I ~ L O W O F A M A X W E L L L I Q U I D F L U I D B E T W E E N
T W O R O T A T I N G C O A X I A L C O N E S H A V I N G
T H E S A M E V E R T E X
BY D. K . MOHAN RAO
(Depattment of Applied Mathematics, lndian lnstitute of Science, Bangalci~-I 2, h;dia)
Received Sr 24, 1962
(Communicated by Dr. P. L. Bhatnagar, r.&sc.)
1. INTRODUCTION
THE motion of a Newtonian fluid and two classes of non-Newtonian fluids, namely, the Reiner-Rivlin fluid and the Rivlin-Ericksen fluid between two rotating coaxial cones having the same vertex has bee~ investigated by Bhat- nagar and Rathna ~ taking into consideration the inertial terms in Oseen approximation. The main characteristics of the mofions ate the introducfion of secondary flows. In the case of non-Newtonian fluids the secondary flow is divided into two distinct domains. The stream lines in the inner region are closed loops. In the outer region the fluid mores more or less parallel to the generator of the cone to which it is closest, moving towards the vertex near one cone and away from the vertex near the other cone. In the present note we shall c o n s i d e r a Maxwell fluid. 2, ~ We find that when the relaxation times are small the secondary flow characterisfics are similar in nature as predicted by Bhatnagar and Rathna. by
2. THE VlSCOSITY DISTRIBUTION
The equation of state for the Maxwell fluid under consideration is given p,ik = 2 f f s bxk e tl~mr (x', t') dt'
(2.1) o o
~b (t -- t') = f N r(r--) e_,t_t
)/~" g r
O where p,ik is the deviatoric stress tensor, e mmr the tate of strain tensor
198
(2.2)

Flow of Maxwell Liquid Fluid between Two Rotating Coaxial Cones 199 x q = x q (x i, t, t') is the position at time t 4 of the material point of the fluid which is instantaneously at the point x ~ at time t, N (7)d~- the total viscosity of all the Maxwell elements with relaxation times between ~- and
-r + d~'.
The sum o f the viscosities of all the Maxwell elements is 7/0 so that o
T f N (~) d r = 70,
T being the largest relaxation time.
(2.3)
We c o n s i d e r a relaxation spectrum in which viscosity S~7o is spread on the portion o f the ~--axis between z ----- O and ~, ----- T, excluding the origin.
N ( 0 = ~ 7 o ( 1 - - S ) 9 1 o
T
-rN (7) dr = aST~/o , O < .e < 1.
O < S < 1 (2.4)
(2.5)
The distribution given above represents the three constant spectrum studied by Walters 3 when a----1.
3. THE PRIMARY MOTION 1
We assume the primary m o t i o n to be t h a t of a Newtonian liquid with the neglect o f inertial terms. Here the stream lines are circles with centres on the c o m m o n axis.
The variables used are rendered dimensionless by m e a n s . o f a standard length L a n d a standard velocity L / 2 , with /2 ---- [/2~ I + I g2 2 [, /2~ and/22 being the angular velocifies of the inner and outer cones respectively.
In spherical polar co-ordinates with the origin at the c o m m o n vertex, the radial and transverse velocities Uo, yo in a meridian plane and Wo the velocity normal to the meridian plane are given by
Uo = Yo = 0
0 0}
+
A2]
(3.1)
(3.2)
A~ - Q ~ - Q~ ( 3 . 3 )

200 D . K . MOrIAN I D o
A~ =
0
+ t2e (log tan ~ -- cot 0x cosec 0x)] (3.4)
K = log t a n ~ }
+ tot 0s cosec 0s -- tot 0x cosec 0x (3.5)
0x and 0~ being the semi-vertical angles of the inner and outer cones respee- tively.
4. THE SECONDARY MOTIONS
We shall now study the modifieafion of this primary flow due to the inclusion of the inertial terms and viseo-elasticity regarding them as seeondary effects.
FoUowing Oseen's scheme, to take inertial terms into consideration in a limited manner, we have the following equations for the determination of the secondary motions.
The equation of continuity
~u 2u 1 ~v v eot 0
- - O. (4.1)
The equations of momentum
- R - ( w J + 2 W o W ) r
---- ~ P + A u -
2u 2 ~v 2v r-~ - - r 2 ~0 ra tot 0
_ 2eS_____TTr
R e o t 0 (woS + 2 Wo w) r
_ ~p 2 ~u
"" - r ~ + / x v + ~ ~ - ~ coseQ s 0
(4.2)

201
2~ST (d~,oh 2 r cos 0 sin 0 k - d r i l
(4.3)
= A w
W r ~ sin ~ 0
(4.4) where b ~ 2 3 t o t 0 3 1 32
A-----3r z + r ~ + r ~ 30 + r 2302, p i s the hydrostatic pressure corresponding to the secondary flows, u, v and w ---- r sin 0 ~o ate the veloeities in the secondary motion in directions parallel to u0, v0, w9, T = ~ T R = L2~/v, v being the kinematie coeffieient of viscosity, and we have taken T to be of the order of the square o f the
Reynolds Number R and neglected terms like R3u, R3v in eomparison with terms like Ru, Rv.
Equations (4.1)-(4.4) are to be solved with the boundary eonditions u - - - - v = w = 0
(4.5)
Following Bhatnagar and Rathna, we assume solutions o f the forro
[
R
~ 0 ~
+ K
1 [ R ~~91 3~b=] v = r sin 0 - ~ + K ~r .I oJ = R~r 4 sin 6~ p = Rpz + Kp~ (4.6) where the suffixes 1 and 2 correspond to the secondary flows in a meridian plane induced by the inclusion of the inertial terms and visco-elasticity respec- tively, and K = 2aST.
Writing
~z = r 5 Fz (O) r = r s F= (0) vi = r, ~ (0)

202 D . K . MOHAN RAO and p~ = c log r q- ~~ (0)
C being a constant,
(4.7) we have the following equations for the determination of Fx (0) and F2 (0) :
D 4 Fx (0) -- 2 cot 0 DaF1 (0) -k (28 -b 3 cot z 0) D z F1 (0)
- - cot 0 (29 q- 3 cot 2 0) DF1 (0) q- 120 Fa (0) r ( 0 ) + A2] (4.8) and
D' F2 (0) -- 2 cot 0 D a F2 (0) q- (8 q- 3 cot 2 0) D 2 F2 (0)
-- (9 q- 3 cot ~ 0) cot 0 DF2 (0) = 16 KA~ 2 cot 0 cosec a 0 (4.9) where d
D=-- dO"
We can easily check that
F1 (0) = a~ [7 cos 5 0 -- 10 cos a 0 q- 3 cos 0] q-a2
[(
7 c o s 5 0 - 1 0 c o s a 0 q - 3 e o s a
) 0
40
-
23 cos ~ O +
163
-}- az [cos ao --cos 0] -{- a~ i( ) 0
+ cos' 0 - 2 ]
[ 0 ( 9 7 5
+ A x z log 2 t a n ] - - 8 c o s 5 0 + 7 4 c o s 3 0 - 8 c o s 0
)
9
- 1 )
63 5 2 211 j A~A2
80c~ 0 + T c o s 3 0 + ~ - 0 c o s 0 + 30 '

Flow of Maxwell Liquid Fluid between Two Rotating Coaxial Cones 203 and
F 2 ( 0 ) : b ~ c o s 0 + b 2 + b 3 (cos 3 0 - - c o s 0)
+ b4 [cos~ O + (cosS O -- cos O) logtan O]
+ K [ ( c o s 0 + log tan ~ ) (1 -- 2 cos' 0)
+ (cos 0 -- cos a 0) log 2 tan 2] where the constants a's and b's have to be determined from the boundary conditions (4.5).
5. THE STRESSES and
The pressure p i s given by (4.6) and (4.7) with
2 Pi (0) = sin 2 00~o ~ - cosec 0 [(21 + t o t ~ 19) DF1 (0)-- cot/9 D~Ft (0)
+ D3F1 (0)] c = 3bx
Pz (0) = 4 A12 cosec 4 0 + cosec 0 DF2 (0) + 6 J" F~ (0) cosec 0 da.
The stress c o m p o n e n t s are given by
Prr ----- -- P -- 6 Rr 2 cosec 0 DF1 (0) -- 2K cosec 0 DF~ (0)
Peo ---- -- P + 2 Rr 2 cosec 0 [4 DF1 (0) -- 5 F1 (0)]
+ 2 K cosec 0 [2 DF2 (0) -- F2 (0) t o t 0] p~~ = -- p + 2Rr ~ cosec 0 [5 Fx (0) t o t 0 -- DF1 (0)]
+ 2K cosec 0 [2 Az ~ cosec s 0 + 3 t o t 0 F2 (O) -- DF= (0)]
Pro = Rr~ cosec 0 [I0 F1 (0) + cot 0 DFx (0) -- DZFI (0)]
+ K cosec 0 [tot 0 DF= (0) -- D~F2 (0)]
P e , = R 2 # sin 0 D~" (0)
Prr = 4 R=r ~ sin 19 ~- (0).

204 D. K. MOHAN RAO
6. NUMERICAL WORK
We have obtained the values o f the constants a's and b's in the case when
~" 7T
~ 1 = 1 , 9 2 = 0 , 0 1 = ) , 0 2 = ~ , a = l , S-----0.5 and
T = 0'01.
The m e t h o d is quite general and other cases may be similarly discussed.
The values o f a, S and T correspond to an idealized visco-elastic fluid stuclied by Waltcrs. 3
The values of the constants a's and b's ate given by al = -- 0.3002735 A1 ~ a2 = 0 . 0 0 3 0 9 1 3 6 5 A 1 ~ a3 = -- 0.02403479 A t 2 a4 = 0"00494618 A t 2 bl = ha = -- 1 "5866184 A1 a b~ = 0 a n d hi = 0" 0467785 A t ~.
The stream lines for the secondary flow ate given by
= R~I + K~2 = constant.
Figure 1 shows the stream lines in the present case for
~k = 4- 10 -s X7"5, 4- 10 -s •
@ ffi
0 0.1 0 " 2 0-:3 0 . 4 O'S 0 - 6 0 7
Fin. 1.
PLATE
Stroam Lines of the Secondaty F]ow wb•n R ---- 0.1 and T = 0"01.

Flow of Maxwell Liquid Fluid between Two Rotating Coaxial Cones 205
7. DISCUSSION OF THE RESULTS
We observe that the flow field is divided into two domains by a stream fine.
The dividing stream line starts normally to the cone and the plate and the radius vector drawn from the vertex to the dividing stream line goes on decreas- ing as we move from the cone to the plate. Ir appears that the circular divid- ing stream line as reported by Bhatnagar and Rathna x is the effect of the assumption of small angle between the cones.
In the region near the vertex the stream lines are closed loops. In the outer region the fluid is drawn towards the vertex near the stationary plate and is thrown away from the vertex near the rotating corte. In general, then, the secondary flow resembles that reported by Bhatnagar and Rathna t i f we make allowance for the smaller angle between the cones.
SUMMARY
We consider the secondary flows arising in the motion o f a Maxwell fluid between two rotating coaxial cortes having the same vertex. We find that in any meridian plane passing through the common axis of the cones, the flow field is divided into two regions. Such a division of flow field was first reported by Bhatnagar and Rathna. ~
ACKNOWLEDGEMENTS
The author wishes to express his sincere gratitudr to Prof. P. L. Bhatnagar for suggesting this investigation and for his kind help and guidance during the preparation o f this paper.
REFERENCES
1. Bhatnagar, P. L. and Rathna,
S . L .
2. Oldtoyd,J. O; ..
Quart. Joutn. Mech. and AppL Maths. (To
Proc. Roy. Soc., 1950, 200A, 523. be published).
3. Waltets, K. .. Quatt. Joutn. Mech. and Appl. Maths., 1962, 15, 76.
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