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402

Fluid Motion in a Curved Channel.B y W. R. D e a n , M.A., Imperial College of Science.

(Communicated by S. Chapman, F.R.S.—Received July 31, 1928.)

Experimental work due to Prof. J. Eustice* has shown th a t there is no marked critical velocity for a fluid flowing through a curved pipe. If the pipe is straight there is a sudden increase in the loss of head as soon as the velocity exceeds its critical value ; below the critical the loss of head varies as the first power of the velocity, bu t above it approximately as the second power. But if the pipe is curved there does not appear to be such a sudden change a t any velocity of flow. One possibility is th a t flow through a curved pipe is stream-line a t velocities much greater than the critical for a straight pipe, but experiment*)* seems to show th a t the critical velocity is smaller in a curved pipe than in a straight one. If then the motion in a curved pipe becomes unstable a t a velocity somewhat less than the critical for a straight pipe, the absence of a sudden increase in the loss of head in this region suggests th a t the stream-line motion in a curved pipe (unlike th a t in a straight pipe) is unstable for small disturbances. A similar problem shows th a t it is not unlikely th a t curvature may have such an effec t: it is believed th a t uniform shearing motion between flat plates is stable for small disturbances,J but Prof. G. I. Taylor§ has shown th a t shearing motion between concentric cylinders can in certain conditions become unstable for small disturbances.

A theoretical investigation of the stability of flow in a curved pipe is certain to be a m atter of great difficulty, and therefore a simplified form of the problem, the stability of flow under pressure through a curved channel (i.e., between concentric cylinders), is here considered. I t is shown th a t the motion can become unstable for a small disturbance of exactly the type found by Taylor to be possible in motion between rotating cylinders.

I t is assumed th a t d, the distance between the two cylinders, is small in comparison with a, the smaller radius, and if this assumption is introduced at the start the equations, which are complicated when exact, become quite simple. As a check on the method seemed desirable, the results it gives are shown to be approximately the same as those found by Taylor by a more

* ‘ Roy. Soc. P roc.,’ A, vol. 84, p. 107 (1910). t J . Eustice, ‘ Roy. Soc. P roc.,’ A, vol. 85, p. 119 (1911). t E.g., R. V. Southwell, ‘ Phil. Mag.,’ vol. 48, p. 545 (1924).§ ‘ Phil. T rans.,’ A, vol. 223, p. 289.

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F luid Motion in a Curved Channel. 403

satisfactory procedure in which an equivalent assum ption is made only after an exact solution of the equations has been obtained.

The critical velocity for the type of disturbance considered is found to be given by

vdlv = 36 (a/d)*,

where v is the mean velocity.This value is small enough to suggest th a t the disturbance m ay (in a certain

range of values of a/d) be th a t which actually arises when stream-line motion breaks down. This, however, could hardly be decided except by experiment, since it would be alm ost impossible to examine theoretically all possible types of small disturbance. W hat the work of the paper does show is th a t a type of small disturbance which could not persist in a stra igh t channel is possible in a curved channel, and th is makes more likely the explanation suggested above of the absence of sudden changes when the velocity of fluid flowing through a curved pipe passes through its critical value.

2. The equations for the steady m otion of incompressible fluid, referred to cylindrical co-ordinates (r, 6, z), are, if the velocity components associated with these co-ordinates are independent of 0,

D W + w 8U _ V ! _or oz r

u |Y + w 3v + uy =or oz r

andu 9w + w a w

or oz

d_ / P \ /02U 1 3XJ SfXT U \dr \ p / V 'dr2 r dr dz2 r2! 9

i a _ / p \ / m , i 3 v , 3 ! v _ v \r 30 \ p / V dr2 r dr dz2 r2/ ’

d / P \ (dJW J_ 0W , 82W \d z \ p ) V \ dr2 r dr dz2 / ’

( 1 )

( 2 )

(3)

where P is the pressure, p the density and v the kinem atic viscosity. The equation of continuity is

0U , _U , 0W dr r dz

0. (4)

Suppose th a t the fluid is contained between the cylinders r = a and r — a + d, and th a t

P /p = * 0 + / ( r ) ,

where k is constant ; the pressure then varies across a section of the channel but falls steadily (if k<C0) along the central surface of the channel, r = a + eZ/2, as 0 increases.

2 d 2



404 W . R . D ean.

There is an we can write

exact solution of the equations appropriate to th is case, which

u = w = o, v = v0, =the actual expressions for V0 and P 0 are not required.

Let r — a + x, so th a t the cylinders are x = 0 and x = d ; then a first approxim ation to V0 is given by

V0 = (*2 - Xd), (5)u 2 va

if term s of the relative order dja are neglected.3. Consider now a steady m otion slightly different from th a t above :

U = u, V = V 0 + ^, W = w, P = P 0 + p ;

u , v, w and p are all assumed to be small and independent of 0.Substituting these expressions in equations (1) to (4), using the relation

between P 0 and V0 and ignoring squares and products of u, v, w, we reach the equations

_ jrv> =a + x

\o / v \ , f S2U , 1 3u , C2U ______ l^ - ' ' l dx2 1 a + x dx dz2 (a + x)4 2 * * * f 9

/0V O , V0 \u V t e + ~ x )

_i___ 1 + 1 L _ 1 , (7)13a;2 a + x dxSz2 (a + a;)2 i

0 =

and

o /p \ , fd 2w , 1 dw , c2w)3 z \ p / 'r\of%% n —l— r)nr. f)?? I

0 = q___ !L_ _i____.dx a + x dz

l dx2 1 a + x dx dz2 J 9

dw

(8)

(9)

4. Now let the assum ption th a t d/a is small be introduced. A fust approxim ation to the equations is then obtained by neglecting such terms as

1 d u _____ua + x dx (a + x)2 ?

in comparison w ith — . Again — 2 V0v/(a + &) can he w ritten 2\ 0vja,

but we have evidently no grounds for neglecting the latter term in comparison. . d2u



Fluid Motion in a Curved Channel. 405

W ith these approxim ations, equations (6) to (9) become

_ 2 V 0v _ 8 (p \ a«e»\a 0 a ; \ p / \8x2 2/ ’ (10)

„ 3 V „ _ v /§ !e + 3 Mdx V \0*2 ' 0Z2,/ ’ (11)

0 II 1

"o

+

qaTq

j

(12)

„ du dwdTx^"d~z'

(13)

We have to find th e condition th a t there m ay be a non-zero solution of these equations subject to th e boundary conditions

u = v = w = 0 ; x — 0, d. (14)

5. Although i t is fairly clear th a t the m ost im portan t term s in equations (6) to (9) are reta ined in equations (10) to (13), the process of approxim ation is no t on a precise basis. As i t seemed desirable to have a check on the method, i t is now in §§ 5-7 applied to th e m otion of fluid between ro ta ting cylinders, since th is problem has been w orked ou t by Prof. G. I . Taylor starting from an exact solution of equations very sim ilar to those above.

Equations (6) to (9) apply to th is new problem provided we write

V 0 = At -j- B Jr — A (a + x) + B + x), .

where A and B are constants whose values depend on the angular velocities of the two cylinders.

In th is case, unlike the last, V0 does no t vanish a t the boundaries, and we have therefore no reason for neglecting V0/(a + x) in comparison with 0Vo/0 * ; in fact, since the expression for V 0 is simple there is no point in making any approxim ation in the first instance in the left-hand sides of (6) and (7). Making the other approxim ations as before, we get

- 2{A + B/(« + x ) > = - 1 . (£ ) + v ( | £ + § £ ) ,

<d2v , 3*tA



406 W . R. Dean.

Assume now th a t u and v are proportional to cos X z and w to sin X z ; and eliminate p from the equations. There results :

' 02

v / * _ X \dx2

dx2

A idw

X2j v = 2Au,

^dx ~ ^ {A + B/(a + XY}and

du \ w = 0'

(15)

(16)

(17)

u, v and w are now functions of x only ; boundary conditions (14) are unchanged.6. Let

r • izx . , • 2nx .u = 6i sm — + 62 sm —— f- . . . ; a d

this form will satisfy the condition u = 0 when x = 0, d. From (15)

2A V v = -------2 sm-MTZXv X2 + m2ic2/d2 d

no arbitrary constants being needed since v also must vanish when x = 0, d.Then

2 {A + Bj(a + x)2} v

= — JA + S l l 2x , a ) } ■ ■ X2 + m2-n2jd2 d 5

rmzx

an approximation is here made in the coefficient of v. Now let

{'Ba2

3z2Y

thena2/

A + ^ ( l — - + ^ ) ! - s i n ^ = 2 sin YTCXd ’

= (a + b \ _ w + b ? / 1 ),

—

We now have

B d2 24mn7t2a4 (m2 — n2)2 ’ B d 16 mn

7i2a3 (m2 — n2)2

a4 \ 2m27i2/

(m + w) even,

1 ------ , (m + n) odd.2 a!

— 9, f B2 |A + ^ ( l - - + 3&2\ \a2 / J

6,

\ v

= 4 A ®v m = i X2 + m2Tz2jd2

= 1 cr sm —— ,= i d

f ■ tzx . • 2m; , 1\a ml sm — + aW2 sm — + ... j

a

(18)



F lu id Motion in a Curved Channel. 407

and we can w rite equation (16)

V / O2 o \ / 0 W I 1 \ V • rT :x

I \ a ^ - x A s + x “ ) = S c ’ s m T -

This gives

v idwA \Sx

and finally

+ Xu) = C'eAX + D V sin-T7ZX

A2 + rV /d 2 d ’

1 1 + D'e-” - s * 7 + *} •

v » ,x = E + ( V + D e - « + j t x ± C0S = <[ W T ^ + a } ■

N ext expand th e com plem entary function as a cosine series. Before doingso it is convenient to change the a rb itra ry constants ; lettt1 , ruu» 1 ta„ - a* -p , rfC, cosh {A (a — d/2)} <ZD, sinh {A (x L + U + D e = E , + 4X sinh (x i/2 ) 4T cT 8h(W /2) ’

(19)

where E x, Cx and D x are new a rb itra ry constants. If the right-hand side is w ritten as th e Fourier Series

, n x . 2nx ,e0 + ei cos — + e2 cos — + . . . ,

it can be seen th a t

&in C,X2 + m 2n 2jd2

We can now write

, m even ; = A .A2 + m27i2/d2 ’

m odd. (20)

. . . I riZXvw/A = e0 + L cos ——r = d

d_ f iy ~ra l A2 + r2iz2ld2+ a | +

There rem ain to be satisfied equation (17) and the boundary condition w — 0, x — 0, d.

The la tte r conditions become, using (17),

t>i + 262 + S&3 + ••• = 0

— b ̂ -|“ 262 — 36g -j- ... = 0

while equation (17) is satisfied provided e0 = 0 and

(21)

m i x 2 + r%2/d2 + v6fj^ + * + 3 5 ^ “ 0, r = l , 2 ,(22)



408 W . R . Dean.

The condition th a t there m ay be a set of non-zero values of Cx, D t, bv b2, ... satisfying these equations can now be w ritten down by equating to zero an infinite determ inant.

7. I t rem ains to show th a t the equation so derived is approxim ately the same as th a t given by Taylor. The la tter* is

0 0 (1 + d 2 (4 + d2X2/7r2) 3 (9 + d2X2/7u2)

0 0 — (1 + d2X2ln 2) 2 (4 + d2X2/7T2) - 3 (9 + d2X2/7i2)

1 — 1 L / 1C2 1C3

2 2 2C1 V 2C3

3 —3 3C1 3C2 W

where

rfim8m w fl, (1 — (x)

(m2 — n2)2

8mn Q, (1 — p) (m2 — n2)2 n 2

, (m n) odd,

(— ■), (w + «) even, \2 a /

V = ~ 0? + (l + 1*)- n,(i - n) + — y4m27r2

Q1? Q 2 are the angular velocities of the inner and outer cylinders, respectively, and jx = Q 2/ Qx.

The determ inant obtained from equations (21) and (22) requires some m anipulation before i t can be set in the; form of th a t in (23). E quation (22) can be w ritten

C, - f ^ (>2 + r27z2Jd2) er4- (X2 + r2n2ld2)2 = 0,d X2

while from (18)

Cr V m=l l 2 + mh

Hence the first two equations of the set (r = 1, 2) are

1 D‘ + 6‘ + T Tand 2tcd

+ • • • —

C. + V “ x, A ;iP + &. { ^ (x * + +V X2 + 4:K2/d2)= 0.

* Loc. cit., p. 308, equation (5 -4 0 ); th e no ta tio n lias been a ltered in some places so as to be nearer th a t of th is paper.



F lu id Motion in a Curved Channel. 409

The condition can now be w ritten

0 0

0 0

1 0

0 2

3 0

-1

2

2

«21

4Av X2 + T?jdd

4A Ojsv X2 + —2jd2

v X2 + i ^ j d 2

4A a 9„v X2 + 4tc2/d2

1

Some simple reductions bring th is equation to th e form (23).M ultiply all rows except th e first tw o b y v/4A, and then divide by this

factor th e first two columns ; then m ultip ly the th ird column by (X2 + 7c2/d2), th e fourth by ( X2 + 4rca/c?2), and so o n ; lastly m ultip ly the first two rows by cZ2/ - 2.

Then the condition is

0 0 1 + d2X2/7T2

0 0 - (1 + <Z2X2/7t2)

1 0 s ^ ( *2 + , w + “11

2 (4 + (f2X2/7r2)

2 (4 + d2X2/TC2)

a21

0 2 a 12 4 ^ (X2 + + «22>

= 0. (24)

The first tw o columns can evidently be reduced to the first two columns in (23), and i t therefore rem ains to show th a t

Q'm rfimn

.“ m + r t - u 1 ( i - r t i ( i + 5 k )

If (m + n) is odd, a ncm provided

£ 2 , ( 1 -3d\2a



410 W . R. Dean.

NowQj( 1 - fj.) = Qx - n 2 = {A + {A + B + d)2}

S B d L __ 3d , 2d? a3 \ 2a a2

Hence if terms of the relative order <L2\a2 are ignored the equality holds. If (to + n) is even, amn = „cm provided

3Bd2 a4 ’

and this is true to the same accuracy as before. Lastly,

-§-1(l + {A) — ^ i ( l — +L a \ 2m27i2 *

if terms of higher order in dja are ignored, and the expression on the right is equal to amm.

Equations (23) and (24) are therefore approxim ately the same, and we can conclude th a t in this case the process of approximation is justifiable. This will be assumed for the problem treated in the present paper.

8. If p is eliminated from equations (10) to (13), and it is assumed that u and v are proportional to cos Az and w to sin Az, the equations become

and(25)

Substituting in the first two equations the approximate expression for V0 given by (5), we have

(s° - xs) ( I f+ x“) = - 3? - rf) ”■and

i k {2x- d)u-

(26)

(27)

The ratio of the coefficient of v in (26) to the coefficient of u in (27) is of order dja, and it may appear a t first sight th a t the former term should therefore be neglected. I t will, however, be seen th a t these two coefficients are



in effect multiplied together, so th a t to obtain a first approximation both terms must be kept in ; further, although we are assuming th a t d]a is small, it does not necessarily follow th a t (roughly speaking) V 0d/a is small.

We have now to find the condition for a non-zero solution of equations (25) to (27) subject to the boundary condition (14).

9. I t is convenient to write the equations in non-dimensional form.Let

y = nxjd, l = dX/n ;


then the equations become

(28)

(29)

and

^ -\-lw = 0, dy

(30)

whereA — — M 3/2TCV a , B = - M 4 (31)

The boundary conditions are

u — v = w = 0 ; y — 0 ,7i. (32)

In the solution of these equations two such expansions as th a t shown in (18) have successively to be made ; further, we have nominally to find the values of A and B for all values of l, so th a t the least possible values of A and B leading to a non-zero solution of the equations can be found. As this would involve rather heavy numerical work a preliminary exploration seemed- desirable. For this purpose (2y]n — 1) in equation (29) was replaced by — cos y, and (2/2/tu2 — y/n) in (28) by — (1 — cos 2y)/8 ; the two expansions required were then, of course, very simple. I t was found th a t AB had a minimum value of roughly 860 when l2 was about 1*5. This result served as a guide in the arithmetical work of the exact solution.

10. In solving equations (28) to (30) we proceed exactly as in §§ 5-7.Let

u = ax sin y + a2 sin 2y + . . . , (33)

(2y!n — 1)^ = 6! sin y + b2 sin 2y + ... ; (34)

then from (29)

v = a a lY^ —2 sin *»y. (35)r -f- m2



412 W . R. Dean.

no arbitrary constants being required since v must vanish when = 0 and when

y = tc.Further, let

B (y2/7t2 — yin) Iv — AB £ cm sin my ;

then from (28)

+ lu = rj'elV + P'*"* - ABl S —i sin m,j>dy l2 + m2so that

= aV" + p'e"*1' — £ sin my + ■p~ ^ 2cOTj

and

w = ae*v + (3e lv + y + £ /^ _|_m \

AB l Z2 + m 2 °m '

The complementary function m ust be [expanded as a cosine series, and as before in § 6 it is convenient to alter the arbitrary constants ; equations (19) and (20) can be used here if l is substituted for X and iz for d.

Then

where

w = e0 + 2 cos my ~1 - ^ L c ] + eZ2 + TO2c» / - i - e-

C= —*> m even>// + m 2

Dl2 + m2

, m odd,

C, D and e0 being arbitrary constants.Equation (30) and the boundary condition w = 0, y = 0, tc, are satisfied if

and

ai + 2a2 + 3 a3 + ... = 0,

cc± -f- 2$2 — 3^3 -f- ... = 0,

1 J 7 , AB l \\ vCf/m „ Cm. ip_j_ m2Cmj + e™ + — °> 1, 2, . . . ) .m

The first two equations of the set above can be written

ax [l2 + l)2 + ABl \ + ID = 0,

di (l2 + 4)2/2 + ABZ2c2/2 + 1C = 0.If then

O0Cm = 2 (X.mr&r>




the condition for a significant solution of the equations is

0 0 1 0 3

0 0 0 2 0

— 1 0 C i + X ocn X 0C12 ><1

CO

0 — 1X a 2i

2n i X a 22

° 2 + 2X a 23

2

__ 1 n X oc31 ^ a 32 p X a 331 V

3 3 ° 3 i - 3

whereX = ABZ2, Cm = (l2 + m2)2/m.

= 0, i (36)

Now divide the th ird column of the determinant by Ci, the fourth by C2, and so on ; then add to the first column the sum of the th ird , fifth, ... columns, and add to the second column the sum of the fourth, sixth, ... columns. There results

Xan | Xa13 , Xai2 i Xa14"TT" + ~~r~ +

Xa2i | Xa23 | Xa22 , Xoc24

2C± “r 2C3 ^ *“ 5 2C2 2C,

1 0

0 _2

1 1 ^ a11 + Cx

C2

X a12c 2

X a212Ci

1 1 ^ a 22 i 2C2

= 0. (37)

11. We have now to find expressions for the coefficients a .

I - £ ( $ - 1) ( 5 ̂ sin ™ my iy -and it can be seen th a t

fttn (TJ? i_ 1 \.. - 22 + m2\12 4m2'

, v / bn f 1 + cos (m — n) tu 1 + cos (m -\- ri)icH ,qo\l2 + n2 l 2 (m — n)2 2 (m + n)2 J J ’

the summation sign is accented to show that the term given by n = m is to be omitted.



414 W . R Dean.

Again,j — 1 1 ^2 an sin n y ) sin my dy ,

so tha t

1 — cos (m — n) 7u _ 1 — cos2 (m — n )2 * 2 (m + w): <39>

where the accent has the same significance as before.The numerical coefficients in equations (38) and (39) must be calculated;

then the series for any a can be written down a t once. I t can be shown for instance th a t

If (m + n) is even = 0. For a13, for instance, is the coefficient of ax m c v and, therefore, the coefficient of in a series containing bv 63, 65 ..., ; from equation (39) it appears th a t the coefficient is zero.

The series for the coefficients a converge rapidly. The series for aie converges a little more slowly than th a t for oc12, but, on the other hand, the former has subsequently to be divided by C6 and the latter by C2. In no case has it been found necessary to take more than four terms to get a result correct to four significant figures.

12. I t is not a t first sight obvious th a t the determinant converges. Conditions sufficient for convergence are th a t the product of the diagonal terms should converge absolutely, and th a t the sum of the non-diagonal terms should converge absolutely.* The diagonal terms, except the first two, are all unity, and therefore the first condition is fulfilled.

I t is clear th a t if we write

and call amn' the coefficient of an in cm', then | amn | <C amn'. If m is even,

a i2 — 48 T0-9533 0-1800 0-0031 0-0003

ti4 U 2 + 1 P + 9 l2 + 25 l2 + 49

For a given value of l any term in the determ inant can be calculated.

1 — cos (m — n) 7i2

7u2 L 22 (m — 2)2 42 (m — 4)2

* W hittaker and W atson, ‘ Modern Analysis,’ § 2*81.



Fluid Motion in a Curved Channel.415

and a mn' (n odd) is obtained from this series by replacing all the coefficients b by 4/tc2 ; amn' is therefore independent of n. I t is then clear th a t

a2„' + a4„' + a « + ••• < J i (<p + p + - ) { e + 2 (§5 + p + - ) } rV*

I t can be seen in the same way th a t if n is even

<*1 n + &Zn + • • • ^ 4 -

Consequently 2 amn' converges to a sum less than J for all values of n.( m )

I t follows th a t the series 2 ocmn, and, a fortiori, the series 2 amn/m converge(m )

absolutely, and in no case can the sum of the moduli equal J. Consequently the double series y y ocmn

mCn/ l \converges absolutely by comparison with the series 2 | — ). This estab

lishes the convergence of the determinant.13. The preliminary work made it likely th a t the value of l2 leading to a

minimum value of AB would be between 1 and 2 ; AB has therefore been calculated for the values 1-0, 1-2, 1*4, 1*6, 1*8 and 2-0 of l2. As a m atter of convenience a new variable x, given by

X = 8X/7T4 = 8AB£2/7C4,was used. The results are

r - 1-0 1-2 1-4 1-6 1-8 2-0

X 83-7 96-4 110-3 125-5 142-1 159-8

8AB/tu4 83-7 80-3 78-8 78-4 78-9 79-9

These are plotted in the figure and show th a t there is a minimum value

AB = ^ (7 8 -4 ) , (40)O

approximately, when l2 is somewhat less than 1-6.



416 W . R. Dean.

The results above were obtained from the first seven rows and columns of the determ inant; two values of x differing by 5 and giving a change of sign were found, the seven-row determinants were calculated for these values, and the value of x which made the determinant vanish was calculated by linear interpretation.

The last figure given is not reliable, because of the method of interpolation, and also because of the slow convergence of the first two rows.

In the case Z2 = 1 • 6 the determ inant given by the first eight rows and columns was calculated.

When x = 125

A8 = (0*0595) (1*0025) (0*2198) (0*6392) (0*6900),

while when x — 130

Ag = (-0 * 0 1 7 3 ) (0*9904) (3*9355) (1*0761) (1*0385);

the two values of A7 are obtained by ignoring in each expression the last factor. Using the values of A8, we find x = 125*4^ using A7, x = 125*5. I t is clear th a t the stage has been reached a t which the calculation of more rows and columns would lead only to unimportant' arithmetical changes in the values of the determinants ; the signs of the determinants evidently would not change. Since the first two rows converge slowly (like the series S 1/^2), accurate numerical results would in any case be difficult to obtain.

14. From (40) the critical velocity for this type of disturbance can be calculated. Let N denote the Reynolds’ number, vd/v, of the undisturbed motion ; v is the mean velocity across a section of the channel. Then

A = 6N /ti2, B = 12dN/7r2a,and

8AB/tu4 = 576 N2d/7c8a.

The critical value of N is then given by

N = 36 (a/d)K (41)

If then, for example, the distance between the cylinders is one-hundredth of the radius of the inner cylinder, so th a t ajd = 100, the critical velocity for this type of disturbance is given by N = 360 ; if ajd = 10, N = 114. The work of this paper will hardly apply to the second of these cases, wherein the value of dja is probably too large ; it has been assumed th a t d/a is small not only explicitly, but implicitly in ignoring the effect on the motion of the ends of the channel.




Some recent experimental work by S. J. Davies and C. M. White* shows tha t the lower critical velocity for flow through straight pipes of rectangular section is about 1400 when the width-depth ratio is large ; and th a t provided this ratio is large enough (100 or over) its exact value is not important. Thus 1400 may be taken to be the critical velocity in a straight channel; compared with this value, those just calculated for a curved channel are small enough to suggest th a t (in a certain range of values of a/d) the type of disturbance considered here may be th a t which actually arises when stream-]ine motion breaks down. But this can hardly be decided otherwise than by experim ent: an exploration of all possible small disturbances is impracticable.

15. I t is perhaps necessary to show th a t the disturbance is damped out if N is less than the critical value, though this seems clear on physical grounds.

Equations (1) to (4) apply only to steady motions ; if the motion is unsteady

(4) is unchanged but au av aw cF J a t * a t must be added to the left sides of

(1), (2), (3), respectively. If it is assumed th a t u , v, w, the components of the disturbed velocity* are proportional to e~ai but do not otherwise depend on t, of the three fundamental equations (28) to (30), the last is unchanged, while the first two are

V2 = —0 lX2 - - ' ) = l2 - (Pcj/7l2V. 7C" \ V /

The equations in §§ 10, 11 are hardly altered ; in some places, but not all, V has to be substituted for l.

I t is easy to see th a t the only changes needed in (37) are these : Cm must be replaced by Cm', where

<V = (l'2 + m2)2/m,

and &mn by amn', the latter being the same function of V as the former is of l ; lastly the (m + 2)th diagonal term is not now 1 but (l2 -j- m2)/(l'2 + m2).

I t is sufficient to take some numerical examples.

vol. cxxi.—A.* ‘ Koy. Soc. Proc.,5 A, vol. 119, p. 95 (1928).

2 E



418 W . R. Dean.

If Z2 __ i .g? (37) becomes

1 x (0-01185) 0 ... = 0, (42)

x (0-00508) 1 x ( — 0-00098) ...

0 x (0-00001) 1 ...

after the first two rows and columns, which contain elements independent of x, have been eliminated. This is an equation to determine x = 8ABZ2/7u4.

Suppose now th a t Z'2 = 1 • 6, and th a t AB has a given value. We have to find the sign of or, and this will be known when Z is found. In this particular numerical case the equation to be solved is the same as (42) bu t for the diagonalterms. We have in fact

Z2 + 1 l'2 + 1

* (0-01185) ... = 0 (43)

x (0-00508) Z2 + 4 Z'2 + 4

from which to determine l ; x as before is 8ABP/714.Equation (42) is satisfied if

x = 125*5, 8AB/tc4 = 78*4.

We have to show th a t for given Z a value of AB less than the critical value leads to a positive value of a.

Assume then in (43) th a t8AB/tu4 = 75,

for instance. If the diagonal term s in (43) were all unity, we should clearly have to take a value of Z2 greater than 1 • 0 to satisfy the equation ; it would in fact be necessary to make

Z2/l-6 = (78*4)/75. (44)

But if Z2 ]> V2 the diagonal terms in (43) are greater than 1, and therefore the value of Z2 given by (44) will not make the determinant zero, but positive, and this requires th a t the value of Z2 should be still further increased. If Z2 is greater than Z'2, cr is positive, and the disturbance in this case is therefore damped out. This argument applies unchanged to any value of Z'2 greater than 1-6. There is a difference in the case of values of lr2 less than 1*6. I t is still clear th a t we must make Z2 > l'2 to make the determinant vanish for




a value of AB less than the critical for V2 ; but there is now the possibility th a t the value of AB taken may be greater than the critical value for Z2, for the critical value is a decreasing function for these values of Z2. The critical value of AB for Z2 is, of course, the significant value ; the value for l'2 is useful only as a guide to the arithmetic. In these cases, then, unlike the last, the two factors operating tend to cancel out. I t therefore seemed necessary to treat one case numerically.

If Z2 = 1*2, (37) becomes

1 x (0-01536) 0

x (0-00662) 1 x ( — 0-00156)

0 ® (— 0-00011) 1

- 0 , (45)

and this equation is satisfied by

z = 96-4, 8AB/7I4 = 80*3.

If Z'2 = 1*2, we have an equation the same as (45) except for the diagonal terms in the determinant, which are the same as those in (43).

Taking 8AB /tu4 = 80, we find tha t

A5 = — 214 X 10“ 4, Z2 = 1 • 23,

A5 = — 108 X H T 4, Z2 = 1-22,

and can conclude tha t the determinant vanishes when Z2 is roughly 1-21. Since Z2 > l'2 we have a positive value of cj, while from the figure it is clear that 80 is less than the critical value of 8AB/tu4 for Z2 = 1-21.

16. An alternative method of showing th a t the disturbance is damped out unless the critical velocity is reached is to assume th a t the disturbed motion is steady as regards the time ; and then to show th a t it decreases exponentially as it progresses along the channel. I t would then be assumed th a t u, v, w were independent of t but were proportional to e~7V, y denoting distance measured in the direction of flow along the centre of the channel. This appears to be a more satisfactory way of approach to the problem ; the fluid must be supposed to enter the channel steadily but in a disturbed state ; what has been found above is the velocity of flow for which t = 0 so that the initial state of disturbance persists along the channel. However, to prove that t > 0 unless the critical velocity is attained requires the solution of a problem algebraically very different from th a t of the special case t = 0,

2 e 2



420 Fluid Motion in a Curved Channel.

and since it appears clear on physical grounds th a t this must be so, it has not been thought necessary to consider the m atter in detail.

17. The work of this paper shows th a t flow in a curved channel may become unstable for small disturbances. This is an effect of curvature : in a straight channel the persistence of a small disturbance of the type considered is impossible. This is offered as a tentative explanation of the known absence of a marked critical velocity of flow in a curved pipe.

In a straight pipe the passage through the critical velocity is accompanied by a sudden increase in the loss of head, but no such sudden change has ever been observed in a curved pipe. A possible explanation is th a t flow in a curved pipe (but not in a straight pipe) may become unstable for small disturbances, for instability so arising is unlikely to lead to other than a gradual change in the loss of head. The corresponding problem in two dimensions which has been worked out here lends some support to this suggestion.

An extension of the analysis to the three dimensional motion of fluid in a curved pipe would be difficult. There is one similarity between the two problems ; stability in a curved channel has been shown to depend on the value of Wd/cc, while the stream-line motion in a curved pipe, and possibly therefore the stability of this motion, depends on N2r/R ,* where r is the radius of the pipe and R the radius of the circle in which it is coiled. However, the stream-line motion in a curved channel does not depend on Wd/a ; th a t is to say, there is, in this case, no scale effect, while in the curved pipe there is. This means th a t (unless the critical value of N2r/R is far smaller than is likely) we m ust not assume the simple parabolic distribution of velocity to hold in stream-line motion in a curved pipe ; and it is necessary th a t the stream-line motion should be simple if the stability problem is to be treated successfully on these lines, for the component velocities of the stream-line motion are coefficients in the differential equations of the disturbed flow.

* W. R. Dean, 4 Phil. Mag.,’ vol. 5, p. 673 (1928).



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