R. K. BEATNAGAB, Flow Generated by the Torsional Osoihtions of a Sphere in 8 Non-Newtonian Fluid 389 ~ .. . . . --- -. .-

Flow Generated by the Torsional Oscillations of a Sphere in a Non -Newtonian Fluid

By 1.2. 1L BIIATNAC:AIL

Far die instutiuniirc B t r h u n g nidit - Newhriecher qmrviskoser und aicrkoeluetischer FCiissigkeiten um einc Kugel, die Torsionsschwingungen awfiihrt, werden die axialsymmetrischen Grenzechidtglei&ungm mit Hilfe der SIWrugasrechnwng integriert, Im Vetglekh zu Neu$onsden. Fliissigkeikn bewegen eich die A’lrGqnhien dee stutioniiren Teile der Strmnfunktion wS fiir S 5 11.3 R und K 5 l / d R con der Kugd fort Bzw. zu ihr hin; SI ~nd, K sind dhe laehs lose Kennmhlen filr dip. Quewiskosiia.t und die V d s k o c h s t ~ z i ~ t , und fi ist dk B e y n o l d s w h l . vm diesen Ci‘wgleiehungcn a d dip. ~~tr6mungsrkhtung in &ner &feridia.n- dene c&. W e Atnp&.de der Schwingung dee imtationairen Teils yf utn yYa u.iichst bia zu einnm geti,<ssen Ab- crtuwd eon det Kugd un ZGnd nimnt dann ab. Dus gleiche Verhulten zeigl die Phase E bei PpGerviskosen Fli6ssig- hi ten, mihrend bei viskoeluatischen FlQisaigkeiten die Phase ntit dem Abstand monoton w i k h t .

T’hs u x b y m ~ w t r d c boundary layer Qqwations of motion for the unsteady f k w o/ non-Newtonicm cross- viaremu u.nd vbco-elastic fluids wa.2 a torsima@ oscillating sphere have been integrated wring method of pr lurh t ion . It i s found tkut if S 11.3 B and K 5 114 R, the stream 1kua for t h steady part ys of the stream /unction shi/t auvy or towards the sphre rp.2alde.e to the correepding crtremh linnes for .A’eti~oniu?a f l w i h , w k ~ e AY c r n d K are the dimen&maless pcruqr&rs charactwising rJbe crocrs-Udscoeity und tktco-eksticity 01 iha fluid u d R i s the R e p o l d s d w . These inequalities dso determine the diredon of flow h. o maridionti1 plune. The ainfi tude of o s c i l l a t h of wnetcady part yf about ys and the p h s e E im.reclse wpto (I. certain d~St4WW lroiri th.e sphwe und t k . n begin to dwease for cr0ss-viscozca fluids. But for viaco-elastic fluids, lhu phme in- creuses as die tam from the sphere increusus.

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>l(Mi1l(OCTefi @a3a BO3lMCTaeT MOIlOTOllIIO C paCCT&lRl3WeM.

- ’LHCJJO YCih 110J1 I~llCa.

y6kJBaCT. To me noeeneHHe meeT @a3a y nonepewIoBmmix IIiWHIiOCTefi, a y BR3I(03JIaCTW¶llblX

1. Iritroductiou ‘l lc iriaiii aiin of this paper is to study a particular type of flow probleni in a general noii-

Sewtonian fluid characterised by the REINER-RTYLIN and HIVLIK-ERICKSEN types of equations in order to find the qualitative effects of non-Newtonian parameters. In general, two types of methods are available: (i) the study of the difference in the stream line patterns between the ideal fluids and the non-Newtonian fluids (ii) by thc measurements of torques and stresses oil the various boundaries immersed in the fluid. Recently HAUERIUNN [l] has ‘studied the flow of a Newtonian fluid between two concentric spheres rotating about a central axis. The corresponding problem for the non-Newtonian fluids has been discussed by BHATXAGAB (P.L.) and ~<AJESWAI~I

121. In bolh these problems the fluid occupies a boundcd domain. LANGLOIS 131 has sliown thitt the cffecl of the inclusion of tlie visco-elastic ternis is lo modify lhe distribution of .the rokary Clow aboul the axis and lo induce tlie secondary flow in a ineridional plane in belween two rotaling sphcrrs. ‘liro~4a.r and WALTICRS 141 haw down that tlic projection ol Ibc slrcani lines and thc torquc acliiiy 011 a sleudily rolating sphero in an elaslico-viscous fluid ilelwnds directly on thc f hid parameters which gives a direcl iiielliod of tletermining lliese parainelcrs. GIBSJCKIJS 153 has shown that the measurement of the frictional force and couple in the siniullaneous traiislatioiial and rotational motion of a sphere in a general incompressible elastico-viscous fluid allows complete deterniination of the material constants of the rheological equation of state up to the third order. In ;I rcceiit communication [(i], the aullior has discussed the secondary flow generated by the slow skady rotation of a spherc in an infinitc!ly exlending lion-New toniaii fluid.

Jn the present paper we discuss the flow generated by a torsionally oscillating syliere in a non-Newtonian fluid characterised by the following constitutive equation

(1.1) Y’.. 13 = - p df j+ q, + Os Dij + Os Eij qi, wl1erc

26

3!)0 lt. li. HH~TNAGAR, Flom Grrieratrd bv tlic Torsioiial Oscillations of a Splicre in a Noii-Newtonian Fluid

is the rate of deformation tensor,

(1.3)

is the acceleration gradient tensor and Ql, Q2, Q3 are the co-efficients of viscosity, visco-elasticity and cross-viscosity respectively. 'I'he fluid is assumed to be extending to infinity in all directions arid is olherwisc a t rest.

111 ;I iion-Newtoniaii visco-inelastic fluid ( I ~ ~ N E I G I Z I V L I N fluid), the coiislitulive equation does not involve tlie time derivatives of llie stress or rate of strain cornponents. But in the 11011-

linear theories proposed by KIVLIN and ERICKSEN tlie stress taken a t any instant can be expressed ;is a polynoriiial in the tensors formed by various order convective derivatives of velocity. The concept of nieniory of the fluid or the visco-elastic properties are introduced through the lime deri\atives of lhe velocity gradieiil tensors. As a first step iri the devclopincrit of this theory- the third arid higher order convective derivatives of tlic velocity are neglected so lliat llie I~IVLIN- k~aicitsmm theory iiitroduccs the elasticity oi the fluid through the tensor L), formed by the gradienl of acceleration u, as g i w n iri (1.3). Also Q2 and @:,in (1.1) do not have the dimensions of viscosity, they :ire called lirre co-efkicienls of visco-elasticity and cross-viscosily, although they may be properly called llie second-order parameters of a non-Newtonian fluid. Auother approach in defining elastico-viscous fluids lias been made by OLDROYD [7] through relaxation times, arid considerable work has been done on these fluids also.

2. Eyaations of the I)rohleni

W e :\bSUllle that the aiigular velocity aboul a11 axis passing lhrough tlie centre of gravity of the sphere is given by

(2.1) (0- I'leul par1 (U C t ' L L ) ,

where ir is the frequeiicy of oscillation and B is its aiiiplitude. Since we are iiiteresletl in studying the effw 1 of lion-Newtoiiiaii parameters close to the surface

of [lie sphere within tlic bouiitl:iry layer we' 1ic.r~. iiiake use of the axis-syiiiiiietric unsteady bountlary hyer qua l ions which are aiiicnable l o Irc~atiiieiil lliuii llie full set of NAVIER-STOKES cyat ions. Following RAJESWAKI 181 llie basic unsteady boundary layer equations in tlie absence of any imposed pressure gradient are : 11 o in c ii l u i n e (1 11 a t i o ii s :

(2.2)

E q u a t i o n of co i i l i i iu i t y :

1:. K. I5IiATNAGAR, Flow Generated by the Torsioildl Oscilhtioris of 391

where a, /3 and y are respectively the kinematical co-efficients of viscosity, visco-elasticity arid cross-viscosity, arid 1 1 , u, in are the components of velocity in the increasing directions of 0, @ and I'. (1 is the radius of llie sphere. Because of axis-symmetry we take a/a@ = 0.

\\'e may also note Lliat in steady-state problems inertial effects as well as cross-viscous and visco- elastic effects of the fluid enter the problem first in second order terms. In our noiisteady problem howeker, there exists a contribution even to the zeroth order approximation through visco-elastic terms. 'The inertial terms and the cross-viscosity terms appear but only in the first order appro- xirnatioii. Jus t as in the associated steady-state problem zerolh order approximation determines the priniary flow, while first order approxirnaliori results in an additional secondary flow, i.c. a flow transverse to the primary flow pattern.

Sphere in LL Noii-Newtonian Fluid - - ~ _ _ _ _ _ -- -________-___--___ - ~~

'I'he boundary conditions of the problem are :

I I I = 0 , u = 1Qe (uw sin 0) , UJ = 0 a t I' = ( I , ('2.5) I r r = 0 , u = 0 a1 I' +oo.

1 1 1 Lrocluciiig llie tlinieiisioiiless parameters througli the lollinving relations :

(2.0)

The momentum equations arid the equatioii of contiiiui ty reduce (on dropping the accents) t o

alld

(2.9) i ? l / iiro 20 - -tz + I 1 c o t 0 = 0 ,

where

are the dimensionless parameters characterisirlg the visco-elasticity and cross-viscosity of Llie fluid. We shall consider the case of large frequency so thal & is small and R is large. Under these assuinplioiis we assume the following series solutions for 11, u and I U :

(2.10) 11 = IT, + & I:1 + &2 F , -+ * * , (2.11) (2.12) w = E HI + + * * 1 ,

v = Go + & G I + &'G2 + - * a ,

392

where the conditions 011 I;, G aiid H are

R. K. BHATNAGAR, Flow Generated hy the Torsional Oscillations of a Sphere in a Non-Newonian Fluid - - ~ - ~ p - - - - ~ - ._-___

- F - * * = 0 , Gn - l ' \C (f"" b i l l o), = G2 = * * = 0 [ F0 - 1 -

j = 0 , 1 , . . . a t z = 0

Substit~itiiig froiii (a.10) -(2.12) i i i (2 . i ) of t l o zero, \ve llU\ c ,

froni (2.7) :

(2.9) a i i c l cqiiatiiig the co-efficieiits of various powers

(2.14)

(2.15)

fro111 (2.8) :

(2.1 ti)

:\11cl frolll (2.9) :

(2.18)

( 2 . l!))

3. Solution of the cyrmtioas C a s e 1 : IIt,re we shall discuss the visco-inrlastic fluids for which K = 0 but S + 0.

Sol\ ing (2.14), with tlic hclp of boundary coiiditioiis (2.13) we have (3.1) F0 = 0 . l'roln (2.lti) alld (2.13)

. siii 0 - li't ( 1 I 1 ) z (3.2) (;, = C"z (, - the rc:d part of wliicli is

(3.3)

R. K. f i l lATiYACAR, 1 ~ 1 0 ~ CenrmtetX by tlw Torsional Osclllnlions of n Splicrr in x X’on-Keutoninn F l i i i t l 393

7‘hus to the zeroth order approximation, the transverse velocity u is giLen by (3.3) and tlicre is no circulatory velocity 11 and no radial velocity I I ) to this order of approximation.

______ ~ - ~- - _______-

ITsing (3.2), Lhe ecluation (2.15) retlriccs l o

(3.1)

which atlrnils the sollition of the form

( 3 J )

Substituting (3.5) in (3.1) and equating the steady and the unsteady parls we have thc eqnations giving /, antl f z :

I;, = [Il(z) + I2(i) sin 0 cos 0 .

(3.6)

whcrc the dash denotes differentiation with respecl Lo r. ‘1’1142 hountlary conditions on f l antl f z arc

( / 1 = / 2 - o n l r = = o , I / 1 = / 2 = o a t r = c o .

(3.8)

Tiilcgral ing (3.6) and using (3.8) w e gc.1

(3.9)

and froin (3.7)

(3.10)

whic.h give

(3.11)

wlicrc

‘1’11~s the first approximation to the circulatory flow velocity is composed of a steady and an unsteady part. lye note that as in the problem of a single oscillating disc investigated by Rosm- m A r r 191 here also the condition a t infinity is not satisfied so tha t the centrifugal and shearing forces a t the sphere give rise to steady radial and circulatory components of velocity.

F i r s t a p p r o x i m a t i o n t o t h c r a d i a l f low: Sul)stiliiling (3.1 1) in (2.19) ant1 inlcgrating with rcspcct t o 7 , nftcr nsing (2.13), we have

which shows tha t the radial flow is also composed of a steady and an unsteady part.

394 R . K. BHATNAGAR, Flox Generated hy the Torsional Oscillations of a Sphere in a Non-Rewtonian Fluid ~ - ~ _ - _ ~ ~ ._._ - - - -

7'1ius we have for the circiilatory and radial compol~ents of velocity

-y!G z ) + (3.13)

Denoting the steady par1 by subscript s and the unsteady part by subscript 1, we have to the first order of approximation

(3.15) -1'-

1 2 , = & (1 - 3 R S ) (1 - -e '' 'I ') sin o coso ;

(3.16)

and

(3.17)

(3.18)

Ifre now introduce the stream funclion y ,

(3.19)

so that the equation of continuity is identically satisfied. The steady and the unsteady parts of y~ are given by

(3.20)

and

(3.21)

t h e real part of which is

F sin2 0 cos 0 IXe yf zzz -- - 8 1 (z, R, S ) cos .( 2 z - C' (2, I<, S ) } , (3 .22)

where

(3.23) ((1 4- 3 R S) cos 11% z + (1 - 3 R S) sin I/R Z ) -

(3.2 I )

2 being the amplitude of oscillation of yf ahout ys antl r' Ihe phase lag.

Wc note the following points:

(i) The stream line area y = 0 consists of the lines 0 = 0, and 0 = x, the plane 0 = nj2 ant1 the surface of the sphere z = 0.

(ii) The steady part of the stream function ys asgiven by (3.20) differs from the corresponding stream function for the Newtonian fluid by a factor of (1 - 3 R S). Consequently, the stream lines for S < 1/3 R shift away from the sphere near the equatorial plane and for S > 1/3 R, Lhey shift towards the sphere, lhe actual magnitude of the shift being proporlional l o S/ii.

Also the direction of the flow can he determined as lollom: Writing (3.15) and (3.1'7) as

we find thal if

(a) S < S, then 11, > 0 antl w, 2 0 according as 0 3 x

SO that the flow is towards the eqiiator, i.e. the fluit1 is drajvn from t h e pole and Ihrown towards t he equalor;

(1)) S > S,. then 1 1 , -< 0 and w, 9 0 according as 0 -' 2 ,

SO tha t the flow is towards the pole i. e. the fluid is drawn from equator and thrown towards the pole. Thus the nature of the steady flow is determined entirely from the intrinsic properlies of Ihe fluid.

(iii) In Tables I , I1 and 111 we have collected the values of I and c' for values of z from z = 0.1 to 0.8 adopting R = 50 and S = 0, 0.005 and 0.1 respectively. \Ye notice t h a t the amplilude I of the vibration of the stream function vf about ys goes on increasing upto a certain distance from the sphere and then begins to decrease. \Ve have stopped the numerical work a t z = 0.8 as beyond this the boundary layer equations may not be valid. T h e phase E' also goes on increas- iiig up to certain distance from the sphere and then hegins l o decrease for s = 0, 0.005 m(l 0.1.

z

0.1 0.2 0.3 0.4 iJ.5 0.6 0.7 il.8

1 cos F I

-0.079259 -0.337627 -0.537053 -0.588980

-0.529869 -0.512314 I - 0.508085

I

-0.563461

1

'0.19CJ712 ' 10.323472 4 0.250557 4 0.134147 +0.068749 ' + 0.053827 1 ,0.060132 A0.068476

0.029258 O.066121 0.083809 0.085428 0.080275 0.075326 0.072952 0.0724933

tall F ' F '

-2.406187 I 12'34' -0.9.58073 136'14' -0.466541 1 .i,j '( )'

-0.227762 l(i7'10 -0.122012 I73"03' -0.101586 174'12' -- 0.1 17373 1 i?,'18'

0 . 1 39773 172 ' % I '

Table 1 I . II =-- 50, AS = 0.005

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

I-O.Od(i998 -0.060815 - 0.223447 - 0.312557 -0.327615 -0.313279 -0.299018 -0.292307

-, 0.100101 -+ 0.369084 $0.418142 4-0.368564 + 0.314461 $0.288786 4-0.284396 i- 0.287706

0.0220 78 0.052896, 0.067052 0.068351 0.0642 19 0.060249 ' 0.058361 1 0.058000

4 5.93010(i

-1.871325 - 1.179189 -0.959849 --0.921817 - 0.951099 -0.984259

-6.068963 80O2.5' 99"21'

118'07' 130O38' 136"l l ' 137'20' 136'19' 135'27'

0.1 0.2 0 .3 0.4 0 .5 0 .6 0.7 0.8

- 2.360693 -5.597068 -6.495550 - 5 3 4 1 023 -5.043667 -4.643073 - 4.564949 - 4.607890

I 10.772333 --0.543136

~ - 2.933567 I --4.219783

- 4.599765 1 -4.410401 1 - 4.200874 I -4.096889

I , 0.111076

i 1.007869 1 1.004877

0.969124

~ 0.877382

l 0.795237

1 0.905869

1 0.872009

--0.327165 t0.097039 t0.451625 4 0.722439 + 0.911989

-/ 0.949479 t0.920247 +0.889103

161'54' 185'36' 204'18' 215"Al' 222"21' 223'31' 222037' 221 "39'

Case I I : Here we discuss in a similar way the hrhaviour of visco-elastic fluids for which

Again solving (2.14), willi the help of coritlilioiis (2.13) we have F0 = 0. 1:rom (2.16) and (2.13)

I< + 0 bllt s -= 0.

(3.26)

'I'tiiis t o tlic zeroth ordrr approximatinn, thr transverse vrlocity 11 is given I)y

(3.2'7)

the real part of wliicli is

(3.28) I ) z7. (1 cos (2 - z siii p) sin 0 , -y/: z 1.05 ,l

where

(3.29)

Again there is no circulatory velocity i i ant1 no radial vrlncily it) to this order of approximalion. ['sing (3.26), (2.15) reduces to

(3.30)

(3.33) 1 R -- f ; + 2 i I< 1; --

R. K. BIIATW ~ ( ; A R , l'lom Gencrntcd by thc Torsional Oscillationr of n Splirrc in A Non-Newtonian Fhiid

'I'hc dash denotes differentiation with respect Lo z and

307 -~ - - ~- - -~ - -

f 2 satisfy (3.8). Tiitegrating (3.32) :iiid (3.33), wc have

(3.3 1 j

(3.30 j

l'h n s

(3 .37)

which shows t h a t for visco-elastic fluids also, the first approxirnation t o the circulatory velocity is composed of a steady and an unsteady part.

F i r s t a p p r o x i m a t i o n t o r a d i a l f l o w :

the rirst approxirnation to the radial flow velocity given by Substituting (3.3'7) in (2.19) and integrating with respect t o z , after using (2.13), we have

Thns l o the first order of approxiinnlion, t h e slcatly ant1 lhr iuistcady parts of circulatory and radial velocity are given by

(3.40)

ant1

(3.4 1)

I

1 7 , l 3

?i ( 1 1 1

Defining the stream function y as

398 R. K. BHATNAGAR, Flow Generated by the Torsional Oscillations of a Sphere in a Non-Newtonian Fluid ~~ - __- -

~ -- - - -

and

The real part of yf can be put in the form

(3.45)

where

K 2 TI2) (4 K R - ~ cos 2 p) {i2 C-az l ln /L ' - cos (p' + 'Y z sinp') - I/R ( 3 . 16)

(1 3- 4 K3 R2) (1 -1- I f I P R2 - 8 I( R cos 2 p - 2 sin 2 p) 2 pi (3.47) Isin&' = - X

and

(3.48) I<' = (1 + 4 K 2 IP) 15 4- 16 K 2 R2 -- 1 (2 K H cos 2 p -+ sin 2 p)]

(3.50) I / z

' Y = 2 R ) , ( 1 -1- 4 K 2 R2

MTc again note tlie following points:

(i) The stream line area y = 0 consists of tlie straight lines 0 = 0 and 0 = n, the plane 0 = nj2 and Ihe surface of the sphere z = 0 as in Case I.

(ii) The steady part of the stream functiony, a s given by (3.43) differs from the corresponding stream function for the Newlonian fluids by a factor of (1 - 4 K R) so Ilia L as long as li < li L f i tlw stream lines are shifted away from the sphere and when K > 114 R, they are shifted towards the sphere. Again, as in the case of visco-inelastic fluids, if I< < K , (= 114 I?) Lhc flow is towardsthe equator and if K > I<, the flow is t o ~ ~ a r d s the polc. Thus here also Ihe nature of the steady flow is determined entirely from the intrinsic properties of the fluid.

(iii) In Table 1 1 7 , we have collected the values of I and E' for values of z ranging from z = 0.1 l o z = 0.8, adopting R = 50 and K = 0.1. We notice tha t the amplitude 1 of the vibration of llie stream function y f about y , goes on increasing up t o a certain distance from the sphere and then begins to decrease as for t he Newtonian and the visco-inelastic fluids but the phase E'

goes on increasing as z increases.

~~~

z ~

0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8

1 ros c ' 1 sin P' 1 ' t a n t ' ~ 8' ~

-t-0.001740 -0.007307 -0.011358 -0.014170 -0.016351 -0.017910 -0.019025 -0.019821

+0.043719 +0.048204 +0.043382 +0.036823

0.030258 + 0.024125 - 0.01 8482 r 0.013311

~~ - ' 0.043749 0.048754

~ 0.044844 0.03948!1 0.034395 0.030049 0.026533 0.023875

~~ ~~~

, 1-25.123872 ~ -6.596962 1 -3.819510 ~ -2.598659

- 1.347013 0.971459

I 1 0.671560

- 1.850529

87"43' 98"38'

104'41 ' 111"03' 118'23' 126O3.i' 135"49' 146'07'

Casc I I I : Since the contributions of both K and S ternis are of the same natnre, similar con- clusions also hold for the case I< =+ 0, S + 0.

Acknowledgment ?'he author is highly indebted to Prof. P. L. RHATNAGAR and Prof. s. HAWA AN for their

kind help and guidance throughout the preparation of this paper. The author is also highly grateful t o the referee for his valuable suggestions which led t o t h e improvement of the paper.

References

1 I\'. L. HABRRM~NN, Physics of Fluids 5, p. 825 (1982). 2 P. L. BHATNACAR and G. K. RAJESWBRI, Indian Joiirnal of Matheniatics 5 , p. 93 (1963). 3 W. LANGLOIS, Quart. dpp. Maths. 31, 61 (1963). 4 ft. H. THOMAS and K. WALTERS, Quart. J. Mech. and App. M.itlis. li , 39 (1964). 5 H. GIESEKUS, Rheologica Acta 3, 59 (1963). (i I?,. K. BHATNAGAR, Proc. Ind. dca. Sciences. Yo1 LX, Sec. L\, KO. 2, p. 99 (1964). 7 J. G. OLDROYD, Proc. Roy. Soc. A 200, 523; 24.5, 278. 8 (;. K. RATESWART, Z,\hlP NIT, p. 442 (1962). 9 S. R n v : w m A T T , Joiirnnl of Fluid Mrrlinnics, 6, p. 206 (1959).

Maniiskriptcingang: 2. 5 . 1964 (rev. Fnssnng 2. 12. 1984)