forms

LT

UJ

8.

{Tn [r'''^ ' r''\

i^^'^^i,o( Lo^^'^'^ /c ^ te^

55
J W

> - T"

Comparing these with (30) for w) = 0, and noting ] ' ^'^o ^ ^ T ^ Co n s 'f'ctn 1

J

we obtain the relations

P. C. _^ B,

cw.-c.j'm;-^ iTr^^^ 4:*

"

_ Cr i iv.^^' to, Ss . /\ ' 4. )

c.

These provide the three relations (3^).

For fvi > 1^ the matching is a little more difficult. The constants i^^,

^ rvxD _, ''r.^. are defined by the / ^/-* oc result : (- ^ ^ '^^ ^ ^ "5"

)

. &U'-^ c'')

Using this in the relations (32) and their auxiliary relations, we obtain

the asymptotic forms

^ 2'"'cnE.A-:'u.:.\ -.

tu

56
For the matching^ it suffices to consider the above relations vith respect

to their counterparts in (30) . Since the exponential terms in <-< ( 63 ^6j ) are correct to a factor { I -h C*" ( ^ )

) while u ( C, ^ . . ._, C_^) in (30) is

correct to a factor ( / f (3'(^')), the best match possible to the level

of approximation taken is in the leading terms only, u ( C, , C-^ ) . The

H ( Cj , C^ ) terms are due to coupling with other functions in the

M ( C , . . . , C^ ) - approximation and should be suppressed for this match.

Then in w, the w( C , C., ) portion is induced by ( ux/ , m ) coupling which

is the part ofw(D , O-,^ D ) not explicitly given. The remaining part,

( Cij , ^t^' ^^ ^°^ coupled to u (the u ( Clj , V- ^ having been suppressed)

and should be matched with the free part w( D ). ¥e obtain then

w

-^.^J^^C - B A'\"-[, * • »

'/3 _ . - r.

fH^ (- Hi

c and

P. a;

For the inviscid parts, [ u (o.)^ '^^i^i/J matching gives, since

Eliminating the tj- from the above lead to (33)^ where

* \ / ; * . *

•^^ /- rti

57
Appendix k

TIETJEN'S AMD RELATED MICTIONS.

In the elements of the determinant A. . occur the Tietjen's fiinction

and some related functions incorporating the viscous effects in the so- called "critical layers". Asymptotic expansions for large 0^ are inadequate

approximations on the neutral stability curves, while the tabulations of

[6] some of them given by Hoistein do not give adequate accuracy in the

crossover range between convergent (small Xci,) and asymptotic representations

A computing routine was developed to compute these functions taking advantage of a routine written by Strand giving Airy's functions CXjl (2 ) and die ( ^ ) to eight decimal digits

.

We introduce the notation:

/

^p(^ ; B, c) =- i

y.(rj

(C --

(C-^->)

where

Then

Putting

.>6-§J

^ ir / S. ^.^-I^J.

58
we have;

e, e. B.

C < -/ - / - 1 o < c< \ - / -/ 1 c > / 1 / 1 and

C = t 8/ ' C = L B.

2 ^^

We also have the Identity:

Y

• n'-]J^^

i Vt

4-

eliminating the need for determining 2-. , /^ . Let also

The elements A., containing {^- , 1.) functions taJse the following forms

U I

?3

^ /-

^6,., 2J... 6.. /J . ^

^3

' r 3*5

^.fefi'^.O

a

59
The following formulae are readily verified:

L

60
when R and I are real. In addition^ we compute the function

X, r^, e c) = J (i, Bc) , i c TCi- B c)

where S, T are real. Then with

7, =- (a^-Jj/.B.

we have may derive:

Q - '

( Q - ^,

^r '^^/j .

These are found from the Airy function properties

(2^ \ f£'\ p*

yJ

r

)

(Recursion formula)

(Airy equation)

giving

^' P'l

(p > =2)

61
X ( 2 ; B , C ) itself is found from the formula:

tMB,c).

(C - /)

(c^-0 J

calculating z j and T^ as follows:

BC = 1 Convergent series: < z < 7*5

Integral evaluation: 7*5 < H < 13*0

Asymptotic series: ? > I3.O

BC = -1 Convergent series: < ^ < 10.6

Asymptotic series: ? > 10.6

Tests indicate a minimum of four significant figures near the joint for

BC = -1, five significant figures minimum elsewhere. The integrals used,

and the basis for the asymptotic expansions, arise from the saddle point

method applied to the defining integrals for ( 2p / « )• They are:

I (T)

P

-i^y^c"'^

62
Appendix 5

THIN FIIM APPROXIMATIONS FOR {H , 77^^ , ^

^u > ^~^ul , ^gJ •

For the detennination of H = x a''^-^ . from the inviscid pressure

equation "

in the core, satisfying ll (f) —> P as f —> ^ , we shall break

the core into an inner core with 7K = Iv^ constant for < f < f and

»n = assumed, and an outer core with linear velocity profile

'^= I

-t yK {t - ^ ) fo^ ^_ < f < X and in which we drop the cylindrical

terms f~^ ^^ J from the equation. The constants fj- and 7^ are determined to give the same gas flow rate as the true profile:

Jy^ / ^ / G'-^ r'J (0< f < X ),

and >; = / ^ 7;^' ( r_^ - x)

gives continuity in 7^^ at f = Tj. . In the inner core,

irfrj - cjLT, U r)

a(T) = TT 'cr) = oc ^x_, (o, r).

Assuming "^ ^ > > 1 as occurs, we shall use leading asymptotic terms at

(X &

^^^^'^'

w.

CIX <x C-"^'

-^x

In the outer core, we have:

TT" - ^

TT' - ^'TT ' c

where T_ is such that

/- c - - ^:{r.'^)

63
Thus f^ is the critical layer location as siiming the same linear profile

is valid to >"= c i.e., if / < c < '^ . If ;k ( f ) = c in the film, ^

is not the true critical point ^. Note "^ = -2 rX. It is ass-umed that

TTCr) ^ A (/- ^(^' ^Jje"^^'"^^^

Joining {l',Tl ) continuously at T = f^. , determines the ratio b/A:

A I ^ ^ o,(r^- f,)

Then we obtain the approximation

hi [y^'. ^)=^

3

[/^-fr:-J]^ ^['-.(^-rjjc'-"^'''^

The mass flow calculation leads to

For the thin film limit n—^ C , another approximation may be made

with m = ^ P

5

We can, to principal order in n , obtain:

6U
The solution "boTJiided as ( ^ - ?c )xa.—> - <^ yields the approximation:

agreeing with the earlier approximation H when o £ =0

B Since the terms with -r are at "best approximate the form we have used is

the first with ex:: set to .<3-

^

The approximation is weakest for non-thin films with P > 0. This

might he remedied by retaining m 7^ in the inner core solution, and using

the asymptotic expansion associated with o:-~¥oo ^^ the solution:

These expansions are very complicated and the result would still he approximate in the outer core even if the broken velocity-profile were exact, so this approximation was rejected.

For the liquid film, the profile is again linear, and the thin film

case again leads to

Tf " '

-fTf^ IT' ^ ^"TT ~ o

as above, where now we have

is the critical layer when < c < 1 . Setting

1, - I' C

and again using

65
the solutions satisfying the requirements:

give on S = L:

^r^' y;/^'^^^^;/'7J"^j,;-^^'-r7.-7x^

Cl^J -^ i^ ^^..^ ^C^^-y J

^ TV = -^^VUsl

(.L

-^ ,<t..^ ^(/7,- yj

1

-tSL II X

Special care must he taken in computing 777. when .-=aL.( <0 - tl ^) Is small

since much cancellation occurs. These thin film approximations cannot he expected to give good results

if gravity causes significant film profile curvature.
Appendix 6

HTVISCID LBDIT

As mentioned earlier, the inviscid stability limit may "be fo^und by

a completely inviscid analysis, or by an order analysis as ^^<-<,—^ o ^ giving the result:

AHr A.,.-fG A

\. ^

PC

^'2

O

The values of these elements for xt-<. = are as follows

A, , = -iH

4r = ' ^'^,

L m, z

0- -)

I -t 3. 3

0(^ A

fr - L X /-c-^

n.

The result is written simply as

ru - -,

^

(jzzy Z-ll^ rf^j-o.

We use the thin film approximations derived in Appendix 5^

H- X ^c^^

/-c / —

Noting also

since 1 - c = - "T^ ( C ~ '^ )•

7x = - ^ C/-^J

67
we have

6? = - ^

l^ .--X ^^

^6L /'O --^^'--^^^ ^ Q^^T)^^ .^-^^^

where '^^^^/c ^ '-'^'^ n. Then

<2

and putting h-^ \ ,

D

f, H = _^:^

-. - tIt^^ f)

Introducing

for small /i , Z A <r

In \2.^

we obtain the inviscid stability curve for thin films:

-n,_^^ -»- '^ P - t'J ' "

(Note that only the disturbance flow is assumed inviscid. The base flow

must be viscous.)

68
When cr = 0^ we have the neutral curve r

rw - m. / f

C- /- . "l^lL .

,-cu-^ (/ ^ ry\

^ ^-t^-yJi -^ ^ J

which agrees with Feldman's result when Q = . When cr ^ 0^ the equation

is quadratic. (Feldman's erroneous formulation of the Interface equation

again leads to a linear equation.) Since concurrent flow implies for the

yr CO ^ ^^ v^ W ^ ^

we have the restrictions on 1'-

h^^y^fK^''

for small h • With the restriction to concurrent flow then, we have

Thus in general , since i^, > 2 rn^ , the sums of the roots ( C, C-^ ) satisfies

Then for waves not too long such that

the roots (l - c) are of opposite sign: one travels faster than the interface_, one slower. For long waves such that

the roots (l - c) are "both positive, each wave travelling more slowly than

the interface

.

For non-long waves, we have for the second solution for small surface

69
tension

C

a h or

^^, --^^ 0^ p""' ^ y

-.'-^(^^ Jt^( ^

"^p n-?y*

70
en

Ui>

cro

<

a>IT

71

CO

UJ>

CE

<

u.

a>

72

V)>

cr

IDO<o

73
1
UJ>

a:

=)ooo

q:

7^

lOO

00o o

II II b CP

en

UJ>

ITO<

o:o

ino

u. I I I L

75

UJ>

q:

=><o

to

76

u.

a>

P~TT

? I

<

S EC O

§£ Al 5 CQ.

ca. m o —

— CO -a- CM _ _:

CO

Ld>

a:3O

<

o

o

b C7>

0)

77

>

<

o

O

OO

u.

a>

78

0)

en

Ld>

a:o

_l<o

r^ lO o O

^^

•—• ro iO

CJ K) OM O1

79

O

>

q:<

X Q

o UJ

u. o

o Q.

<1> 5 q:

CC

LxJ QD

UJ> ccLU

CL

tn

Z) _l

z < a>

CO n Q 1— 2 —Jo o

<

> O)

llJ _i

(T < cn K LU Q Z q: 3

UJ 3

2 e>

_J ora

u. X X

li.

dl 'y313IMVyVd SS3N>10IH1

80

» te-i,

Table 1. ct = C ^<'^ a a.. 3 rtfo 1^-

^a

I

.00100 93 5 1.91 583 te(-3) -.60 (+2) .22 (+1) 66(-l)

.001+32 92 9 2.01 589 33(-3) -•22 (+3) •27(+l) 80(-l)

.00898 91 9 2.01 589 35(-3) -•22 (+3) .26(+l) - 78(-l)

.0154 90 5 2.01 588 37 (-3) -.20 (+3) .21+ (+1) 71(-1)

.0227 89 1 2.01 588 39(-3) -19(+3) .22 (+1) 67(-l)

.0320 67 3 2.01 587 43(-3) -l7(+3) .20 (+1) 60(-l)

.01+28 85 2 2.12 592 37(-3) -.21 (+3) .22 (+1) 68(-l)

.051+6 82 6 2.12 591 1+2 (-3) -.11 (+3) .19(+1) 58(-l)

.0697 79 8 2.12 590 W(-3) -.17(+3) .16(+1) l^9(-l)

.081+8 76 5 2.23 591+ '*'*(-3) -.19(^3) .17(+l) - lt9(-l)

.103 72 9 2.3U 598 l*2(-3) -.2l(+3) .i6(+i) lt9(-l) .119 69 5 2. 31+ 595 52 (-3) -.2l(-h3) .12 (+1) - 39(-i)

.138 65 5 2.1+7 598 5l(-3) -19(-i-3) .11(+1) 3'+(-l) .160 60 7 2.59 597 61+(-3) -.12 (+3) .69(0) 21(-1) ,'

.2 .00100 95 2 1.90 585 38(-3) .22 (+1) 69(-i) -•51(-it) .21 (+2)

.1+ .00100 100 1.81 585 36(-3) .21+ (+1) 7i+(-i) -.56(-l+) .l+0(+2)

.6 .00100 109 1.72 569 28(-3) .30(-t-l) 3H-^) -.8o(-i+) .71* (+2)

.001+32 106 1.72 589 3l(-3) -.7^(3) .27(+l) 86(-i) -.32C-3) .50(-3) 59(2)

.00898 102 1.72 588 35(-3) -•70(3) .23(1) 76(-l) -.58(-3) .9i*(-3) .^1(2)

.0151+ 97 2 1.81 59't 33(-3) -.81(3) .2i+(l) 8l(-l) -.13 (-2) •19(-2) .1+0(2) .0227 90 9 1.81 593 i+2(-3) -.73(3) .18(1) 66(-i) -.ll+(-2) .2l+(-2) .12(2) .0319 83 1.90 596 ^6 (-3) -.78(3) .ll*(l) ' - 6l(-l) -.19(-2) .36(-2) -.56(1)

.01+28 73 1+ 1.99 598 58(-3) -.79(3) .8l+(0) 51{-1) -.12(-2) .l+6(-2) -.31(2)

.051+6 63 2 2.09 597 85 (-3) -.70(3) .20(0) 37(-l) .2l+(-2) .53(-2) -.50(2)

81

(Kl

Table 2

.

•e^e

CT = 0.5(10)

"3 z*!. 3S^

^3

•3fep

.00432 91.6 2.00 .586 35 (-3 -•'^3(3 • 23(1 -•72 (-1 -•23(-3) •49(-3) .42(1) .00896 89.5 2.00 .582 38(-3 -•34(3 .20(1 -.62(-l -•34(-3) • 92(-3) .24(1)

.0151+ 87.3 2.10 .581 34(-3 -•31(3 .21(1 -.64(-i -•44(-3) •19(-2) •23(1) .0227 85.4 2.10 .574 40 (-3 ) -•13(3 .16(1 -•47(-l +^34(-4) 2l(-2) •53(0) .0320 84.9 2.16 .567 47 (-3 +.46(2 .11(1 -•32(-l .10(-2) •25 (-2) -.19(0)

.0U28 86.1 2.10 • 559 5t(-3 ) ^17(3 .71^(0 -.21(-1 .22 (-2) .22 (-2) -35(0)

.05U6 88.7 2.10 •553 59(-3 •25(3 •50(0 -•13 (-1 • 36(-2) 25 (-2) +.16(0)

.0697 92.6 2.00 543 70 (-3 •26(3 .26(0 -.6l(-2 .42(-2) •15(-2) -•54(0)

.0848 96.9 1.91 • 536 77 (-3 25(3 .14(0 -.20(-2 .49(-2) •l8(-2) -•12(1)

.103 102. 1.91 .531 77 (-3 •26(3 • 68(-; ) +.24(-2 • 59(-2) •76(-3) -•37(0)

.119 107. 1.82 .524 Bo (-3 • 24(3 -•70(-:-) M(-2 .66(-2) .19 (-2) -•11(1)

.138 112. 1.82 .521 78 (-3 .25(3 -.17(0' .81 (-2 .73 (-2) -•10(-3) -.28(0)

.160 118. 1.73 •514 79 (-3 •23(3 -.24(0 .ll(-l .84(-2) + .29(-2) -•95(0)

.185 124. 1.65 .508 77 (-3 .21(3 -.32(0' .13 (-1 .8l(-2) -.32(-3) -•17(1)

.214 131. 1.65 •503 73 (-3 • 20(3 -.44(0' •17(-1 .98(-2) H.10(-2) +.24(0)

.248 139. 1-57 .496 69 (-3 • 18(3 -•5lt(0 .20(-l • 95 (-2) -.24(-2) .40(0)

.273 144. 1-57 .492 65 (-3 16(3 -.63(0; •23 (-1 .ii(-i) • 69(-3) .31(1)

.301 149. 1.50 .484 6l(-3 • 11(3 -•70(0 .26(-l .lO(-l) -37(-2) •35(1)

.332 153- 1.49 .477 56(-3 • 69(2 -.79(0' •29(-l •12(-l) -.25 (-3) .86(1)

.366 156. 1.37 .463 54(-3 • 74(0 -.96(0) •34(-l .13(-1) +^50(-2) .40(2)

.404 155- 1.37 .447 WTT -•I4(r -.10(1 .36(-i • 7^-2) -.l8(-l) .60(2)

.445 143. 1.30 .415 39(-3 -•47(3 -.10(1) .29(-l -.15(0) -.60(0) •13(3)

.491 102. 1.24 .326 39(-3 -.11(4 -.20(0) .44 (-2 -.51(-1) -.12(0) .21(3)

.501 78.8 1.24 .264 76 (-3 -.70(3 •55(0) .2l(-l .28(1) • 55(1) .49(2)

.511 63.8 1.34 .197 78 (-3 -.86(3 -•43(-l ) -.l6(-2 .30(-2) .i6(-i) .76(2)

.521 52.6 1-37 .123 13 (-2 -•96(3 -.l4(o) +.33(-4 -.13(0) -.19(0) .13(3)

.532 43.2 1.4o .050 20 (-2 -.44(4 -.12(2) -.69(0) -•13(3) -.16(3) .80(2)

.6 .00432 104. 1.73 .586 3l(-3 -.10(4 •26(1) -.8l(-l -•27(-3) .49(-3) • 54(2)

.00898 99.6 1.73 .582 38(-3 -.81(3 •20(1) -.65(-l -.34(-3) •87 (-3) .32(2) .0154 93-4 1.82 .579 39(-3 -.67(3 •16(1) -•55(-l' -.l4(-3) •15 (-2) .27(2)

.0227 89.1 1.91 .574 44 (-3 -.34(3 •10(1) -.39(-l +.10(-2) • 2l(-2) .32(2) .0320 87.4 1.82 .561 70(-3 -•17(2 • 35(0) -.18(-1' .22(-2) •l4(-2) .16(2) .0428 88.3 1.82 .551 87(-3' +.18(3 -.11(-1 ) -.72 (-2 .40(-2) •13(-2) •26(2) .0546 90.8 1.73 .541 11 (-2 .23(3 -.24(0) -.l4(-2' .50(-2) •72 (-3) .26(2) .0697 94.6 1.65 •531 13 (-2 • 25(3 -.45(0) H.33(-2' .62(-2) •39(-3) .29(2) .0848 98.1 1.50 .521 14 (-2 •21(3 -.64(0) .45 (-2 •67(-2) -.36(-4) .21(2)

.103 102. 1.29 .509 15 (-2 .13(3 -.96(0) • 33(-2 .74(-2) -.52(-3) .14(1) .119 103. 1.12 .498 15 (-2 .48(2 -.i4(i) •88(-3 .82 (-2) -.l4(-2) -.25(2)

.138 95.6 .592 .451 80 (-3 -.21(3 -•33(1) .10(-l' .25 (-2) -.lo(-l) -.11(3)

.160 91.9 .622 •431 11 (-2 -•57(2' -.25(1) .79(-2' •69(-2) -•45(-2) -.59(2)

.185 91.2 .622 .411 13 (-2 + .22(2 -.21(1) .79(-2' .3l(-2) -.88(-2) -•29(2)

.214 92.6 .592 .388 14 (-2 .74(2' -.18(1) .94(-2' .l4(-2) -.98(-2) -.82(1)

.248 96.1 592 •373 15 (-2 •12(3 -•18(1) .87 (-2 15(-2) -.ll(-l) +.66(1)

.273 99.9 .592 • 365 15 (-2 • 16(3 -.19(1) .82 (-2) .79(-2) -•55(-2) .16(2)

.301 106. .622 .368 15 (-2 .22(3' -.22(1) .66(-2) .66(-2) -•9l(-2) .30(2)

1.2 .103 94.8 .568 .402 15 (-2 .74(2' -.16(1) •96 (-2) .44(-2) -.28(-2) .42(2) 119 93-6 .568 .384 17 (-2 .13(3' -.14(1) •lO(-l) .59(-2) -.l4(-2) .48(2)

.138 99-2 • 541 .364 18 (-2 .15(3) -•13(1) .12(-1' .40(-2) -32(-2) 51(2)

.160 103. •541 • 350 18 (-2 .18(3' -.13(1) • 13(-l) •63 (-2) -.l4(-2) .60(2)

.185 108. .515 •333 18 (-2 • 20(3 -.13(1) .14(-1 32(-2) -.49(-2) • 64(2)

.214 115. •515 .214 l8(-2 ) .23(3 ) -.13(1 .14(-1 •7l(-2) -.22 (-2) 77(2)

82

Table

g P Kl

2. 0- = 0.5(10)^ Cont '

d

%^

.2U8 124. .491 313 • 17(-2) •26(3) -.14(1) •14(-1) •9i(-3) -•ll(-l) 85(2)

.273 132- .445 .298 .l6(-2) •26(3) -•14(1) •15 (-1) -•27(-2) -•13(-1) 80(2)

.301 l4l. .445 .302 .l4(-2) •32(3) -•16(1) •13(-1) -.42 (-2) -•19(-1) 97(2)

• 332 154. .404 .294 •12(-2) •36(3) -•18(1) 13 (-1) -.82 (-2) -.24(-l) 94(2)

.366 170. .4o4 .309 10(-2) •52(3) -.25(1) •90(-2) -•92 (-2) -•37(-l) 12(3)

.kok 197. .366 .317 •76(-3) • 75(3) -•35(1) .65 (-2) -•ll(-l) -.48(-l) 13(3)

.41+5 244. .349 •341 •47(-3) .14(4) -.64(1) .6l(-3) -.20(-l) -•99(-l) 18(3) 0.5(10)5 .00163 112. 1.73 •575 .37(-3) •29(5) •26(1) -.8o(-l) -•54(-4) •83(-3) 46(1)

.0021+1 142. 1.42 • 555 •37(-3) •43(5) •30(1) -.93(-i) -.38(-4) •17(-2) 43(1)

.00339 201. 1.06 .526 •29(-3) .70(5) • 45(1) -.14(0) -.22 (-4) •36(-2) 46(1)

.00454 313. .719 • 493 •l8(-3) .18(6) .88(1) -.28(0) -.12 (-4) •lO(-l) 52(1)

-0.5(10)5 .00163 23.0 3-97 .524 •30(-2) -.60(6) -•18(2) -•12(1) -13(-2) -.84(-3) - 50(3)

83

Table 3. a = 0.1(10)^ Kl 1^ t) r- 5-^

.00432 90.5 1.99 .582 .38(-3) -56(3 ) .21(1 ) -•65(-i) -.i8(-3 •34(1) .00898 88.1 2.09 .580 34(-3) -.40(3 ) .21(1 ) -.64(-l) -.22 (-3 .36(1)

.0151* 86.8 2.09 .569 •4l(-3) + .44(2 ) .14(1 -.42(-l) + .28(-3 .21(1)

.0227 88.4 2.09 .560 •48(-3) •36(3 ) -91(0 -.26(-l) .ll(-2 .23(1) .0320 92.3 1.99 .548 .60(-3) •45(3 ) -50(0 -•13(-1) .17(-2 .17(1) .0428 98.0 1.90 .538 .69(-3) ^6(3 ) .27(0 ) -.59(-2) •23 (-2 .15(1) .0546 104. 1.81 .530 .73 (-3) .44(3 ) .12(0 -.10(-2) 27 (-2 .13(1) .0697 112. 1.72 .522 .75 (-3) .1^2(3 -•25(0 + .38(-2) •32(-2 •12(1) .0848 119. 1.72 .518 • 72(-3) .44(3 -.18(0 .88 (-2) .39 (-2 .24(1)

.103 128. 1.64 .512 .7i(-3) .^0(3 -.30(0 .13(-1) .46(-2 .23(1) .119 135- 1.56 .506 .70(-3) .37(3 -•39(0 .16(-1) .46 (-2 .17(1) .138 142. 1.49 .501 .68 (-3) .35(3 -.49(0 .19(-1) • 54(-2 .12(1)

.160 151. 1.49 .498 .63(-3) .35(3 -.66(0 •25(-l) .55 (-2 .29(1)

.185 160. 1.42 .492 59(-3) .32(3 -.80(0 •30(-l) .77 (-2 .30(1)

.214 171. 1-35 .486 .54(-3) .28(3 -.99(0 .36(-l) .62 (-2 .32(1)

.248 182. 1.29 .479 .49 (-3) .2'^(3 -.12(1 .44(-l) .12(-1) Mil)

.273 189. 1.28 .475 .44(-3) .22(3 -.14(1 .52(-l) .67 (-2 .84(1)

.301 196. 1.22 .468 .hl{-i) .15(3 -.16(1 .59(-l) .18(-1 .10(2) 332 202. 1.17 .458 .36(-3) .64(2 -.19(1 .68(-l) .82 (-2' •13(2)

.366 204. 1.11 .446 .3l(-3) -•95(2 -.22(1 .78(-l) .12(-1' .19(2)

.404 197. 1.11 .427 .26(-3) -.39(3 -.24(1 82(-l) .63 (-2 •36(2)

0. 6 .00432 103. 1.73 .583 .33(-3) -.11(4 .24(1 -.75(-l) -.2l(-3) .48(2) .00898 97.9 1.82 .579 .33(-3) -.83(3 .20(1' -.65(-l) -.l5(-3) .47(2) .0154 94.6 1.82 .567 .t5(-3) -.66(2 .10(1 -.35(-l) .67(-3) .36(2) .0227 95.9 1.82 .556 .57(-3) +.39(3 .43(0] -.17(-1) .18 (-2) .44(2) .0320 100. 1.73 .51*3 .75 (-3) .52(3 .36(-J-) -.50(-2) 26(-2) .i*3(2) .0428 106. 1.57 .531 .9l(-3) M{3 -.20(0' +.39(-3) 29(-2) .35(2) .0546 112. 1.50 .522 .98(-3) .1^5(3 -.42(0) .56(-2) .36(-2) .40(2) .0697 119. 1.36 .513 .10(-2) • 39(3 -.65(0' .87 (-2) .40(-2) .35(2) .0848 126. 1.29 .506 .ll(-2) • 38(3 -.88(0) .13(-1) .'+7(-2) .40(2)

.103 132. 1.12 .497 .ll(-2) .28(3 -.13(1 .13(-1) .50(-2) .17(2)

.119 137- 1.01 .490 .ll(-2) .23(3 -.17(1) .13(-1) •56(-2) .20(1)

.138 125. .464 .455 .48(-3) -.32(3' -.57(1) .23(-l) .87 (-3) -.20(3)

.160 120. .464 .432 .65 (-3) -.67(2' -.41(1) .20(-l) .h5i-2) -.11(3)

.185 120. .464 .^13 .78(-3) .59(2' -.34(1) .18(-1) -.19(-2) ^.66(2)

.214 123. .421 .388 .85 (-3) .15(3' -.32(1) .16(-1) .47(-2) -.40(2)

.248 130. .464 .385 .92(-3) .25(3' -.33(1) .11* (-1) -.19(-2) -.24(2)

.273 137. .442 .375 .89(-3) .32(3' -.3^(1) .16(-1) -.12(-2) -.18(2)

.301 149. .442 .372 .84(-3) .1^5(3] -.40(1) .15 (-1) -.26(-2) -.15(2) O.U(IO)^ .001 96.1 1.91 .583 .38(.3) .82(4 .23(1) -•70(-i) -.46(-4) .£6(-3) ^43(1) 0.2(10)5 .001 93.9 1.91 .582 .i*l(-3) .15(U) •21(1) -.64(-i) -.42 (-4) •l7(-3) ^36(1) -0.2(10)5 .001 92.0 2.01 .587 3H-3) -.34(41 .24(1) -.75(-l) -.6l(-4) • 62(-4) .51(1)

-0.1*(10)5 .001 89.2 2.01 .586 .38(-3) -.11(5) .21(1) -.67(-l) -.54(-4) .76(-4) .42(1)

Ok

Table h. a = OA(lO)^

? p ^ (KL % C

^(^P

-^^i.

.001 91-3 1.99 .583 .36(-3) 22(1) -.68(-i) -.45 (-4) .1+3(1)

.001I32 88.6 2.09 .586 .38 (-3) .66(3) .22(1) -.64(-i) .99(-^) .81(1) .00898 95.6 1.99 .5^7 .5i^(-3) .19(1+) .57(0) -.15(-1) .49(-3) •1+6(1)

.0151^ 108. 1.81 • 530 .65 (-3) .n(h) .15(0) -.l8(-2) .73(-3) .46(1) .0227 121. 1.56 .517 •70(-3) .12(4) .72(-3) .32 (-2) .77(-3) • 3l+(l) .0319 135. 1.14-9 .509 .67(-3) .12(4) -.26(0) • 12(-1) .l0(-2) •45(1)

.0lf28 li*9- 1.1*2 .502 .63 (-3) • 11(1+) -.51(0) .21(-1) •13 (-2) • 56(1)

.051*6 163. 1.35 .1*97 .59(-3) .10(4) -.76(0) •29(-l) •15 (-2) .64(1) .0697 179- 1.22 .1*91 .55(-3) .88(3) -•97(0) .36(-l) •15 (-2) •56(1)

.O8I18 194. 1.17 .1+86 .50(-3) .82(3) -.12(1) .1+5 (-1) .20(-2) .62(1)

.103 210. 1.11 .1*82 M(-3) .77(3) -.16(1) .57(-l) 19(-2) .69(1)

.119 224. 1.06 .1*78 .U2(-3) .72(3) -.19(1) 67(-l) •30(-2) .71(1)

85

Tat)le 5. a = 0.8(10)''

3 ? fel -3 H 0/«„ '2^ op

.001 89.8 1.99 .577 •39(-3) • 19(1 -.58(-l) -•25 (-4) • 35(1)

.00432 95.1 1.99 .548 .53(-3) 37(4 ) .64(0 -.17(-1) .22 (-3) .20(-3) '+9(1) .00899 112. 1.72 .525 .67(-3) 31(4 ) .96(-:L) +.98(-4) .38(-3) .lM-3) • 1*9(1) .015U 133- 1.56 .511 .65 (-3) 26(4 ) -.26(0 .12(-1) .55(-3) .69(-4) 63(1)

.0227 151. 1.42 .502 .62 (-3) 22(4 -.55(0 •22 (-1) .66(-3) -.36(-4) 71(1)

.0319 170. 1.28 .494 .57(-3) 18(4 -.85(0 .32 (-1) •75(-3) -.l9(-3) • 76(1)

.0U28 190. 1.17 .489 .52(-3) 15(4 -.12(1 .42 (-1) 83(-3) -.37(-3) 77(1)

.o^kS 208. 1.06 .484 .^7 (-3) 13 (^ -.14(1 .5l(-l) .87(-3) -.56(-3) 70(1)

.0697 230. 1.01 .479 .'^2 (-3) 12(4 -.19(1 .68(-i) •ll(-2) -.56(-3) .86(1) .0848 250. .960 .475 • 37(-3) 11(4 -.24(1' .84(-l) • l2(-2) -.l4(-2) • 97(1)

.103 272. .870 .471 .33(-3) 98(3 -.28(1 .97(-l) .ll(-2) -.l7(-2) .81(1)

• 119 290. .829 .468 .3l(-3) 92(3 -•30(1 .11(0) .l7(-2) + .l2(-3) • 85(1)

.138 311. .789 .464 .26(-3) 87(3 -39(1' .mo) .13 (-2) -.28(-2) • 93(1)

.160 334. .752 .460 .24(-3) 82(3 -.48(1 .16(0) 29(-2) + .39(-2) • 11(2)

.185 360. .752 .455 .l9(-3) 78(3 -.62(1 .21(0) .15 (-2) -.50(-2) • 14(2)

.214 388. .682 .449 .15 (-3) 73(3 -.83(1 .27(0) .88(-2) .3l(-l) .24(2)

.248 422. .648 .443 .l0(-3) -.13(2 .44(0) .l6(-2) •34(2)

.260 434. .623 .440 .90(-4) 68(3 -.15(2 •50(0) .30(-2) -.75 (-2) .37(2)

.273 447. .611 .437 .78(-4) 66(3 -.17(2 .60(0) .15(0) .81(0) .50(2)

.287 465. .587 -.434 .55 (-4) 78(3 -.26(2 .87(0) .67(-2) .3l(-3) .62(2) ~ .301 486. .576 .i^33 .39(-4) 11(4 -.36(2 .13(1) .11(1) .59(1) 12(3)

2 .001 91.4 1.99 • 579 •35(-3) .21(1' -.64(-i) -.30(-4) 23(2)

"o! 4 .001 96.3 1.90 .579 .32(-3) .23(1' -.70(-i) -.33 (-4) 43(2)

6 .001 104. 1.72 .577 .33(-3) .22(1' -.69(-l) -.30(-4) 49(2)

.00432 110. 1.72 .548 .^7(-3) 44(4 .65(0' -.18(-1) .30(-3) .l9(-3) .68(2) .00898 130. 1.49 .524 .6l(-3) 36(4 -.88(-£ ) +.23(-2) .t*7(-3) .12(-3) .70(2) .0154 152. 1.29 .509 .63 (-3) 27(4 -.40(0 .14(-1) .59(-3) .44(-4) 76(2)

.0227 173. 1.17 .500 .60 (-3) 23(1* -.78(0' .26(.l) .73 (-3) -.76(-4) 89(2)

.0320 194. 1.06 .492 .56(-3) 19(4 -.12(1' .38(-l) .86(-3) -.26(-3) 10(3)

.0428 2l4. .960 .486 .52 (-3) 16(4 -.17(1; .50(-l) 97(-3) -.50(-3) • 11(3)

.0546 234. .870 .481 .'^8(-3) 14(4 -.22(1; .62 (-1) .ll(-2) -.7M-3) 11(3)

.0697 256. .789 .476 M(-3) 13 (^ -.28(1) •75 (-1) •12(-2) -.12(-2) 11(3)

.0848 276. .716 .472 .h2{-3) 11(4 -.34(1) .86(-l) .l3(-2) -.15 (-2) 10(3)

.103 299. .650 .468 .38(-3) 11(4 -.42(1) .10(0) .m-2) -.25 (-2) 97(2)

.119 318. .619 .464 .35 (-3) 11(4 -.50(1) .12(0) 19(-2) -.l5(-2) •11(3)

.138 340. .561 .460 .33(-3) 10(4 -.60(1) .13(0) .17 (-2) -.33(-2) • 94(2)

.160 365. .485 .457 .3l(-3) 10(4 -.75(1) .13(0) .20 (-2) -.47(-2) .40(2)

.185 31^3. 1.92 .^31 .17 (-3) 10(4 -.20(2) .10(0) .27 (-2) -.14(-1) -•68(3)

86

fej.

Table 6. a 0.12(10)"^

"3 01, -3 »-

.0001 88.9 2.04 .573 .38(-3) -.94(4 ) .18(1 -.54(-l) -.87 (-6' 12(-4) .42(1) .000161 89.40 2.o4o • 5639 .43 (-3) .25(5 ) .13(1 -.38(-l) .38(-5 .l4(-4) .42(1) .000259 93.62 1.981 .5508 .53(-3) •49(5 ) .74(0 -.2i(-l) .12(-4) •isx-i*) .46(1)

.0001+18 102.6 1.867 .5366 .6l(-3) .53(5 .34(0 -.79(-2) .20(-4] .ll(-4) .52(1) .000673 116.3 1.708 .5228 .65 (-3) .46(5 .34(-]L) .23(-2) .26(-4) .75 (-5) .60(1) .001 131. 1.54 .512 .66(-3) .37(5 -.20(0 .lO(-l) .34(-4) .34(-5) .65(1)

.002 164. 1.28 .497 .59(-3) .24(5 -.67(0 .27(-l) .45 (-4 -.82 (-5) .61(1) .004 207. 1.05 .486 .48(-3) .16(5 -.14(1 .52(-l) .58(-4) -.46(-4) .10(2) .007 251. .880 .479 .40(-3) .11(5 -.22(1 .78(-l) .68(-4) -.ll(-3) .12(2)

.015 324. .692 .472 .30(-3) .65(4 -.36(1 .13(0) .84(-4) -.3l(-3) .15(2) .030 415. .562 .465 .22 (-3) .45(4 -.56(1 .19(0) .ll(-3) -.74(-3) 18(2) -0.13(10)5 .0001 8C.9 2.04 573 •38(-3) -.94(4 .18(1 -.54(-l) -.87 (-6) •12(-4) .42(1) .000161 89.40 2.040 .5639 .43 (-3) .25(5 .13(1 -.38(-l) .38(-5) .l4(-4) .42(1) .000259 93.62 1.981 .5508 .53(-3) .49(5 .74(0 -.21(-1) .22(-4) .13 (-4) .46(1)

.0001+18 102.6 1.867 .5366 .6l(-3) .53(5 .34(0 -.79(-2) .20(-4) .ll(-4) .52(1) .000673 116.3 1.708 .5228 .65 (-3) .46(5 .34(-]L) .23 (-2) .28(-4) .75 (-5) .60(1) .001 131. 1.54 .512 .66{.,) .37(5 -.20(0 .lO(-l) .34(-4) .34(-5) .65(1)

.002 164. 1.28 .497 .59(-3) 25(5 -.72(0 .28(-l) 45 (-4) -.95(-5) .61(1) .004 210. 1.03 .486 .48(-3) .18(5 -.16(1 .57(-l) .59(-4) -.36(-3) .10(2) .007 274. .796 .482 3'*(-3) .24(5 -.37(1 .12(0) 73(-4) -.4l(-2) .11(2) .010 4i6. .556 .488 .l4(-3) .87(5 -.13(2 .44(0) .77(-4) -.35(-l) .71(1) .011 529. .446 .489 .89(-4) .14(6 -.23(2' .78(0) .62(-4) -.67(-l) .72(0)

.032 700. .340 .487 .57(-4) .20(6' -.38(2' .13(1) .38(-4) -.11(0) -.59(1)

.013 924. .256 .485 .4l(-4) .24(6 -.54(2' .16(1) .17 (-4) -.14(0) -.97(1)

.014 1190. .198 .463 .3l(-4) .28(6 -.68(2 .23(1) .l4(-4) -.17(0) -.11(2)

.015 1500. .158 .483 .24(-4) .32(6 -.91(2 .30(1) .40 (-5) -.23(0) -.14(2)

.016 i84o. .127 .482 .20(-4) .36(6 -.11(3^ .37(1) .94(-5) -.28(0) -.16(2)

.017 2240. .105 .461 .l6(-4) .41(6 -.14(3 .46(1) -.ll(-5) -.33(0) -.19(2)

.018 2680. .0877 .481 .l4(-4) .46(6 -.17(3 .55(1) .27 (-5) -.40(0) -.22(2)

.019 3170. .0742 .481 .ll(-4) •52(6 -.20(3 .66(1) .48(-5) -.46(0) -.26(2)

.020 3720. .0629 .460 .98(-5) .56(6 -.23(3 .76(1) .13(-4) -.54(0) -.32(2) 1.0 .0130 875. .265 .487 .5l(-4) .22(6' -.44(2) .11^(1) .72 (-4) -.13(0) -.90(1) 1.1 874. .265 .487 .52 (-4) .21(6' -.42(2) .11^(1) .83 (-4) -.12(0) + .17(2) 1.0 .0131 896. .257 .487 .49(-4) .22(6 -.45(2) .15(1) .69(-4) -13(0) -.19(2) 1.1 894. .257 .487 .5l(-4) .21(6' -.43(2) .14(1) .8l(-4) -.12(0) +.59(1) 1.1 .0134 960. •239 .486 .46(-4) .23(6' -.49(2) .16(1) .69(-4) -.14(0) -.22(2)

1.2 958. 239 .487 .47(-4) .22(6' -.47(2) .15(1) .8o(-4) -.13(0) +.28(1) 1.4 .0140 1090. .208 .486 .4i(-4) .24(6' -.55(2) .18(1) .88(-4) -.16(0) -.35(1)

1.5 1090. .208 .486 .42(-4) .24(6' -.53(2) .17(1) .97 (-4) -.15(0) + .22(2) 1.8 .0150 1350. .166 .486 .34(-4) .26(6' -.67(2) .22(1) .97(-4) -.19(0) + .23(0) 1.9 1350. .166 .486 .35 (-4) .27(6' -.66(2) .21(1) .12(-3) -.18(0) .22(2) 2.2 .0160 i64o. .137 .485 .27(-4) .32(6] -.85(2) .27(1) .12(-3) -.24(0) .12(1) 2.3 l64o. •135 .485 .29(-4) .31(6] -.61(2) .25(1) .13 (-3) -.22(0) .19(2)

87

Table 7. ct = 0.23 (lO)'

3 f h Kl "r c 0^

*4, -2 <r- 0^

.0001 90.4 2.03 560 .46(-3) .71(5] .11(1) -.32(-l) .34 (-5) .84 (-5) .44(1) .000161 96.4- 1-95 546 .55(-3) .10(6^ .59(0) -.16(-1) .78(-5) .72 (-5) •49(1) .000259 107. 1.79 531 .65 (-3) • 92(5' .24(0^ -.44(-2) .ll(-4) .54(-5) .51(1) .000418 122. 1.63 518 .66(-3) .79(5) -.64(-].) .56(-2) .l6(-4) .33(-5) .62(1) .000673 l42. 1.41 506 .66(-3) .57(5) -.32(0) .14(-1) .l8(-4) .7l(-6) .64(1)

.001 162. 1.29 498 .60(-3) .47(5^ -.64(0^ +.25(-l) .23 (-4) -.33(-5) .79(1)

.002 204. 1.06 456 .49(-3) .30(5] -.13(1) .49(-l) .29(-4) -.2l(-4) .10(2)

.004 257. .854 479 .39(-3) .19(5) -.23(1' .82(-l) .36(-4) -.66(-4) .13(2)

.005 277. .797 477 .36(-3) .16(5) -.27(1' .95 (-1) .38(-4) -.90(-4) .13(2)

.006 294. .749 475 .33 (-3) .14(5) -.29(1' .10(0) .38(-4) -.ll(-3) .13(2)

.008 323. .690 473 .30(-3) .12(5^ -.36(1' .13(0) .44(-4) -.17(-3) .15(2)

.010 348. .644 471 .27 (-3) .10(5) -.ind) .14(0) .46(-4) -.22(-3) .16(2)

.014 389. .583 468 .24 (-3) .83(4^ -.50(1) .17(0) .52 (-4) -.34(-3) .18(2) -0.36(10)5 .000161 96.4 1.95 546 .55(-3) .10(6^ .59(0) -.16(-1) .78(-5) •72 (-5) .49(1) .000259 107. 1.79 531 .65 (-3) .92(5' .24(0^ -.44(-2) •ll(-4) .54(-5) .51(1) .000418 122. 1.63 518 .66 (-3) .79(5) -.64(-] ) .56(-2) .l6(-4) .33(-5' .62(1) .000673 l42. 1.41 506 .66 (-3) .57(5' -.32(0' .14(-1) .l8(-4) .7l(-6) .64(1)

.001 162. 1.29 498 .60(-3) .49(5' -.68(0^ .26(-i) .23(-4) -.23 (-5) .81(1)

.002 206. 1.04 487 .49(-3) .35(5) -.15(1' .54(-l) •29(-4) -.ii(-3) •99(1) .004 297. .743 484 .29(-3) 65(5) -.48(1^ .17(0) .4o(-4) -.27 (-2) .12(2)

.005 399- .579 489 .16 (-3) .16(6) -.12(2^ .40(0) .4l(-4) -.ll(-l) .81(1) .0051 4l6. .560 489 .14 (-3) .18(6] -.13(2) .45(0) .40 (-4) -•13(-1) .72(1) .0052 434. .540 489 .13(-3) .20(6^ -.15(2' .50(0) .39(-4) -•15(-1) .62(1) .0053 1*55

.

•515 489 .12(-3) .22(6 -17(2) .56(0) .37(-4) -•17(-1) .49(1) .0054 477. .495 490 .ll(-3) .24(6) -.19(2' .63(0) .35 (-4) -•19(-1) .36(1)

.0055 502. .470 489 •98(-4) .26(6) -.21(2^ .70(0) .33 (-4) -.2l(-l) .22(1) .0056 529. .450 489 .88(-4) .29(6' -.24(2 .80(0) .32 (-4) -.25 (-1) .93(0) .0057 558. .426 489 .8i(-4) .31(6) -.26(2] .87(0) .29(-4) -.27(-l) -.53(0)

.0058 590. .404 489 .73 (-4) .34(6^ -.29(2' .97(0) .26(-4) -30(-l) -.19(1)

.0059 624. .384 488 .66(-4) .36(6^ -.34(2 .11(1) .25 (-4) -.35(-l) -.32(1)

.0060 660. .360 488 .62 (-4) .38(6^ -.35(2' .12(1) .2l(-4) -.36(-i) -.42(1)

.0062 739. .324 487 .52 (-4) .42(6) -.42(2^ .14(1) .l6(-4) -.42(-i) -.61(1)

.0064 826. .291 486 .45 (-4) .43(6^ -.45(2' .15(1) .12(-4) -.45(-l) -.69(1)

.0066 919- .258 485 .4l(-4) .48(6^ -.54(2) .18(1) .lO(-4) -.52(-l) -.80(1)

.0068 1020. .234 485 .36(-4) .53(6' -.62(2^ .21(1) .72(-5) -.6o(-i) -.88(1)

.0070 1120. .210 484 .33(-4) .54(6^ -.66(2^ .22(1) .5l(-5) -.63(-l) -.89(1)

.00720 1240. .192 483 .30(-4) .58(6^ -.75(2' .25(1) .40(-5) -.70(-l) -.93(1)

.00740 1350. .175 483 .27(-4) .62(6^ -.83(2^ .28(1) .45 (-5) -.76(-l) -.95(1)

.00750 i4io. .167 483 .26(-4) .62(6^ -.85(2' .29(1) .35(-5) -.78(-l) -.96(1)

.00770 1540. .153 462 .24(-4) .65(6 -.93(2' .31(1) .16 (-5) -.85(-i) -.99(1) .00790 1680. .l4o 482 .23 (-4) .67(6' -.99(2 .33(1) .24(-5) -.89(-l) -.10(2) .00810 1820. .129 482 •2l(-4) .70(6 -.11(3 .36(1) .59(-6) -.97(-l) -.10(2) .00800 6460. .520 257 .ll(-5) -.58(6 .68(3 -.20(2) .89(-5) .91(-1) .24(3) .0100 652. 1.92 365 .l4(-3) -.33(4 -.11(1 -.68(-i) -.l6(-3) -.23 (-2) -.17(2)

1 .0060 660. .360 488 .62(-4) .37(6 -.34(2 .12(1) .21(-4) -.35(-l) -.12(2)

2 658. .364 488 .62 (-4) .38(6 -.35(2 .12(1) .22 (-4) -.36(-l) -.18(2)

3 657. .364 488 .63 (-4) .38(6 -.34(2' .11(1) .25 (-4) -.35(-l) -.19(2)

4 655. .364 488 .65 (-4) .37(6 -.33(2 .11(1) .27(-4) -.34(-l) -.15(2)

5 653. .364 489 .67(-4) .36(6 ) -.32(2 .11(1) .30(-4) -.33(-l -.55(1)

88

Table ?. tr = 0.23(lO)' Cont'd

3 P Pi (^.L "Cr c

-3 (Q-p ^3 °.>

.6 653. .360 I468 .70(-i+) .33(6) -.30(2) .10(1^ .34(-4) -3l(-l) + •92(1)

5 .0061 . 689. .3'^2 koa .62(-i+) .37(6) -.34(2) .11(1 .26(-4) -.36(-l) -.19(2)

.6 687. .3^*3 1^88 .6k(-k) .36(6) -.33(2) .11(1^ .3l(-4) -34(-l) -.60(1)

.7 687. .3^3 488 .66{-k) .35(6) -.32(2) .ii(r .35 (-4) -.33(-l) +.13(2)

.6 .0062 12k. .326 km .59{-k) .39(6) -.37(2) .12(1 .28 (-4) -.38(-l) -.20(2)

.7 722. .326 kS8 .6l(-k) .37(6) -.35(2) .12(1 .33 (-4) -.36(-l) -.42(1)

.8 722. .326 1+88 .63{-k) .36(6) -.34(2) .11(1 .37(-4) -.35(-l) + .17(2)

• 9 .0065 836. .280 1*88 .52{-k) .41(6) -.41(2) .14(1' .32(-4) -.42(-l) -.15(2) 1.0 834. .280 1+88 .5k{-k) .1+0(6) -.40(2) .13(1 .38(-4) -.4l(-l) +.76(1)

1.2 .00675 937. .250 1*88 .l+8(-l+) .l+i+(6) -.46(2) .15(1 .43(-4) -.47(-l) .10(2) 1.3 937. .2W 1+88 .50(-l+) .42(6) -.43(2) .14(1 .48(-4) -.44(-i) .35(2)

l.lt .0070 1050. .223 1+87 M(.-k) .1+8(6) -.51(2) .17(1 .47(-4) -.53(-l) .12(2) 1.5 1050. .221 1+87 .k3(-k) .45(6) -.46(2) .16(1 .53(-4) -.50(-l) .35(2)

89 GPO 850 - 099