I~LOW OF A MAXWELL
LIQUID FLUID BETWE
TWO ROTATING
COAXIAL
CONES
HAVING
THE SAME VERTEX
EN
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 Bhatnagar 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.
2.
THE VlSCOSITY DISTRIBUTION
The equation of state for the Maxwell fluid under consideration is given
by
f
p,ik = 2
s bxk
~b(t -- t') b~7-m
?x, r e tl~mr (x', t') dt'
f
(2.1)
oo
~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 of the viscosities of all the Maxwell elements is 7/0 so that
T
f N (~) dr = 70,
(2.3)
o
T being the largest relaxation time.
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=~7o(1--S) 91
O < S < 1
(2.4)
T
-rN (7) dr = aST~/o, O < .e < 1.
(2.5)
o
The distribution given above represents the three constant spectrum
studied by Walters 3 when a----1.
3.
THE PRIMARY MOTION1
We assume the primary motion to be that of a Newtonian liquid with
the neglect of 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 g22 [, /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 Wothe velocity
normal to the meridian plane are given by
Uo = Yo = 0
(3.1)
w o = r s i n O[Al { l o g t a n 20- - c o t O c o s e c 0} + A2]
(3.2)
A~ -
(3.3)
Q ~•-K Q~
200
D . K . MOrIAN IDo
1 [t21(cot0~cosec02--1ogtan~)
~-K
A~ =
0
+ t2e (log tan ~ -- cot 0x cosec 0x)]
K = log
+ tot 0s cosec 0s -- tot 0x cosec 0x
(3.4)
(3.5)
tan~}
0x and 0~ being the semi-vertical angles of the inner and outer cones respeetively.
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-(wJ+2WoW)
r
---- - - ~~rP + A u -
2u
2 ~v
r-~ - - r 2 ~0
2v
ra tot 0
[ d%'~ I
_ 2eS_____TTr
sin2 0 ~,-d-ff,/
(4.2)
R e o t 0 (woS + 2 Wo w)
r
_
"" -
~p
r~ +/x
2 ~u
v + ~ ~ -
~ coseQ s 0
Flow of Maxwell Liquid Fluid between Two Rotating Coaxial Cones 201
2~ST
(d~,oh 2
cos 0 sin 0 k-dril
r
rR ( vdw~
dO +2uw~176176
)
=Aw
(4.3)
W
r ~ sin ~ 0
(4.4)
where
b~
2 3
tot 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 of 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
when
( 0 = 00 1= 02
(4.5)
Following Bhatnagar and Rathna, we assume solutions of the forro
1
U=--r~.sinO
1
v = r sin 0
[ R ~ 0~ + K ~-02]
[R ~~91
3~b=]
- ~ + 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 respectively, 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)
(4.7)
C being a constant,
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 (klOg tan 0 -- eot 0 cosec 0 ) + A2]
= 4A1 I.A1
(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]
[( 7cos 5 0 - 1 0 c o s
q-a2
-
-
a0q-3eosa
) l o g t a n 20§
40
23 cos ~ O + 13
163
-~
-}- az [cos ao --cos 0] -{- a~
i( cos a 0 -- cos 0) log tan ~_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
+ l o g t a n 0 ( - - 7 4 c9o s 4 0 + ~ c o s ~ 0
63
5
2
211
80c~ 0 + T c o s 3 0 + ~ - 0 c o s 0
-1)
j
A~A2
+ 30 '
)
Flow of Maxwell Liquid Fluid between Two Rotating Coaxial Cones 203
and
F2(0):b~cos
0+b2+b3
(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 pressure p i s
THE STRESSES
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
and
Pz (0) = 4 A12 cosec4 0 + cosec 0 DF2 (0) + 6 J" F~ (0) cosec 0 da.
The stress components 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)]
Pe, = R 2 # sin 0 D~" (0)
Prr = 4 R=r ~sin 19~- (0).
204
D. K. MOHAN RAO
6.
NUMERICALWORK
We have obtained the values of 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 method 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 At 2
a4 = 0"00494618 At 2
bl = ha = -- 1 "5866184 A1 a
b~ = 0 and
hi = 0" 0467785 At ~.
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
07
90 ~
PLATE
Fin. 1. 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 decreasing as we move from the cone to the plate. Ir appears that the circular dividing 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 of 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 of this paper.
REFERENCES
1. Bhatnagar, P. L. and Rathna, Quart. Joutn. Mech. and AppL Maths. (To be published).
S.L.
2. Oldtoyd,J. O;
.. Proc. Roy. Soc., 1950, 200A, 523.
3. Waltets, K.
A4
..
Quatt. Joutn. Mech. and Appl. Maths., 1962, 15, 76.