Alison Hooper, B.R. Duffy and H.K. Moffatt- Flow of fluid of non-uniform viscosity in converging and diverging channels

8/3/2019 Alison Hooper, B.R. Duffy and H.K. Moffatt- Flow of fluid of non-uniform viscosity in converging and diverging chan

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J . F l u i d Mech. (1982), vol. 117,pp. 283-304Pr in ted in Great Britain

283 wFlow of fluid of non-uniform viscosity in converging anddiverging channels

By ALISON HOOPER,School of Mathematics, Bristol University

B. R. D U F F Y A N D H. K. MOFFATTDepartment of Applied Mathematics and Theoretical Physics,Silver Street, Cambridge(Received 17 February 1981 and in revised form 9 July 1981)

It is shown th at the well-known Jeffery-Hamel solution of the Navier-Stokes equationsadmits generalization to the case in which the viscosity ,U and density p are arbitraryfunctions of the angular co-ordinateo. When IRaj < 1 ,where R is the Reynolds numberand 2a the angle of divergence of the planes, lubrication theory is applicable; thislimit is first treated in the context of flow in a channel of slowly varying width. TheJeffery-Hamel problem proper is treated in $93-6, and the effect of varying theviscosity ratio h in a two-fluid situation is studied. I n $ 5, results already familiar inthe single-fluid context are recapitulated and reformulated in a manner th at admitsimmediate adaptation to the two-fluid situation, and in $ 6 t is shown that the single-fluid limit ( A +- 1) is in a certain sense degenerate. The necessarily discontinuousbehaviour of the velocity profile as the Reynolds number (based on volume flux)increases is elucidated. Finally, in $ 7, some comments are made about the realizabilityof these flows and about instabilities to which they may be subject.

1. IntroductionFlows with non-uniform viscosity, particularly two-fluid flows with a viscosity jump

a t the interface, arise in many processes of technological importance. Analysis of suchproblems is complicated (i)by the fact t ha t t he interface geometry in general changeswith time even if the boundary conditions are steady, and (ii)by the fact t hat, if theinterface intersects a solid boundary, the no-slip condition is (in general) inadequate inan immediate neighbourhood of the contact line, which may be observed to moverelative t o the solid surface.

I n attempting t o approach problems in this category in a general way, it seemsnatural first to study situations in which these particular difficulties are avoided.Figure I shows one such situation, viz the flow of a fluid of non-uniform viscosityalong a two-dimensional duct of slowly varying width. This elementary problem isanalysed (under the lubrication approximation) in $ 2 , in order to provide motivationfor the subsequent study ( $ $ 3-7) of the Jeffery-Hamel configuration (figure 2 ) , whichis the main content of this paper. I n $ 3, we show that the well-known exact solution ofthe Navier-Stokes equations (Jeffery 1915, Hamel 1916; see Batchelor 1967, 5 5.6)can be generalized t o the situation in which the viscosity ,U and/or the density p are

10 F L M 117

www.moffatt.tc


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284 A . Hooper, B . R. uffy and H . K . Moffattarbitrary functions of the angular co-ordinate 0- conditions that are clearly com-patible with a steady purely radial flow. A continuous variation of p and p can beinduced by heating one boundary, and in this case ( $3 ) an exact solution of the couplednonlinear equations of motion and heat transfer may be obtained.

In $ 4-6, we specialize to the two-fluid problem of figure 2 (c) , n which the density iscontinuous, but the viscosity is discontinuous across 0 = 0. In 5 4, the low-Reynolds-number analysis for this situation is presented, and the upper limit of the angle 2 a forwhich purely radial flow may be expected is obtained. In $ 5 , results that are well-known for the uniform viscosity case are obtained by a numerical procedure and areinterpreted in a new light. In 5 6, the same numerical procedure is used to analyse theeffects of the viscosity jump, and it emerges that an important effect of the viscositystratification is to resolve the degeneracy implied by the possible existence of steadyasymmetric solutions which (in the uniform-viscosity case) are mirror images of eachother. Resolution of this degeneracy also implies important qualitative changes in thebehaviour of the flow as the Reynolds number increases.

2. Flow of viscosity-stratified fluid in a slo\trly varying channelIn the situation of figure 1 (a), et a be a typical wall slope, and let R = Qp/ j l , where2Q is the total flux of fluid, and jl is the average viscosity across the channel. Then we

may expect that the lubrication approximation will be valid provided thata < , R a < 1. ( 2 . 1 )

The natural transverse co-ordinate is

and, in the lubrication approximation, the streamlines are given byA steady flow is possible ifp is constant on streamlines, i.e. if = const.P = Pu('I)*

The velocity u(7) long the streamlines then satisfies

where $ ( p g ) = - G ( x ) ,G ( x )= - p / a ~ , U ( X ) = al(x)+u,(x).With boundary conditions u(0)= u(1)= 0, (2 .4 ) integrates to give

where

(2 .3)

In the two-fluid case shown in figure 1 ( b ) , he interface y = h(x)becomes 7 = H ,where


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Flow of &id of non-uniform viscosity 28 5

(U ) ( b ) (C )FIGURE.Flow in a slowly varying channel with viscosity stratif icat ion: ( a ) ontinuous variationof viscosity; ( b ) wo-layer model; ( c ) wo layers entering a lowly diverging section; the interfacepositions itself in the diverging section along a plane 0 = const.i.e. the ratio of widths of the two layers is constant. The viscosity function is

In the lower fluid (viscosityp2), 2.6) givesa2GP2= uz(7)=- 77-W ) ,and the flux in this laver is.I Q2 = a j r u 2 ( q ) d q= a3G( 3 ? j - H ) H 2 .

6PZSimilarly,a3G1 - ( 3 ? j ' - ( l -H ) ) ( l -H ) ' ,- 6P lwhere (cf. 2.10)

l+(A- l - l ) ( l -H)21+ A-1- 1)(1 H) .7' =

(2.9)

(2.10)

(2 .11)

(2.12)

(2.13)

(2 .1 4 )Clearly, since Q1 and Q 2 are both constant, we must have

a3G= const. (2.15)Moreover, elimination of a3Gfrom (2 .12) and (2.13) yields

AqH'[AH' + (1-H) (3+H ) ]= (1-H)' [( 1-H ) 2+A H ( 4 H ) ] (2 .16)(whereq = Q1/Q2), a quartic equation for H , which has a unique real root H ( q ,A ) in thephysically relevant range 0 < H < 1.

We have neglected surface tension in the above analysis. The pressure jump due tosurface tension y is approximately y a2h/ax2,and the associated jump in pressuregradient along the interface is y a3h/ax3. f, for example, h(x)= h, coskx,with a = h,k,this jump has order of magnitude ya3/h& and this is negligible compared withlap/ax]N jiQ/ai (where a, is a typical value of a(x)) rovided that

(2.17)For very slow flow rates, surface tension inevitably becomes important, and will tendto keep the interface plane in the situation sketched in figure 1 b).

For the particular case of a cha,nnel that diverges at a constant angle (figure I c ) ,10-2


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286 A . Hooper, B. R . Duffy a n d H . K . M of f a t tthe condition (2.8) implies that the streamlines in the diverging section are purelyradial (0 = const. in polar co-ordinates with origin r = 0 a t the (virtual) ntersection ofthe diverging planes). Hence, in the diverging section, ,U = ,u(0),and ucc f ( 0 ) / r , rommass conservation. The above simple analysis is of course valid only if the conditions( 2 . 1 )are satisfied.

In 9 3 , we relax these conditions, and consider the general Jeffery-Hamel problemfor arbitrary a and R , and arbitrary ,U(@. We shall also allow the density p to be afunction of 6 ;since inertia is, in general, important, this will also have an influence onthe velocity field.

3. Jeffery-Hamel flow with ,U = p(O) ,p = p ( 0 )Consider now the flow configuration of figure 2 ( a ) ,with p = U(@),p = p (8 ) .Let

and let j i ( 0 )= p/p, p(0)= p / p . In polar co-ordinates ( r ,0, z ) , letu = 2 = - , o , o .P 1

Then the radial and transverse components of the Navier-Stokes equations are

where p ( r ,6 ) s the pressure field. From the second of these,

where c is constant, andpo s a t most a function of r . Substitution in ( 3 . 3 ~ )hows th atp o may be taken to be constant, and we obtain(jif ) f + 4jif + 2p f 2 = c . ( 3 . 5 )

This is the required generalization of the standard Jeffery-Hamel equationf f + 4 f + 2 f 2 = c .The boundary conditions are

and it follows from ( 3 . 3 )and ( 3 . 5 ) hatf = O on 8 = + a ,

The constant c therefore provides a measure of the wall pressure gradient.If f (0,a, , A ) is any solution of ( 3 . 5 ) , 3 . 7 ) ,where the parameterA summarizes the

dependence on the functions j i (8)and p (O ) , f then we may construct a correspondingReynolds number

R = Q P / p = f ( e , a, , A )de = ~ ( a ,, A ) , ( 3 . 9 )11where R > 0 corresponds to (net)diverging flow, and R < 0 corresponds to (net) con-verging flow. For fixeda, 3.9)provides a relationship (in effect) between wall pressuregradient (i.e. c ) and flow rate (i.e.R).

t Strictly, of course, this is a functional dependence.


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Flow of Jluid of non-uniformviscosity 287

FIGURE. The Jeffery-Hamel configuration: ( a ) ource flow (& > 0) or sink flow (& < 0) withangular stratification of ,U an d p ; ( b )angular stratification caused by differential heating of th eboundaries 8 = f a; c ) the two-layer flow studied in $1 and 6.

I n the well-studied case of a single fluid of homogeneous properties, the parameter Aof course disappears and we have a relationship

R = R(a, ), (3.10)which defines a surface (with possibly many folds and branches) in the space of thevariables ( R ,a, ). The geometry of this surface provides important information bothabout the multiplicityof steady-state solutions and about he manner in which a solutionchanges as a or R is slowly varied. A classification of velocity profiles in terms of thenumbers of maxima and minima that they exhibit was provided by Rosenhead (1940)and refined by Fraenkel (1962), who then sought to determine the boundaries in the( R , )-plane of regions in which solutions of a given class may exist. This informationis in fact all contained in the function (3.10); for example, the important barriercurves as, -, of Fraenkel (1962) may in principle be obtained by eliminating cbetween (3.10) and the equation

g(a,c) aR/& = 0. (3.11)I n 0 5 below, we shall recapitulate some results of the single-fluid analysis, and pre-sent these in a manner t hat admits simple extension to the inhomogeneous situation.

Before specializing to the two-fluid problem, let us briefly note one situation inwhich continuous angular stratification of viscosity and density may be expected, vizth at in which the walls 0 = & a are maintained a t different temperatures Tl and T,,as depicted in figure 2 (b) .A steady temperature distribution T(r,0) is then establishedin the fluid. Neglecting buoyancy forces, the velocity field is still given by (3.2))and,neglecting heat produced by viscous dissipation,t T satisfies

U. T = K V ~ T , (3.12)where K is the thermal diffusivity of the fluid. The solution is simply

T = (Tl-T,)e/2a+~(Tl+T,), (3.13)and sop and p , being functions of T, are functions of 0 via (3.13).When AT = Tl-T2is not too large, a linear variation o f p and p with 0 may be expected.

The neglected buoyancy force is of order aT AT, where aT is the coeficient oft The heating effect associated with viscous dissipation in the uniform-viscosity case wasstudied by MIllsaps & Pohlhausen (1953).


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288 A . Hooper , B. R.Duffy nd H . K . Moffattthermal expansion, and this is indeed negligible compared with the viscous force (oforder v Q I/ r 3 )provided r < (vlQI/a,gAT)f = Ib. (3 .14)Neglect of viscous heating CD is justified if the extra temperature Tv(r, )generated inthis way is small compared with AT. Considering only the case of converging flow( Q < O ) , T, is given (in order of magnitude) by

- - w @ n 4 - -aTv P Q2r ar pcp r4

(3 .15)Both inequalities (3 .14 ) and (3 .15) can be satisfied provided that

FIQl 4g2/@gAT< 1 , (3 .16)and, under this condition, ,U and p will be functions only of 0 throughout th e ranger d < r < rb .4. Discontinuous variation of ,U; behaviour near R = c = 0for a discontinuity on the plane 0 = 0 (figure 2 c ) :To simplify matters, suppose now that p is uniform, and th at ,U is uniform except

. (0 < 8 < a ) ,.=( #U2 ( - a < 8 < O ) .Then ji = $(pl+,uz), nd, writing(4 .2)

(4.3)

(4 .4 )[ f ] : = [pf ]? = o on 8 = 0, (4 .5)

f i (@ (0 < 8 G a ) ,f ( 8 )= ( f , n ( - < e < o),(3 .5) becomes

and we have the boundary conditionsP l ( f ;+ 4.fJ + 2ffl = c = P 2 ( f i+ 4f2) +

fib)= 2 ( - a ) = 07the latter ensuring continuity of velocity and tangential stress across the interface.It may be noted that t,he normal stress is

coo - p + 2 p e o o = - p + 2 j i j i (4 .6 )T))sz.and this is automatically continuous across 8 = 0 since the constant c in (3 .4 ) is thesame throughout the flow domain.

We consider first the behaviour near R = c = 0, where the terms 2f : , 2ffzof (4 .3)may (presumably) be neglected. The solution of (4.3)-(4.5) is then(4 .7)c - l f i ( 8 ) = A(cos2a- cos2 0 )+B(sin 2a -sin 28) ,c-f2(8) = hC(cos2 a - cos28)- hB(sin 2a+ sin 269,


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Flow of Jluid of non-uniform viscosity 289270'

251.5'

180'

90

I ,0.23 0.3 1.O x(a )

( h )FIGURE. (a) urves given by (4.11) (solid) and (4.12) (dashed). Above the dashed curve,reversed flow occurs in the less viscous fluid. On the solid curve 2cr,,(A), the inertia-free solution(4 .7) is singular; for a > a,,(h), eigenfunction solutions dominate over the Jeffery-Hamelsolution; the point S ( A = 1, 2a N_ 257.5') is singular in this respect. The change in the characterof the ( R ,c)-diagram along the segment E+F is discussed in 6. ( b ) Sketch showing the low-Reynolds-number streamline pattern tha t may be expected at a point such as X of figure 3 (a)(where reversed flow occurs) when a channel diverges at an angle 2a.


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290where

A. Hooper, B . R. u . y and H . K . Moffutt(1 A ) (1 -cos 201) 2- 1 - A ) cos 201, c =h+ ( 1 - A ) cos 2a 8h cos2a '= , B =8hcos 2a 8h sin 2a

( 4 4and h = p1/p2 = f i l / f i 2 ) . The Reynolds-number relation (3.9)may then be calculatedin the form

(4.9)/c = (8h)-l[ ( A- 1)2(a tan a)+242a- an 2a)l = g(a, ) , say,and we may infer th at the general function R(a, , A ) satisfies

(aR/aC),=R=O= g (a ,4.There is a singular behaviour when g(a,A) = 0, i.e. when

tan 2a- a ( A---t a n a - a 2h *(4.10)

(4.11)For any h > 0, here are two angles a,,(h) and ac2(h)atisfying (4.11); hese are shownby the solid curves of figure 3(U).Only the lower curve 2a,,(h) is in general physicallyrelevant,f since the similarity solution loses physical significance for a > a,,(h); inthis regime, the flow may be expected to be sensitive to the precise geometry and ent ryconditions near 0 (Barenblatt & Zel'dovich 1972; Moffatt & Duffy 1980).As the angle 201 increases from zero, the wall stress as given by the solution (4.7)decreases, andvanishes first (in the lessviscous fluid) when (af/aO),,, = 0; this condi-tion determines an angle a b given by - 2hcos 2ab =-l + h ' (4.12)The variation of 2ab with h is shown by the dashed curve of figure 3 ( a ) .For 2a > 2ab,reversed flow occurs in the less-viscous fluid. The streamlines may then be expected tohave the form sketched in figure 3 ( b ) .

5. Recapitulation and reformulation of results for the single-fluidsituation ( A = 1 )5.1. Anulytical results

There are three independent analytical results that provide useful checks on thenumerical procedure that follows.(i) Inertia-free Zinzit.When h = 1 , (4.9)gives

R / c = t (2a- tan Za), (5.1)from which we infer that

(aR/ac),=,=, = i (2a- tan 2a).Note th at the slope of the (R,c)-curve a t the origin changes sign as 2a increasesthrough in.

t The single-fluid limit A 3 1 is peculiar here; the symmetric low-R flow breaks down(Fraenkel 1962) a t 2aCz( l ) 257.5". For the two-fluid flow ( A $. l ) , he velocity is necessarilyasymmetric about 8 = 0, and it is the antisymmetric ingredient that produces the singularityon the lower curve when A == 1. (The limits h --f 1 and 2a + l do not commute.)


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Flow of Jluid of non -un iform viscosity 291(ii) Rosenhead's exact solutio n. A particular exact solution of (3.6) and (3.7) (Rosen-

head 1940, equation (5.11)) s given byf (8)= 2H - 1- 3H tanh2H48, (5.3)

where H is a parameter related to a byaHB = artanh (-I )* 9 (5.4)the corresponding values of c and R then beingc = 2(H2- I ) , R = 2[3(28- 1)]*-2a(H+ 1). ( 5 . 5 )

Equation (5.4) has two roots Hl(a)nd H 2 ( a ) or a < a, N 38.23". When a = a,, hesesolutions merge, and there is no real solution for a > a,.Equations (5.4), (5.5) define(parametrically) a curve in the (a , , R)-space which lies on the surface R = R(a, ) inthe region R > 0 ,a 6 a,.Figure 4shows the projection of this curve on the ( c ,R)-plane,the trajectories of the two solutions as aincreases from 0 toa,being as indicated. (Whena = a,,H 2: 1.198, R E 1-159,c E 0.868.) The projection of the same curve on the(R, a)-plane s the curve denoted g1 y Fraenkel(l962).

When H $ 1, (5.4) givesaH4 N artanh(8)B= 1.146 .. = /3, say, (5.6)

(5.7)

f (8)N H{2-3tanh2[H4(a-8)-P]}, (5.8)

and (5.5) givesER N 2p(2/6-/3) = 2.988..., H N (=&)& $ I .

Equation (5.3) may then be written

which has the character of a positive jet trapped between the two walls.(iii) Boundary-layer behaviour a5 c, -+ 00.When c is large, (3.6) may be solved by

boundary-layer techniques. For a boundary layer on 8 = a, he appropriate boundary-layer variable is

and f admits the asymptotic expansiont= K ( a - 8 ) , K = (+$) (5.9)

f (0) K Y o ( r ) f i (T ) +K-Yz(r)+ * . * (5.10)The appropriate boundary conditions are

f , (O ) = 0 , f A ( r ) + O as r+oo ( n= 0,1,2 ... . (5 .1 1 )Substitution of (5.10)in (3.6) gives, a t leading order,

f : = 2(1- f i ) , f O ( 0 ) = 0, f l (0O) = 0, (5.12)with the well-known solution (see e.g. Goldstein 1938, 5 56)

f , = 2-3tanh2(r+/3), ( 5 . 1 3 )t An expansion of this kind was proposed by Bulakh (1964),bu t he did not obtain the solution(5 .15 ) forfi obtained here.


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292 A . Hooper, B. R. Du ffy and H . K . Moffatt

2 -

-2 -1 0 1 2 3 4 5 CFIGURE. Locus of the Rosenhead solution (5.3)-(5.5)as a ncreases from 0 to a,,.

where /3 2~ 1.146 as above. At the next order,

and th is has the solution

From (3.9)we now have

(5.14)

(5.15)

or, in terms of c ,R - ~ ~ ( 2 ~ ) 4 + 6 ( 1 - ( $ ) 4 ) ( & ) f - 2 ~ ~ + ( $ ) 4 +O(C-)). (5.16)

Higher terms in this expansion may be obtained, but the expression (5.16) is quitesufficient to provide a good check on the accuracy of the numerical procedure (seefigures 6 and 7 below).

Two other types of asymptotic solution may be constructed, and these are alsorelevant in cataloguing possible behaviour for large c. These are the 'wall-jet boundary-layer ' solution (cf. (5.8))

f N K2[2-3tanh2(7-/3)]-[3($)4tanh(7-/3)sech2(7-/3)+ 1]+O(K-2), (5.17)

(B"

and the 'interior-jet' solution,fJ N K2[3sech2K(8-81)-1]-P+O(K-2). (5.18)The latte r makes a contribution to R given by

SR, ( f J + K 2 + 1)dO - 6K, (5.19)- &


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Flow of JEuidof non-uniform viscosity 293so that, for example, a velocity profile exhibiting a single central jet with conventionalboundary layers on both walls will yield an asymptotic relationship only slightlydifferent from (5.16), namely

R N - a ( 2 ~ ) * + 6 ( 2 - ( ; ) * ) ( & ) 4 - 2 a + ( $ ) + - +O(cd).(c), (5.20)5.2. Numerical results

Accurate determination of any of the characteristics of the surfaceR = R(a,c) requiresnumerical work, either using the exact elliptic-function solution of Rosenhead ( 1940))or by direct numerical integration of (3.6). The la tte r procedure is more easily extend-able to the inhomogeneous situation, and it is therefore the procedure th at we adopted.The equation was integrated by a shooting method, using a fourth-order Runge-Kutta-Merson procedure. For givenc,f ( - a) as guessed, and theequationintegratedacross to 0 = a. According to the value off (a ) , ( - a ) was then modified, and theintegration repeated. The procedure was iterated until the condition f (a )= 0 wassatisfied to a specified accuracy.

The linearized solution provided a guide in the initial guess-work for small c; andthen, for each new value of c, linear extrapolation gave a first estimate of the newf ( -a ) . For each solutionf (0,a,c), the corresponding R = R(a, ) was calculated from(3.9). Figure 5 shows the form of the resulting curves near the origin (R = c = 0) for201 = +n, 71and n.The slope of the tangent a t the origin satisfies (5.2)) nd the point Xcorresponding to Rosenheads solution for 2a = in is as indicated. Points on thesecurves (which we shall denote as branch A curves) correspond to solutions which, ina sense elaborated by Fraenkel (1962)) are analytically dependent on R and a ina neighbourhood of R = 0 and a = 0.

Figure 6 shows the branch A curve for 2a = in computed out t o c = 300. This showsthree important features, which are present also in similar diagrams for all other valuesof a < a, (for the behaviour when a > a,, see 3 5.3 below):

(i) The solid curve corresponds to solutions symmetric about 0 = 0, while thedashed curve that branches off from the point P, corresponds to solutions that areasymmetric about 0 = 0. At P,, ( f- ) 0, and as we pass through P, on the solidcurve, reversed flow appears on both walls. In Fraenkels notation, for different valuesof a, he value of R, = Rp(a) ay, at P, provides the curve 9Y2(for 2a < n) nd B-l (for2a > n). he bifurcation a t P,from symmetric to asymmetric flows is continuous (butnot analytic) in the parameter R.

(ii) The solid curve shows a maximum a t the point R, (R, N 5.5 in figure 6). ForR > R,, a steady solution analytically related to the branch A solutions is not possible.Since aR/ac = 0 a t R,, it is clear tha t, as stated earlier, the corresponding barrierin the (R,a) -plane s given by eliminating cbetween (3.10) and (3.11).If Ris increasedthrough R,(a), discontinuous behaviour must occur (see below).

(iii) For R < 0 and c large, the asymptotic expansion (5.16) is relevant; the 1-term,2-term) 3-term and 4-term approximations are as indicated in figure 6; the 4-termapproximation is indistinguishable from the exact numerical curve for c 2 0.Similarly, the circled points on the upper solid portion of branch A (figure6) have beencalculated from (5.20) to order c-i, and the accuracy here is good for c 2 00; theprofiles here do indeed have the character of a forward central je t with reversed flownear the walls. For c, N 200 ( K E 3.16) and a = in, he second term of (5.20) evidently


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294 A . Hooper, B. R. Du ffy and H . K . Moffatt

FIGURE. Branch A curves near R = c = '0 for 2a = n, in,+n.Rosenhead's solutioncorresponds to the point X on the curve 2a = in ( c = - .131, R = 0.217).

h5

C

-5

- I (

-11

FIGURE. Branch A of the curve R = R(tn, ) , together with successive approximations to thelower portion based on the asymptotic expansion ( 5 . 1 6 ) . The four-term approximation isindistinguishable from the exact result for c 2 0 ( - R 2 ) . The circled points are valuespredicted by the asymptotic results of $5.1, correct to order c- i .


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Flow of Jluid of non-uniform viscosity 295Brancli U

FIGURE(a, , c ) . For legend see p. 296.


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296 A . Hooper, B. R. uf fy and H . K . Moffattt

1 t( d )

FIGURE. ( a )The function R = R(gn, ); he solid portions correspond to flows symmetric about0 = 0, an d th e dashed portions correspond to asyrnmetric flows. The circled points a re valuespredicted b y the asymptotic analyses. ( b )Velocity profilcs corresponding to the points A, , A , ,A , , B, and B,. ( c ) Velocity profiles corresponding to points A, , B,, B,, B,. Only one of the twoasymmetric profiles B, is included here. ( d )Asym metric velocity profile A (c = 23 , R = - .5 )an d points calculated from one-term (m ) and two-term (0)ounda ry-layer analysis.still dominates over the first, and soR > 0. The points on the dashed portion of branchA have been evaluated through combining a normal boundary layer on one wall witha wall-jet boundary layer (equation (5.17))on the other.

The question of what may happen when R > R, is to some extent answered byfigure 7 (a), hich shows the (R,)-diagram for 2a = 7 ~ . his shows a second computedbranch, denoted branch B , which extends to higher positive values of R . The velocityprofiles corresponding to the points A,, . ,A , on branch A , and B,, . ,B, on branch Bare shown in figures 7 ( b , c ) . The circled points in figure 7( a) again correspond toasymptotic evaluation correct to order c-i. The points on the lower portion of branch Bwere obtained using a wall-jet boundary layer on each wall (see e.g. the profile labelledB, in figure 7 ( c ) ) . The profile A, (figure 7d) (a wall-jet on one boundary and a con-ventional boundary layer on the other ) is poorly approximated by the leading-orderboundary-layer analysis, but accurately represented when the O (1) erms of (5.10) and(5.17) are included. Jf R increases through R,, then clearly a jump to branch B ispossible, such a jump involving a 30-fold increase in c! This is a transition, in Fraenkels(1962) notation, from a profile of class 11, to a profile of class 111,; the fact that it isnecessarily discontinuous was recognized by Fraenkel, but may be more clearlyapparent in the visibly separated branches of the (R,)-diagram than in the intricaciesof his elliptic-function treatment.

The qualitative picture tha t emerges from these results and from a large number offurther profile computations, is summarized by the sketch of figure 8, which is typicalof any angular separation in the range i n < 2a < 2ac.(If 2a < in , he only qualitativechange is that branch A passes through the origin with negative slope-cf. figure 5 . )The velocity profiles on the various sections of branches A and B are as indicated,together with the code used by Fraenkel(lQ62) o distinguish the various classes. The


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Flow o JEuid of non-uniform viscosity 297

FIGURE. Qualitative character of velocity profiles on various sections of branches A and B, forin < 2a < n. The Roman numerals correspond to Fraenkel's (1 9 6 2 ) classificat,ion.The classeschange in a continuous manner at 0 and at PO,Pb, ., and in a discontinuous manner a t jumpssuch as R , .+S , .dashed portions are drawn doubled, to indicate that there are two profiles (mirrorimages of each other) for each point.

Although we have numerically located only branches A and B, it is clear that theseare merely the first two of an infinite sequence of branches A , B, C, ... on which theprofiles become progressively more complex. It is widely believed that the profiles ofbranches B ,C, . are unstable, and attention is generally focused on the branch Aprofiles. The question of stability of branch A profiles may be related to the behaviournear the point R,, nd the fact t hat , if R s increased through R,, jump to a totallydifferent profile is inevitable.

5.3. Behaviour of branches A and B for 0 < 2 a < 277Figure 9(a) shows a sketch of the variation of branch A as 2a increases from 0 to2a , EJ257.5". The locus of the maximum R, s indicated by the dashed curve.

For 2 a > 2ac, as already observed, the linearized solution for small c is physicallyirrelevant, and the same may be true of the solution of (3.6) also. There is neverthelesssome interest in continuing the sequence of curves of figure 9 ( a )beyond 2ac, in orderto provide a further point of contact with the results of Fraenkel( l962).This variationis shown in figure 9 ( b ) ,which includes branch B also. When 2a N 266") a fold developsin branch A , and, a t a value 2apbetween 266.7" and 267") t merges with branch B, hetopology of the branches changing a t this critical angle. Defining R, always as theReynolds number a t the maximum on the lower branch, R, ncreases discontinuouslya t 2ap , hen decreases to zero a t 2a = 2 4 Y 282") and then continues to decrease as 201increases to 27r.


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298 A . Hooper, B . R. Duffy and H . K . Moffatt

R I // ranch B

( b )

( C )

R,FIGURE. Sketch of the variation of branch A and its maximum R,( b ) change in topology of branches A and B as 2a passes throughvariation of R, s a function of a.

with a : (a) 0 < 261 < 2 a , ;2a , 2: 266.8"; (c ) inferred

The inferred qualitative variation of R, over the full range 0-2n is shown infigure 9 (c), in which the portions denoted g3,%9-2 correspond to Fraenkel's (1962,figure 5) notation.

6. Numerical results for the two-fluid situationWe now return to the situation of $4, n which there is a viscosity jump across6 = 0. We may suppose, without loss of generality, that h = p1/p2< 1. In the discus-

sion that follows) we focus attention on the change th at occurs in branch A of the(R ,c)-diagram ash decreases from unity. Figure 10( a ) hows what happens to branch Afor the case 2a = +7r7h = 0.9;note that the abscissa here is the modified constant

C .(1+h)24h2c1 =-The most striking effect is th at we now have two intersecting branches) A' and A" say,where A' passes through the origin, and A" does not. Figure 1 0 ( b )shows the profilescorresponding to the points A;,A ; , A I , A ; (c1 = 100).A ; and A; are nearly (but notexactly) symmetric about 6 = 0. The 'degeneracy ) associated with the asymmetricflows when h = 1 is clearly resolved when h + 1.

The points P ' , P" in figure 10 ( a )correspond to solutions for whichf ( a )= 0. As Rincreases through R(P' ), the flow changes from purely divergent flow to a flow of


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Flow of Juid of non-uniformviscosity 2996

R4

2(

-2

-4

-6

-8

-10

-12

.100 200 c1 300

( b )FIGURE0. (a )Modification of branch A for 2a = when h = 0.9: the branch splits into twobranches, A' and A"; he points P' and P" correspond to profiles with zero gradient on R = U.( b )Velocity profiles for h = 0.9 corresponding to the points A; ,A; ,A:, A; on branches A' and A".type A ; (divergent bu t with inflow near the wall in th e less viscous fluid). Similarly,asR increases through R(P"),he flow changes from type A ; (inflow n the more viscousfluid) to one of type A ; with inflow on both walls. It seems likely that a t least one of theflows A ; , A ; , A : (probably A ; ) s structurally unstable.

On branch A', there is again a maximum value of R,Ri say, and now if R is increasedslowly from 0 through Ri, he flow will (presumably) jump from branch A' to branch


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300 A . Hooper, B. R.Duffy and H . K. MoffattR4

2

0

-2

-4

-6

-8

-10FIGURE1. Change in branch A as h decreases.The abscissa is c1 = a(1+h-1)2 c.

A . f R s increased further, a second jump will occur (presumably to branch B)whenR = R:, the maximum on branch A.Figure 11shows the effect on branch A of a further decrease in A. As h decreases,RA decreases (and the value of c1where the maximum occurs increases quite rapidly).

The value of R at which reversed flow appears also decreases as indicated in thefigure.

When R s large and negative, we again expect the solutions (on branch A) to havea boundary-layer character near each wall, with boundary-layer thicknesses pro-portional to p! and ,ut in the two fluids. For the region near 8 = a, n analysis similarto that in Q 5 .1 shows that, for large c,

where K = (hc), andf,, andf, are given by (5 .1 3 )and (5.15). There is a similar structurenear 8 = -a.At leading order, the flow outside the boundary layers is effectively theinviscid flow associated with a line sink ( Q < 0 ) .Across 8 = 0, the O ( K 2 ) olutionsin the two fluids match with zero shear stress: the effectively inviscid flow is indifferentto a jump in viscosity a t 8 = 0. Continuing to next order, we have, well away from the

f ( 8 ) K2fo(r)+Zlfl(r), r = ZTJK ( a - 4 , (6.2)

walls,

indicating that there must be a weak (i.e. O(1 ) ) shear layer a t the interface. A straight-forward analysis shows that, near 8 = 0 ,

(6.4)- K 2 - j i 2 ( 1+A2e - 2 82 ) , 7, = -,ii;dK8 ( 8 < 0)).K2-,1Z~(l+A~e-~~1),l = ,Z;iKS (8 2 0))f ( 8 ) [with A, = A - I - A-4, A, = h- A:. Figures 1 2 ( a ,b ) show velocity profiles for c1 = 105,1 0 6 for a range of values of h (for 2a = 10). As expected, the velocity gradient is


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Flow of jluid of non-unifo rm viscosity 301

Ih\ h = 0.1

\ \ .9 //

( b )-700 J

FIGURE2. Velocity profiles for 2a = loo,and various values of h. ( a ) 1 = 106;( b )The jump in velocity gradient at 0 = 0 is scarcely detectable for h > 0.5. c1 = 106.

almost continuous across 8 = 0, consistent with the above argument. (The strongdecrease in flux as h decreases is associated with the corresponding reduction in c,given by ( 6 . 1 ) :e.g., when h = 0.1, c = 3.3 x 10-2c,.)

Finally, we have investigated the behaviour of the branches A and A as h decreases(for fixed a ) o the value for which a = ac,(h) E -f F in figure 3a) .With 2 a = 150,the critical value of h given by ( 4 . 1 1 ) s 0.228; figure 13 shows the computed behaviouras h decreases from 0.27 to 0.228 . For h = 0 .21 , the branches A and A still cross (atthe point. V in the figure), but a t h = 0.265 the curves have pinched off ,forming twonew branches A and A.At this value of A , there is no solution (on these branches)


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302 A . Hooper, B. R. uf fy and H . K . MoffattR

1 2 c ,

-0.2

-0.4

FIGURE3. Change in branches A' and A" for 2a = 150' as h decreases from 0.27 to 0.228 (cf.figure 3 ) . At a value of h between 0.27 and 0.265, the branches split (near V ) orming two ne_wbranches A' and A" which move apart with further reduction of A. For h = 0.228, branch A'touches the axis R = 0 at c = 0.

with - .27 c R < - .08. As h is decreased further, 2' and 6"move further apart,until (when h = 0.228) 6" ies in R < - 0.66 and 2' lies in R > 0, with a minimumat 0 (consistent with (4.11)at the critical value of A ) .

7 . DiscussionThe foregoing analysis enables us (in principle) to determine the position of the

interface in such situations as those sketched in figure 14 (a , ) , even when 1Raj is notsmall. We have seen in 3 2 that,when 1Ral < 1, the interface positions itself so that theratio of fluid widths is constant. When JRalB 1, the situation is very different. Forexample, in the situation of figure 14(a),when lRal B 1 , the two fluxes Q1 nd Q2 areequal (apart from boundary-layer corrections) and so the position of the interface inthe straight section is determined by (2.16)with q = 1 . As h + 0, H -+1, i.e. the inter-face moves towards the upper boundary.

Similarly if Q1 nd Q2 are specified, then the position of the interface in the straightsection is determined by (2.16))and its position 8 = p in the adjacent converging ordiverging section (as, for example, in figure 14b) is, in principle, determinate fromnumerical solution of the Jeffery-Hamel problem.There is of course an important question concerning the stability of the varioussteady flows described in this paper. Instabilities may be of three types. First, there isthe conventional instability of flows exhibiting points of inflexion, which may beexpected at quite low Reynolds numbers (R 2 0). The methods of Eagles (1966)


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Flow of Jluid of non-uniformviscosity 303

FIGURE4. Determination of the asymptotic position of the interface : ( a )a case of convergingflow entering a uniform section; (b)a case of flow entering a diverging section. I n both cases thedownstream position of the interface is determined in principle by the values of q = Q1/Q2 andh = pl/p2. n bo th cases, the figure is schematic, and is not intended to imply th at a solutionhas been found which describes the flow through the transition region, which may be extensivewhen the Reynolds number is not small.

could perhaps be extended to cover the two-fluid situation. Secondly, there is theinstability associated specifically with viscosity stratification, discovered by Yih(1967) ; this is a long-wavelength instability, involving inertial effects, but whichnevertheless persists at arbitrarily smallR. he physical mechanism of this instabilityis as yet obscure, and it is by no means clear how it may be affected by weak con-vergence or divergence of the fluid boundaries.

Finally, there is the question of structural stability (in the sense of bifurcationtheory) already alluded to in 0 6 . If we imagine the steady problem (3.5) imbedded inth e wider class of unsteady problems (with the same boundary conditions)

af+ ( p f ) +4pf+ 2p f = c,atthen the steady solutions obtained above may be unstable to perturbations Sf (13, ) .On general grounds (Benjamin 1976) one would expect approximately half of themultiplicity of flows a t a given Reynolds number to be structurally unstable; anassociated hysteresis behaviour in increasing and decreasing the Reynolds numberthrough a range of positive values may be anticipated.These various stability aspects perhaps deserve further study.

This work has been partially supported by the Science Research Council underResearch Grant no. GR/A/5993.4. We gladly acknowledge the penetrating commentsof L. E.Fraenkel on an earlier version of this paper.

R E F E R E N C E SBARENBLATT,. I. & ZELDOVICH,YA.B. 1972 Self-similar solutions as intermediate asymp-

totics. Ann. Rev. Fluid Mech. 4, 285.BATCHELOR,. K. 1967 An Introduction to Fluid Dynamics. Cambridge University Press.BENJAMIN,. B. 1976 Applications of Leray-Schauder degree theory to problems of liydro-BULAKH,. M . 1964 On higher approximations in the boundary-layer theory. J . A&. Math.

dynamic stability. Ma,th. Proc. Camb. Phil. Soc. 79, 383 .Mech. 28, 675.


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304 A . Hooper, B. R.Du& and H . K . MoJfuttEAGLES,. M. 1966 The stability of a family of Jeffery-Hamel solutions for divergent channel

flow. J . Fluid Mech. 24, 191.FRAENKEL,. E. 1962 Laminar flow in symmetrical channels with slightly curved walls. I. On

th e Jeffery-Hamel solutions for flow between plane walls. Proc. R . Soc. Lond. A 267, 119.GOLDSTEIN, . 1938 Modern Developments in Fluid Dynamics, vol. 1 . Clarendon.HAMEL, . 19 16 Spiralformige Bewepngen ziiher Flussigkeiten. Jahresbericht der DeutschenJEFFERY,. B. 1915 Steady motion of a viscous fluid. Phil. Mag. Ser. 6,29, 455.MILLSAPS,K. & POHLHAUSEN,. 1953 Thermal distributions in Jeffery-Hamel flows betweenMOFFATT,H. K. & DUFFY, . R . 1980 Local similarity solutions and their limitations. J . FluidROSENHEAD,. 1940 The steady two-dimensional radial flow of viscous fluid between twoYIH,C.-S. 1967 Instability due to viscosity stratification, J . Fluid Mech. 27, 337.

Math. Vereinigung.25, 34.

non-parallel plane walls. J . Aero. Res. 20, 187.Mech. 96, 299.inclined walls. Proc. R. Soc. Lond. 175, 436.

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Alison Hooper, B.R. Duffy and H.K. Moffatt- Flow of fluid of non-uniform viscosity in converging and diverging channels