FLOW
OF AN ELECTRICALLY
CONDUCTING
NON-NEWTONIAN
FLUID
BETWEEN
TWO
ROTATING
COAXIAL
CONES
IN THE
PRESENCE
OF EXTERNAL
MAGNETIC
FIELD
DUE TO AN AXIAL
CURRENT
BY D. K. MOHAN RAO
AND
P. L. BHATNAGAR, F.A.Sc.
(Department of Applied Mathematics, Indian Institute of Science, Bangalore-12)
Rer
October 24, 1963
ABSTRACT
Bhatnagar and Rathna (Quar. Journ. Mech. Appl. Maths., 1963, 16,
329) investigated the flows of Newtonian, Reiner-Rivlin and RivlinEricksen fluids between two rotating coaxial cones. In case of the last
two types of fluids, they predicted the breaking of secondary flow ¡
in
any meridian plane. We find that such breaking is avoided by the application of a sufficiently strong azimuthal magnetic field arising from a line
current along the axis of the cones.
~OTATION$
B =
magnetic induction vector
E =
electric intensity
H =
magnetic field
D =
displacement vector
J =
current density rector
pe =
excess charge density
=
electrical conductivity
e =
j
Al
=
dielectric constant
axial current
V =
velocity vector
T =
stress tensor
269
270
D . K . MOHAN RAO ~
P. L. BHATNA~AR
e =
rate of strain tensor
D =
aeceleration gradient temor
I
idem tensor
=
P =
pressure
p =
'7 =
density
eoeffieient of viseosity
~Te =
coetª
~v =
coetficient of viscoelasticity
12 =
characteristic angular velocity
of eross-viscosity
1. INTRODUCTION
THE flOWS of Newtonian, Reiner-Rivlin and Rivlin-Ericksen fluids between
two rotating coaxial cones having the same vertex have been recently investigated by Bhatnagar and Rathna. tlj They have predicted the breaking of
secondary flow in case of the last two categories of fluids. In the present note
we include the effect of external magnetic field produced by current along the
axis of the cortes to examine its effect on the secondary flows.
The flow is characterized by the tire non-dimensional parameters.
R-
PL~~2, the Reynolds number
,/
S = ,je~,
the cross-viscosity parameter
,/
K = ,7,~~,
the viscoelasticity parameter
,/
M = oLaf2,
j'N = ,7L~~2,
where
and L is a standard length.
We find that a large axial current suppresses the breaking of the seeonda~y
flow in case of non-Newtonian fluid. The axial current necessary for this
purpose is more when the angular gap between the cones is small.
Flow of an Electrically Conducting Non-Newtonian Fluid
2.
271
BASIC F_~UATION,~
The equations governing the steady flow of an electrically conducting
liquid aro:
MaxwelI's equations
V • H = J (neglecting displacement eurrent),
(2.1)
V • E=0,
(2.2)
V . D = pe, T
.B=0,
(2.3)
D = 27E, B = H (taking magnetic permeability to be unity),
(2.4)
Current equation
J = o lE + V • 13) (neglecting convection current),
(2.5)
Continuity equation
V . V----0,
(2.6)
Momentum equation
pWV)V=
V. T+(V
• H) • H,
(2.7)
and the rheological equation of state:
T = - - p i + Te + ~ee 9 e + ~vD,
(2.8)
where {Tij}, {eo} and {Dij } denote the stress tensor, the rate of strain tensor
and the aeeeleration gradient trensor.
3. THE PmMAgY MOTION
We have taken the primary motion to be that of a Newtonian fluid with
the neglect of inertial terms. The stream lines and the lines of magnetic
induction are circles in planes perpendicular to the axis of the cenes, with
their centres lying on it.
We shall work in spherical polar co-ordinates r, 0, ,~ with the origin at
the cerumen vertex of the cenes, £ being measured from the axis of the cenes
and 4' from a convenient meridian plano.
The variables ate rendered dimensionless by means of a length L, a
velocity L ~2 with Q = I~21[ + [~21, ~21 and ~22 being the angular velocities
of the inner and outer cenes respectively a n d a magnetic field j/L, w~ere
j is the applied axial current. In our sc•eme, hydrostatic pressure will be
given by p,~S2.
272
D . K . MOHAU RAO AND P. L. BHATNAGAR
Denoting the quantities of the primary motion by the suffix zero, we have
2
R o - (O, O, ~i¡
'
Yo = (0, 0, r sin 0Ojo),
O~o = Al ((log tan 0 -- eot 0 cosec 0) + A2,
01 -- 92
Al = - - - ~ - - - ,
1 [Ql(cot02cosec02--1ogtan~)
A2 = D-K
(
)]
o~
+ f22 log tan ~ -- eot 01 eosee 01
,
01
0~.
K = log tan ~ -- log tan ~ + cot 02 cosec 02 -- cot 0x coso: 01,
where 0~ and 02 ate the semi-vertical angles of the inner and outer cones
respectively.
4.
SECONDAgYF t o w
Wo now consider the effect of inclusion of the inertial terms in Oseen's
approximation and the first-order effect of cross-viscosity and viscodasticity
retaining only the first powers of S and K. Denoting the perturbation velocity
by (u, v, w) and the induced magnetic field by (/h-, h0, he), the linearized
equations determining tho porturbod magnetic field and velocity aro
E4x = 0 ,
(4.1)
1
ax
hr = -- r~
s--f~-0 ~ ,
1
~x
h0 = r s i n 0 ~r '
E4 =
Ahr
[~2
sinO0(
~ ~t]s
~/~ + - T r - k ~ /
J,
hr
2M [" ~
v
ra s i n r0 = --~ L ~ Csi~0) + ~0 Crs---sin 0 ) ] '
(4.2)
Flow of an Electrically Conducting Non-NewtOnian Fluid
- - Rw~ _
r
A u - - 2 u - - 2 c o t 0v
~P +
br
~
rz
2 ~v
r ~ ~0
--r2__aN . s i 1¡ 1 7 6
sin'O
2K [Ojo'+ (dO,o3"l.
r
273
(4.3)
kdO] J'
- - Rwo = c o t 0
= --
r•0
v
2 bu
r 2 sinO0 + r ~ ~0
-k A v
2N 1
r 8 s~
&~176
~0 (r sin Ohm) -- 4S
-r- sin 0 cos 0 (k-d0-)
6 K s i n 0 c o s 0/'dc~176
r
\-d0]
R[vbw o
]
i.
~ -t- 2uwo + VWo cOt O = A W - r 2 s i n 2
~u
~
~v
3z
2 ~
(4.4)
'
w
0,
v cot 0 _ 0,
(4.5)
(4.6)
where
A =~-2 + i~
cot 0 ~
+ -W
1 b~
~0 + ~ ~ 0 ' '
Equations (4.1)-(4.6) have to be solved under the boundary conditions :
when 0 = 01, 0~
hr = h 0 = he = 0 (continuity of the magnetie field),
(4.7)
u = v = w = 0 (no slip eondition).
(4.8)
With the help of equation (4.1) a.nd boundary conditions (4.7) we can
easily show that
hr = h o = O.
The induced magnetic ¡
in the azimuthal direction is given by (4.2),
* The details of these equatiom ro.ay be sr162in [1].
274
D.K.
MOHAN RAO AND P. L. BHATNAGAR
As in [1], wo assome solutions of the form
1 ~~
r ~ sin 0 ~0 '
u=
1
~~b
r sin 0 ~r '
r = Rr~F ( 0 + Sr~~ (0 + Kr~P (0,
h i = Rr3V (0) + Sr 9 (0) + Kr9 (0).
It is evident that the successive terms in the above cxperessions for ~b
and h~ denote the contributions of inertial terms, cross-viscosity and viscoelasticity respectively. Equations (4.3)-(4.5) then give the following set of
simultaneous equations for the determination of the functions F, F, P,
~', 9 and 9:
(D z - c o t 0 D + 6 ) ( D
e-cot0D++20)
F(0)
+ 4 N (D -- 3 cot O) ~, (0) = 4 AlOjO,
(4.9)
1
sine0 ( D e + cot 0 D + 1 2 - - sm-~-~) ~, (0) = 4M (D -- 5 clot 0) F (O),
(4.9 a)
(D e -- t o t 0 D + 6) (D e -- tot 0) 1~(0) + 4 N ( D -- tot 0) 9(0)
= 32A12 cot 0 coseca0,
sine0
(%D ~ + tot 0 D + 2 - - - -
(4.10)
= 4M (D -- 3 tot 0) F (0),
(4.10 a)
(D ~ -- cot OD)(D ~ -- tot 0 D + 6) F(8) + 4 N ( D -- tot 0) 9(0 )
= 2 sinO0 [4 cos 0 Do~aZ - - sin e~oaDeo0 -- cos 0O~o~],
(4.11)
and
[-----4M (D -- 3 t o t 0) F (0),
(4.11 a)
Flow of an Electrically Conducting Non-Newtonian Fluid
275
whcrr
d
D ~ ~r0-.
5.
SOLUTION FOR SMALL ANGULAR GAP BETWEEN THE CONES
When the angular gap between the eones is small, say ac, we obtain
solutions of the pairs of equations (4.9)-(4.11) in the form
F=
Z an~n, F =
S an~n, F = Z, ~n~n,
9
0
oo
oo
v = s b,~r '~, ~ = 2: b,~~", 9 =
O
0
0
oo
~ b,~~ '~,
Q
where
~----- 0s-- 0.
In view of the boundary conditions (4.7) and (4,8) we have
ao=a~=ao=a~=~o=~~=bo=bo=bo=O.
Equating the coefficients of various powers of ff in equations (4.9)-(4.11),
we get
(b2, b~, ~~) = 89tot 0~ (bx, b~, bx),
(5.1)
6 sin~0z b3 = -- 8Ma2 + (3 cos20s + 1 -- 12 sin20~) b•,
(5.2)
12 sin~0z b6
= -- 12Ma3 -- 20Ma2 tot 09. -- hi [1 -- 12 sin 20~ -- 5 cosa02
+ ~ ~ot 0~(3 co:O, + 1 - 12 ~in~0~],
(S.3)
6 sin~0~ (bs, b3) = -- 8Ma= + (3 cosZ0~ + 1 -- 2 sin~0z) (bx, bl),
(5.4)
12 sin*0,. (b4, B4)
= -- 12Ma3 -- 12Ma~ cot 0z) -- [I -- 12 sinO02 -- 5 cos20~
+ 2 cot 0~ (3 cos20s + I -- 2 sin~Oa) (bl, bl) ,
,
1
(5.5)
276
D.K.
MOHAN RAO AND P. L. BHATNAGAR
for Newtonian liquid:
24a4 + 12 cot Ozaa + (28 + 3 cot20z) 2az = 4At~o~, + 4Nbx,
(5.6)
120a 5 + 48 cot 02a4 + 30 (6 + cotS0z) aa + 2a2 t o t 0 s (35 + 9 cotS0s)
= -- 8A12 cosecZ02 + 4N (2bs + 3 t o t 02bx),
(5.7)
with the boundary conditions
oo
oo
ah ~ n = S han c-n-l= z~ bn c : = 0;
2
2
(5.8)
1
for Reiner-Rivlin fluid:
24a4 + 12 cot Osa3 + 2 (8 + 3 cotS0z) a2
= 4Nbx + 32A1 s cot 02 cosec30s,
(5.9)
120a 5 + 48 t o t Osa4 + 30a a (2 + cotS0z) + 6a2 t o t 0~ (5 + 3 cot*0~)
= 4N (2b~ + c o t Ozbl) + 32A1 z cosec302 (4 cosec*02 -- 3),
(5.10)
the boundary conditions being the same as in (5.8). The corresponding
equations for Rivlin-Ericksen fluid are obtained by including
-
-
4A~oJs -- 2 sinSO~ cos0soJs ~
and
8A12 cosec30s + 2 (2 sin 0s -- 3 sin30z) w2~ + 8Alws cot 0s
respectively in R.H.S.'s of (5.9) and (5.10).
It was found difficult to get general information about the flows from
these equations and consequently we have studied numerically the following
cases :
77"
0s = ~ ,
7/"
a =45"
Case (i)
= 20,
Ha = l, 100, 1000, 2000
Case (ii)
12z = O,
Ha-----1, 100, 1000, 2000.
the method of determining the coeffieients being the same as in [1].
values of the constants a's are recorded in Table I.
The upper values correspond to 12~q
The
= 20 and the lowr cones to 0 3 = 0,
Flow of an Electrically Conducting Non-Newtonian Fluid
277
TABLE I
H~=I
H~ = 1 0 0
H,
=1000
Ho=2000
-- 0.000217029
-- 0"00022202
- - 0"01269267
0"01573677
-- 0.00023403
-- 0.00019757
-- 0.0148364
0.018944
0.00785146
0.00793821
0.00631353
0"00593
0.00930075
0.0094035
0.00747894
0.007
0.0100069
-- 0.009929
0"007912
0.00735
0-011626
-- 0.01157112
0.0078441
0.461
0.00643617
0.00654667
0.3475
-- 0-555
0.0050927
0.00426336
0.405482
-- 0.11347
aa
--
0.168513
-- 0.171786
-- 0"1246
-- 0.1344
--
0.199618
-- 0.202555
-- 0.147554
-- 0.14163
-- 0.216569
-- 0.214352
-- 0.1567
-- 0.16807
-- 0.249523
-- 0.247112
-- 0.15476
3.5212
0.050796
-- 0.051993
-- 2"1422
4.2457
0.00054595
--
0 - 0 0 0 5 2 4 1 2
--
0.0351338
-- 0.3177
-- 0-47059
0.006202
0.0146767
-- 08"3764
-- 0.4881
0.0447197
0-0291716
tia
--
a4
-- 0.0045355
2 " 4 8 4
--
0"39726
ti4
--
--0381
-- 0.595306
--0.3947
-- 0.461032
--3.309
-- 2-0942
a4
--
0.0077524
-- 0"0249
0.0448826
0.0487536
-- 0.3478678
-- 0.2865747
11.57494
11.41298
11.553
11.554
13.80023
13.71302
13.6855
13-687
a•
~5
~5
4.0112
-- 2.80634
14.38455
14.3814
14.3546
14.347
17.13943
16.406
14"3536
17"099
278
D . K . MOHAN RAO AND P. L. BHATNAGAR
6.
STREAMFUNCTION
The stream lines to our approximation are #ven by
~b ~ Rr 5 F (0) + Sr 3 F (0) + Kr 3 l~ (8) = constant.
In the cone-plate arrangement considered above, F (0) is negative for
Ha = 1 and 100 but is positive for Ha ~ 1000, whereas l~ (0) and l~ (0) are
positive for all values of Ha. Figures 3 and 4 show the stream lines for
H a = 100 and
H a = 1000 respectively.
/
~t'Q.
0
91
;
i~)
;= 1-
- m2u ,(3.5.7. ~5, 2~, 35 )
R .0-1. S . K . O . H;I00
Fin.
l.
/.
Stream Lines in the Cone-P]ate .kznmgement.
w
Ha,100, R,S ,K,O.I
-~o=~,..r =, s, v, ,s, ==,.3
Fzo, 2, Str)am Lines in the Cone-Plate Arran8ement,
Flow of an Electrically Conducting Non-Newtonian Fluid
279
n
2
4
~lO0
Fzo.
3.
6
8
10
12
14
9
R.S.K,O-I, ~7~ 9 t-(3,5.7)
Stream IAnce in the Cone-Plate Arrangement.
90
1
,o%. [z.~.7..~5.2~.35]
" ' ~ , 0 0 0 ~ Rm$=X~ 0 "
FIG.
4.
Stream Lince in the Cone-Plate Arrangement.
It is interesting to note that in the case of flow separation (Fig. 3)in
the presence of magnetic field, the dividing stream line is circular as was the
case in [1].
7. LARGE AXIAL CURRENT
In this section we shall assume the axial current and hence the Hartmann
number Ha is large for any angular gap between the cones.
Writing
F (0) = sinSO~ (O),
P (o) = sin'e,~ (e),
1~ (8) = sinSO~ (0),
280
D.K.
MOHAN RAO AND P. L. BHATNAGAR
and eliminating D~(0),
(4.9)-(4.11), wc havc
D~(0),
D~(0)
betwcen the pairs of equations
---Ax (D -- cot 0) coo
N
--
1
-- 1-6-H~a~
[sinS0 D 4 + 22 sin46 cos 0 D 3 + (147 sinS0
- - 167 sin50) D 2 + ( 3 3 3 sin20 - - 506 sin40) cos 0 D
+ 192 sin 0 - - 672 sin3O + 504 sinS0]
+ (D -- t o t 0) (D -- 3 t o t 0) v (O),
(7.1)
8A~~ ( D + t o t 0) t o t 0 cosec30
'lq'-
-1
[sin30 D ~ + 14 sinO0 cos 0 D s
-- l~-H~-a~
- - sin 0 (20 -- 71 cos~0) D ~ -- cos • (48 -- 154 cos20) D
+ 24 sin 0 (1 -- 5 cos20)]
• [ s i n ~ ( D * + eot 0 D + 2
silo0)~(0)]
+ (D + t o t 8) (D -- t o t 0) ~ (0),
(7.2)
1 (D + eot 0) [sinO0 {4 cos 0 (DoJ0) 2 -- sin 0o~oD~oo -- cos 0oJ0z}]
-1 [sin30 D 4 q- 14 sin20 cos 0 D a
-- 1--6-H~-az
- - sin 0 (20 -
71 cos20) D ~ -- cos 0 (48 -- 154 cos~0) D
+ 24 sin 0 (1 -- 5 cos20)]
-]- (D q- t o t O) (S -- cot 0) ~ (0).
(7.3)
Flow of an Electrically Condueting Non-Newtonian Fluid
281
For large Ha we consider solutions of the above equations of the form
1 ,~
7 (0) -- N
7i(0)
Ht) 2(i-1)'
(7.4)
Fi(O)
~ ) ,
F(O =
(7.5)
with similar expressions for ~ (0), F (0), ~ (0) and 1~ (0).
The equations giving the leading terms 71 (O), ~a (0) and ~a (0) namely,
(D -- cot 0) [(D -- 3 tot 0) Yx (O) -- A~oJo] ----0,
(7.6)
(D + tot 0) [(D -- tot 0) 91 (0) -- 8A1 a tot 0 cosec30] = 0,
(7.7)
and
(D + c o t 0) [(D -- cot 0) 9~ (0) -- 2 sina0 {4 cos 0 (DO~o)z
-- sin 0too Dcoo -- cos 0Woa}] = 0,
(7.8)
admit the solutions
~'a (0) = sina0 [b# tot 0 +
b21 -~- 89 tOoa],
(7.9)
[
_
2A1 a ]
9a (0) = sin 0 ~~1 cot 0 + b~1 -- si-¡
(7.10)
and
I
_
~a (0) -----sin 0 ha1 tot 0 + b21
2Al
sin4 02
~ a ] .
(7.11)
Similarly, the equations giving Fx (0), Fa (0) and ~'x (0) are
(D~ -- cot OD + 6) (Da _ c o t 0 D + 20) Fa (0) = 4bl x sin 0, (7.12)
4bll0 '
(D ~ -- eot 0 D ) (Da _ tot 0 D + 6) -Fa (O) = sin
(7.13)
(D a -- eot 0 D ) ( D a -- r
(7.14)
attd
0 D + 6) f~l (0) = sin 0 '
282
D . K . MOH~d~ RAO ~
P. L. BHATNAG~d~
which admit the solutions
Fa (0) = a 91(7 cos 6 0 -- 10 cosS0 -t- 3 cos 0)
2a
-t- a=1 I7 cos'0 -- 3 c~
16
q- 13 -t- (7 cos60 -- 10 cosS0
+ ~oos 0)lo~~o~ § o~~~oo~,0_ oo~ o,
q-a~ [(cosSO-- cos O)logtan2 q-cos=O--2 ]
4b91
+ -45- sinO0,
1~1 (o) = a~ cos o + a=1 + a31 (cosS0 - cos o)
q- a 9 1[cos=0 -t- (cosS0 -- cos 0) log tan 0 -- 4 sin30b 91
3
and
1~~(0) = ~ 91cos 0 + ~d + ~~~ (co ssO - cos O)
§ ~,~ [oo~,o § ~oos~o-oos O~,o,,an~]-'~-~-91 sln" , ~ u.
For i > 1, we have
Fi (0) ----a 91(7 cosS0 -- 10 cosa0 + 3 cos 0)
-t- a=t9 I 7 cos'0 - - 23
~- cos=0 + ~16 + (7 cosS0 - - I0 cosS0
+ 3 cos 0) log tan ~ -t- as s (cosS0 -- cos 0)
~1
9
0
+ a: [~cos~0 -- co~ 0) iog ~ ~ 2 + oo~'0 --
~]
4b 91
-t- - ~ - sinS0 -/- sinS0 L i (0),
~,i (0) = sinS0 [b 91c o t o + b~.i -
{1~ sin=0 Ds + 16 sin 0 cos 0 D i
+ (77 -- 63 sin=0) D + 147 tot 0} L i (0)
9--12
s-~~dO ,
Flow of an Electrically Conducting Non-Newtonian Fluid
283
F i (0) = a~i cos O + a=i + a3 =(cos~0 -- cos 0)
_ 4 sin30bxi + sin80E i (0),
3
9i (0) = sin 0 [ ~xi cot 0 + b, i -- 1 {sinO0 D a + 8 sin 0 cos 0 E 2
+ (15 cos20 -- 2) D + cot 0} L • (0)
-
]
,
Fi (0) = ~ i cos 0 + ~~~ + a3i (cosS0 -- cos 0)
+d.i[cos'0+
(cosS0 -- cos 0) log tan 0]
_ 4 sinS0b i + sinSOi~i (0),
3
and
9i (0) = sin 0 [ hi i cot 0 + bzi -- 1 {sin'e D ~ + 8 sin O cos OD ~
+ (15 cos=0 -- 2) D + t o t 0} ~i ( 0 ) ] ,
where
lE
DL i (0) -- sin80 D z + cot 0 D + 12 -- s ~
1]
-.
1[
~]
-.
1[
~]
7i-x (0),
DL ~ (0) = s--~-0 D2 + tot 0 D + 2 -- s-~mT0 ~i-~
DL z (0) = s-~--0 D~ + cot 0D + 2 -- s-~f0 ~i-i (0).
The constants a's, b's ate detcrmined from the boundary conditions
(4.7) and (4.8).
Figuras 1 and 2 gire the strr
Case (a):
R=0"I,
linr
S=K=0.
and
Case(b): R = S = K = 0 " I .
in the following two cases:
284
D . K . MOl-tAN RAO Am~ P. L. BrIATNAGAR
8.
CONCLU$ION$
We find that for small axial eurrent the nature of the seeondary flow is
similar to the one predicted in [1], but ir we have large axial current that
is high azimut¡
magnetic field, this breaking is avoided. It is interesting
to note that
(i) the effects of cross-viscosity and viscoelasticity ate similar inasmuch
as they flatten the velocity profile in any meridian plane (Figs. 1, 2).
(ª for large angular gap, the flow separation is suppressed at smaller values
of Hartmann number (Figs. 1, 3, 4). (ª there is no secondary flow
separation in case of Newtonian fluids and for small values of Hartmann
number the flow is similar to the one described in [1] but for large values
of Hartmann number the sense of flow is reversed.
ReFeIt~CE
1. Bhatnagar, P. L. and Rathna,
S.L.
Quar. Journ. Mech. Appl. Maths., 1963, 16, 329.