H.K. Moffatt and B.R. Duffy- Local similarity solutions and their limitations

8/3/2019 H.K. Moffatt and B.R. Duffy- Local similarity solutions and their limitations

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J . Fluid Meoh. (1980), vol. 96, part 2, p p . 299-313Printed in Great Britain

299

Local similarity solutions and their limitationsBy H. K . MOFFATT A N D B. R. DUFFY

School of Mathematics, University of Bristol, England(Received 16 March 1979)

Two problems exhibiting breakdown in local similarity solutions are discussed, andthe appropriate asymptotic form of the exact solution is determined in each case. Thefirst problem is the very elementary problem of pressure driven flow along a ductwhose cross-section has a sharp corner of angle 201. When 2 a < +T, a local similaritysolution is valid, whereas, when 2a > &r, the solution near the corner depends on theglobal geometry of the cross-section. The transitional behaviour when 2 a =5 +T isdetermined.

The second problem concerns low-Reynolds-number flow in the wedge-shapedregion 1/31 < a when either a normal velocity proportional to distance from the vertexis imposed on both boundaries, or a finite flux 2Q is introduced or extracted a t thevertex (the Jeffery-Hamel problem). It is shown tha t the similarity solution in eithercase is valid only when 2 a < 201, z 257 .6"; a modified problem is solved exactlyrevealing the behaviour when a > a,, nd the transitional behaviour when E = a,(when the similarity solutions become infinite everywhere), An interesting conclusionfor the Jeffery-Hemel problem is that when a > a,, nertia forces are of dominantimportance throughout the flow field no matter how small the source Reynoldsnumber 2 Q / v may be.

1. Introductionordinates Oxyzwith Oz parallel to the flow, the equation for the velocity w(~,) sConsider Poiseuille flow along a straight duct of cross-section 9. hoosing co-

V2w = - G / p in 9 , (1.1)whereG s the applied pressure gradient, and ,U the viscosity of the fluid. The boundarycondition is

w = O on 8 9 . (1 .2)1f the boundary 8 9 has any sharp corners (as is the case if, for example, it has

square or triangular cross-section), then i t is very natural to enquire into the natureof the solution of the problem (1 l ) , 1 . 2 )near the corner. Choosing polar co-ordinates( r ,8) ith origin at the corner and with 8 = 0 along the bisector of the angle, so thatlocally the boundary consists of the two planes 8 = ? a (figure l ) ,one might reasonablyconjecture that w must have a local similarity form depending only on r , 8, G/,u anda (i.e. independent of the 'remote' geometry of the duct) . If this is so , then ondimensional grounds,

w (G IP ) "f (@ (1 .3)0022-1 120/80/4360-8100 $02.00 0 1980 Cambridge University Press

www.moffatt.tc


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300 H . K . Moffatt and B . R . Duffy

wV w = - GIP0=- - (YFIGURE. Poiseuille flow along duct whose cross-section3 has a corner of angle 2a.(the dependence on a being implicit), and the only possible form of f(0) ompatible

with (1.1) and (1.2)is

This solution explodes when 201 = $ 7 ~ and also when 2a = $n), nd this leads oneto question the validity of the above conjecture.? It is physically clear that thesolution must remain finite when 2 a = in, nd that the similarity form (1.3) cannottherefore be correct for this angle. The difficulty is in fact quite an elementary one,and we shall resolve it in 9 2.A similar difficulty, which may be described as an inapplicability of a local similaritysolution, can however arise in more complex physical contexts, when the means ofresolution of the problem may not be so readily apparent. A second example, whichwe shall treat in $9 3-5, is the following. Suppose that viscous fluid is contained in thespace 101 < a between two planes 8 = fa which are hinged at their intersection (sothat the angle a may be varied), and suppose that the normal velocity is instan-taneously prescribed on the planes, by rotating the planes about 0 as indicated infigure 2(a) , s

U , = T r w on 8 = +a. (1.5)t A similar breakdown was noted by Fraenkel (1961) in the context of inviscid corner flow

with constant vorticity; Fraenkel correctly resolved the behaviour when 2a = +T ,but, by limitingattention to acute angles, the inapplicability of the similarity solution for 2a z &I (see 5 2 below)was not explicitly appreciated.The problem has been touched on in a number of other studies, but the particular difficultydiscussed here does not appear to have been explicitly recognized. For example, Collins & Dennis(1976) study flow along a curved duct of triangular cross-section, and they consider particularlythe case of a ight-angled triangle. They sta te that the solution of (1.1)is O(r2)near r = 0 regard-less of the angle of the corner ,but, as suggested by (1.4) , and as demonstrated in $ 2 below, this isnot true when 2a > (although the subsequent argument of Collins & Dennis remains sub-stantial ly unaffected).A second example is provided by the work of Allen & Biggin (1974) who studied the flow of aliquid filament down an inclined plane, the shape of the free surface being controlled by surfacetension and the contact angle a t the 3-phase contact line. When th is contact angle is a, ocalanalysis using a zero stress condition on the free surface8 = 0 and a no-slip condition on the solidboundary 0 = a again yields (1.4) , which breaks down when a = $77. Alleii & Biggin in fact chosea = in in their numerical computation of the velocity distribution over the cross-section of thefilament; a inite element procedure provided an apparently well-behaved solution. It was in factthrough attempting to understand the structure of this solution th at we were led to investigatethe paradox presented by (1.4) when a = 4


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Local similarity solutions 301

FIGURE. (a )The hinged plate problem, w r 2 / vQ 1.( b )The Jeffery-Hamel problem, Q / v < 1.Sufficiently near 0, inertia is clearly negligible, a.nd the stream function $ ( r , 0 )satisfies the biharmonic equationand the boundary conditions V4$ = 0 (1.6)

Again it is tempting t o seek a similarity solution for 9 epending only on CL) , r , 0 anda ;on dimensional grounds this must have the form

$ = h r 2 f ( 0 ) , (1 .8)and the unique functionf 0 )compatible with (1.6) and (1.7) (Moffatt 10640) is

sin 20- 20 cos 2af ) = sin 2a - 2a cos 2a *

This solution explodes when tan 201 = 2 a , i.e. when 2a = 257.45". For this criticalangle,f 0) apparently becomes infinite for all (01 4 a, and the corresponding velocitycomponents derived from (1.8) become infinite also. The assertion tha t the Reynoldsnumber is low near the corner therefore requires re-examination, as does the relevanceof a similarity solution of the form (1.8).A closely related difficulty has been noted (Fraenkel 1062) in the contest of thelow Reynolds number limit of the Jeffery-Hamel problem (Batchelor 1067, $5.6).Suppose that the planes 0 = f are fixed and t hat fluid is introduced at a volume rate2& (per unit length) at the intersection (figure 2 b ) ; in this case, the low Reynoldsnumber limit of the Jeffery-Hamel stream function (which provides an exact solutionof the Navier-Stokes equations) is

sin 20 - 20 cos 2a'= Qsin2a-2acos2a' (1.10)

with precisely the same angular dependence as (1.8) and the same pathologicalbehaviour a t 2a z 257'.


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302 H. . Hoffatt and B. R.DzGffv

= a

( U ) ( b )FIGURE. The sector geometry considered in 5 2.The same type of difficulty has been encountered and resolved in an entirelydifferent physical context by Sternberg & Koiter (1958). When an elastic wedge is

subjected to a couple that is concentrated a t the vertex, the dilsplacement field (whichmay be derived from a function satisfying the biharmonic equation) involves thesame factor (sin2a- a cos2a) - l which blows up at 261 x 257". Sternberg & Koiterresolved this apparently paradoxical behaviour by spreading out the force fieldproviding the couple overa small neighbourhoodof the vertex and solving the resultingproblem exactly. We shall follow a similar procedure in the fluid context in $$3-5,and shall in this way providea simultaneous resolution of the hinged plate and Jeffery-Hamel paradoxes.

2, The Poiseuille problem at a sharp cornerConsider the problem defined by ( 1 . 1 )and (1.2) when9 =Ba)s the domain bounded

by the lines f3= f and the circular arc T = a.We may consider the full range of angles0 < 261 < 2n, he limiting case 2 a = 2 n representing a circular pipe with a radial' lange (figure 3 ) .Suppose first that 2 a ?= an or #n.Then the general solution of (1.1) symmetricabout 8 = 0 and satisfyingw = 0 on 8 = f swhere

Gr2 cos28w =- --I) + 5 AnrhnC0sA,8,4p cos2a n- 0A,a = 4 ( 2n+ l)n, n = 0 , 1 , 2 , ... .

The coefficientsA, are determined by the condition w = 0 on r = a (101 < a); singthe orthogonality

we find

conditioncosA, ecoaA, eae = ad,,,!


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Local similarity solutions 303Now A , = nr/2a, so that A, >< 2 according as 2 a 5 in.Hence the particular integral.

is the dominant part of the solution (2 .1) near r = 0 only i f 2 a < )R. If 2 a > in, hesolution is dominated near r = 0 by the leading term of the series of eigenfunctions,namely

The particular value of the coefficient A , depends on the 'global' geometry of 9,ndso the asymptotic form ( 2 . 6 )cannot be obtained from purely local considerations.When 2 a > &r, A, and A, are both less than 2 , and so the first two terms of theseries in ( 2 . 1 ) both dominate over w , ( r , 0 ) for small r . In the extreme case 201 = 2 n(figure3 b ) , we find

The cases 2 a = in and $R require special attention. If we let 2a = +n+c in theexpression ( 2 , 1 ) ,we find

so that, in the limit E -+ 0 (i.e. 261 = tn),

2u Leading terms in expansion of 'U: near r = 0r a ,thenr2 log r , then r2 , then T srn lza , then r2 , hen r3nlzad, hen r2 log r , then ra, then rL8rn12a, then r3n/2a,then r2 , hen r6n12a

TABLE1

A similar treatment of the neighbourhoodof 201 = $71 shows that, when 2 a = $n,

1= -E "-(log:) cos20+0sin203np 4In this case, the singularity in wo(r,O) is compensated by an equal and oppositesingularity in the second term (n= 1)of the series in ( 2 . 1 ) .The above results concerning the leading terms in the asymptotic expansion ofw ( r ,0) near T = 0 are summarized in table 1 .

As noted in the footnote on p. 300, the same type of mathematical problem has


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304 H . K . Moffatt and B . R.Du#y

(C )FIGURE. Flow gmerated in domain ga y rotation of bounding cylinder keeping 6 = f ixed:( a ) assuming no separation; ( b )and ( c ) , likely streamline pa tte rns allowing for boundary-layerseparation.been considered by Fraenkel (1961) n a different physical context. An example ofthe sort of situation envisaged by Fraenkel (who limited attention to acute angles201 < &r) is indicated in figure 4( a ) ,which shows a method of generating a flow withclosed streamlines in the domain ga , y rotation of the cylinder r = a with angularvelocity Q , keeping the radial boundaries 8 = 1 fixed. If R = Qa2/v B 1 , and if theflow does not separate from the boundary a t any point, then the vorticity w is uniformthroughout the core region (Batchelor 1956)- w = wo say - and the stream function$ ( r , 8) is determined by

V2$ = -wo in ga , $ = 0 on 8ga. (2.10)The solution for $ is then precisely analogous to the solution for w( r ,O)describedabove. The above proviso concerning non-separation is however physically un-realistic. I f 2a < T , the flow will separate at some point on 8 = a where the pressuregradient impressed on the boundary layer is adverse (figure 4 b ) i ; and if 2a > T , theflow will separate at the sharp vertex (figure4c) . In either case, the vorticity will takedifferent constant values in different subdomains of $Ba separated by thin layers inwhich viscous forces are non-negligible; the specification (2.10) is then clearly in-applicable.

t The seoondary eddy so formed will then also presumably separate when R is sufficientlylarge; and indeed, when R -+CO, the limiting steady solution presumably involves an infinitesequence of eddies, within each of which the vorticity takes a different constant value.


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Local sim ilarit y solutions 305

FIGURE.Modified Jeffery-Hamel problem treated in $ 5 3-5. The hinged plate problem is obtainedin the limit a + 00, and the conventional Jeffery-Hamel problem in the limit a+ 0, w +CO ,Q = +was = constant.

3. A modified Jeffery-Hamel problemConsider now the following modification of the Jeffery-Hamel problem (figure 5):

suppose that viscous fluid is contained in the space 181 c a between the planes 8 = k 01on which the radial velocity is zero and the normal velocityuo s specified as

The total flux introduced into the region 181 c a s then

The hinged plate problem described in 3 1 is obtained in the limit a+m, and theJeffery-Hamel problem is obtained by the limiting process a+O, w+m, with Qconstant.

The stream function $ ( r , 8)satisfies the biharmonic equationV4$ = 0

and the boundary conditions (3.3)

This problem may be solved by means of the Mellin transform (cf. Tranter 1948,Sternberg & Koiter 1958, who considered problems involving elastic wedges, andMoffatt 19643 in which a similar problem with specified tangential velocity on 0 = fwas treated). This is defined by

with inverse relation


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306 H . K . Moffatt and B . R. DuffyIn (3 .5 ) , p may be complex, its real part being such tha t the integral exists; clearly,provided

where e > 0, the integral (3 .5 ) will certainly exist in a neighbourhood of the lineRep = - 1 in the complex p planet. We shall find that c = - is an appropriatechoice for the real constant c in ( 3 . 6 ) .The various operations tha t follow may all bejustified under the conditions (3 .7 ) ,which may be verified retrospectively.

Under these conditions, equation (3 .3 ) transforms to

m d the boundary conditions ( 3 . 4 )transform toi n 0 = +a. (3 .9 )The (unique) solution of (3 .8 )and (3 .9 )is

(3 .10)W(p) = (p+l)sin2a-sin2(p++)a. (3.11)

(3 .12)(3 .13)

- wa*+2 COS ( p+ 2 ) a sinp8 cospa sin ( p+ 2) 0$(P,O) =-(P) -[ P P+2whereNote thatand tha tIt follows from (3 .6 ) (with c = - 1) hat

W (-P) = - W ( P- )p( p , e ) = a---l)iJ(p- 2, e ) .

@( 8) = (r/a)2lW/,, 8). (3 .14)It will therefore be sufficient in what follows to restrict attention to the outer regionr > a; the stream function in the inner region r < a may then immediately be inferredfrom (3 .14) .

Substitution of (3.10)in (3 .6 ) now gives] d p . (3 .15)P 1 cos(p+2)asinpB- ospasin(p+2)8(--)-( r ,e) =512m -1-t- W(P) P P + 2

When r > a , this integral may be evaluated by closing the contour by a large semi-circle in the right-hand half-plane R e p > - 1. The integrand has a removable singu-larity a tp = - 1, and poles at the other zeros of the function W ( p ) .We therefore haveto locate these zeros, and calculate the corresponding residues. Further progress

t The condition of zero forcing a t infinity implicit in (3.1) may be relaxed somewhat withoutaffecting the structure of the solution. In particular, if (3.1) is replaced by uB= 7P(r)whereP(r)= O (T - 8 ) a t 03, o ( r 8 ) as + O , then the same method of solution is applicable with very similarresults.


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LoCal similarity solzctiom 307

-2 0-2 0I

FIGURE(a ) .For legend 8ee next page.requires rather careful consideration of the function W ( p ) ,which is analysed in thefollowing section.

4. Zeros of the function W(p),Re p >- 1pConsider first the real zeros of W ( p ) .Let p + 1 = [ / 2 a ; thenand the real zeros of W ( p ) are given by the intersections of the straight lineLa: = m(a)[ and the sine curve S : = sin


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308 H . K . Moffatt and B . R.Duffy

FIGURE. (a)The intersections of the straight line y = m(a) and the sine curve y = sin5 givethe real zeros of W ( p ) . b)- (e) Solid curves are the family defined by (4.4) and dashed curves arethe family defined by (4.5). The intersections marked 0 give the complex zeros of W ( p )withi+iv = 2 a ( p + 1 ) . Double zeros are indicated by 0 . 6) 2 a = 135'; (c) 2 a = 159'; ( d ) 2 a = 168';( e ) 2 a = 257'.

(iv) When 2 a = 2ac w 257.45" there is a double root of W ( p )at p = 0 (6 = 2 a c ) .(v) For 2 a c 2ac, the smallest root of W @ ) is p = 0; for 201 > 2ac, however, thesmallest root p , satisfies(with equality only at 201 = 2 n ) .

- 1 2 < p , < o ( 4 . 2 )Consider now the complex zeros of W @ ) .Let

P + 1 = ( E+ 7 ) / 2 a , (4.3)and suppose that 7 + 0. Since W @ * )= ( W @ ) ) * ,he complex zeros occur in complex


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Local similarity solutions 309

5

4

7

i

1

0

- 4

1

FIGURE. Sketch of variation of the first few zeros of W(p ) s functions of 2a.Where the curvesare solid, p , is real; where the curves are dashed, p , is complex, and only the real part of pn srepresented.conjugate pairs, and i t will be sufficient to consider those in the first quadrant 5 > 0,7 > 0. The real and imaginary parts of the equation W ( p )= 0 may be written

sin5 sin2a- - -6 2acoshq (4.4)

For given a, quations (4.4) and (4.5) represent two families of curves in the (5 , 7 )plane, and the intersections give the complex roots of W ( p ) . The intersections of(4.4) with the real axis 7 = 0 give the real roots treated above.] These curves aresketched in figures 6 ( b - e ) for the four particular choices of the angle 2a represented


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310 H. . Boffatt and B . R.Duffyby the linesLa in figure 6 (a),From such curves, we may infer the following propertiesof the zeros of W @ ) .(i)For 0 < 2a c 1, n eaoh stripSan,where8, is defined by

8, 3 ,g;mn< ,g < (m + l )n , rn = 1,2, ...},there are either two real zeros of W ( p ) as discussed above)or two complex conjugatezeros. We shall denote the two zeros in the strip S,, by pznand pZn f l ;f they are realthen pznQ pZn+,and if they are complex pZn+,=p & (and we may adopt the con-vention Imp,, > 0,mp,,+, c 0).

(ii) For 7r < 2a c 271, in each strip S2,+, (n= 1,2,3, ...) there are either two realzeros of W ( p )(232n+l, ,,+~) or two complex conjugate zeros (p2n+aP&+~).n thestrip S,, there are always two real zeros ( p l , p 2 ; p l p , ) which coalesce to form adouble zero only for the critical anglea = a,.

The behaviour of the roots p,, p, , p 3 , . (or their real parts where they are complex)is indicated in figure 7, over the full range 0 < 2a Q 2n. Note the position of thecritical angle 2a, and the denumerable sets {2a,}, (2a3 , ( 2 a 3 (with limit points 71,n,27r respectively) on this figure.

5. Asymptotic behaviour of the stream function (3.15)W e are now in a position to evaluate the integral (3.15). When r > a, the contour,

closed by a large semicircle in the half-plane Rep > - 1, encloses all the poles pndescribed in the previous section. Suppose first that a s such that all of these poles aresimple (i.e.a {a,}U {a;} U {a:} U {ac})then standard evaluation of (3.15) gives, forr > a,

and in this ormula, since the polesere simple,W(p,) = sin 2a- a cos 2(pn+ 1 )a + 0. (6.2)

Note that the complex terms of the series (5.1)occur in complex conjugate pairs. Notefurther that, from (5.1),for r > a,

since W ( p n )= 0 and limp-l W ( p )= W(0).This provides a check for consistency with(3.2), since the total flux introduced through the boundaries is

(5.4)P+O

$(r , a) (r , -a) = waa = 2Q.Equation (5.1)has the form

CO+(r,e) = c (:)pnjn(e), r > a,n= 1 (5.5)

wheref,(6) depends implicitly on a.The result (3.14) then gives the correspondingexpansion


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Local similarity solutionsFor 2 a c ac = 267.46', p1= 0 and

where ( 0 ) s as given by ( 1 . 9 ) ,and so (5.5) ives$(r , e ) N imaf(e) Q f ( @ for r B a,

i.e. the Jeffery-Hamel solution ( 1 . 1 0 ) .Similarly (5 .6 )gives$ (T , e ) * J

which is just the similarity solution given by (1 .8) .For 2 a > 2ac,however, as noted in $4 , the smallest root p1 ies in the range

where nowfl(e)= (Wl(pl))-l[pi1 os(pl 2 )a inpl 8- (pl 2)-l cospl a sin (pl+ 2 )81,

311

( 5 . 7 )

(5 .8 )

( 5 . 9 )

( 5 . 1 0 )( 6 . 1 1 )

( 5 . 1 2 )and clearly we do not recover the similarity solutions in this situation. The forms(5.10)and (5.11)do however satisfy the basic requirements ( 3 . 7 ) , incep, 2 -4.

In the Jeffery-Hamel limit (a+O, w+m, Q constant), ( 5 . 1 0 )gives, for any fixed T ,$(r, 0 ) 2 & ( r / 4 - ~ ~ f l ( e ) . ( 5 .1 3 )

From this, we may construct a local Reynolds numberR - Q / v ) T/u)-% ( 5 . 1 4 )

Since p , < 0 when U > a,, t would therefore appear that R > 1 (in the limit a+ 0, rfixed) no matter what the magnitude of the source Reynolds number 2 Q l v may be.Inertia forces are then of dominant importance for all T , and a low-Reynolds-numbertreatment of the Jeffery-Hamel problem for a > a, would appear to be quite un-tenable, no matter how small Q / v may be.As mentioned in the introduction, the breakdown of the Jeffery-Hamel solutiona t a = ac (when R = 2 Q / v+ 0 ) was noted by Fraenkel ( 1 9 6 2 ) who also described asimilar behaviour at finite R (see also Fraenkel 1973) . From Fraenkel's discussion, i temerges that there exists a domain 9 : c a , ( R ) in the a - R plane (with ac(0) 129"),within which the Jeffery-Hamel stream function ?,hJ(8;, ) s a continuous functionof a, nd tha t ?,hJ --t 00 (for all 8 )as a f a , ( R ) . t may be conjectured from the discussionof the present paper that when a > a , ( R ) , the Jeffery-Hamel solution is irrelevant,and that the flow for such angles must depend on the manner in which the sourceQ is distributed in the neighbourhood of the vertex, as well as on its net strength.Returning now to the low-Reynolds-number limit, when 201 = 2ac, as noted earlier,W has a double zero at p = 0, i.e.

W'(0)= sin 2 e- a cos 2 a = 0, (5.15)and so the integrand in ( 3 . 1 5 )has a double pole at p = 0. The leading term of the


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312 H . K . Hoffatt and B . R. Du&asymptotic expansion of $ for r 9 a comes from the residue at this double pole and,from (3.15), s given by

a inpe cospa sin ( p+ 2)8P + 2-

1w22a,2sin 2a,{logk) e cos2a, - sin 281+ [ge cos2e- sin 20+ac sin 2a,l(5.16)[which again satisfies $ ( ~ , a ) &m2,consistent with (3.2)]. Once again, as in thesimpler problem treated in 3 2, a term logarithmic in r appears when the angle 2a iscritical. Here, the situation is closely analogous to that resolved in the elasticitycontext by Sternberg & Koiter (1958). For r < a , the correspondence (3.14) showsthat$ N r210grasr+Owhena = a,.

We now have the additional interesting complication that W(p) has a double zero[and so the integrand in (3.15)has a double pole] whenever a akes one of the specialvalues (of which there is a countable infinity) for which La s tangent to S (see thediscussion of $3). For example, when 2a = 201, z 159", there is a double pole atp, = p 3 z 1.78 and the second and third terms of the expansion (4.1) have to be re-placed by

(5.17)

giving a contribution proportional to ( a / r ) p 2 og ( a / r )for r > a . This contribution ishowever dominated (when r 9 a ) by the leading term of ( 5 . 1 ) , which is (5.8) (with2a = 159").Similarly when a akes any of the special values a,, a;, a: for which Latouches S, evaluation of the residue at the corresponding double pole, p , say, givesrise to a contribution to $ proportional to ( a / r ) p mlog ( a / r ) ; his is, however, alwaysdominated by the terms of the series (5.1)with n < m.

- wa2 d a p cos ( p+ 2)a inp8 cospa sin ( p+2) 0-+-k)"(P2) dP [ P P + 2 l l p = p 2 ,

2 a < 2 a , z 257'r2 , hen r p * +2ro, hen r-nr

2 a = 2 a , 2 a > 2 a ,rp1 f 2 , then r2, hen r ) r + 2r - p l , then To , then r -va

Hinged plate limitJeffery-Hamel limit

TABLE . Asymptotic dependence of @(r ,0) on r as r l a + 0 and r l a+ 0 for the problem de-fined by (3 .3) and (3.4). The exponents p , , p , and p 3 are the roots of W ( p ) = 0 ordered so that- < p 1 < Rep, < Rep, (figure 7 ) .For 2 a > 2 a , , - < p , < 0.

rla + or / a -+ 00

r2 og (al.), then r2 , hen rps+2log ( r / a ) , hen ro , then r-*a

The conclusions of the above analysis are summarized in table 2, which shows theradial dependence of the leading terms in the asymptotic expansion of $ ( r , 6 ) forr/a+O and r la- tco for the problem defined by (3.3)and (3.4). t should perhaps beemphasized that we have assumed throughout that the boundary conditions andresulting flow fields are symmetric about 0 = 0. If antisymmetric ingredients areincluded, then the conclusions of table 2 require modification (see concluding footnoteof Sternberg & Koiter 1958). The antisymmetric ingredients can be important in


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Local similarity solutions 313certain circumstances, and in particular in someof the situations envisaged by Jeffrey& Sherwood ( 1 980).

The 'explosion' in the solution (1.8) at 2a ;=: 257' was originally drawn to ourattention by Dr T. J. Pedley, whose provocative observation is gratefully acknow-ledged, as is a stimulating discussion of the phenomenon with Dr J.M. Rallison. Thework is supported by S.R.C.Research Grant no. GR/A/5993.4.

R E F E R E N C E SALLEN, R. F. & BIGGIN,C. M. 1974 Longitudinal flow of a lenticular liquid filament down aninclined plane. Phys. Fluids 17,287-291.BATCHELOR,.K. 1956 On steady laminar flow with closed streamlines at large Reynoldsniimber. J . Fluid Alech. 1, 177-190.BATCHELOR,.K. 1967 Int?oduction to Fluid Dyrtamics. Cambridge University Press.COLLINS,W. M. 8; DENNIS,S. C . R . 1976 Viscous eddies near a 90" and a 45" corner in flowFRAENKEL,. . 1961 On corner eddies in plane inviscid shear flow. J . Fluid Mech. 11,400-406.FRAENKEL,.E. 1962 Laminar flow in symmetrical channels with slightly curved walls. I.On the Jeffery-Hamel solutions for flow between plane walls. Proc. Roy. Soc. A267,FRAENKEL,. E. 1973 On a theory of laminar flow in channels of a certain class. Proc. Camb.HARDY,G. H. 1902 On he zeroes of the integral func tionz-sinz = E$ - l)n-1z2n+1/(2n+)Messenger of Math. 31, 161-165.JEFFREY,D. J . & SHERWOOD,. D. 1980 Streamline patterns and eddies in low-Reynolds-

number flow. J . Fluid Mech. 96, 15.MOFFATT, . K. 1964a Viscous and resistive eddies near a sharp corner. J . Fluid Mech. 18,MOFFATT,. . 1964b Viscous eddies near a sharp corner. Arch. Mech. Stosowanej 16,365-372.STERNBERG,. ; KOITER,W. T. 1958 The wedge under a concentrated couple: a paradoxTRANTER, . J. 1948 The use of the Mellin transform in finding the stress distribution in an

through a curved tube of triangular cross-section. J . Fluid i'fech. 76, 417-432.

119-138.Phil. SOC. 3, 361-390.

1-18.

in the two-dimensional theory of elasticity. J . Appl. X e c h . 25,575-581.infinite wedge. Quart.J . llech. Appl. kf at h. 1, 125-130.

H.K. Moffatt and B.R. Duffy- Local similarity solutions and their limitations

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