SUPERPOSABILITY
FLOWS
IN
OF $TEADY
AXI-SYMMETRICAL
A NON-NEWTONIAN
FLUID
BY (MISS) S. L. RATHNA
(Department of Applied Mathematics, Indian Institute of Science, Bangalore.12)
Rr
November 2, 1959
(Communicated by P. L. Bhatnagar, F.N.L, F.A.SC.)
LEw (qa, Pa, Dl) and (q2, P~, D2) be the two flows of an incompressible fluid
of uniform density p, kinematic viscosity v and kinematic cross-viscosity
ve, where q, p, D denote the velocity vector, the presssure and the potential
from which the external forces are assumed to be derive& The two flows
are said to be superposable, if a pressure (Pi + P2 + 7r) can be found such
that (qa + q2, Pa + P2 + % ~2~ + D2) is also a solution of the Stokes-Navier
equation with the necessary modification in the initial and boundary conditions.
The equations of motion for a non-Newtonian fluid are given by,
(~ui + ui, juj) = t~j,j
Pk-~" +fi,
uj,j = 0
(0.1)
(0.2)
ti i being the stress tensor given by
tji = _ p~ji + Fadji + F2didj,~,
(0.3)
where f i is the body force,
di i = uj,~ + ui, j
(0.4)
is the rate of deformation tensor, Fa and F2 are functions of some material
constants and the second and third invariants of the rate of deformation
tensor di i.
By taking F a---/~ and F2 =/~e we recognize Fa and F~ to be the
coefficient of viscosity and coeiticient of cross-viscosity.
1. For the problem on hand it is convenient to use cylindrical coordinate system (r, 0, Z) in which z-axis is taken to be the axis of symmetry
and r is the distance from the axis. The velocity components at any point
~,s
155
156
(Mtss) S. L. RATHNA
are given by uf, u o and Uz, where the velocity vector q does not necessarily
lŸ in the meridian plane, so that
q = udr + Uoio + Udz,
where for axial-symmetry the velocity components are functions of r and
z only besides the time t.
(a) Equations of motion.wThe equations of steady motion in cylindrical
co-ordinates, with terms of azimuthal variation neglected, are,
[
~ur
~ur
uo' ,
Okur ~r + uz ~z - r - - ] ( ?uo
~uo u~U_o)
bUz~
~r
~Prz
-5 - ~
q-
Prr--Poo
r
'
?Pro
~Poz
2Pro
-- ~r + - b z - - + - - r - '
O Ur~r + U z ~ z +
( ~Uz
~Prr
(1.2)
3Prz
~Pzz Prz
~r + ~ z + r '
P Ur yr + U z T z - / -
(1 . 1)
(1.3)
along with the equation of continuity
~Ur
Uf
~-r- + r - +
bUz
~z
-
-
O,
(1.4)
where the stress matrix PO and the rate of strain matrix e0 in the case of
non-Newtonian fluids are related as (Braun and Reiner 1)
( PrrPrOPrz~
(!
PorPooPoz) = - - p
Pzr Pzo P z z /
0 i)/errer~
1
q-I~~eoreooeoz )
0
\ e z r ezo e z z /
/erreroerz~
-~ t% ~eoreooeoz )
\ e z r ez Oe z z /
letrero erz)
~k eoveooeoz
,
(1.5)
\ e z r ezo e z z /
where eij = eji
err = 2 -3r
~ur
e r o - 3u
~ro
uro
eoo = 2 uf
r
e r z - bur
~z + 3Uz
~--r
ezz = 2 ~Uz
•z
e~
(1.6)
3u
bzo
Using the relations (1.4), (1.5) and (1.6), the equations of motion (1.1),
(1.2) and (1.3) for axially symmetric steady flow become,
Superposability of Steady Axi-Symmetrical Flow~ h~ Non-Newtonian Fluid 157
~ur + Uz ~u~
Uo2
-~ r
u~-ff
~%.2
pl @
~r + v ( V~u r - r ~u~)
" +2ve { 4 ?ur
~~ . -~r
+
2r
§ Ve { 2 ck~ ar
uO-?)
~r ('5u~
~ \(bu,
\ ~ r - - ~ ) - t - Y z5Uo ~z
~ r --U-r~
1 (b_Uo'~Z§ I'~"0
rk~z)
~bl0
)U 0
uf & + Uz ~z +
k~r
~~"o~
(1.1 a)
u;) bz~-)'
UrU 0
r
p ~0 + v
V~Uo-- rZ
~Uo ~ (~ur
+ Ve
~Uz'~
~2u~.
~Uo (~~~ ~.q
~z + k ~z -F ~r ]
k ~z -q- ~r ]
- - 2 ~ z - ~r \ ~r
-- 2 bzuz ( bu~
b,'bz k 91
--2
;o)
~z ~
5ur
br
4~Uz{~_uo__~)
r bz k br
( 1 . 2 a)
-I--~ k-~z -q- ~r ] -~z ) '
and
~Uz
bu z
uf ~ . + uz ~~r_
1
P
@ + vV2Uz + 2ve .f4 ~Uz . b~Uz
. . . . uf. 39ur
oz
( - ~z
bz 2
r 5rbz
+ \ ~ z + ~r] \ ~ z ~ + ~r~z]
1
~ur
~Ur __ 1 ~Ur . 3U_z~
~2tŸ0 ( 3 U o
+ ~.,~
- ?)}.
u r ~U2z
r br ~
f~u~ (2 ~~2u~q- ~-~-2u~
ve ,(Yz
bre)
(1.3 a)
(MIss) S. L. RATHNA
158
Expressing the velocity components in terms of the
function ~, we have
uf -- r ~z ' uo = Z (r, z), Uz . . . .
Stokes" stream
r ~r ,
i.e.,
,=
(~ ~)'~ § ~'. § C - -r1 ~~r) iz"
(1.7)
(b) The conditions of integrability.--The conditions of integrability o f
the equations o f motion in temas o f the Stokes' stream function ~b turn out
to be,
rJ(~zr-'Er~~)--2X bzbX
--~ vE'~bq-2ve{2J(~zr-2E'~)--2j(~r-ZEr2~b
--(~
bz
E~~
)' 1~~
r "zbz
E4~ +
2rJ
- _
)
_
r bZ br
3X ~X
r ~ ~z
~~ }
(1.8)
and
1
riJ(~z
rXr)
(~~ ~)+ {~(~~~~+~r~r~(~~~~)
J lkzrJ
wherc
E 2 _ ~~
~2
=~+~--r~r
1
-b r ~ ~r ~r~z q- r z ~ ~zZ~ '
(1.9)
Superposability of Steady Axi-Symmetrical Flows in Non-Newtonian Fluid 159
and
3'z
b~
1
V~ -------3r" -~- ~ + q ~r"
(c) Conditions for superposabilio' o… two motions.--Taking the two
motions given by
qs
= ( l b~bs~
( 1 3~bs"~
k-r ~ ) ir + Xsio + \ - - r ~r-) iz
where s = 1, 2 corresponds to two motions. Using the conditions of
integrability (1.8) and (1.9), we obtain the following conditions for superposability of two motions:
bX2-- ~~, bX_.
r~ (~ r"E~~9 + r~(~~ r-~y~9 -- ~~~-~
~zx
-- 2ve [ 2 j (~ar-2Er'~b2) + 2r J (~r-*Er*~bx)
--
~~,~ ( ~ )
'
+ - -r- -~z- ~r
+~ ~-~~r
r12 ~~b2E4r
,~z
~,~~,.
--~-~
q~~
ve[r2J(E2x~X*) q- r~J ( E ~ 2 ! l )
z
-~~~~,~
3 X~.bXt
r ~ bz
3X~ bX2
r ~ 3z
-
~Xj E~X2 -- 5 ~3X~2 E2Xl t
5 -~-
and
J ( ~ rXr~')§ J (~~ rrX9
+ 2J
r
A4
z
0. lo)
(M~ss) S. L. RATHNA
1~0
+2J\
z );/+2J~b;
+ 2 a~b1 3~X2
~)+2J
2 ~6z ~~Xl
2 ~~b~3~X2
r ~r ~rbz + r ~r ) r 3 z -+- r bz
~r)
2 ~_~~_~2X1)"
bz ~ + r ~z ) z - ) "
(1.li)
We can easily show that the eonditions (1.10) and (I .11) are both
neeessary and sufficient for the two axially symmetrie flows of a non-Newtonian fluid eorresponding to the two stream functions ~bxand ff~ to be superposable.
2.
CONDITIONS
FORSELF-SUPERPOSABILITY
OFA FLOW:
Using the conditions (1.10) and (1.11), it follows that the flow with
stream function ~b is self-superposable if and only ir,
rJ (d/ r-~E2~b'~ - - 2X ~X
xz
r
]
bz
-----2ve [2 J (~ r-~Er 2~b) -- 2J (b~
~ r-2Ez~) q _ _2
r bb~b
zbr(E/r
z
r z ,,z
_]
r~J
3x ~x
r ~ ~z
5 ~X
1
~.- E'2x
E"-XzXŸ
~X. X
(2.1)
/
and
J(z~rrX)= !Ve[)J (V2~b
r z
+ 2 b~b ~~X
--Ÿ2J(~ ~)-~-2J (3~ el)
2 ~~ ~~X3
r ~r ~rbz + r b z
3.
~z-2
"
A
(2 2)
PARTICULAR
CASES
in the conditions of superposability and selfsuperposability by putting Ve = 0, we obtain the conditions of superposability
(a) N e w t o n i a n f l u i d s . - - I f
uperposability of Steady Axi-Symmetrical Flows in Non-Newtonian Fluid 161
and sdf-superposability f o r a Newtonian fluid as obtaincd by Lakshmana
Rao. ~
(b) Velocity rector lies in the meridian plane.--Wc can find the conditions of supcrposability and sclf-supcrposability in stcady axi-synunetrical
flows when the velocity rector q lies in the mcridian plane,
Le.,
(3. l)
q = u,i, + uzi,
or in temas of the Stokes' stream function
~ / iz
and the vortir
(3.2)
is given by
= 1-E*~.
(3.3)
r
In this case the equations of motion ate given by (1.1 a), (1.2a) and
(l.3a) with X-~O.
The condition of integrability (1.8) becomes
rJ(~ r-'rE'~b)
= vE*~b-t-2ve{2J(~zr-SEr'~b)--2J(~fr-'Erg~b
)
while the condition (1.9) is automatically satisfied.
The condition of superposability (1.10) for two axi-symmetrical flows
with stream functions r and ~2 is given by,
~~~
r'~r~/
162
(Mm) S. L. I ~ T ~ A
r~
r ~ ~z
r 3z ~-r k,-r-=-1
(3. 5)
-]- r ~z br \ r ~ J ( "
The eondition (1.11) is automltically satisfied. The condition for
self-superposability of this flaw is given by:
~(~rT~)
= 2"e
j
r-~ 2~ -- 2J
r-S
r
-- lrz (~~-~zE4~ ~- 2 bz ~-~br(E~~~~)I "
(3.6)
From the condition (3.5), we scc that a rotatŸ
flow with suifax 2, is superposable on an irrotational flaw with sttflix 1, if
,(~, r-T~')
--2ve
r
J (~1 r-2Era~b~)- 2J ( ~
r - ' ~'~b2
Je
Using the conditions (3.5) and (3.6), we can prove the following theorems.
THEO~M 1.--AII irrotational motions are self-superposable and any
two irrotational motions are superposable on each other.
THEOREM Z--The two flows will be superposable if the vorticity of each
flow is proportional to r. Also ah axi-symmetrical flow will be self-superposable if its vorticity is proportional to r.
TH~Oed~M 3.--If oJl/r is constant along the stream lines of flow 2, while
~=/r is constant along the stream lines of flow 1, then the two flows are
superposable if in addition the velocity components and vorticities satisfy the
condition
2 (uxvx -t- u2v2) q- r (vx~ x -k v=~z) -- O.
(where ah and aJ~ are the vorticities of the two flows).
Superposability o f Steady Axi-Symmetrical Flows in Non-Newtonian Fluid 163
TaEOREM 4.--An axi-symmetrical flow is self-superposable, if the vorticity a, is of the forro oJ = f (4')r and if the stream function 4' satisfies the
condition
~ (~ ~~)
E4ff = 2r ~
Whilo the condition of integrability reduces to
vE4~b = 2ve ~ t-4 ""
7r~
t~ ~~.
Also a rotational flow with stream ftmction ~2 is superposable on ah
irrotational flow with stream function ~b~, if in general
~s ; f(~ 91 r
~d
The corresponding theorems for a Newtonian fluid have been given
by Bhatnagar and Verma. s
I ana deeply grateful to Prof. P. L. Bhatnagar for suggesting the problem
and for Iris kind guidance throughout this work.
R~RL~CE$
.. Q ~ . Jo~,. Mech. a~t App. Math., 1952, 5, 42--53.
.. Proc. Ind. Acad. Sct., 1957, 45A, 418-23.
3. Bhatnagar, P. L. and Verma, lbid., 1957, 45A, 281-92.
1. Bmunand Reiner, M.
2. LakshmanaRao, S.K.
P.D.