Slip flow past slender pointed bodies

T H E C O L L E G E O F A E R O N A U T I C S

C R A N F I E L D

SLIP FLOW PAST SLENDER POINTED BODIES

by

REPORT NO. 160 May. 1963.

THE C O L L E G E OF A E R O N A U T I C S

C R A N F I E L D

Slip Flow Past Slender Pointed Bodies by

-Fit. Lt. C. B. Stribling, B . S c , D . C . A e .

SUMMARY

Using Oseen's approximation to linearise the Navier-Stokes equations, a solution to the problem of axisymmetric slip flow past slender bodies of revolution at zero incidence has been obtained, in a manner similar to that used by Laurnxann in his paper on the slip flow past a flat plate^"'.

The drag coefficient has been evaluated for both incompressible and compressible flows, and this has been compared with normal boundary layer values. It is found that the skin friction drag is the dominant component of drag in the incompressible case and in the compressible case.

This work was conducted at the College of Aeronautics, Cranfield, in partial fulfilment of the requirements for the Diploma of the College.

List of Symbols Introduction

The incompressible case The compressible case References

Appendix A - Alternative forms for the functions b,(x) and bg(x)

Appendix B - Fluid velocity, p r e s s u r e , temperature and density in linearised viscous flow in t e r m s of the functions <p^ .^^and x Appendix C - The inverse transform of J

LIST OF SYMBOLS free s t r e a m speed of sound functions of x

Fourier transforms of a (x), a (x), etc. o • functions of x

^ M'' - 1 constants

D / i p U S - drag coefficient in slip flow skin friction drag coefficient in continuum flow wave drag coefficient

specific heat at constant p r e s s u r e D + D = D + D = total drag

p V 1 * ^ UL/2i/

thermal conductivity molecular mean free path length of body

U/a - free stream Mach number

co-ordinate in direction normal to body surface p r e s s u r e

transform variable

component of velocity parallel to body surface velocity vector

components of q heat transfer rate radial co-ordinate

r

S 2Tr L r (x) dx s u r f a c e a r e a of body S(x) TT j r (x) I body c r o s s - s e c t i o n a l a r e a t 2 6 T t e m p e r a t u r e AT t e m p e r a t u r e j u m p at s u r f a c e T p e r t u r b a t i o n t e m p e r a t u r e of g a s adjacent to s u r f a c e s T t e m p e r a t u r e of s u r f a c e w U f r e e s t r e a m v e l o c i t y u, V, w C a r t e s i a n v e l o c i t y c o m p o n e n t s U s l i p v e l o c i t y V r a d i a l v e l o c i t y component X, y, z C a r t e s i a n c o - o r d i n a t e s a t a n d r / d x j3 i T l - M ^ y r a t i o of s p e c i f i c h e a t s r .5772 , . . E u l e r ' s constant 6 m a x i m u m r a d i u s of body X y M V 2 k o -H coefficient of v i s c o s i t y t> IJI p k i n e m a t i c v i s c o s i t y p d e n s i t y (T LJC / K P r a n d t l n u m b e r P <p , <p , <p v e l o c i t y p o t e n t i a l s X v o r t i c i t y function -kx

List of Symbols (Continued)

Suffix oo denotes free s t r e a m value

Asterisk * denotes sum of free s t r e a m and perturbation values

No suffix etc. denotes perturbation values ofu, v, w, p, p, ju, K and T Fourier transforms oi <p , x etc. are denoted by ^, x etc.

in which slip effects occur at the boundaries of the fluid and even, to a certain extent, into free molecule flows where, however, they eventually cease to describe the physical conditions correctly and have to be replaced by other methods based on kinetic theory.

It is the object of this paper to investigate an axisymmetric solution to the Navier-Stokes equations in the slip flow regime, employing Oseen's approximation to simplify the equations, and because it is suited to dealing with the velocity discontinuity which appears at fluid boundaries.

The Oseen method linearises the equations by considering perturbations on a uniform s t r e a m , and allows the problem to be solved throughout the entire flow field, but the perturbations on the free s t r e a m must be everywhere small.

Bodies must therefore produce only minor changes from free stream conditions, and the method is thereby restricted to slender configurations where the body cross-section is small compared with its length; in particular, the fluid velocity at boundaries, with such slender bodies, must be similar in magnitude to the free s t r e a m velocity, a condition which is not satisfied in boundary layer flows at normal densities but which, in slip flow, is permitted by the fluid velocity discontinuity at surfaces, which is characteristic of this r e g i m e .

Use of Oseen's approximation, therefore, demands that the slip velocity be of the same order of magnitude as that of the free s t r e a m . Since this slip velocity is a function of molecular mean free path and body shape which, as noted above, is itself r e s t r i c t e d , the problem is confined to one in which the mean free path is limited to a small range of values and must, in fact, be large compared with the dimensions of the body, i . e . the Knudsen nunnber is large. This is outside the range of Knudsen number normally associated with slip flow, but theoretical work in the kinetic theory of gases, by Wang Chang and Uhlenbeck*°', shows that the slip form of boundary condition is valid for all values of mean free path, although numerical coefficients may change; there is also some experimental evidence that this is so.

This work then, although ostensibly concerned with the slip flow regime (.01 < Knudsen number < 1.0), is actually a description of fluids at a somewhat lower density, which is approaching that of free molecule flows, but in which it is assumed that the Navier-Stokes equations still hold and in which the slip boundary condition is valid.

2 . T h e i n c o m p r e s s i b l e c a s e (a) T h e e q u a t i o n s

T h e full N a v i e r - S t o k e s e q u a t i o n s for s t e a d y i n c o m p r e s s i b l e flow in t h r e e d i m e n s i o n s a r e

={c 8U* , * 8U ^ * a u 1 9P _L n2 *

dx 8y 8z p 8x

U * 8 v " * av"^^ * av"" 1 8P _. „a *

+ v -— + w - — = - — r * - + v V v 8x ay az p 8y >K aw* * a w * ^ * 8w*_ 1 8 P * ^ „i * u T— + v ' r — + w — - = - - - ^ + i ' 7 ' w 8x 8y a z p a z T h e equation of continuity i s 8x a y a z >k ]jc 9k jk

If u, v , w and p a r e r e p l a c e d by U(l + u), vU, wU and p „ ( l + p), w h e r e u. v, w and p a r e s m a l l n o n - d i m e n s i o n a l p e r t u r b a t i o n s on a uniform s t r e a m U, t h e s e e q u a t i o n s b e c o m e , n e g l e c t i n g second o r d e r t e r m s and u s i n g a n o n - d i m e n s i o n a l C a r t e s i a n c o - o r d i n a t e s y s t e m 8u a x 8v 8x 8w 8x pU* 9 ^ pu'^ öy 1 R e 1 R e 1 R e V* u V* V V« w

w h e r e R = and L i s the l e n g t h u s e d t o n o n - d i m e h s i o n a l i s e the c o - o r d i n a t e s y s t e m .

8u ^ 8v , 8w _ _ 8x By 8z (5)

L a m b s h o w s t h a t , w i t h c y l i n d r i c a l s y m m e t r y about the x - a x i s , the n o n - d i m e n s i o n a l p e r t u r b a t i o n v e l o c i t y v e c t o r (j^= (u, v , w) m a y be s p l i t into two c o m p o n e n t s , one i r r o t a t i o n a l b e i n g the g r a d i e n t of a s c a l a r p o t e n t i a l function ^, and the o t h e r r o t a t i o n a l b e i n g d e r i v e d f r o m a s c a l a r v o r t i c i t y function x, s u c h t h a t

u = | i + | i - 2 k x ( 2 . 1 ) a x ax

^ 8 £ ax "" ^ 8y ay

and also w = p = V = r

az

Pco 80 ^ 8r 8x 8z

ax

ax

8r ( 2 . 2 ) (2.3) where v is the non-dimensional radial velocity component in axisymmetric flow.

<f) and X satisfy the equations

V^0 = 0 (2.4)

( v - 2 k | ^ ) x - 0 (2.5)

with 2 k = — = R

V e

The problem is reduced to finding solutions to equations (2.4) and (2.5), which are derived from second order partial differential equations; four independent boundary

conditions will therefore be needed, and will be provided by conditions on the normal and tangential fluid velocity components at the body surface, and by conditions at infinity.

Inviscid flow equations, which are first order equations, are obtained when k = oo ( u = 0); equation (2.5) is then reduced to x = 0, and equation (2.4)

gives Adams and Sears solution'^' using the normal fluid velocity boundary condition only. In the following analysis, the solution will tend to the inviscid solution as k •» oo , except inside layers of order 1/k in thickness adjacent to the body surface; i . e . as Reynolds number becomes l a r g e , the viscous effects are confined to a boundary layer. The case k = 0 is of no practical interest, but provides a check on the solutions for 0 and x , which should then be identical.

(b) General solutions

Equations (2.4) has been solved, using the exponential Fourier transform, by Adams and Sears^^' as follows

If 0(p) = ƒ ^e'P^'dx

_ oo then V*0 = 0 , 2 - , becomes d_^ 1_ 8 j j -ar^ r 8r " P ^ 0

This is a modified Bessel equation of zero order, giving 0 = A (p) K ( | p | r) since ^ = 0 at r For small values of r

?• = - A^(p) ( ^ r + l o g i | p | r j + O(r^) ( 2 . 6 ) Applying the i n v e r s e t r a n s f o r m

1 oo

(p = -^ f ({, e dp and using the F a l t u n g t h e o r e m , then

• - oo / • 8 a , 8a 0 = a ^ ( x ) l o g | - i / ~ l o g ( x - y ) d y + i / " g ^ log (y-x)dy ( 2 . 7 ) — 00 V 1 *" w h e r e a (x) = - — /" A (p) e ^ dp i s an a r b i t r a r y function, to be * - o o d e t e r m i n e d f r o m the b o u n d a r y c o n d i t i o n s . <j) m a y c o n v e n i e n t l y be e x p r e s s e d 0 = a (x) log I + b (x) ( 2 . 8 ) O 6 o X 8a «> 8a

w h e r e b j x ) = " i ƒ -9^" log (x-y) dy + i ƒ " g j " log ( y x ) dy ( 2 . 9 )

J _ o o X E q u a t i o n ( 2 . 5 ) b e c o m e s i 2 2 8 f , 1 ai*- , a 'k 2 „ + — r-^ + ; - k ^ = 0 after m a k i n g the s u b s t i t u t i o n 2 J. Qj. „ 2 o 8 r ^ o r 8x" kx X = e f Applying the F o u r i e r t r a n s f o r m 8 % ^ 1 a ^ , 2 , 2^ — - -r- - <P + k i \f' 81? r 9 r giving _ ^

1^ = A,(p) K L^p^ + k" r j

F o r s m a l l v a l u e s of k r

7 = - A^(p)(^r +log [ i ^ ^ n ? r ] J +0(k'r') (2.10)

A s long a s the p r e v i o u s r e s t r i c t i o n on r , r e q u i r e d for equation ( 2 . 6 ) , i s o b s e r v e d , t h i s r e s t r i c t i o n on k r d o e s not l i m i t the r a n g e of v a l u e s of k that m a y be

c o n s i d e r e d , k m a y a l w a y s be a s s m a l l a s we p l e a s e , but l a r g e k is e q u a l l y a c c e p t a b l e , p r o v i d e d r i s s m a l l enough.

I n v i s c i d conditions a r e , h o w e v e r , excluded s i n c e , a s k •• 00 ( v i s c o s i t y -«O), r m u s t tend to z e r o i . e . only e x t r e m e l y thin b o d i e s m a y be c o n s i d e r e d and, in the l i m i t , the t r i v i a l flow of a uniform s t r e a m p a s t a 'line' i s obtained.

00

t o give >p = a,(x) l o g - + b,(x) • oo w h e r e a^(x) = " n™ / -^i^P^ e ^ dp - c o and b^(x) = - i ƒ ^ l E i [ - k ( x - y ) ] - T - log k j dy

+ è r ^ ' [ Ei[-k(y-x)] - r - logkjdy

X

^(x)rr+logk j - i ƒ g-! Ei r-k(x-y)"]dy

OO

+ i j ^ ' Ei [-k(y-x)] dy (2.11)

f r dt the e x p o n e n t i a l i n t e g r a l function. = a

e t

w h e r e E i ( e ) = *" ^ kx H e n c e x = e i^ = a^(x) e l o g J + b,(x) e = a^(x) log I + b^(x) ( 2 . 1 2 ) kx w h e r e a (x) = a (x) e ( 2 . 1 3 ) b^(x) = b , ( x ) e * ' ' ' ( 2 . 1 4 ) b,(x) and bj(x) m a y be e x p r e s s e d in the a l t e r n a t i v e f o r m s ( d e r i v e d in Appendix A)

-kx x 8 ^ kx 1 / 2k \

^i^^^ " " I I Q^ ^°e (^-y) <iy + ^ / gr: y ^2<y>e' •^J log(y-x) dy

•'o X ( 2 . 1 5 ) ? 8 a 2kx 1 a / o^. \

b^(x) = - 5 ƒ 8 7 log(x-y)dy + | j ~ (^ a,(y)e"''*'y J log(y-x)dy ( 2 . 1 6 ) o X F r o m e q u a t i o n s ( 2 . 1 ) and ( 2 . 3) we find u = r ^ + — - - 2 k x 8x 8x

= r a' (x) + a' (x) -2k a {x)~\ log ^ +rb' (x) + b'(x) -2k b (x) 1

[ _ o a ' J ^ L o a " J ( 2 . 1 7 ) ' a^(x) log I +b^(x) s a y . ( 2 . 1 8 ) and V = Q + s~~ r 8 r a r = - f a (x) + a ( x ) " l ( 2 . 1 9 ) r L o 2 J a ( x ) " — s a y ( 2 . 2 0 )

6

-(c) T h e b o u n d a r y conditions

T h e b o u n d a r y conditions will be applied to a s l e n d e r body of r e v o l u t i o n , pointed at both e n d s , of length L , and m a x i m u m t h i c k n e s s 2 6 L , m a k i n g the u s u a l s l e n d e r body a s s u m p t i o n s c o n c e r n i n g the body r a d i u s , slope of the body s u r f a c e , e t c . L e t the r a d i u s of the body be L r^(x), with the a x i s of s y m m e t r y along the x - a x i s and u p s t r e a m end at the o r i g i n .

S(x) >: j r [ r ^ ( x ) ^ ^ and 2 6 = t T h e n o r m a l v e l o c i t y condition g i v e s

d r

V = : r - (1 + u) ( 2 . 2 1 ) r dx

and the t a n g e n t i a l v e l o c i t y condition

U = ( l + u ) c o s a + v sin a ( 2 . 2 2 ) s r w h e r e U = n o n - d i m e n s i o n a l s l i p v e l o c i t y d r , and tan a = —— dx d r

Since -p^ = 0(t) by the s l e n d e r h y p o t h e s i s , equation(2. 21) s h o w s that

V = 0(t) ( 2 . 2 3 ) r We find t h e r e f o r e , that a (x). a ( x ) , a j x ) = 0(t''), f r o m ( 2 . 1 9 ) and ( 2 . 1 3 ) o ' * 0 , x = 0(t^log t ) , f r o m ( 2 . 8 ) and ( 2 . 1 2 ) while u = O ( t ^ o g t ) , f r o m ( 2 . 1 7 ) E q u a t i o n s ( 2 . 21) and ( 2 . 22) b e c o m e d r V = — + 0 ( t ^ o g t ) ( 2 . 2 4 ) r dx Us = 1 + 0 (t^log t) ( 2 . 2 5 ) Hence d r a ( x ) T-' = V = - V - r . f r o m ( 2 , 2 0 ) and ( 2 . 2 4 ) dx r r (x) ^ W = ^ ^ (S(x)) = ^ ( 2 . 2 6 ) F r o m k i n e t i c t h e o r y , the s l i p v e l o c i t y U = c, ^ r-^

w h e r e c^ = a c o n s t a n t , ( a p p r o x i m a t e l y equal to unity)

T = the n o n - d i m e n s i o n a l m e a n free path of the m o l e c u l e s

= Knudsen n u m b e r q = component of the fluid v e l o c i t y p a r a l l e l to the body s u r f a c e n i s in the d i r e c t i o n n o r m a l to the body s u r f a c e .

Now 8 £ _an 8u 2 -— c o s a _{8 r} 8u , 8Vr

8r ar

^

Ur

d r dx —1 c o s a s i n a -8 x / + 0 ( t ' l o g t)

av

r 8x sin^ a

- f a "i

t h e r e f o r e U = c . ^ ^ + - - ^ . - - M = 1 , f r o m ( 2 . 2 5 )

s 1 L ar ar dx J

f a ' ( x ) + a'(x) - 2k a (x) a (x) " T O 2 Z 4 / , > = c ^ r-T - — r (x) 1 L r. ( x ) 2f . 1

rr(x)

f r o m ( 2 . 1 7 ) and ( 2 . 2 0 ) .

Since the b r a c k e t c o n t a i n s t e r m s of o r d e r t, it follows t h a t I m u s t be of o r d e r t i . e . the Knudsen n u m b e r m u s t be l a r g e . T h i s r e s t r i c t i o n i s a

c o n s e q u e n c e of the u s e of O s e e n ' s a p p r o x i m a t i o n , which r e q u i r e s the fluid v e l o c i t y a t a l l p o i n t s t o b e of the s a m e o r d e r of m a g n i t u d e a s the f r e e s t r e a m v e l o c i t y .

T h e l a r g e Knudsen n u m b e r d o e s not p r e v e n t the u s e of s l i p b o u n d a r y c o n d i t i o n s . T h e second b o u n d a r y condition t h e r e f o r e g i v e s

a ' ( x ) + a'(x) - 2k a (x) a (x)

o 2 2

"^i^^) r,^(x) ' c, e and it follows that

r ' ( x ) = - ^ ( 2 . 2 7 ) a,(x) = = a^(x) -1 2k 1 2k S'(x) 27r S"(x) 2ir 2 r (x)

[ r ; ( x ) ] - —::: J + o(tnog t)

1 1 r (x) " ' r (x) r " ( x ) - ~ j ( 2 . 2 8 )

:,7 J

r,(x) 2k ^ l^r^(x) r ; ( x ) - - ! — J + O d - ' l o g t ) ( 2 . 2 9 )

a (o), a (o), a (1) and a (1) a r e a l l z e r o , p r o v i d e d r (x) r "(x) = 0 at x = 0 and

o 2 o 2 1 1

X = 1 ; t h i s condition i s s a t i s f i e d if the body i s pointed at both e n d s . T h e n , from ( 2 . 1 8 ) and ( 2 . 2 0 ) , we find

\ "^ ^x) 2 i

u = [ - ^ + [ r ; ( x ) ] J log I +b^(x) ( 2 . 3 0 )

c, e

8

-and _{2 j r r} S'(x) (2.31)

F o r a g e n e r a l pointed body, hj,x) will contain l o g a r i t h m i c s i n g u l a r i t i e s at X = 0 and at x = 1, and the l o c a l v e l o c i t y and p r e s s u r e at the n o s e and t a i l will be p h y s i c a l l y untenable and c o n t r a r y to the s m a l l p e r t u r b a t i o n h y p o t h e s i s . The effect i s , h o w e v e r , confined to a r e s t r i c t e d r e g i o n and i t s net influence on d r a g f o r c e i s n e g l i g i b l e .

The s i n g u l a r i t i e s m a y be r e m o v e d by the i m p o s i t i o n of the additional r e s t r i c t i o n t h a t t h e n o s e and t a i l should be c u s p e d , i. e. no s t r e a m w i s e

d i s c o n t i n u i t i e s in c u r v a t u r e , but s i n c e t h i s p a p e r i s u l t i m a t e l y c o n c e r n e d with d r a g f o r c e , the m o r e g e n e r a l a n a l y s i s of a pointed nose h a s been followed. (d) D r a g

The d r a g m a y be evaluated by c o n s i d e r i n g the p r e s s u r e , momentuna flux and v i s c o u s s t r e s s e s at the s u r f a c e s of a c y l i n d e r of length L and r a d i u s R which j u s t e n c l o s e s the body. Let S, and S3 denote the u p s t r e a m and d o w n s t r e a m plane f a c e s of t h i s c y l i n d e r , and S^ the c u r v e d s u r f a c e .

The d r a g due to the v i s c o u s s t r e s s e s at the cylinder s u r f a c e s is D 4M i p U ^ p U L

UifL-,^^.4(i^

av, 8x r=R

^^

%-t

/ f e ) ^ ^ . •'s, x = l 1 r r (x) o ,.1 r r (x) , ~|

ƒ [ ^ * [ « € h

2ir 0(t = (2.32)

T h e d r a g due t o p r e s s u r e and m o m e n t u m change through the c y l i n d e r is D

JL =

i p U "

I - 2 — + 2(1 + u)^l dS, - 2 [

ig^ Li PU'' JX=O I,

d, -2 1^ [ V - " > ] ^ - - /

r- *

s,

Ï P U

+ 2(1 + u)^ dSj

Jx=l

Using the continuity equation, and equation ( 2 . 2 ) t h i s b e c o m e s

_JE_ = /• f- 2 1 ^ + 2ul dS - 2 / f u v 1 dS2 - [ r- 2 1 ^ + 2JI dS, + 0(t*log't)

CRANFIELD

Slip Flow P a s t S l e n d e r Pointed Bodies

by

F i t . L t . C, B . S t r i b l i n g , B . S c , D.C.Ae.

ERRATA

PAGE 9. Line 5: D. should r e a d T-r-2

1 iPO

Da

Line 15: D, should r e a d 1—7^2

9

-.' . T o t a l D r a g D i s given by

i»u" i p u ' k J L c, « ^ ' -I -I

O

T h i s t o t a l d r a g i s due to the n o r m a l and t a n g e n t i a l v i s c o u s s t r e s s e s at the body s u r f a c e .

2 w f

T h e f i r s t t e r m , D , = _ / r^(x) dx, m a y b e c o m p a r e d with L a u r m a n n ' s

k c, e i

o r e s u l t for a flat p l a t e (Ref. 6 ) .

1

It g i v e s a d r a g coefficient, b a s e d on body s u r f a c e a r e a S = 2w / r^(x) d x ,

c = A . . ^

° ' i p U ^ S k c , -^

which i s e x a c t l y the d r a g coefficient for one s i d e of a flat p l a t e . T h e s i t u a t i o n i s s i m i l a r to that of l a m i n a r b o u n d a r y l a y e r t h e o r y w h e r e skin f r i c t i o n d r a g

coefficient for one s i d e of a flat p l a t e 1 ••^28 ^g ^-j^g s a m e a s that for a s l e n d e r e body of r e v o l u t i o n (Ref. 3). C ^ w i l l r e m a i n finite a s R e y n o l d s n u m b e r i n c r e a s e s to l a r g e v a l u e s b e c a u s e T IS p r o p o r t i o n a l t o _ . ^ 1 O 2 T h e second t e r m , D^ = — / r r j ( x ) l dx, is a v i s c o u s t e r m which b e c o m e s o s m a l l at high R e y n o l d s n u m b e r s . It i s , t h e r e f o r e , a low R e y n o l d s n u m b e r t e r m , due to t h e t h r e e d i m e n s i o n a l body s h a p e . 3 . T h e c o m p r e s s i b l e c a s e (a) T h e e q u a t i o n s

T h e s t e a d y t h r e e - d i m e n s i o n a l N a v i e r - S t o k e s e q u a t i o n s and the continuity equation for c o m p r e s s i b l e flow a r e * /^ * a u , * au * 8u U r + V - — + W - — 8x 8y 8z 8 p * ^ * oa •<' X 1 * 8 /^ au* , a v * aw*N - —*- + u V u + — u — ( — + — -^ ) 8x ^ 3 ^ 8x V a x 9y az y 2 8Ai*/au*_^ 8 v * ^ 8 w * \ aM* Bu*

3 8x \ 8 x 8y Sz / ax ax ^JL(^^

+

^JL')+

iü" ( 8ü''+aw*\

8y \ 8 y 8x / 8z \ az ax / and two s i m i l a r e q u a t i o n s . * P _{V ax ay az / \ ax ay az /}

T h e s e s m a l l , and the p e r t u r b a t e q u a t i o n s of e n e r g y and ions f r o m conditions in u * V * w p * * p * = = = = = = U (1 + Uv Uw Pood + p j i +

^'J^ +

s t a t e , m a y be l i n e a r i s e d a uniform s t r e a m defined u) P) p) iu) by by c o n s i d e r i n g giving w h e r e M Bu 8x U_ aoc

J,

7 M , 8 E + J _ n2,j + J _ / 8 J L + 8 ' v . 8 j v _ A * ax R '' 3R V 8x 8x a y ax azy • N

the f r e e s t r e a m Mach n u m b e r and the c o - o r d i n a t e s y s t e m i s n o n - d i m e n s i o n a l i s e d with r e s p e c t to L . V*v + av 1 ap ^ 1 „2 i f d'u d'v a^w '\ \ 8 x a v a „ * 8 y 8 z / 8x 7M 8 y R 3R aw 1 8 D _— = _ —i— + a x _TM az R V w +

1 _ ("

ifu

3R \ 8 x a w 8 y 8 ^ 8yaw 8y85 a'w a z " Bu 8v aw _ 8x 8y az 8p ax

y.3.

1) ( 3 . 2 ) T h e s e e q u a t i o n s m a y be combined into 8_!P 8 x ' — 2 V p = ^ ^ V* I - ^

rM

^^e

8x ( 3 . 3 ) T h e n o n - d i m e n s i o n a l i s e d l i n e a r i s e d e n e r g y equation b e c o m e s 8 T ^ , i \ (^ 8u ^ 8 v ^ 8w \

87 ^ <^-^>

Va^

^ 8? ^

8 7 /

=

aR _ 1 _ e V'T w h e r e T i s the n o n - d i m e n s i o n a l p e r t u r b a t i o n t e m p e r a t u r e and P r a n d t l n u m b e r K, ( 3 . 4 ) T h e equation o b e c o m e s f s t a t e * P = P» P * : P Poo = p i T_ T ( 3 . 5 )

11

-F r o m (3.2) and (3.4)

e Eliminating T from (3. 5) and (3. 6)

e which, with (3.3) gives

8!E. . J_ ^2 Jï_ ^^M + J. { ± . l) v<8£^ (3 8)

3

If it is assumed that <T = T (it is .72 for a i r ) , the last t e r m of the above equation vanishes and there remains the p r e s s u r e equation

ax* M e E l i m i n a t i o n of p and p f r o m ( 3 . 5 ) , ( 3 . 6 ) and ( 3 . 9 ) g i v e s \ 8x o-R y V a x * M J L - v * 8 T o-R 8x e

T h e s o l u t i o n of t h i s , the t e m p e r a t u r e e q u a t i o n , will be a l i n e a r c o m b i n a t i o n of the s o l u t i o n s t o

8 Ü i ^ v^T - - ^ 7^ ^ = 0 ( 3 . 1 0 ) and

1 ^ - ^ V*T = 0 (3.11)

As in the incompressible case, with cylindrical symmetry the perturbation velocity vector may be split into two components, derived from a potential function and a vorticity function. Now, however, the potential function 0 is divided into two pax'ts, </>^ and <t>^, each associated with one of the two temperature equations (3.10) and (3.11).

Thus q = q, + qg + Qj = (u, v )

r

= V d + V d + f(x)

This splitting of the linearised flow equations was first carried out by L a g e r s t r o m , Cole and Trilling^'*). They neglected the effect of thermal conductivity and so did not divide the potential function into two p a r t s , but they obtained separate solutions in the form of waves for 0 and x, which they termed 'longitudinal' and 'transverse' waves respectively. T r i l l i n g ^ ' ' showed how thermal conductivity

could be included, to give a method of solving a l i n e a r i s e d flow field p r o b l e m in t e r m s of the t h r e e functions 4>^ , (p^ and x defined above; L a u r m a n n ^ " ' follows t h i s m e t h o d in h i s p a p e r on s l i p flow o v e r a flat p l a t e , calling the t h r e e s o l u t i o n s of 0 , , <l>2 and x , p r e s s u r e , t e m p e r a t u r e and v i s c o u s w a v e s r e s p e c t i v e l y .

D e t a i l s of t h i s s p l i t t i n g p r o c e s s a r e given in Appendix B and r e s u l t in the following e q u a t i o n s for 0, , 0^ and x , and the a s s o c i a t e d n o n - d i m e n s i o n a l p e r t u r b a t i o n v e l o c i t y v e c t o r s , p r e s s u r e s , t e m p e r a t u r e s and d e n s i t i e s in a x i -s y m m e t r i c flow. P r e s s u r e wave V'^. 2 B^p, 7M_ ^2 8 i , _{ax^ 2kcr 8x} a^, 8 r ( 3 . 1 2 ) / 8 ^ , p = ym'\^ ^' - ^^ <p ^1 ' |_3k ax J ^1 with f ,

, . - DM- Si

= M ' 3_ 4 2 7 ~2 3k V -8 ax T e m p e r a t u r e wave V^ 0 - 3k 8_0, ^2 2 8 x ( 3 . 1 3 ) 8^2 8^2

ar

Pa 3k T , = -TT 0 2 *^2

13 -V i s c o u s wave

v%

2k | i = 0 8x ( 3 . 1 4 ) . - ( 8JL ax o, 8 " 2k X, -— 8 r P3 = 0 T = 0 3 P, = 0 T h e v i s c o u s wave i s t h e r e f o r e independent of c o m p r e s s i b i l i t y . (b) G e n e r a l s o l u t i o n s

(i) T h e p r e s s u r e wave equation ( 3 . 1 2 ) i s

( ' " L e t T h e n - M* 7 ' M ' ' 2k or ' ( a 8 x * = X ' d'^ 2ko- a

V)^

= 0 ax / 8*0, 1 8<p 8*0 \ 3*0 8*0 1 80 \ a r * r 8 r g ^ V 8x* 8 r ^ 8 r 0 ( 3 . 1 5 )

J2)

C l a r k e s o l v e s a s i m i l a r equation in h i s p a p e r on r e l a x a t i o n effects on s l e n d e r b o d i e s ; he u s e s F o u r i e r t r a n s f o r n n s to get equation ( 3 . 1 5 ) in the f o r m

8*95 1 85 — 1 + - —1 8 r » ^ ^^ 2 (1 - M - i pX) - _ P ;—••—r~- 0 = 0 1 - 1 pX 1 A g a i n , t h i s i s a B e s s e l equation of z e r o o r d e r giving 0 = A (p) K I 1 - M* - i pX 1

_ P ^ i 1 - i p X ^ ^ J

F o r s m a l l v a l u e s of r ,

ï,= -^,P,[r.io,[lp.JII^Ï=^^j;

+ 0(r=) and t h e r e a r e no r e s t r i c t i o n s on M, k o r X in m a k i n g t h i s a p p r o x i m a t i o n . T o obtain the i n v e r s e t r a n s f o r m , it i s n e c e s s a r y to c o n s i d e r two c a s e s d e p e n d i n g on w h e t h e r M < 1 o r M > 1.

C a s e 1 M < 1 Now, 0, m u s t be e x p r e s s e d a s w h e r e A,(P) ^^ = 1 - M* r + log _IPI + 1/3 /X ] + i/X J and, f r o m C l a r k e ' s p a p e r , the i n v e r s e t r a n s f o r m i s x

0^ = a^(x) log I " 2 a 7 / ^5^y^ ^ ° ^ ^^"^^ '^y "^ 2 8x / ^s*^^ ^°^ ^^'^^ ^^

2 \dx X / /

^ 2 U " xj ƒ

s<y^

• - o o ^'^ '^ , . -^*(x-y)/X aj(y) e "^ log(x-y) dy - (x-y)/X . . e -^ log(x-y) dy If a (x) = 0 for X < 0 at X > 1 s 8a r l r " " I r s 0^ = a ^ ( x ) l o g - - 2 ƒ —^ log(x-y)dy + - Ï ^ l o g ( y - x ) d y 2 ^ 2 X a , ( y ) e " ^ < ^ ' y ' ^ \ o g ( x - y ) d y + ^ ƒ a,(y) e "^^^'^^^^ log(x-y)dy o ' "^o 8% - ^ * ( x - y ) / X , / v^ 1 f ^^5 - ( x - y ) / X , , . , —5- e ^ •^ ' log(x-y)dy + g j gTT ^ log(x-y)dy 8y o = a, 1^ 2

r 1 / 8a, i f aa

.<^)l°g| - I i ^ log(x-y)dy+2 ] ^log(y-x)dy

^^(y) r -^'(x-y)/X -(x-y)/X

(x-y)

I ^

dy a^(x) log 2 + b^(x) ( 3 . 1 6 )

1 /

a a 1 ƒ 8 a j w h e r e b _ ^ ( x ) = - ^ / g - ^ log(x-y)dy + - j — log(y-x) dy a,(y) ( x - y ) - ^ ' ( x - y ) / X _ ^ - ( x - y ) / X 1 ( 3 . 1 7 )

15

-Case 2 M > 1

-0, - - A,(p) [r + i log [ pr i ^ ^ ^ JJ

where B = M - 1

Again, from Clarke, the inverse transform is

X "

0^= a ^ ( x ) l o g | - ^ ƒ a^(y) log(x - y)dy + i ( ^ - | - ' ) ƒ a^(y)e-^<y"^>^ log(y-x)dy J - c o ^ ' X

^ K a T ^ l ) f a,(y)e-<^-y)/^ log(x-y)dy

. , , r r '^» 1 , v^ i f ^=^^^ -B*(y-x)/X^ . i F ^»^^^ -(x-y)/X ^

= a^(x) log 2 - j — log(x-y)dy - ^ j ^—^ e dy + ^ j ^^^ e d

o x o

= a (x) log ^ + b(x) (3.18) 9 Z 5

X 1 X K/ . / ^ , / .^ i / ^«^^^ -B*(y-x)/X^ ^ 1 /• a,(y) (x-y)/x

where b^(x) = - / r-^ log(x-y)dy - i ] — ^ e dy + 2 j "T^v ^ o ^ x •'^ o •'^

(3.19) (ii) The temperature wave equation (3.13) is

{ » 3k 8 V

V V - - 7 r r— 70 = 0

This equation was solved in section 2(b) giving

0, = a ( x ) log ^ + b(x) (3.20)

where b^(x) = " i f g-r log (x-y) dy + | / ay I ^ ^ a^(y)l log(y-x)dy

° ^ (3.21) (iii) The viscous wave equation (3.14) has the solution

X = a (x) log ^ + b (x) (3.22) 7 2 7

i 8a 2kx f

where b^(x) = " i j gT log(x-y)dy + | J 8^ I ^ ^ a^(y) |log(y-x)dy

° ^ (3.23)

The two slender body solutions above r e s t r i c t k to finite values; this restriction was discussed in section 2(b) and the same conclusions hold h e r e .

80, ^ 8^, ax

= ra5(x) + a^(x) + 4 ( x ) - 2k a,(x) "1 log | + rb^(x) + h[{x) + b'^(x) - 2 k b^(x) ] ( 3 . 2 4 )

= a^(x) log I + bg(x) say ( 3 . 2 5 ) 90 90

v = _ ! 2 ax r — + — + —

a r 8 r 8 r

= raj,(x) + a^(x) + a^(x) 1 ^ ( 3 . 2 6 ) a,(x)

= - — - s a y ( 3 . 2 7 )

and 2 8 0 , QU

( , - 1) M g^ + f 0,

= - (7- 1)M* ra;(x) log I + b;(x)| + ^ ra^(x) log | + b,(x)J (3.28)

= a^^(x) log |- + b^^(x) s a y ( 3 . 2 9 )

(c) The b o u n d a r y conditions

T h e s a m e s l e n d e r body of r e v o l u t i o n a s t h a t d e s c r i b e d in s e c t i o n 2(c) will be c o n s i d e r e d h e r e . T h r e e b o u n d a r y conditions a r e now a v a i l a b l e , v i z . n o r m a l and t a n g e n t i a l v e l o c i t i e s , and t e m p e r a t u r e j u m p at the s u r f a c e .

T h e n o r m a l condition g i v e s d r

v = - r ^ (1 + u) r dx and the t a n g e n t i a l condition gives

U = (1 + u) cos o + V sin a

s r T h e t e m p e r a t u r e d i s c o n t i n u i t y AT i s given by

AT = (1 + T ) - T s w

w h e r e T i s the n o n - d i m e n s i o n a l p e r t u r b a t i o n t e m p e r a t u r e of the fluid adjacent to the wail and T i s the n o n - d i m e n s i o n a l wall t e m p e r a t u r e .

w By c o n s i d e r i n g o r d e r s of m a g n i t u d e , t h e s e e q u a t i o n s can be simplified to d r V = r-^ ( 3 . 3 0 ) r dx U = 1 ( 3 . 3 1 ) s and AT = 1 - T ( 3 . 3 2 ) w

17 -( 3 . 2 7 ) and -( 3 . 3 0 ) show t h a t , . S'(x) a (x) = — » 2ir ( 3 . 3 3 ) A s b e f o r e , _{" s = ^=1 * v § 7 + 87" • d 7 ;} / 8 u 8 v ^ d r \

i g n o r i n g the t e r m in t e m p e r a t u r e g r a d i e n t along the s u r f a c e . F r o m ( 3 . 2 5 ) and ( 3 . 3 1 ) , r ( x ) a,(x)

[r;(x)]

T h e t e m p e r a t u r e j u m p AT = c^ ë ( — j + O(t*log t) ( 3 . 3 4 ) c , e r = r F r o m ( 3 . 2 9 ) and ( 3 . 3 2 ) , So<^) (1 - T ^ ) ^ , < ^ ) c , I ( 3 . 3 5 ) F i n a l l y , f r o m e q u a t i o n s ( 3 . 24) to ( 3 . 29), ( 3 . 3 3 ) , ( 3 . 34) and ( 3 . 35), u , v and T m a y be deduced and a (x), a (x) and a (x) m a y be e x p r e s s e d

r ( x )

" =(7fH<^^>]0^°^ I ^ V->

_{c , *} r 2 i r r (1 - T ) r (x) T = ^^—^ log I + b,„(x) ( 3 . 3 6 ) ( 3 . 3 7 ) ( 3 . 3 8 ) a^(x) 2k c , *

f Sic

L 2w x) r ( x ) c , *

t H '

]

r . ( x ) _ j^ r,(x) r , M - — j \ -3kx/2(7-l)M* a (x) = » (7 - DM* X / 3 k x / 2 ( 7 - 1 )M* 3k ( S^JO ^M> <<^) ^ r,(x) \ 2 \ 2w- 2k 2 k c . T / 2k c , T (1 - T ) r (x) w 1 c * dx

, . r

s'(x) ^<^> <<^)

^

^<^>

] ^,

%<^^

=1

"21;^

2 k — ^

2 r ^ f

J - %'

(x)

(d) Drag

As before, the viscous and p r e s s u r e drags, associated with a cylinder of length L and radius R enclosing the body, will be considered separately.

The drag due to the viscous s t r e s s e s at the cylinder surfaces is

7^. = è / (If ^'^^ ^) -. - 1 / (If) -. 4 ƒ (If ^'^)

2pU "^s s '^ /• f 8 u ^ 8_v, V^X ^ / g.X

[r:(x)]

dS,

'^ n M^') ,rr'(x)1%^''<^>

k I ^ Z L 1' J 23r o • = • ' dx + O(t*log t)

^' •' f •••'^' [.,-,.,]' , d.

•o = • • k = 0(t*)

which is identical with the incompressible case.

The drag due to the momentum flux and the pressure at the cylinder surfaces is O(t*log t), as in the incompressible case, giving for the total drag

A

1

r,(x)

(dx

(e) Heat transfer rate

By F o u r i e r ' s law, the rate of heat transfer to the body surface is

q K* Too f 8T

w L \ 8n wall

^ ~ 11» I ( dT

I ? ) ^^ .O(t*logt)]

wall

Thus the Stanton number = ——7; r—; with h = C T

P» U(h^ - h j p K.O

C e Lp U Cn a 'oo p

19

-which is a constant for all regions of the body surface and decreases with increase in mean free path 3 .

(f) Drag - an example

Finally, a few calculations will be made in order to compare the drag of a slender body under continuum flow conditions with drag under slip flow conditions.

A specific example of a body of length L with meridian shape given by r,(x) = 46 (x -X*) , a parabola, will be considered. 6 is small.

(3)

For boundary layer flow , Goldstein gives

= L32L

' « e '

for the laminar skin friction drag coefficient, based on surface area S, of a slender body of revolution-, form drag may be neglected.

At a typical continuum Reynolds number of 10 , Cj = .0013

At supersonic speeds, the wave drag is D , 1 w i p U ip ƒ ƒ S" (x) S ' (y) log I X - y I dx dy giving C^ ^w D

i p u ' s

= 32 5' for our example.

Thus at normal densities, the supersonic p r e s s u r e drag coefficient is appreciably l a r g e r than the skin friction coefficient for values of 6 of order . 1. This is the r e v e r s e of the situation in slip flow where, as we have seen, p r e s s u r e drag is small compared with viscous drag.

For slip flow

dx

^D ^ 2» k

i p U *

/[^^[^«r}

which, for the parabolic body, gives

It has been shown in section 2(c) that Ï = 0(t_ ) i . e . the mean free path is large compared with the body length and so I , for the purposes of this example, will be taken to be 1. ; also c — 1

5 ' C ^ ^

^D — R _e

For a typical slip flow Reynolds number of I ,

(3.39)

D IB 6

Using the relation M = ^ R . 1 - ^ , which is derived from kinetic theory, equation (3. 39) may be expressed in the form

C M =£i- 12

These results are plotted in the following graph of log C against log R .

LOG,oCo or - 3

1^

N

y = 32 « M X c . ifi

k J

N

« • • 1 t ' 05 i m 0 3 -< } = 1

s

sS = OS . j - 0 3 L;

12a/

1. C*'-'-

t)

SLIP FLOW UXS^R,

3 4 5 6 CX)NTINUUM BL. FU3W

THIS GRAPH SHOWS CLEARLY THE DIFFERENCE BETWEEN THE ORDERS OF MAGNITUDE OF THE DRAG COEFFICIENTS.

21 -4. References 1. Adams,M. C. , S e a r s , W.R. 2. C l a r k e , J . F . 3. Goldstein, S.

Slender body theory - review and extension. J . A e r o . S c i . , Vol.20, 1953, pp 85-98.

Relaxation effects on the flow over slender bodies. C. o. A. Note 115, 1961.

Modern developments in fluid dynamics. O . U . P . , 1938.

4. L a g e r s t r o m , P . A. Cole, J . D . , Trilling, L. 5. Lamb.H.

Problems in the theory of viscous compressible fluids.

Guggenheim Aero. Lab. Report, CALTEC, 1949. Hydrodynanriics, pp 609-611. C . U . P . , 1932. LauriTiann, J. A. Trilling, L. Wang Chang, C.S. Uhlenbeck,G.E.

Slip flow over a short flat plate.

P r o c s . of the F i r s t Int.Symp. on Rarefied Gas Dynamics, ed. by F.M.Devienne, pp 293-316, Pergamon P r e s s , 1960.

On thermally induced sound waves.

M . l . T . Fluid Research Group peport No. 54-2, 1954. On the behaviour of a gas near a wall.

A P P E N D I X A

A l t e r n a t i v e f o r m s for the functions b,(x) and h^{x)

U s e i s m a d e of the b o u n d a r y condition t h a t a (x) = a (x) = 0 f o r X < 0 , X > 1 . F r o m e q u a t i o n ( 2 . 1 1 ) X i b^(x) = a ^ ( x ) [ r + l o g k j - i ƒ ^ ^ E i [ - k ( x - y ) ] dy + i ] ~' E i [-k(y-x)] dy "" , , - k ( x - y ) -jx-e x - e a (y) e •' "] a ^ ( y ) E i [ - k ( x - y ) ] + / ' ^ -y ' ' y j o = a , ( x ) [ r . log k J - i ^^^Q 1 Lt 2 6..0 a ( y ) E i [ - k ( y - x ) ] 1 ,1 a / y ) e X - y k(y-x) y - X dy

]

= a ^ ( x ) [ r + log k J - ^^^^ [ a,(x - e ) ( r + log k + loge) + / ^^^^ ^ ^^ j

T + r 1 a (y) e y k(y-x) dy

]

c+6 X ~6 X ~ €

i^J^Q I a^(x - Ê ) l o g e + j - log(x-y) a (y) e ' ^~^ \ + e " ^

T

log(x-y) ~ [ a ( y ) e ^ ] d y j dy

+ i ^ \ [ - a(x + 6) logs - riog(y-x) a,(y) e'^^^'""^ 1 + e

kx x+6 log(y-x) ^ [ a , ( y ) e " y ] dy J e 2 - k x _{d ^ ( y ) e ' ' ' '} j log(x-y) dy + | / log(y-x) — [ a , ( y ) e ' ^ j d y k / ' d a 2kx r ^

23

-A P P E N D I X B

F l u i d v e l o c i t y , p r e s s u r e , t e m p e r a t u r e and d e n s i t y in l i n e a r i s e d 1 v i s c o u s flow in t e r m s of the functions 0 , , 02 and x

q = q + q + q , 2 >2l ^ 2 .3 3 W0 + V0 + f ( x ) 1 2 ' ^ f (x) m a y be e x p r e s s e d in the f o r m ' A U w h e r e w i s a v e c t o r function r e l a t e d to x 8 p ,

V q = 0 , and so f r o m the continuity e q u a t i o n , ^ ~ ^ It m a y be a s s u m e d that p = 0 , and p and T m a y a l s o be t a k e n t o be z e r o . E q u a t i o n s ( 3 . 1 ) then give for q^

8u , 8x R 3 e 8v , 8x R ' e 8w , —' = J - V«w 8x R » e

which a r e the s a m e e q u a t i o n s a s t h o s e obtained for the r o t a t i o n a l c o m p o n e n t of q in i n c o m p r e s s i b l e flow (Ref. 5). X t h e r e f o r e s a t i s f i e s

('^•--è)

w h e r e 2k = ^ i ^ = R u = ^ - 2kx 3 8 x 8x V = r— r j 8 r F r o m e q u a t i o n s ( 3 . 1 0 ) and ( 3 . 1 1 ) »*T , , -, a T — Ï - -^ ' T = - ^ V* - — L (1)

8 T 2 I _2 8x oR 2 r T (2) and f r o m ( 3 . 6) 8 T dp — i . (.y - 1) _ ï = _ 1 _ 7=" T (3) 8x ^ 8x orR ^ e C o m b i n i n g (2) and (3) ftT 8p 8x ax T + p = c o n s t a n t (4) 2 "^a p = c o n s t a n t = 0 s a y . ' 2

After r e m o v a l of t e r m s dependent on x f r o m the x - c o m p o n e n t of e q u a t i o n s ( 3 . 1 ) , we h a v e ' e e •' w^ + Wj) j (5) 8 i ^ 8 7 < and s i n c e q = 7 0 and q = V 1 ' 1 ~ 2 2

F r o m the continuity equation ( 3 . 2 ) 8p « « „ - - + V 0 = 0 8x ^z dp

87 ^ T 8-7 = ° ^^°-<«>

P - - -5- 0 _{2 « 2} and f r o m (4) T = ~ 0 2 2 2

25 -T h e 0 d e p e n d e n t t e r m s of e q u a t i o n (5) a r e 8**, = _ J _ 8p, 2 — + J _ 7" E l l 8 x ' 7M 8x 3R 8x e 2 L 8x ^«1 ' - ' t 4 ^g Pi = - ^ ^ ^ 1 8 7 - 3 R - ' K J e But p s a t i s f i e s ( 3 . 9 ) 1

8_% . J L _ , . p .- ^ ,^p

a , . , , 3R / JL . i _ 7* . JL_ya J _ \ / ^2. __e A \ . = 0 0 s a t i s f i e s the s e c o n d b r a c k e t .

("-"•5^.f^%^)^

E q u a t i o n (7) g i v e s p a s P,

^'^"[ir ''^- ^']

p = M* 1

r^v«0 - '-^1

{_3k ' ^ 8x J (7) F r o m ( 3 . 6 ) 8T, dfi — = (v - 1) —* + -Ï— V* T (8) 8x ^^ 8x a R 1 * ' e and 8p, 8 T dp 8 7 ^ 8-7' = 8 7 ' ^rom(3.5) (9) E l i m i n a t i o n of p and p f r o m (7), (B) and (9) g i v e s 2 8?^. T = - (7 - D M — ' (10) F i n a l l y , f r o m (7) (9) and (10) .

A P P E N D I X C T h e i n v e r s e t r a n s f o r m of ip E q u a t i o n ( 2 . 1 0 ) g i v e s

^ = - A^(p)fr + log I -fp^ + k'

T h e i n v e r s e t r a n s f o r m i s 1 " - - i p x , i/- = 2 7 / i/- e ^^ d p

i r A / p ) ( r + l o g i r + ilog(p*+k*) j e ' ^ P ' ^ d p

» - CO

= a , ( x ) l o g i r - ^ I A,(p)(^r+ilog(p*+k*)J

e "^ d p - i p x . w h e r e a ( x ) = - -— / A , ( p ) e ^ d p 2n

L

N o w 27

L

' A,(p) (^r+i log (p* + k*) j e "'P''dp

2» OO i p A ^(p)

_ r + è i o g ( p ^ + k ' ) l -ipx

- i p '

aa^(y)

,, 8 y w h e r e f ( x - y ) d y b y t h e F a l t u n g t h e o r e m - i p x e d p

' 2» L " ^ p

J _oo

= _Lr i° r r + iiog(p*+k*) L-ipxjp ^ r r i i J j 2 Ê t

J-oo n p* + k*) - i p x , e ^ d p C h a n g e t h e d u m m y v a r i a b l e p t o - u i n t h e f i r s t i n t e g r a l a n d t o +u i n t h e s e c o n d . k * ) ' f(x) =

_ i _ j , ° r r + i l o g ( u * + k - ) l iux^^ ^ / i j L l i £ ^ i i l ± J i : ) l e-^"^duj

JoO Q

L-.\ r f r f ilOg(u*-f k')1/-iuX _^iuX J ^^ j

2i 27ri

°r IM- \ l o g ( u ' ' + k')"]

27

-^

ƒ[•

r+ i iog(u* + k*)

u

sin u [x I dx sgn x

- sgn X ( r + log

OO s i n u X

_{i ] l o g ( ^ l + - j — ^ d u}

i r 1 A . u \ s i n u Ixl ,

where E i ( 0 )

^ sgn x j^ (r + log k ) I + I -TT Ei [-k | x | ] J

T

-• - O 0

i sgn X l r + log k - Ei j - k |x|J J

= - i sgn xj Ei f- k jx n - r - log k

7 8a r p -,

.-. ^ = a^(x)log - - i sgn (x-y) / —' [ E i [ - k | x - y | J -T - log k

= a,(x)log I - i / g ^ ' ( E i [ - k (x-y)] - r - l o g k j d y

- 0 0 oo

^ I Jx 5 7 ( ^ ^ [-k(y -X) ] - r - log k ) d y

Thus l/l = a^(x) log 2 + b,(x)

where

_{'^ f 8 a / -, \}

b,(x) = - i j ö 7 \ ^ i [ - k ( x - y ) J - r - logkjdy

- O O "^ g o

+ 1 £ ^ ( E i [ - k ( y - x ) | - r - l o g k ) d y

X 00 _ X

^ j- 8a, /• 8a I f 8a

= i ( r + l o g k ) ] - ^ d y - ] ë y d y j - i ] 8 ^ E i [ - k ( x - y ) ] d y

_ • — OO X " • • " 0 0

+ i / g^' Ei [-k(y-x)] dy

X oo

= a^(x) ( r + log k) - i j — ' E i [-k(x-y)] dy + i J — Ei[-k(y-x)] dy

giving equation (2.11).