FLOW
OF NON-NEWTONIAN
FLUID BETWEEN
TORSIONALLY
O S C I L L A T I N G DISCS
B'r (Mlss) G. K. RAJESWARI
(Department of Applied Mathematics, lndian Institute of Science, Bangalore-12)
Received May 30, 1961
(Communicated by Prof. P. L. Bhatnagar, F.^.sc.)
I.
[NTRODUCTION
A Vr~EOREa'ICALanalysis is made o f the flow of an incompressible non-Newtonian viscous ¡
contained between two torsionally oscillating infinite
parallel discs on lines similar to those of the corresponding Newtonian case
eonsidered by Rosenblat. We study the two cases where (i) one dise only
is oscillating torsionally while the other is at rest and (ii) both the discs
oscillate with the same frequency and amplitude but the motions at any
instant are in opposite directions. Assuming that the amplitude of the
oscillation ¡
(O/n) is small and neglecting the terms of 0 (S2/n) 2 in the
equations of mo.tion irt comparison with terms of 0 (O/n) we solve exactly
for the velocity field and pressure satisfying all the boundary conditions
completely. This approximation gives the same transverse velocity field as
in the Newtonian case obtained by Rosenblat. We find that the radialaxial flow has a mean steady component a n d a fluctuating component of
frequency twice that of the oscillating disc. We observe that the approximations made gives the velocity field valid only for small Reynolds numbers
unlike the Newtonian case, where the asymptotic expansion of the velocity
field for large Reynolds numbers has also been obtained. We have studied
the non-Newtonian effects on the streamlines of the steady part of the
radial-axial flow and on the steady component of the radial veloeity in both
the cases through a dimensionless parameter S = (ve/d~), where ve is the
kinematic coefficient of cross-viscosity assumed as constant and d is the
distance between the plates. We have compared the flow of the Newtonian
case (S = 0) with the non-Newtonian case considering both negative and
positive values of S. We observe the continuity of the effect as S changes
from negative to positive values. We have derived the expressions for the
shearing stresses acting on the plates for small Reynolds numbers.
1. We consider a body of non-Newtonian fluid bounded by two infinite
parallel plane discs which are represented by the planes z = 0 and z = d
188
Flow of Non-NewtonŸ
Fluid between Torsionally Oscillating Discs 189
in a cylindrical polar co-ordinate system. The discs perform torsional
oscillations about the axis, r - 0. If w, v, w are respectively the radial,
transversc and axial velocity components, p, the prcssure, p, the density,
t~, t~e, the coefficients of viscosity and cross-visccsity and v, ve, the kinematic
coefficients of viscosity, the equations of motion of a non-Newtonian fluid
in cylindrical polar co-ordinate system ate:
3u
bu
bu
~z
~7 + u ~ . +
W~z
r
1
rk v
p
"~
(~r ~
+
+
1 3u
u}
r ~r
~4 au b~u
2 /'~_u'~~
2u").
.~2/'~_v~ .v~
~~ ~ (~~ ~~_ l(~o~'
av +
~-7
_
-
~~~
~v
(~~ v) ~/~)':vl (~. ~)
uv
u T; + w ~ + -7
~,a2v
~' ( ~ ~
§
~~v , 1 av
+ -~-z~ -
r ~,"
~-:~+~r)
v}
>
~.~2v ,?u
~w)
~,~;- + 5 7
+ "c ( ~ T ~
-~r~---zTs + ~ + ~
v__'~
:w ~ l ~
- - 2 ~-~ ~. \ ~ r
4 ~w (~v
r~~7
A~
_
vA
r/
- - 2 az 2 ar
~2w ( 91
2 ~rTz ~F
~)
(I .2")
190
(MISS) G. K. RAJESWARI
3w
bw
3w
SF + u ~ + WSz-
1 ~p
~~Aw
~,w
p bz + V(~r2 +-~z =. + r
l aw}
u b~u
u ~2w
1 ~u s
+ 2Ve ( 4 b w bZw
t ~~ 5z ~ - - r ? r ~ z - r ~ 7 ~ - r ~ r ? z
1 3u 3w
r br br
"~ ('~ ;)}
~-=- + Si= ) ' aTa-z ~
(,3,
and
bU
U
~W
~% + r + ~z = 0.
2.
frr
(I .4)
ONE DlSC O$CILLATING
We considr the disc at 2-----0 to perform torsional oscillations o f
n and angular speed /2, while the disc at z = d, remains at test.
Following Rosenblat, we assume the veloeity field a s
.Q2 b
u ---- r - - F ( y , r),
(2. t)
v = rI2ei~g ( y ) ,
w = --
2d Qs F ( y , r ) ,
n
where
Y = ar
Further
and
r -- nt.
we assume that
l~_- .~2r2
P -- ~
P (Y, -) -r 2~=d"K (y, r).
(2.2)
The equation of eontinuity (1.4) is identically satisfied by such a choice
of velor
fir
The b o u n d a r y conditions of the problem are
u = w -= O,
v = R e ( r O e ira)
at z = 0,
(2.37)
Flow of Non-Newtonian Fluid between Torsionally Oscillating Discs
191
and
u = v =-- w = 0
at
z = d.
(2.4)
Substituting (2.1) and (2.2) in the equations of motion (1.1) to (1.3)
we have
~~F + ( ~ ) :
>y~~.
rf~F~,
~2F
?~F1 -- [eS~'g(y)]:
= -- P(Y, ~') + ~l~:F
-~s + S
(~f- r[ ( ~-~'F-"
-~)
-- S [et" dg] ~
~F ~'F 1
~y ~yS
j
(2.5~
~F
(2.6)
and
9 _~F _ 4
- b-r
r
(7)'
F-by
>~
2 [1 bP
--~~ ~~y
~"F + 2 b K
(~_)2 ~F bZF
+ _R2 ___
bv"
- ~ -- 28S
.
~3' by ~ ,
(2.7)
where
S = d~
and
R =
nd----2"v
The b o u n d a r y conditions transform to
~F
F . . . .~y
...
0,
g=l
at y = 0,
(2.8)
and
F
~F
by
g = 0
at y = 1.
(2.9)
(MIss) G. K. RAJF_.SWARI
192
On the assumption that the amplitude of the oscillations namelj I2/n
is snall, retaining only the first order terms in ~2/n, we have for the trata ;verse
velocity component
1 d2g
ig (Y) = R - ~
(2.10)
with
g (0) ----- t
and
g ( l ) = o.
The solution of (2.10) is
g (y) :
sinh Vi-R (l -- y)
sinh ~//R
(2.11)
s'tme as obtained by Roscnblat in the Newtonian case.
For small Reynolds numbers we have
v
=
(l -- y)
1 --
3 91
R y (2 -- y) sin
+ 91
y (8 + 8y -- 19.2),2 + 3y a)
nt} +
Cos ni
0 (RZ).
N:glecting the terms of order (O/n) 2, we have from (2.5) for the radial-axial
component of velocity
b2F
~ybr
[gei~] 2 =-- -- P (y, T) +
1 ~3F
R ~y3
S [e~'-~] ~
(2.12)
From (2.7) we have
F
1 ~~F
~K
~--+ -~~. = N ?y--a~y,
(2.13)
and equating the coefficient of r 2 to zero we have
1 ?P
2 ay
2Semi, dg d2g
which determines
dy dy ~ -- O,
(2.14)
PO', ~').
Let
F (y, 7) -= f ( y ) + h (y) e 2~',
(2.15)
Flow of Non-Newtonian Fluid between Torsionally Oscillath~g Discs
193
and
(2.16)
K (y, ~') = K o (y) + K, (y) e zi'.
From (2.14) we have
P
,, 2i7. [dg "~2 _ - , (~).
(2.17)
Substituting (2.15), (2.16) and (2.17) in (2.12) and (2.13) we have
2ih' (y) e 2i~"-- [gei~] "
= -- 2a (-r) -- 3S [ eir dy.]dg]2 + 11< [ f ' " (y) -f- h'" (y) e2i'],
(2.18)
and
d Ko 0')
2ih (y) e 2i~ = ~I [ f ' " (y) § h"' (y) e ~i~'] + ~,
+ ~d K1 (y) e 2i'.
(2.19)
These equations are sufficient to determine the unknowns f(y), h (y), K 0 (y)
and KI(y). The above equations indicate that the radial-axial flow has a
mean steady component and a fluctuating component of frequency twice
that of the oscillating plate. From (2.11) we have
l rcosh2ty,--cos~yl] + l [cosh2t(l +i)Y~--l]
[g(y)]~e~i~" = 2 [- ~-sh-~-- cos
2 [ -cosla~(l~ 7qS~-)SL- j
e 2ir,
(2.20)
and
~J
fi~ [cosh )tY~+ cos ~Y!1
- - cos
egi~ = -4- [ ~ - 0 - ~ ~
ia 2 rcosh•(1 + i ) y , + 1] e ~i1",
§ 4- L--cosh a (1 § i) ----/- J
where
;~ = ~/2---R
and
Yl = 1 - - y .
(2.21 )
(MIss) G. K. RAJ~SWAm
194
Sabstitution of (2.20) and (2.21) into (2.18) yields
f ' " (Y) - Lo + 3S
a ~ r c o s h Aya + cos aya]
1 ]'cosh ayl--cos ay~]
(2.22)
and
1 h"' 0 ' ) - _,h
9"'
R
0')
['cosh A (1 q- i) Ya q- t ]
= La + 3S ~ L ~-sfi-~(i 7k7-/)2---] ]
I [-cosha(l 9 1
--92 [. c o s h a ( i - l - i ) _ 2 ]
1]
]"
(2.23)
where we have taken
2~ (~') --'-- Lo -9 Lle ~i'.
The boundary conditions (2.8), (2.9) reduce to
0
at y = 0
and
y = 1,
(2.24)
h = h' = 0
at .v-~ 0
and
y = 1.
(2.25)
.f= f'=
and
The solution of (2.22) for the mean steady c o m p o n e n t under the conditions (2.24) is found to be
1
1
4~ (1 --
q- ~1y (1 -- y)~ (cosh h § cos )t) -• (sinh h + sin )t) -- / ya (l - y)
]
+3Sg
3y ~ + 2y s)
.
• [(2y a - 3y ~ § 1) (sinh ~ -- sin )9 -- (sinh ay, -- sin ~Yl)]}
3S -8- y (1 -- y ) ' ,
(2.26)
Flow of Non-Newtonian Fluid between Torsionally Oscillating Discs
195
and
f'(Y)=
[
1
] {[
1 (cosh Aya -+-cos Ay~)
coshA--cosA
--4
+ 2~ y (1 -- y) (sinh X ,+ sin A) + 1 (1 -- 4y + 3y ~)
• (cosh ~ + cos )t) -- I y (2 -- 3y) ]
A2 [
6
+ 3S -ff . cosh Ay1 -- cos AyO + X y (y -- 1)
• ( s i n h a - - s i n h)]} + 3 S 8 ( 4 Y - -
1 -- 3y2).
(2.27)
For small Reynolds numbers we have
9f(Y) ~]20R Y2 (I -- Y)~ {(3 -- Y) + ~
R~
[-- 5Y5 + 3 5 y 4
-- 105ya + 175y 2 - 133y -- 21]
+_R_~ [ y a _ 5y~.+ 1 0 y - 10)]} + 0(RS),
(2.28)
and
R
{ (6--
.f'(y) ~ l ~ Y ( 1 - - y )
1 5 y + 5 y 2)
R~
3780 [42-- 1365y2+45 (--y<-k7yS--21y4+35ya+7y)]
R2S
7
[7 (v 4 -- 5y 3 + 10y 2 -- 10v) + 201} + 0 (RS).
-
(2.29)
The solutions for S----0, corresponding to the Newtonian case are in
agreement with those of Rosenblat.
The unsteady component of the radial-axial flow is given by the equation
(2.23). We have
4 [cosh A (1 + i) -- 1 ] [cosh a (1V,2
+ i) _ 1 _ ~_(12V
~ ~ + i)
x (~yV/~--+ i)].] h (y)
sinh .....
196
(MIss)
=
+
(3S--~
G. K.
RAJ[]w~a~x
(1 4- i)] [ y ( l
~) {[,_ cos,,,,,_~_,
§ cosh ~(1 § i))
x/2
a (1 § i)
"~/2 y (cosh ~(1 § i ) + A ( 1 + i ) sinh
1)
2
• (l 4- i) sinh A (1 4-. i) cosh a (tv,24- i) .v
2
;~(1 + ~)
x [(-- 1 " cosh )~(1 -4- i) cosh ~ (1 + i)
1
,
- x/2
- - V,--~
• sinh A(l + i) sinh
A (1 § i) sinh ~ (1 § i)
~/2
§
a (1 § i)
)t(l t i )
x cosh a(la/2+ i ) 4_' a (lv'2+ i) sinh
v'2
sinh A (1 § i)]
a(t + i) J
X (1 --cosh A(1 § i))
§
['~
x
sinh ;~(1~/2§ i)
sinh A(1 -4- i) sinh A(I + ii
'
'
y } __
"V'2
{
Y
v/2
A(l + i)
1
~-2
sinh ;~-(1
- / 2§- - i)
[
x ~ -- sinh a (1 4- i) Ti § sinh )~(1 x_ i) cosh
A (1 § i)
y
t
-~/2
(2.30}
For small Reynolds numbers we get,
h 0') ~ ~R Y.o0 . v) .~-{(3.
y)(l
3iRS/
iR (30y 3 -- 150y 2 § 1673, + 29)}"1 + 0(Ra).
,J
210
(2.31)
Flow o f Non-Newtonian Fluid between Torsionally Oscillating Discs
197
The shearing stresses for the velocity field (2.1) are given by
~u [(q
Trz = ~-~
T0z = ~
~v[
]
/~--2~e~
~u]
,
and
3u ~v
For small Reynolds numbers the shearing stresses acting on the plate
z = 0 are given by
r ~2~ R [ 1 + c o s 2 n t q - R ( 23 9S)q. - ~
sm2nt ] -k-0(Ra),
Trz = q d n 20
Tez--
t~ dl2
1§
cosnt--~-sinnt
+O(RZ),
and
r2 0 8 R [
T r a = - - / ~ e d 2 n 40 cos3nt+3cosnt
+ P. ( 3 s - 6--Y6)
181,..tsm
3nt 4- sin
,,,)] + o ~R~~.
The shearing stresses acting on the disc z = d are
9)
R [ 1 + c o s 2 n t + R ( 13S-/-1~~
sin2nt ] +0(Ra).
Trz = ~ dr f2n~ 30
Tez
=
--
'
q ~ f2
[( 1 - - ~~~~
3 6 0 1 cOS nt
.
+ g sin nt
]
+ 0 (R3),
and
r ~ Q3 R [
Tr0=--t~ed 2 n 60 cos3nt+3cosnt
q- R(3S § ~10)(sin 3nt + sinnt)] q-0 (Ra).
198
(Mlss) G. K. RAJESWAm
Figure 1 represents the effect of non-Newtonian term on the typical
streamlines of the steady component of the flow. We havc drawn the curves
for S = - 0-5, - - 0 . 1 , 0 and 91
We observe the continuity of the
effect as the parameter 'S' changes from negative to positive values. The
negative non-Newtonian coefficient flattens the streamline while the positive
coefficient makes it more curved. Figure 2 shows the dimensionless steady
0'4
02
o
I
2
FIG. 1. One dise oscillaling:
S-:-0.5
- - 0 - 1 , 0, -i-0"05.
3
4
5
Streamlines of steady radial-axiŸ
f~/
,i
f.(~
Icr R = = 5
-I-0
(
.O.B
'o'e
0 -
"~".".,,
f'
I
l
-'0"6
Flo. 2.
One dise osr
1
~
b
I
"-0"2
l
|
O'O
Steady radial vr
0"2
I
~
I
- 0"6
f ' for S --- -- 0.1, O. -t- 0.05.
and
F l o w o f N o n - N e w t o n i a n Fluid b e t w e e n Torsionally Oscillating Discs
199
c o m p o n e n t of the radial vclocity, namely, f ' for S . . . . 0-1, 0 and + 0.05.
We find.f' = 0 at y = 0.45 for S = -- 0"1 and at v = 0.48 for S =- 4- 0.05.
BOTH DISCS OSCILLATING
3.
We now consider the case where the two discs are oscillating with same
frequency and a n g u l a r speed but in opposite directions. The b o u n d a r y
conditions in the present case are
u = w --- O,
at z = 0,
v = R e ( r O e int)
and
(3.1)
u = w=
O,
at z = d .
v = -- R e (rl2e int)
The equations o f m o t i o n (1.1) to (1.3) can again be t r a n s f o r m e d by the
choice o f the velocity field (2.1) in a similar way and for the transverse flow
we have
d~g -- i R g = 0
as previously but
(3.2)
with g ( 0 ) =
1, g ( 1 ) = -
1.
The solution n o w is
g (y)
sinh ~ f R Yl - sinh a/i-R Y
sinh V'iR
'
=
(3.3)
once again the same as that o b t a i n e d by Rosenblat.
For small Reynolds numbers this yields the velocity profile
r-~ ~ (I -- 2y)
1 -- )-6-0 y (1 -- y) (1 + 3y -- 3y z) cos nt
+ ?i- y (l -- y) sin 1 t
+ 0 (113).
(3.4)
F r o m (3.3) we have
~~
A
+ c~
[g (Y)]zeZi~ -= cosh ~ -- cos
I
[~os~~(, _ ~,_ ~o~~,l_ ~,]
c o s h ~a(i + i ) +coshA(l
+i)-I
Ir
|cosh
L
a
(1 4 - i ) ( 1 - - : y ) - -
"1
1)/
d
(3.5)
200
(Mlss)
G.
K.
RAJESWARI
and
A
A
[dgl2e2i~ ~ z c o s h ~ + c o s ~ [
A(l_2v)+cos~(l_2y)]
l_~J
=2coshA--cosA
cosh~.
.
_
A
-§ l a 2 cosh ~ (1 + i) + 1
2 coshA(1 + i ) - - 1
' i)(1 -- 2 y ) + l ] e 2i"
• [ cosh ~A(1 -~-
(3.6)
_1
Substitution of (3.5) and (3.6) in equation (2.18) yields the cquations
necessary for determining the steady and the fluctuating componcnts of the
radial-axial flow under the same boundary conditions (2.24)and (2.25).
The steady components of the axial and radial flows are
A
A
1 cosh2 + cos~.
f(Y)=--2 cosh~-- cos A
3S
--I
(3y ~
{("2 )C
2v 8)
--
• (2~ sinh ~A -- cosh ~) + ~1 sinh ~A(1 -- 2y) + yeosh A
-~sinh2] + ( 3 S ~
--
A 9 A
+ 1) I(3y2-- 2ya) (cos~ -- ~sin~)
1 sin
" ~.
A(1 -- 2v)
--ycos~ +~sin
~,
.
(3.7)
and
A
A
1 cosh ~ + c o s )
f'(Y)=2coshA--cosA
A~
[
(2
{ ( 3 S ~ - - - 1 ) 6(y 2 - y )
sinh~
A
• [6(y'--Y)(C~
A
-oos~(~_~~~]}
~(l
2y)]+
5+
- 2 sin
" 2) + cos ~.A
(3.8)
Flow of Non-Newtonian Fluid between Torsionally Oscillating Discs 201
For small Reynolds numbers we have
R
f(Y) ~ 6 b Y ~ ( 1 . y) 2. (1
R 2
. 2y) ~9. -- 1 + 3 •
[-- 3 -F 20y
R2S
(3,8 -- 2y 2 91 2y -- 1)] -- -28- [4y= -- 4y + 3]} § 0 (R~),
(3.9)
and
r
R 2
.f'(y) ~.. J~,@~Y(I-- y) ,/(I -- 5y q- 5y 2) +-~.~ [60(y~--3yS+4y'--3y 8)
+ 5y(13y -- 1) -- 1]
R2S
28 [ 7 ( 4 y ' - - 8y 3
(3. ]o)
§ 7y" -- 3y) + 3] } -k 0 (Rn).
The fluctuating component of the radial.axial flow is given by
a(1 + 0
- - - ~ U - h (y)
=
iLt [~(1
- - q-i) .1' -- sinh a (l.v,2H- i) y ] --F D A (1.V,2
q- i)
T
V'2
L
A~
i
3S~--k~
x [ l _ c o s h ~ ( l V'2+ i ) Yj + coshA
(1 + i) -- 1
x
{
-
a ( l + ~)
L
v'2
Y § ~-2 sinh a (1 + i) cosh ~ (1 § i)
x cosh •(1v 7-~
+ ' t) y__ sinh 2 •(1v'2+ i ) sinh a(1"V'2
+ " t) y
- v'---2•osh 7 2
( 3S --4-~~-- ~)
§ i) sinh~ (1 q- i)(1 -- 2y)
+ cosh A(I + i ) - - 1 "
~ (1 + i) cosh ~ (1 -F "
{@2 sinh2
x/2 t)y
a (1 + i)
;~(1 q- i)
.~~
y eosh ----~
1
~
}
V,-~ sinh (l--Fi) (1 --2y) ,
(3. li)
202
(Mlss) G. K. RAJESWARI
where
11"
F
+0
t~S [cosh a (I 4- i) -- 1] 12
cosh aO~/2
L
2
2
=
;~ (1 + i)
2t (1 + i)]
a/2
sinh a / f - - I
~2 -- b2
i
3S-4-
~r sinh ~_ (1 ~-, i) sinh ~
(1 4- i)9
A(l+i)~__2_cosh~A(l.+i)sinh~_2h (1 + i ) }
~, +i)cosh a (1V2-+ i)
+(3S~+ ~) {x/2sinh~(1
A (1 + i) + "~sinh 2 a ( l + i ) { l
• sinh ~-~
2
\
--cosh A(I
A(l+
x/2
A(1~/2+ i ) sinh ~--~
A (1 + i ) }
(3.12)
and
D [cosh ~(1 + i ) - --2
II [2
c o s n.)t(1
- - - - - --+-i)
-~~
2 t (~-21 - i) sinh a (lv/T~
2 i)]
- ---a
(I
A(1 + i))
1 sinh ~A(1 +i) sinh ~-2
A (1 ~-i)
+ ~-2
'
--
3S -$ +
• {cosh*~(I + i ) ( 1 - - c o s h A(I____+V,2/--))
+ ~
sinh
(1 + i) cosh ~ (I
i) sinh ~--2 (1 + i) .
(3.13)
In the present investigation
sin:dl Reynolds numbers only.
term on the typical streamline
are oscil!ating in the opposite
we find that our approximations are valid for
Figure 3 shows the effect of non-Newtonian
of the steady flow when both the discs
directions. The flow is symmetrical about
Flow of Non-Newtonian Fluid between Torsionally Oscillating Discs 203
the plane y = 89 Curvesare d r a w n f o r R = 1 0 a n d S = - - 0 . 5 ,
--0.1,0,
-F 0.05. The effect ofnegative non-Newtonian coefficient is to flatten the
streamline and that of the positive non-Newtonian coefficient is to make it
more curved. Figure 4 represents the steady component of the radial flow
I0
y
0.8
0"6
(
0"4
0-2
f
I
2,,2'-s=-o.,
2'
I
Fto. 3.
S ~--0-5.
f
4
2
Two discs oscillating:
- - 0 . 1 , 0, - F 0 . 0 5 .
3
4
5
6
Streamlines of steady" radial-axiat flow for R = 10 and
L7
1,0
~~..
L
-0"06
F ~ . 4.
I
L
-0"04
t
I
-002
I ~ * I ) ~
0
I
0"02
").04
l"wo discs oscillating: Steady radial "~c('cts. / l't:~ S =
~-
I f'(Y)
0"06
--O'1. O, -'- 0.C5.
204
(MIss) G. K. RAJESWARt
for S . . . . 0.1, 0 and -i-0.05. For S ~ - 0.1 it is zero on the planes
v - - - 0 . 2 6 and y = 0.72. There is a radial outflow between these and the
walls due to the predominance o f ccntrifugaI action. For 0"26 < y < 0-72
there is an inflow as the pressure gradient is then the rnain factor. For
S = -t-0.05 itis zero on the plane y-----0.29 and y-----0.71.
The shearing stresses acting on •hc discs _----0 and z-----d are equal
in magnitude but opposite in sign.
For small Reynolds numbers we have the shearing stresses,
Trz = t~ dr f22n 1R20[4 4- R (12S - !) sin 2nt] _L 0 (R~),
Toz . . . .
2/~~/s
+O(R~),
1 ' ~/2-0 c o s n t - - - ~ s i n n t
and
1.2 Da R [
T r e = - t Z e d 2 n 120
8cosnt
R(12S--)
5
sinlTt
-- R(12S - 1) sin 3nt I + 0(RS).
SUMMARY
The flow of an incompressible non-Newtonian viscous fluid contained
between two torsionaily oscillating infinite parallel discs is invesligatcd.
The two specific cases studied are (i) one disc only oscillates w~ilc tl~r, other
is at test a,-td (ii) both discs oscillate with the same frequency and amp]itude
but in opposite directions. Assuming that the amplitude of oscillation,
O/n, is small and neglecting the squares and higher powers of I2/n, the
equations of motion have been solved exactly for velocity and pressure
satisfying all the boundary conditions. The effect of both positive and
negative coefficients of cross-viscosity on the steady componen•s of the
ftow has been represented graphically.
ACKNOWLEDGENENT
The author is highly indebted to Prof. P. L. Bhatnagar for his censtant
guidance and kind help t h r o u g h o u t the preparation of this p3per.
REFERENCE
Rosenblat, S.
711-61.
dourn, of Fluid Mecll., 1960, 8, 388.
Printed at the Ba.ngalore Press, Bangalore City, by T. K. Balakrishnan, Supermtendent,
aud Published by The Indian Ar
of Sciences, Baagalore.