Helical flow of an annular mass of visco-elastic fluid

Chemical Engineering Science, 1900, Vol. II. pp. 252 to 250.

Pergamon Press Ltd., London.

Printed in Great Britain

Helical flow of an annular mass of visco-elastic fluid A. G. })epnrtment of Chcrnical

li~nginccring,

FREDRICKSON

Univcrsity of

~Iinnesota,

l\linneapolis ]4,

~IinJlesola

(llcccived 1 Alf1!J 1050)

Abstract-An analytical solution of the equations of change is presented for the steady flo\v of nn arbitrary visco..elastic fluid through a concentric cylinder annulus. It is assurncd that Illotion is irnparted to the fluid by an impressed pressure gradient and/or a gravitational tion and by the steady rotation of one or the other, or both, of the annulus (~ylindcrs. IJndcr such conditions, the paths traced out hy individual fluid particlcs \vill he cireular hclices. The solution is presented in tcrrns of definite integrals whieh contain nn arbitrary funetion F (Y) ; this function rnay be called an H apparent viscosity." It is sho,vn ho\v the funetion Ii' (Y) may be deterrnincd frorn standard experirncnts \vith capillary, or rotational, viscorneters. Suggestions nrc advanced for pcrfornling the required intcgration~, a.nd it is sho,vn ho\v the equations Illay be applied to other cases of interest. Resume-L'auteur presente une solution analytique des equa.tions de In. vurintion, pour l'ecoulcment pennanent, el'un fluide arbitraire viscopclastiqlle, it tra.vers un aoncall entre deux cylindres concentriques. II est suppose quc Ie rnouvcrllcnt cst conununiquc au fluide pur I'npplication d'un gradient de pression ou par l'aceeleration de la pcsanteur, ou par les dcux actions sinlultances, et par In rotation pcrnlanente de run nu l'autre, ou des deux cylindrcs (~onstituant l'anneau. Dans ces conditiolls, les trajcctoires de chaquc partieule du fluide seront des helices circulaires. La solution est presentee en fUllction d'integrales dcfinies qui rcnfcrnlcnt une fonetion nrhitraire F (Y). Cette fonction peut ctre appelcc H viseositc llppnrcntc." l,'autcur rnontre conlffient In. fonction I? (Y) pcut ctrc dctcrrnince par des experiences standard avec viscosirnctre (~apillaire ou rotatif. Iles suggestions sont presentees pour cffectucr Ies integrations neccssaires ct il cst Illontre comnlent les equations pcuvcnt ctre appliquecs a d'autrcs cas inlcressants. ,vird cine aWl.lytischc Uisung filr die Verforrnungsglci(~hungen ciner stntionaren Strornung einer bclichigen viskoclastischcn Fliissigkcit durch cinCH kOllzcntrisellcn zyJindrischen Ringspnlt mitgeteilt. Ilie Bewcgung ka.nn der Fliissigkeit cntwerdcr durch cinco angelegrcn Druckgradienten undjodcr durch cin Beschleunigungsfcld und durch die stationii.rc Rotation des inneren, des iiusscrcn oder hcidcr Zylinder des Ringspalts aufgedriickt ,verden. Unter diesen Bedingungen sind die Stromlinien cinzelncr FHissigkcitsteilehen I{reisspiralcn. Ilie I.iisung \vird in Porln bestilllnlter Integ-rale nlitgcteilt, die cine heliebigc Funktion F (1 enthlllten; diesc Funktion sci" scheinbnrc Viskositat "gcnannt. Es \virdge7.cigt, \vic dic Funktion /1' (Y) nus Stundardversuehen rnit I{npillnr- oder Ilotationsviskosimctern hcsthnlnt \VertICil kann.~ Zur Ausfiihrung der verlnnbrten Integrationen \verden Vorsehliige ~enlacht urul cs \virfl gezcigt, wic die Gleiehungen auf andere intcrcssicrcntlc Fiille ungc\\~auclt \verden ki>nnen. ZU8ammenfassun~-Es

7

)

IN

INTRODUCTION

years, a number of articles concerning the flow of non-Newtonian fluids in annular spaces have appeared. VOLAltOVICI-I and GUTKIN [1], SIICIIIPANOV [2] and VAN OLPHEN [3] published approximate solutions for axial flow of Binghaln . plastic materials in annuli, and MORI and OTOTAKE [4] published a more general solution RECENT

which unfortunately is in error. The exact solution for the flow of a Bingham plastic in an annulus ,vas published by I~AIRD [5] and by FItEDHICKSON and BInD [6]. F'UEDRICI{SON and BIun [6] also derived a solution for a so..called " po,ver Blodel " fluid; their results for both the Bingham plastic and the power rnodel fluid were given in tcrlTIS of diJnension]css variables. The

252

IIelical flo'v of an annular Illass of viseo.. elastic fluid

tabulated solutions of have been extended by

and BIHD et ai. [7] and by

FREDRICI\:SON l\IELUOSE

n=

.

+

SAVINS [8]. FREDRICKSON

[9] has derived a solution for the combined axial and tangential flo,v of an arbitrary, inelastic, non-Ne,\\rtonian fluid in an

CONSTITu'rIVE

EQUATIONS

FOR

VISCO-ELASTIC

TIlE TUEOltV OF RIVLIK AND EUICI{SEN

RIVLIN and ERICKSEN [II] have published a phenomenological theory of stress..defornlation relations for isotropic materials. On the basis of that theory, RIVLIN [12] has sho\\rn that if one 'assumes that in a visco-elastic fluid vlhich is isotropic in its state of rest, the stress components, 'IT", at a point in the fluid are expressible as '1 • polynomials in the gradients of velOCIty, acceI eration, second acceleration, ... , ('n - l)st acceleration at that point and then the stress matrix. n (= 7T may be expressed as polynomial in 1l kinematic lllatrices D 1 , D 2 , ••• , D n · The elements of the matrix D 1 (= are just twice the elements of the usual velocity-strain matrix : D .. (l) = ~vi oVi (1)

II iill)

IIDij(l)II)

()X;

1)

+

~a~i

••• ,

11-

IIDij(r) II) with l' = 2, 3, are defined by the recursion formula

n.,(r)

=

~ D ..(I'-l)

and the matrices Dr (=

t)

~t

+

+ Ev 2.. D .. (r-l) + l=l 'ox, lJ E(D .(r-I) ()vm + Dm.(r-I) ()Vm) 3

1}

m=l

mt

~~"

UWJ

J

DX'I

CJ: 5

+

annulus. His equations are actually a solution to a special case of the system of differential equations derived by IlIVLI~ [10] to describe the combined axial and tangential {lO\V of an annular mass of visco.. elastic fluid. It is the purpose of this paper to show how the equations of FUEDUrCI{SON may be generalized to give the solution of the problem posed by RrvLIN; viz., for the helical flow of an annular mass of viscoelastic fluid.

FLUIDS;

-

(2)

If the geometry of the flow situation is such that Dr = 0 for r > 2, then RIVLIN [12] has

shown that the relation between the stress matrix and the kinematic matrices is given by

1)1 + 0: 1 D 1 + (l2 D 2 -t- Cl 4 D 2 2 + (D 1 D 2 -t- D 2 D}) + D 2 D 12) + lX?

+ lla D 12 +

+ (to (D 12 D 2 +

(D 1 D 22 + D 22 D 1 ) -tCl S (D}2 D 2 2 + D 22 D 12)

+

(8)

for incompressible fluids. The coeflieients (J.,k nre not constants, but n.rc expressible as polynolllials in the ten* scalar invariants tr D l , tr D l 2, tr DI 3, tr D 2, tr D 22, tr D 23, tr D 1 D 2, tr D 12 D 2, tr D 1 D 22, tr D 12 D 22. In the above expressions, it is to be understood that the usual rule [13] of matrix Inultipication is applied and that D rk == D rk - l Dr. ~'he trace (" tr ") of a nlatrix is sinlply the sunl of its diagonal elements. For inelastic fluids, the stress is a function only of the velocity-strain (Le., up to an arbitrary hydrostatic pressure, - p), so that the matrices Dr with l' > 1. do not affect the stress. For such fluids, RIVLIN'S equation becomes

Equation (4) is the equation derived by REINElt [14,15] and by RIVLIN [16] and applied ~y RIVLIN [16, 17] to explain the so.. called "Wcissenberg effect" [18]t. The coefllcicnt (Xl Inay thus be called the "viscosity" and the coefficient lcxa has been called [19] the " normal stress coeOlcient.' The derivations of I?UEDUICKSON cited above utilized the constitutive equation (4) with the further hypothesis that the coeflicient IXa could be set equal to zero. OLDROYD [20] has fornlulated an alternate theory of visco.. clasticity. The constitutive equation of OLDROYD contains terms not only in the gradients of velocity, acceleration, ... , (1~ - 1)8t acceleration, but also in the convective derivatives of the elements of n. In this paper, however, it ,viII be assumed that the rheological behaviour of the fluids in question is given by equation (8). *For incornpressiblc fluids, tr D 1 = 0, 80 \VC need eonsicler only nine sculnr invariants. tOLDHOYU (2()) has suggested that the Wcisscilberg cft'cct nU1Y he due to l1uid elnstieity ; this view scenlS to be borne out by the expcrilncntn) \\'ork of Il.oJu~nTs [21].

258

A. G. EQUATIONS

}"ou.

HELICAL

FLOW

IN

FnEDIUCKSON

ANNULI

Consider the steady, combined axial and tangential flo'v of a mass of visco-elastic fluid in a concentric cylinder annulus. Axial motion is imparted to the fluid by an impressed pressure gradient and/or a gravitational acceleration; tangential motion is imparted by causing one or the other or of both the annulus cylinders to rotate with constant angular velocity. By hypothesis, we are dealing with a region far enough removed from the entrance (or exit) of the annulus so that the angular velocity, w, and the axial velocity, u, at any point of the region depend only on the radial distance of that point from the axis of the annulus.

o

DIRECTION OF GRAVITY

z

Let the z..axis of a rectangular Cartesian co-ordinate system lie along the axis of the annulus, and point in the direction of axial flow (Fig. 1). Then, if one chooses- the direction of the x-axis so that y = 0 at the point of fluid considered, the kinematic matrices Dr are given by [10]: D1 =

2 (r 2 w'2

o rw'u' TW'

u'

D2 =

0 0 0 0

Dr = 0,

+ U'2)

0

o r

>2

tr D 2

=

tr D 12

Y, tr 0

2 2

=

Y, tr D l 3 = 0,

= y2,

tr D 23

= 0, tr Dl~ D 2 = ! y2, tr D 1 D 22 = 0, .and tr D 12 D 22 =

tr D 1 D 2

=

!

y3,

Y3.

(7)

Hence, it appears that all coefficients (J.,k in equation (8) are functions only of Y. But Y is a function only of r so equations (7) show that the a.~ are functions only of r. Substitution of equations (5) into equation (8) yields the expressions for the stresses:

1)

Cl. 3 )

=

(13

rw' u'.

(8)

However, by integration of the equations of motion, RIVLIN [10] sl10wed that C 2 1T12 = B/r ; ?T13 = ! Pr + (9)

(5)

'1"

where Band C are constants of integration and P (another constant) is defined by

p

[10].

tr D 1 = 0,

=-

cx'8

7723

Then the invariants of D 1 and D 2 loay be written

1. llcHenl flo'" in un nunulus.

+ l (2C(2 + Y + (at + as) Y2 -t+ y3, 1T22 = - p + IXa r 2 w'2, 1733 = - P + a3 U'2, 1712 = rw' [(11 + OCs Y + r:t.7 y2], 1Tl3 = u' [ell + el5 Y + (17 Y2] 1Tll

0 0 0 0 0 0

where primes denote differentiation \vith respect to r. Let (6) Y = 2 (r 2 w'2 + U'2) as

FIG.

= _

07T33

oZ

=

(1) _

oZ

Pg

..

.

(10)

In equation (10), p is the density of the fluid and gz is the z..component of the gravitional acceleration g. (It is assumed that gm and gy may be neglected). The quantity P nlay be called the frictional pressure gradient. Define the dimensionless radial co-ordinate ~ by

g = r/R

(11)

in which Ii is the radius of the outer annulus cylinder. Then combination of equations (8), (9) and (11) gives

254

I-Iclical flow of an annular luass of vist·o"phtsti(· 1I11id

R

1712

=

fJ/e 2 = e~;

1713

=

PR2

2

(e

2

[
+


IT

+


P]

(12)

>.2)

e

-

= -tie

elt.t [ (Xl

+ (X5

conditions (19) and (20) are the statemCJlt~ that the fluid does not" slip" at the solid boundaries of the flo,v systcln. ~~rom equations (17), (18) and (20), there result 1

Y

+ rJ.

7

Y2]

(13)

Qo -

Qi =

fJ f '3 ]I' (IT) d{

(17a)

wllere the constants ~ and A may be derived from B, C and R. Equations (12) and (18) arc identical to equations (4.9) of RIVLIN'S paper

'[10].

Since the

rJ.k

are functions of Y, we can set

F (Y)

= (Xl + (X5 Y + IX? Y2

(14)

so that (12) and (13) become

dw dg

=

fJ

du _ PR2 d, -

(15)

~3 F (Y)

2

(e

Z

-

A) 2

g

1

F(Y)

.

(I6)

The separation of variables represented by equations (15) and (16) is permissible, since Y, ,and therefore F (Y), is a function of r (or f). The velocity distribution is then obtained by integration: (17)

f ('2 - >.2) 1

U

= - P R2 2

~.

' I t T (Y)

E

u=o

(19)

where Do is the angular velocity of the outer cylinder of the annulus. The boundary conditions at the inner cylinder of the annulus are co

=

Q,

u"=o

at r

=

1<11 (~

=

K)

Equations (17a) and (18a) are the determining equations for f3 and A. A method for perfornlillg the integrations called for by these equations ,viII be presented below. The volulnetric flo,v rate, Q, is 271

Q=

JJ

o

1

R

1t

(r) ,.,/1' tIO =

KR

217R2

J

u (e)

ede

(21)

K

or, because of (18):

(22) .

Interchange of the order of integration and rearrangenlcnt gives finally

(18)

In equations (17) and (IS), we have used the boundary conditions that at r = R (, = 1)

( 18U)

(20)

where KR is the radius and D j is the angular velocity of the inner annulus cylinder. The

Equation (28) is the desired expression for the flow rate througll the annulus. '!'hc values of the torques required to Inaintnin the cylinders of the annulus in steady rotation may no,v be computed. The torque on the inner cylinder (per unit length of cylinder) is given by the expression:

*Equutions (17n), (IBn) nnti (:!:l) shu,,' thl\t fot" a given fluid, the relations bct\\'ecn 4(~/ 17 11 3 , P 11/2, K and Q o - Dj nrc independent of the scale of the nppnratuH· upon which those relations nre detcrnlincd. 'J'his ,,"ould he un itnportant (l,ot\sideration in uny experinlcntnl progrutUnlc designed to test t.he validity of the theory developed herein.

255

A. G. Of,

FREDRICI{SON

(18a) are performed by SOlne numerical Incans. If the assumed values of fJ and" satisfy equations (17a) and (18a), those values are correct and equations (17), (18) and (23) Inay be integrated (numerically) ,vithout difliculty. If the assumed values of f3 and " do not satisfy equations (17a) and (18a), new values must be assumed and the entire process repeated. It is obvious that the amount of computational work necessary is rather formidable, and the Inethod is best handled by a digital computer. Finally, let it be noted that the values of f3 and " so conlputed will hold for a particular set of values of - Pll/2, K and Q o - !ti. If any of these quantities are changed, then fJ and " will change also.

because of (12), Gi =

211

R2 {3,

(24)

Since in a steady state or rotation, the saIne couples must act on every lamina of the fluid, it follows that equation (24) also gives the torque acting on the outer cylinder (per unit length). DETEU,.lINATION

OF

OF EQUATIONS

CONSTANTS;

INTEGRATION

(17a), (I8n) and (23)

In general, the constants f3 and " must be determined by trial-aod-error numerical inte.. gration of equations (17a) and (18a). In order to perform these integrations, it is necessary to have a graph, table, or analytical expression describing the variation of F (Y) with Y. It will be shown below how such a graph, table, etc. may be prepared by analysis of viscometric data. For "pseudoplastic" fluids, the graph of F (Y) would have the general form of Fig. 4 in the article of JOBLING and RODERTS [22] [for "TJ (apparent) "]. * At any rate, suppose viscometric experinlents have established the dependence of ]? (Y) on Y. Then to perform the integrations, the following procedure is sllggested. Reasonable values of fJ and " are assulned. One then computes the values of 1712 and 1113 from equations (12) and (18) for a series of values of , between g = K and ~ = 1. At a given radius, a value of F (Y) is assumed and rw' and u' are computed from equations (8). The value of Y is then computed, and from the graph, table or equation for F (Y) vs. Y the value of F (Y) corresponding to the computed value of Y is found. If the assumed value of F (Y) checks with the computed value of F (Y), one then proceeds to the next value of in the series and computes F (Y) at that radius; otherwise, nc\v values of F (Y) must be assumed and the calculations repeated until assumed and calculated values of F (Y) agree. Once the variation of F (Y) with corresponding to the assumed values of f3 and A has been established by the process described above, the integrations called for by equations (17a) and

e,

e

e

*In that article, the abscissa is (1

}T)1/2 =

rate of shear

DETEH~lINATION

F (Y)

OF

FltOl\1

VISCOl\IETRIC

DATA

'fhe equations describing the flow of a viscoelastic fluid in a rotational (~~ Couette") type viscometer may be derived from the equations given in this paper by setting 1-£ = o. Hence, for the Couette viscometer, the kinelnatic matrices become

o

2r 2 w'2 0 0

0 rw' () 0 ; D 2 = o 0 0 1·W'

o () 0 o 00

; Dr

=

0 (25)

r>2

The ten invuriants of D 1 and D 2 arc again given by equations (7), but the invariant Y assumes the simpler form Y == 2r2 w'2 (Oa)

For the stresses, \ve find

+ ! 2(cx + Y + + ((,(4 + a) Y2 + et s J 3, 1T22 = - P + i (.(a Y, 1133 = - 1) == rw' [ell + Y + Y2] = rw' J? (Y) 1Tl1

= - p

2

(,(3)

T

fX

. 1712 Tr 13

==

1T23

(12a)

= ()

lIenee, the fUllction ]? (Y)

The stress equations

256

(/.7

ttl)

1T 12

]1'

==

(Y) is sitnply 7T 12

rw'

Couette.

(26)

Inay be deternlined from the

I-Ielieal 1T12

f10\V

== f3/ g2

of an annular Blass of viseo-elustit· Huitt

(27)

Gi = Go = 21T R2 f3

(24a)

and the measured torques on the cylinders*. l'he quantity rw' is the usual rate of shear, and nlay be determined from experinlental data by the standard means [28]. Equation (26) sho,vs that F (Y) is nothing more than the viscosity (appa1-ent viscosity, really, since the usual Couette viScolneter does not distinguish between elastic and inelastic fluids) as determined by the Couette viscometer. Hence a plot of viscosity, as given by equation (26) against twice the rate of shear squared (= Y) gives the plot of F (Y) vs. Y required in solving the problem of helical flow in an annulus. The equations describing the flow of a viscoelastic fluid in a tube may be derived frOln the equations for helical flow in ullnuli by setting' w = o. l'hus, the kinematic nlatrices nlay be written as D1

=

o0

U'

0 0 0 u'O 0

D2

==

Dr = 0,

In the capillary tube viseOlneter, the rate of shear is not uniform (as, to a first approxinlation, it is in a COllette ViSCOlllcter \vith a small gap) but is a muxirnlun at the ,vall of the tube and zero at the centre of the tube. lIowever, the shear rate at the ,vall may be determined fronl the RABIXO""ITscII-l\{OONEY equation [24, 25], and the corresponding shearing stress at the wall is given by

7T131

o

= p~

·

(80)

1T13 /U'lwall (again, an apparent viscosity) vs. twiee the rate of shear at the ,vall squared (= V). 'fhe data nlust be eorrected for entrance and exit losses, kinetie energy effects, "'all effects, etc_ l\'1ethods of eorreetion have been discussed by OLDROYD [26] and by l?REDRICKSON [27].

DISCUSSION AND CONCLUSIONS

It ,vas possible to solve

0 0 0 0

2

Hence, the function ]1' (Y) nlay be deternlincd frolu capillary ViSColIlcter data by plotting

2U'2 0 0

0

waH

ItlVLIN'S

equations

for helical flow in annuli because of the relatively. simple geometry of that type of 00\\'. l'hu8, in

helical flow, there are only t\VQ (u' and rw') non-vanishing, independent elements of the for flow in a capillary tube. Because of equations velocity-strain matrix, and these are related to (28), the ten invariants of D 1 and D 2 are again the stress (7T12 and 1713) by the sanlC relation. given by equations (7), but here, the invariant There is the further consideration that for the Y is case considered, the coefficients OCk in the con(6b) y == 2U'2 stitutive equation (3) are functions of only one invariant quantity, Y. As shown above, these For the stresses, we find two circumstances lllake it possible to separate the variables in RrvLIN's equations and so to 7Tll = - P + 1 (2cx 2 + cxs) y + + (lX4 + lX6) 1'2 + CXs lT3, affect a solution_ In more complicated flo,v situations (such as, 1T22 = - p, 7TS3 = - P + t CXs Y, say, flo,,' through a diffuser, devclopnlcnt of flow in a pipe, etc.), the simplifications noted above ,vould not result, and any attenlpt to arrive at (12b) 1718 = U' [cx l + ('ls Y .+ rJ. 1 Y2] = u' F (Y). an analytical solution would be faced vlith grave difficulties. Hence, the function l' (Y) is given by In this respect, let it be noted that the type F (Y) = 7T 1J CapiIlnry (29) of flow which prevails in the usual visconletric u experiments (with eapillary tubes or rotational viscometers) is not very general, in that the *The experinlcntnl data Illust of (!ourse be (~orr('('tcd for end effects, non-uniforulily of shear rate, etc. The rheological coefficients OCk bee-ollle functions of only one invariant (luantity (Y). In a Inare Ilec cssnry corrections have been disl'usscd by "rOllS [2:l]. 1"

>

2

(28)

257

A. G..FUEUJUCKSON

general flow situation, the coefficients ':I..k ,\'auld appear as functions of other invariants. lIenee, the usual data obtained from capillary tube or rotational viscometers nlay be inadequate to predict the rheological behaviour of nonNewtonian fluids in fairly general flow situations. The method of solution adopted herein nlay also be used to solve the problem of eOlnbined axial and tangential flo\v of Bingham plastic materials in concentrie cylinder a.nnuli. r\ccording to OLDHOYD [28], the Binghaln plastic is described by the constitutive equation

n= -

pI

with

'1

+ "7 D 1

if

7

D 1 = 0 if

7

=

2

>

2 70 ,

2

~

"0 2

NOTATION

IJ, C = constnnts of integration Dr = r th kineJnatic 1l1ntrix (elelllcnts

Dii r» torques acting on unit lengths of irmer and outer cylinders of annulus ~ == accelera.tion due to bJTllvity (components gx' gil' gz) J.' (1") = function of Y (" apparent viscosity") - defined by equation (14) I = identity nlatrix (elenlcnts 8ij ) P = frictional pressure gradient - defined by equation (10) ]J = hydrostatic pressure Q = volumetric flow rate Il = radius of outer annulus cylinder r = radial co·ordinate u = axial velocity vi = i·component of velocity iI:, y, Z = cartesian co-ordinates 1· = invariant quantity - defined by equation (H) (/.k = k lh rhcoI()gi(~aI coeflicient in equation (a) ft. A = constunts of integration Sij = I{ronecker delta (= 1 ir'i = j; = 0 if i =F j) , = dummy variahle of integration 'rJ = U viscosity" of llingharn plastic 1l1uteriuldefined by equation (:32) K = ratio of rudius of inner annulus cylinder to radius of outer annulus cylinder fLo = plastic viscosity n = stress matrix (eleJJlcnts 1Tij) ~ = dirncnsionlcss radia.l co-ordinate p = density of fluid T 2 = defined by equntion (as) TO = yield stress of Bingharll plastic Jnaterial Q i = angular velocity of inner annulus eylinclcr Do = angular velocity of outer annulus cylinder w = angular velocity of fluid nt x, y, z. Gi , Go

(81)

fLo

(32)

/V-r 2

1 -

expression for the stresses and integrated to give the velocity distribution.

'TO

In these equations, "0 is the yield stress, flo is the (constant) plastic viscosity, and the invariant quantity ,.2 is defined by 8

,.2

8

tEE

=

(7Tij

i=lj=l

+ IJoij)2.

(38)

For helical flow of a Binghanl plastie lllaterial in an annulus, oue can easily solve the equations of motion and so obtain expressions for the stresses (equation 9) and hence for the yield condition (,.2 = 'T0 2 ); this will be a polynomial of sixth degree in the reduced radius 1'he solution of this polynomial for the two roots lying hetween K and 1 will give the boundaries of the "Illug flo\v" region. The constitutive equations (31) may then be combined with the

e.

=

11.EFE UENCES

[1]

M. }'. and

VOLAROVJCII

[2]

SUCUJPANOV,

[3J

VAN OLPJlES

Y. and

[4]

~JonI

LAUU)

[6]

Fn:EDlucKSON ~I«;LJu)sE.1.

[8]

SAVISS

[0]

FREDIUCKSOS

[10]

lllVI"lN

()'l'OTAKJ';

Induslr.

~I.

[7J

G~

N. Chent. Engng, (Japan) 1H5:1 17 22.j..

l~llg"g.

ClteJrl.. 10.')7 49 ]:18.

A. G. and It. ll. BInI> ITlduslr. 1~llg"g.

C.,

.J.

A. 1\1. Zit. f('cltu. It';z. ID46 16 a~1.

P. !{. Zit. techno Fiz. 1949 19 1211. II. J. 1n,..'I. ])clrol. 1950 36 22:1 (1950).

[5]

\V.

GUTJUS

SAVIN'S

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