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The stability of two stratified non-newtonian liquids in couette flow
Journal of Non-Newtonian
161
Fluid Mechanics, 24 (1987) 161-181
Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
THE STABILITY OF TWO STRATIFIED NON-NEWTONIAN IN COUEl’TE FLOW
LlQUIDS
N.D. WATERS and A.M. KEELEY Department of Applied Liverpool (Ct. Britain)
Mathematics
and Theoretical
Physics,
The University
of Liverpool,
(Received May 16, 1986; in revised form August 21, 1986)
Summary
Consideration is given to the stability of the interface between two Oldroyd liquids with shear-dependent viscosities, flowing in distinct layers while undergoing plane Couette flow. Results are presented as regions of stability in the plane determined by the logarithms of the viscosity and depth ratios. The work of previous authors for two Newtonian, power-law and constant-viscosity Oldroyd liquids is revealingly presented in a similar fashion. It is found that the dependence of the viscosities on shear-rate can drastically affect the regions of interfacial stability in a way over and above that due to just a change in the effective viscosity ratio. It is also found that for the Oldroyd liquids this viscosity variation affects the stability when it is present in the less-viscous layer.
Introduction
There are many industrial and manufacturing processes in which two or more immiscible liquids flow in contact but in distinct layers. Examples range from the use of water to reduce the pumping costs of crude oil to the coextrusion of two or more polymers to produce materials with unique properties. The possibility of instability at the interface of such flows is well known. Apart from the more obvious effects due to density differences Yih [l] and Hickox [2] have found that for simple flows viscosity differences alone are sufficient to cause instability. Hooper and Boyd [3] have shown that unbounded Couette flow of two viscous liquids of equal density and negligible surface tension is always unstable. They also show that surface tension is always stabilizing (while density difference can have either effect). 0377-0257/87/$03.50
0 1987 Elsevier Science Publishers B.V.
162
They suggest that such instabilities have a slow growth rate and are sensitive to surface tension and may not always be obvious in experiments. Li [4] considered the Couette flow of two constant viscosity Oldroyd [5] B liquids and found that the presence of elasticity can stabilize or destabilize the flow. Khan and Han [6,7] carried out theoretical and experimental investigations for the flow of two liquids in plane Poiseuille flow. In their analysis they employed two Coleman and No11 [8] second-order fluids, which exhibit constant viscosities, and found that two-dimensional interfacial instabilities occuring in the plane of the flow depend only on the viscosity ratio and not the elasticity ratio, whereas two-dimensional instabilities in the plane perpendicular to the flow depend on both ratios. Recently Williams and Williams [9] have considered a two-dimensional stratified flow of two liquids in a converging channel. The liquid in the centre of the channel undergoes planar extensional flow with the liquid near the walls acting as a lubricant. The flow is regarded as locally parallel and that of the lubricant as approximated by Couette flow. They acknowledge that, apart from the effects of density and surface tension differences, interfacial instabilities may be caused by viscosity difference or the presence of elasticity In 1983 Waters [lo] extended the analysis of Yih [l] for two viscous liquids in Couette flow to two inelastic “power-law” liquids. He demonstrated that shear-thinning can have a significant effect on the stability of the interface. In order to assess the combined effects of shear-thinning and elasticity, commonly exhibited by non-Newtonian liquids, we consider in this paper the Couette flow of two liquids which can both be characterized by the four-constant model of Oldroyd [5]. Similary we are able to demonstrate that the effect of shear-thinning can be significant, and that this effect is not solely due to the fact that the viscosity changes but also to the ratio of the power-law indices for two power-law liquids and to the interaction between shear-thinning and elasticity for the Oldroyd liquids. We also demonstrate that for the Oldroyd liquids it is the shear thinning of the less-viscous layer that affects stability. Theory
The primary jlow
We consider two different elastico-viscous liquids, both characterized by Oldroyd four-constant models, flowing steadily in distinct parallel layers between two horizontal parallel planes when one plane is at rest and the other is moving with constant velocity U in its own plane, and there is no pressure gradient.
163 We refer to rectangular Cartesian coordinates xi such that the interface between the two liquids is x2 = 0. The plane x2 = d, moves in the xi direction and x2 = -d,, is the stationary plane. We assume that the velocity distribution has the form (
uj(x2>T
OY
‘)7
(1)
which satisfies continuity immediately, where j = I for the upper layer and j = II for th e 1ower layer. Both liquids are characterized by equations of state of the form TiA= -jgik
+ Pil, 7
(2)
(3) where rik is the stress tensor, pik is the extra stress tensor, Z$’ is the rate of strain tensor, p is the excess pressure, gik is the metric tensor, 77 is the viscosity at zero rate of shear and A,, A, and p0 are material constants, each with the dimensions of time and 9/9t is the Oldroyd [ll] upper convected time derivative. We shall append the suffix j denoting the appropriate layer when necessary and drop the overbars when we non-dimensionalize using a typical speed U, typical length d, and the density pi of the upper layer as follows x = xi/d, p =P/p$J?
u = U/U,
y = x2/d,,
Pil, = PiX_/PIu2
S, = Uh,/d,,
(4
9
S, = UX,/d,
M=
U/+,/d,.
It is easy to show that the only non-zero 2P(S,
stresses are given by
- S,)(du/dy)=
(5)
‘11 = R(1 + MS,(du,‘dy)=)
’
1 + MS,(du/dy)= 1 +Ms,(du/dy)= where R = Up,d,/qI
is a Reynolds
Pi = 1 and Pit = ~~/ni With no pressure
1’
number
based on layer I,
= P.
gradient
the equations
of motion
reduce
to
llp12/ay = 0 for both layers which on using (5) gives d=u = 0 in each layer, dy=
(6)
164 subject to the boundary conditions ut(1) = 1,
u,,(-6)=Owhere6=d,,/d
I
(7)
and the continuity of velocity and stress at the interface u1 (0) = uil (0) and ( pi2 >i = (f& at Y = 0. The solution of (6) satisfying (7) and (8) is
(8)
ui=aiY+b
(9)
(j=I,
II)
where b = 1 - a, = a,,8
(10)
Equation (1 l), on using (lo), becomes a quintic in a, or a,, which is readily solved using, for example, the Newton-Raphson method. Stability analysis
We follow Yih [l] and Waters [lo] and consider a two-dimensional infinitesimal disturbance for the above steady solution without implying that Squire’s theorem [12] (cf. Yih [l]) necessarily holds +. For each layer we assume a perturbed velocity distribution of the form (u(y) + u*(x, Y, 0, 0*(x, Y1 t)) (12) and a pressure p +p*(x, y, t), where u(y) is given by (9). (10) and (11) and p is a constant. We also assume that the relevant non-zero stresses have the form Pll +P,*,(%
Y7 4
P&(X,
Y, 09 Pn+P:2(x,
Y1 d
(13)
where the starred variables are all small and non-dimensional. The equation of continuity permits the definition of a stream function +(x, y, t) where
u*_aICl aY
’
u*=
3.
(14)
As is customary we assume without loss of generality that all perturbation quantities contain an exponential time factor (real part understood) of the form ($9 P*, Pl:, PL
p&J=
[e(u),
f(Y),
Xexp[ia(x
C(Y),
- ct)]
G(Y)9
4CY)l (15)
+ Squire showed that for a Newtonian liquid the minimum Reynolds number for amplified Ssturbances occurs for a two dimensional disturbance; Yih [13] extended this to include two stratified Newtonian liquids.
165 where t = Ui/d is the non-dimensional time, a is a real positive number proportional to the reciprocal of wavelength and c is complex. Following Yih [l] and Waters [lo] we shall seek solutions for cp(r) of the form + = &, + ‘YC#I~ + a2~2 + . .b for each layer
06)
and c = c0 + (act + a2c2 + . . . , which is a non-singular perturbation around the case (Y= 0 which corresponds to very long waves. cuR is assumed small compared with unity, and, as pointed out by Yih [l], no matter how large R is, there is a range of (Yfor which the perturbation procedure is valid. On linearizing in the perturbation amplitude and making use of the equations for the primary flow, the equations of state (2) and (3) with -eqn. (15) give to order (Y
w
F, =
2cPt(N, - &)
aRMN,’
+ (Y 2+;‘( 4 - A$) + i$$N,NzMa
11
-$k(~-C&N,-N&4-N,)
,
1
F2 = --$
#,‘(N,N,-2N,+2N2)+a
&‘(N,N,-2N,+2N2)
1
-
r(y;;;)
(2u+&N; + Cp;(u - c,)(4 - 3N,)) 1
2icuN2q0 RN ,
&=-
11 ,
07)
1
where Nr = 1 + a2MS,,
N2 = 1 + a2MS2, aI = aI1 = a, PI = 1, PII = P
and a dash denotes differentiation with respect to y. Thus the equations of motion for each layer give the following differential equations for c#+,and &:
Q
plJ
iv
=o,
$9; =iy+&‘(u -
(1% c,,) - 6ie0
(4 - N2)(4 - NNl - 4) ,
where Q = (N,N, - 2N, + 2N,)/N:,
(1%
aRMN:
y, = 1 and yIr = p,,/p,
= y.
166 The perturbation
velocities
are zero on the boundaries
which implies
~*(l)=~~;(l)=cp,,(-s)=~~;,(-s)=O, and continuity
(20)
of u* at the interface
(see Yih [l] and Waters
[lo]) gives
@r(O) =%1(O).
(21)
The other kinematic
condition
(a/&+u(O)a/ax){=u*
at the interface
= -i@r(O)
is
exp[icu(x-cl)],
so that {, the deviation of the interface from its mean position, is given by 2 = (&(0)/Z) exp[ia( x - ct)] w h ere c”= c .- u,(O) = c - b. The condition that u* is continuous at the interface gives
+;(o) - #II(O)=
Jq+Y,, - UI).
The shear-stress
condition
(22)
that p& + Ipi,
is continuous
yields
I$ = F&r at y = 0. Finally, ( -p*
(23)
the condition -p’s
for the normal
+ p&z)1 - ( -p*
in which T = T/pU2d, written
stress is expressed
by
- p’c + ~$2)~~ = - TC12~/tlx” at y = 0,
is the non-dimensional
surface
tension.
This can be
{ -cYY(@’ + a+) - aF, f iF; + ~IF,}~ - { -a~(?+’
+ a+) - aFl + iF; + aI;;>,,
= q[(y-
1)F-2+a2T] (24)
evaluated YI=l,
at y = 0, where F = ( U2/gd1)1/2 is the Froude
YII = PI/PI
number
and
= Y*
It is worth noting that in order to keep the equations as simple as possible we have defined p the viscosity ratio, and y the density ratio in such a way that they take the value unity when they occur in expressions involving the suffix I associated with the upper layer. Zero-order solution Equation (18) subject to (20)-(24) to that given by Yih [l], that is C/+)= 1+ By + cjJ* + Dy3,
to order zero in o[ has a solution
similar
(25)
167 in which (m + 63) + a2B, m8 -
B = _ (m + 3a2 + 4a3) I
,
2S2(1 + S)
m +
C, = mCII,
C,, =
D, = mD,,,
D,, =
B,,=2
m
’
S3
ma2(1 + 6)
’
a2-m
2m8*(1+
8) ’
where m is given by m =
PQII/QI-
The eigenvalue Z. = co - b =
c0 is given by i 2ma2(1
+ 6)( aI - uII)
m2 + 2m8(2 + 36 + 2a2) + S4.
First order solution On using (25) and (26) subject to the conditions (20)-(24) the solution to eqn. (19) to order (Yhas a form similar to that given by Yih [l] and the eigen value to order (Y, c1 is found to be given by Re c1 =O, Im c1 =RI,
+ WJ,,
(27)
where
--q;(1) q(y)
= $$f
+J4;1W) + &#v) - (Y+
(UC - 3zp)y5 60
-_
l$q]>
(28)
E&y” 12
andG=(ZJ*)-‘-&,BI-aI;
+ The term 2~18 occurring in the denominator of the expression for &, is misprinted as 3m8 in [lo].
168 the elastic contribution
J2= mQr(1 +(3*
J2 is given by
w2
(m;82+K,-kK,,)
+ S)( a, - alI) ’
+
62 +
46)kg,,(-S)]],
(29)
where k=
(‘~-%h, = h-Y2)U 0,
-
S2),
(Yl - Y&
’
K=${C&(4-3N,)-@N:), H= -${~DZ,,(~-~N+ZC(ZN;-~N,+~)}, 1
%aD(N, - 1)(4 - N,)y4 g(v) =
3N;)
and W = (S, - S,), is a Weissenberg number for the unperturbed layer I.
flow of
According to the theory of hydrodynamic stability Im ci < 0 implies that the small perturbations given by eqn.. (15) will decay and the flow remain stable whereas the perturbation will grow for Im c1 > 0 and give rise to unstable flow. In eqn. (27) we see that Im c, consists of two parts, RJi a viscous component, and WJ, an elastic component. Shear-thinning affects both these components. Obviously, in order to ensure that our perturbation scheme (16) remains valid we must also impose the condition that CYWis small compared to unity. As pointed out by Yih [l], the small (Yanalysis presented here predicts the onset of instability in the form of long waves. These waves are unlikely to result in turbulence when R and W are small but will probably lead to finite waves, the growth of which can only be analysed by considering non-linear effects. Thus we predict that instabilities, when they exist, exist for all Reynolds numbers in the form of finite waves due primarily to the difference in viscosity of the two liquid layers. However, there may well exist
169 other unstable modes, corresponding to (Ynot small, that are not predicted by this work. Before discussing the implications of the results given above we shall first review some of the results of previous authors mentioned in the introduction. In all cases and in the present work we shall follow Yih [l] and only give consideration to the case when the liquids have equal densities (y = 1). Two Newtonian liquids
In Fig. 1 we encapsulate the results of Yih [l] for two Newtonian liquids (obtained from (27)-(29) by putting S, = S, in both layers) onto one diagram in which the sign of Im cr and hence the stability or instability of the flow is indicated in the (log,,& log,,/3) plane. Obviously there is no contribution from II/J2 in this case. It is easily observed that the interface is unstable for the viscosity ratio j3 > 1 and the layer thickness ratio S < 1 or /I < 1 and 6 > 1. This is consistent with the stability of the interface being unchanged if the layers are completely interchanged, whereby (30) as noted by Yih [l].
Fig. 1. Regions of stability in the (log,,, 6, log,,, #I) plane for two Newtonian
liquids.
170 Two Power-law liquids For two power-law liquids with consistency constants K~, ~~~ and powerlaw indices n, and n,, respectively Waters [lo] found that eqn. (27) reduces (31)
2.0 b)
UNSTABLE 1.0
0.0
I ‘1 I
-1.0
UNSTABLE
Fig, 2. Regions of stability in the (log,, 8, log,, .$) plane for two power-law liquids N = (i) 0.1 - - - - - -; (ii) 0.5 -- -- --; (iii) 1.0 (b) N=(i) 1.0 -; --_; (i;i) 5.0 ___.
when (a) (ii) 2.0
171 where R is a suitably modified Reynolds number N = n,,/n,
and .$= (U/d,)N-‘(~,,/~,)““‘.
5”’ represents the ratio of the reference viscosities of the two liquids and 5 is equivalent to /?, the viscosity ratio in the Newtonian case, when n I = n II = 1. In Figs. 2(a) and 2(b) we see the regions of interfacial stability and instability in the (log,, 8, log,, 4) plane for two such liquids for different values of the power-law index ratio N. They illustrate the dramatic effect that the power-law parameters can have on the regions of interfacial stability and can obviously be chosen to stabilize or destabilize a given flow. For two Newtonian liquids there is a dividing contour at j? = 1 which corresponds to the case of equal viscosities and shear rates. The distortion in this contour for two power-law liquids can easily be predicted by putting c1i= a,, which leads to the relationship 5 = (1 + 8)N-1 which can be confirmed by comparison with Figs. 2(a) and (b) (a, = an * c; = 0 * Im c1 = 0 by eqns. (26)-(29)). In Couette flow the shear rates in both layers is constant and thus shear-thinning liquids exhibit constant apparent viscosities different from their reference viscosities. It is interesting to ascertain whether the changes in the regions of interfacial stability for the two power-law liquids are due solely to the variation in the viscosities with the shear rate. In Fig. 3 we plot
STABLE
UNSTABLE
Fig. 3. Regions of stability in the (log,, 6, log,, &) plane showing the changes due to using the actual viscosity ratio & for two power-law liquids when N = (i) 0.1 - - - - --; (ii) 1.0 - - --; (iii) 5.0 -.
172
the regions of interfacial stability in the (log,, 6, log,, /3,) plane where & is the ratio of the apparent viscosities. It can readily be shown that J1 = J1( N, /I,, 8, y) and we observe that although the regions are now much more like the Newtonian case they are still dependent on N, the index ratio. The shape of the dividing contour at j3, = 1 re-appears and is unaltered as N changes but the other contour changes shape as well as being shifted, commensurate with a scaling of 8.
(a)
UNSTABLE
STABLE
l.O-
l.O-
UNSTABLE
UNSTABLE
173
to-
UNSTABLE
STABLE
I?
UNSTABLE
Fig. 4. Regions of stability in the (log,, 6, log,, /?) plane for two constant viscosity Oldroyd liquids when (a) k = 0.1 and W/R = (i) 0.001 -; (ii) 0.5 - - -; (iii) 10.0 -----. (b) k =l.O and W/R = (i) 0.001 -; (ii) 0.5 - - --; (iii) 10.0 -----. (c) k-10.0 (ii) 0.05 ----; (iii) 1.0 -----. and W/R = (i) 0.001 -;
Two elastico-viscous
liquids with constant viscosity
On putting Mr = i&f,,= 0 in eqns. (27)-(29) we obtain the expression obtained by Li[4] for two Oldroyd B liquids, which exhibit no variation in viscosity with rate of shear. In this case WJ, is not zero and eqns. (27)-(29) become Re ci =O,
Im ci =RI,(S,
/3, y) + WJ,(S, p, k).
(32)
A relationship similar to eqn. (30) can be derived when the layers are completely reversed. In figs. 4(a), (b) and (c) we illustrate the regions of interfacial stability in the (log,, 8, log,, p) plane for small, moderate and large values of W/R when the elasticity ratio k = 0.1, 1.0 and 10 respectively. (W/R = [ 7j( A, -A2)/pd2]I represents the ratio of the time scale associated with the elasticity with that of viscosity in layer I, and can take any value without being inconsistent with aR and aW both being small compared to unity). The small and large values of W/R correspond to the contours due to Ji and J2 respectively and these figures present the results given by Li-[4] in an economical fashion. Complete agreement is found with the results of Li. We conclude that elasticity can stabilize or destabilize the interface but that it has no effect on the stability, for the range of the viscosity ratio j? shown, if either layer is much wider than the other. This is
174 illustrated by the distortions to the regions of stability being confined to a fairly narrow band of values of 6. Two elastico-viscous Equations (27)-(29)
liquids with shear-dependent viscosities are of the form
Re c,=O,
j = I, II.
(33)
Unfortunately the relative algebraic simplicity of the previous cases has now disappeared; the products MS, and MS, from both layers occuring independently in Ji and J,. However, they only contribute to J1 through the shear rates a, and a,, and m = /?Qrr/Qr, JI being independent of k, the elasticity ratio. We find that the presence of shear-thinning in one layer usually only affects the stability of the interface if that layer is the less (or potentially the less) viscous one. Thus, for example, if layer II is shear-thinning the stability contours in the (log,, 6, log,, p) plane are mainly affected in the region j3 < 1. If both layers are shear-thinning only the shear-thinning in the less-viscous layer has much effect on the stability contours. Thus we
-I+,
-VE
+VE
Fig. 5. Regions of sign of J1 in the (log 1” 8, logIcj /3) plane for two Oldroyd liquids when (ii) 10.0 ------; Mu = 0, (s, - S,),, = 0.5, (S,/&),, = 0.125 and MU = (i) 0.1 -; (iii) 1.0 - - -.
175 confine our investigation to the case when only one layer is shear-thinning and we choose that layer to be layer II (MI = 0, Mu # 0). The parameters (MS,),, and (MS,),, only distort the regions of sign predicted by J1 from those for the viscous case when we approach the maximum shear-thinning possible for an Oldroyd liquid (0 < l/9 < AZ/X, < 1). In Fig. 5 we illustrate this for (S/S,),, = (h,/X,),, = 0.125, remem-
2.0 (b)
0.0_-
-VE
+VE
l.O-
-
-
-
-VE
C-N \
+VE .\ \ \
-l.O-
\ \ \ \ \
-24
- *o
I 3.0
dil
I.0
AI
176
Fig. 6. Regions of sign of J2 in the (log,, 8, log,, /3) plane for two Oldroyd liquids when M,=O, (S,-S,),,-0.5 and (i) MII=O.l and (S2/S1)II=0.125 ---, (ii) MII=O (constant viscosity in both layers) -, for k = (a) 0.1, (b) 1.0 and (c) 10.0.
bering that this represents the factor by which the viscosity in layer II is reduced at high shear rates. We see that the effect changes dramatically with increasing Mu and is destabilizing in that the interface is stable for a smaller range of 6. The apparently stabilizing effect which we observe is the distortion of the contour corresponding to a, = a,, from the line /3 = 1. This is due of course to the change in the actual viscosity in layer II by shear-thinning. In order to interpret the more complicated diagrams which follow it may help to remember that, for any contour or contours drawn for particular values of the relevant parameters, crossing a contour changes the sign or the stability behaviour from that already indicated. It is impractical to indicate on the diagrams the sign or stability of every region. In Figs. 6(a), (b) and (c) we show how the regions of sign of J2 vary for elasticity ratios k = 0.1, 1.0 and 10.0 respectively. In each case we take (S, - S,) II = 0.5, M, = 0, Mu = 0.1 and (S/S,) = 0.125 and show for comparison the case with no shear-thinning (M, = M,, = 0). We observe that shear-thinning can affect the regions of sign of J2 quite dramatically in the region p c 1, S < 10. Even more dramatic effects can be achieved by increasing (S, - S,) rI to the much larger value 5.0 and taking M,, = 1.0 for the same values of the other parameters. This is illustrated in Figs. 7(a), (b) and (c) where we observe that the contour which is at p = 1 for the case MI = M,, = 0 is now noticeably deformed. This also occurs for J1 with these
177 values of the parameters, and the shape of this deformed contour can be predicted by putting u1 = a,, in eqns. (10) and (11). These larger values of the parameters are included for illustration but we doubt whether they are consistent with the smaII parameter expansion used earlier. It is of interest to note that when we redraw Figs. 5 and 7 with p replaced by & the apparent viscosity ratio, the central dividing contour & = 1 is
178
Fig. 7. Regions of sign of Jz in the (loglo 8, log,, /3) plane for two Oldroyd liquids when M,=O, (S,-S,)II-5.0 and (i) M,,=l.O and (S~/S1)II=0.125 ---, (ii) MII=O (constant viscosity in both layers) -, for k = (a) 0.1 (b) 1.0 and (c) 10.0.
re-established in a similar fashion to when we carried out the same exercise with two power-law liquids. For example Figs. 5 and 7(a) become Figs. 8(a) and (b) and we note that the other contour distortions do not disappear but only change shape somewhat. This implies again that the effect of shearthinning on the stability is not solely due to the variation of the viscosities with shear rate. In order to assess the combined effect of the variation of J1 and J2 with shear-thinning we consider the regions of sign of J1 + (W/R) J2 given by eqn. (33) using (28) and (29) for the typical case M, = 0, M,, = 0.1, (S, S,) II = 0.5 and (SJS’,) II = 0.125 for specified values of W/R when k = 0.1 and 10.0 in Figs. 9(a) and (b) respectively. Decreasing W/R gives contours which tend to those given by J1 alone (cf. Fig. 5(a)(i)) and increasing W/R tend to those given by J, (cf. Fig. 6(a)(i) and (c)(i)). The contours in the region j3 > 1 are affected only by the elasticity ratio, in line with the behaviour predicted in fig. 4. This is consistent with our previous statement that shear-thinning usually affects the stability only when it is present in the less viscous layer. In part, this must be due to the fact that the shear rate is usually higher in the less-viscous layer. Conclusions (1) In Fig. 1 we have encapsulated the results of Yih [l], showing the regions of interfacial stability for two Newtonian liquids.
179
1.0-
f
VE
-VE I
.*' <
0.0
-1.0-
-2.0 -24
/
,A
!
- VE
+VE
-1.0
o!o
l!O
z’“‘“’
-2.0
-1.o
0.0
1.0
2.0
Fig. 8. Regions of sign in the (log,, 6, log,, 8,) plane showing the changes due to using the actual viscosity ratio 8, for two Oldroyd liquids: (a) J1 corresponding to Fig. 5, (b) J2 corresponding to Fig. 7(a).
(2) We have presented the results of Waters [8] for two power-law liquids in a similar fashion in Fig. 2 indicating the possible dramatic effect of the power-law parameters. (3) The results of Li [4] for two constant viscosity Oldroyd liquids are similarly presented in Fig. 4, and we have shown that the presence of
180 2.0-l
(a)
la-
UNSTABLE
STABLE
STABLE
UNSTABLE
-l.O-
2.0
lb)
1.0
UNSTABLE
STABLE
0.0 STABLE
UNSTABLE -1.0
-2.0+ 1 -2.0
-1.0
0.0
1.0
0
Fig. 9. Regions of stability of the (loglo 6, logI, 8) plane for two Oldroyd liquids when -0.125, (a) k-0.1 and W/R -0.1, (b) MI = 0, MIX= 0.1, (S, - WII = 0.5, (s2/&)11 k = 10.0 and W/R = 0.01.
elasticity can stabilize or destabilize the flow but appears not to effect the stability if one layer is very much wider than the other. (4) We find that for two shear-thinning Oldroyd liquids the presence of shear-thinning in layer I mainly affects the stability for /3 > 1 and similarly shear-thinning in layer II affects it mainly for j3 -C1. We thus have the
181 interesting conclusion that shear-thinning usually affects the interfacial stability only if it occurs in the less-viscous layer. This allows us to restrict our investigation to the case when shear-thinning is present in only one layer. (5) For liquids with shear-dependent viscosities the true (or apparent) viscosity ratio & is obviously different from p which is calculated from the reference viscosities. When we use & instead of /3 to draw our stability regions we might expect them to reduce to those for the constant viscosity case. Although this does occur for one of the dividing contours (this is the one that corresponds to Im ci = 0 in the case of equal viscosities) other distortions still remain. This indicates that the stability is not only influenced by the fact that the viscosities change but also by the mode by which they change. Acknowledgement The authors would like to thank one of the referees for spotting an algebraic error and for making helpful comments and suggestions. References 1 2 3 4 5 6 7 8 9 10 11 12 13
C-S. Yih, J. Fluid Mech., 27(2) (1967) 337. C.E. Hickox, Phys. Fluids., 14 (1971) 251. A.P. Hooper and W.G.C. Boyd, J. Fluid Mech., 128 (1983) 507. C-H. Li, Phys. Fluids, 12 (13) (1969) 531. J.G. Oldroyd, Proc. Roy. Sot., A245 (1958) 278. A.A. Khan and C.D. Han, Trans. Sot. Rheol., 20 (4) ((1976) 595. A.A. Khan and C.D. Han, Trans. Sot. Rheol., 21(l) (1976) 101. B. Coleman and W. Noll, Rev. Mod. Phys., 33 (1961) 239. P.R. Williams and R.W. Williams, J.N.N.F.M., 19 (1985) 53. N.D. Waters, J.N.N.F.M., 12 (1983) 85. J.G. Oldroyd, Proc. Roy. Sot., A200 (1950) 523. H.B. Squire, Proc. Roy. Sot., Al42 (1933) 621. C-S. Yih, Quart. Appl. Mech., 12 (1955) 434.
161
Fluid Mechanics, 24 (1987) 161-181
Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
THE STABILITY OF TWO STRATIFIED NON-NEWTONIAN IN COUEl’TE FLOW
LlQUIDS
N.D. WATERS and A.M. KEELEY Department of Applied Liverpool (Ct. Britain)
Mathematics
and Theoretical
Physics,
The University
of Liverpool,
(Received May 16, 1986; in revised form August 21, 1986)
Summary
Consideration is given to the stability of the interface between two Oldroyd liquids with shear-dependent viscosities, flowing in distinct layers while undergoing plane Couette flow. Results are presented as regions of stability in the plane determined by the logarithms of the viscosity and depth ratios. The work of previous authors for two Newtonian, power-law and constant-viscosity Oldroyd liquids is revealingly presented in a similar fashion. It is found that the dependence of the viscosities on shear-rate can drastically affect the regions of interfacial stability in a way over and above that due to just a change in the effective viscosity ratio. It is also found that for the Oldroyd liquids this viscosity variation affects the stability when it is present in the less-viscous layer.
Introduction
There are many industrial and manufacturing processes in which two or more immiscible liquids flow in contact but in distinct layers. Examples range from the use of water to reduce the pumping costs of crude oil to the coextrusion of two or more polymers to produce materials with unique properties. The possibility of instability at the interface of such flows is well known. Apart from the more obvious effects due to density differences Yih [l] and Hickox [2] have found that for simple flows viscosity differences alone are sufficient to cause instability. Hooper and Boyd [3] have shown that unbounded Couette flow of two viscous liquids of equal density and negligible surface tension is always unstable. They also show that surface tension is always stabilizing (while density difference can have either effect). 0377-0257/87/$03.50
0 1987 Elsevier Science Publishers B.V.
162
They suggest that such instabilities have a slow growth rate and are sensitive to surface tension and may not always be obvious in experiments. Li [4] considered the Couette flow of two constant viscosity Oldroyd [5] B liquids and found that the presence of elasticity can stabilize or destabilize the flow. Khan and Han [6,7] carried out theoretical and experimental investigations for the flow of two liquids in plane Poiseuille flow. In their analysis they employed two Coleman and No11 [8] second-order fluids, which exhibit constant viscosities, and found that two-dimensional interfacial instabilities occuring in the plane of the flow depend only on the viscosity ratio and not the elasticity ratio, whereas two-dimensional instabilities in the plane perpendicular to the flow depend on both ratios. Recently Williams and Williams [9] have considered a two-dimensional stratified flow of two liquids in a converging channel. The liquid in the centre of the channel undergoes planar extensional flow with the liquid near the walls acting as a lubricant. The flow is regarded as locally parallel and that of the lubricant as approximated by Couette flow. They acknowledge that, apart from the effects of density and surface tension differences, interfacial instabilities may be caused by viscosity difference or the presence of elasticity In 1983 Waters [lo] extended the analysis of Yih [l] for two viscous liquids in Couette flow to two inelastic “power-law” liquids. He demonstrated that shear-thinning can have a significant effect on the stability of the interface. In order to assess the combined effects of shear-thinning and elasticity, commonly exhibited by non-Newtonian liquids, we consider in this paper the Couette flow of two liquids which can both be characterized by the four-constant model of Oldroyd [5]. Similary we are able to demonstrate that the effect of shear-thinning can be significant, and that this effect is not solely due to the fact that the viscosity changes but also to the ratio of the power-law indices for two power-law liquids and to the interaction between shear-thinning and elasticity for the Oldroyd liquids. We also demonstrate that for the Oldroyd liquids it is the shear thinning of the less-viscous layer that affects stability. Theory
The primary jlow
We consider two different elastico-viscous liquids, both characterized by Oldroyd four-constant models, flowing steadily in distinct parallel layers between two horizontal parallel planes when one plane is at rest and the other is moving with constant velocity U in its own plane, and there is no pressure gradient.
163 We refer to rectangular Cartesian coordinates xi such that the interface between the two liquids is x2 = 0. The plane x2 = d, moves in the xi direction and x2 = -d,, is the stationary plane. We assume that the velocity distribution has the form (
uj(x2>T
OY
‘)7
(1)
which satisfies continuity immediately, where j = I for the upper layer and j = II for th e 1ower layer. Both liquids are characterized by equations of state of the form TiA= -jgik
+ Pil, 7
(2)
(3) where rik is the stress tensor, pik is the extra stress tensor, Z$’ is the rate of strain tensor, p is the excess pressure, gik is the metric tensor, 77 is the viscosity at zero rate of shear and A,, A, and p0 are material constants, each with the dimensions of time and 9/9t is the Oldroyd [ll] upper convected time derivative. We shall append the suffix j denoting the appropriate layer when necessary and drop the overbars when we non-dimensionalize using a typical speed U, typical length d, and the density pi of the upper layer as follows x = xi/d, p =P/p$J?
u = U/U,
y = x2/d,,
Pil, = PiX_/PIu2
S, = Uh,/d,,
(4
9
S, = UX,/d,
M=
U/+,/d,.
It is easy to show that the only non-zero 2P(S,
stresses are given by
- S,)(du/dy)=
(5)
‘11 = R(1 + MS,(du,‘dy)=)
’
1 + MS,(du/dy)= 1 +Ms,(du/dy)= where R = Up,d,/qI
is a Reynolds
Pi = 1 and Pit = ~~/ni With no pressure
1’
number
based on layer I,
= P.
gradient
the equations
of motion
reduce
to
llp12/ay = 0 for both layers which on using (5) gives d=u = 0 in each layer, dy=
(6)
164 subject to the boundary conditions ut(1) = 1,
u,,(-6)=Owhere6=d,,/d
I
(7)
and the continuity of velocity and stress at the interface u1 (0) = uil (0) and ( pi2 >i = (f& at Y = 0. The solution of (6) satisfying (7) and (8) is
(8)
ui=aiY+b
(9)
(j=I,
II)
where b = 1 - a, = a,,8
(10)
Equation (1 l), on using (lo), becomes a quintic in a, or a,, which is readily solved using, for example, the Newton-Raphson method. Stability analysis
We follow Yih [l] and Waters [lo] and consider a two-dimensional infinitesimal disturbance for the above steady solution without implying that Squire’s theorem [12] (cf. Yih [l]) necessarily holds +. For each layer we assume a perturbed velocity distribution of the form (u(y) + u*(x, Y, 0, 0*(x, Y1 t)) (12) and a pressure p +p*(x, y, t), where u(y) is given by (9). (10) and (11) and p is a constant. We also assume that the relevant non-zero stresses have the form Pll +P,*,(%
Y7 4
P&(X,
Y, 09 Pn+P:2(x,
Y1 d
(13)
where the starred variables are all small and non-dimensional. The equation of continuity permits the definition of a stream function +(x, y, t) where
u*_aICl aY
’
u*=
3.
(14)
As is customary we assume without loss of generality that all perturbation quantities contain an exponential time factor (real part understood) of the form ($9 P*, Pl:, PL
p&J=
[e(u),
f(Y),
Xexp[ia(x
C(Y),
- ct)]
G(Y)9
4CY)l (15)
+ Squire showed that for a Newtonian liquid the minimum Reynolds number for amplified Ssturbances occurs for a two dimensional disturbance; Yih [13] extended this to include two stratified Newtonian liquids.
165 where t = Ui/d is the non-dimensional time, a is a real positive number proportional to the reciprocal of wavelength and c is complex. Following Yih [l] and Waters [lo] we shall seek solutions for cp(r) of the form + = &, + ‘YC#I~ + a2~2 + . .b for each layer
06)
and c = c0 + (act + a2c2 + . . . , which is a non-singular perturbation around the case (Y= 0 which corresponds to very long waves. cuR is assumed small compared with unity, and, as pointed out by Yih [l], no matter how large R is, there is a range of (Yfor which the perturbation procedure is valid. On linearizing in the perturbation amplitude and making use of the equations for the primary flow, the equations of state (2) and (3) with -eqn. (15) give to order (Y
w
F, =
2cPt(N, - &)
aRMN,’
+ (Y 2+;‘( 4 - A$) + i$$N,NzMa
11
-$k(~-C&N,-N&4-N,)
,
1
F2 = --$
#,‘(N,N,-2N,+2N2)+a
&‘(N,N,-2N,+2N2)
1
-
r(y;;;)
(2u+&N; + Cp;(u - c,)(4 - 3N,)) 1
2icuN2q0 RN ,
&=-
11 ,
07)
1
where Nr = 1 + a2MS,,
N2 = 1 + a2MS2, aI = aI1 = a, PI = 1, PII = P
and a dash denotes differentiation with respect to y. Thus the equations of motion for each layer give the following differential equations for c#+,and &:
Q
plJ
iv
=o,
$9; =iy+&‘(u -
(1% c,,) - 6ie0
(4 - N2)(4 - NNl - 4) ,
where Q = (N,N, - 2N, + 2N,)/N:,
(1%
aRMN:
y, = 1 and yIr = p,,/p,
= y.
166 The perturbation
velocities
are zero on the boundaries
which implies
~*(l)=~~;(l)=cp,,(-s)=~~;,(-s)=O, and continuity
(20)
of u* at the interface
(see Yih [l] and Waters
[lo]) gives
@r(O) =%1(O).
(21)
The other kinematic
condition
(a/&+u(O)a/ax){=u*
at the interface
= -i@r(O)
is
exp[icu(x-cl)],
so that {, the deviation of the interface from its mean position, is given by 2 = (&(0)/Z) exp[ia( x - ct)] w h ere c”= c .- u,(O) = c - b. The condition that u* is continuous at the interface gives
+;(o) - #II(O)=
Jq+Y,, - UI).
The shear-stress
condition
(22)
that p& + Ipi,
is continuous
yields
I$ = F&r at y = 0. Finally, ( -p*
(23)
the condition -p’s
for the normal
+ p&z)1 - ( -p*
in which T = T/pU2d, written
stress is expressed
by
- p’c + ~$2)~~ = - TC12~/tlx” at y = 0,
is the non-dimensional
surface
tension.
This can be
{ -cYY(@’ + a+) - aF, f iF; + ~IF,}~ - { -a~(?+’
+ a+) - aFl + iF; + aI;;>,,
= q[(y-
1)F-2+a2T] (24)
evaluated YI=l,
at y = 0, where F = ( U2/gd1)1/2 is the Froude
YII = PI/PI
number
and
= Y*
It is worth noting that in order to keep the equations as simple as possible we have defined p the viscosity ratio, and y the density ratio in such a way that they take the value unity when they occur in expressions involving the suffix I associated with the upper layer. Zero-order solution Equation (18) subject to (20)-(24) to that given by Yih [l], that is C/+)= 1+ By + cjJ* + Dy3,
to order zero in o[ has a solution
similar
(25)
167 in which (m + 63) + a2B, m8 -
B = _ (m + 3a2 + 4a3) I
,
2S2(1 + S)
m +
C, = mCII,
C,, =
D, = mD,,,
D,, =
B,,=2
m
’
S3
ma2(1 + 6)
’
a2-m
2m8*(1+
8) ’
where m is given by m =
PQII/QI-
The eigenvalue Z. = co - b =
c0 is given by i 2ma2(1
+ 6)( aI - uII)
m2 + 2m8(2 + 36 + 2a2) + S4.
First order solution On using (25) and (26) subject to the conditions (20)-(24) the solution to eqn. (19) to order (Yhas a form similar to that given by Yih [l] and the eigen value to order (Y, c1 is found to be given by Re c1 =O, Im c1 =RI,
+ WJ,,
(27)
where
--q;(1) q(y)
= $$f
+J4;1W) + &#v) - (Y+
(UC - 3zp)y5 60
-_
l$q]>
(28)
E&y” 12
andG=(ZJ*)-‘-&,BI-aI;
+ The term 2~18 occurring in the denominator of the expression for &, is misprinted as 3m8 in [lo].
168 the elastic contribution
J2= mQr(1 +(3*
J2 is given by
w2
(m;82+K,-kK,,)
+ S)( a, - alI) ’
+
62 +
46)kg,,(-S)]],
(29)
where k=
(‘~-%h, = h-Y2)U 0,
-
S2),
(Yl - Y&
’
K=${C&(4-3N,)-@N:), H= -${~DZ,,(~-~N+ZC(ZN;-~N,+~)}, 1
%aD(N, - 1)(4 - N,)y4 g(v) =
3N;)
and W = (S, - S,), is a Weissenberg number for the unperturbed layer I.
flow of
According to the theory of hydrodynamic stability Im ci < 0 implies that the small perturbations given by eqn.. (15) will decay and the flow remain stable whereas the perturbation will grow for Im c1 > 0 and give rise to unstable flow. In eqn. (27) we see that Im c, consists of two parts, RJi a viscous component, and WJ, an elastic component. Shear-thinning affects both these components. Obviously, in order to ensure that our perturbation scheme (16) remains valid we must also impose the condition that CYWis small compared to unity. As pointed out by Yih [l], the small (Yanalysis presented here predicts the onset of instability in the form of long waves. These waves are unlikely to result in turbulence when R and W are small but will probably lead to finite waves, the growth of which can only be analysed by considering non-linear effects. Thus we predict that instabilities, when they exist, exist for all Reynolds numbers in the form of finite waves due primarily to the difference in viscosity of the two liquid layers. However, there may well exist
169 other unstable modes, corresponding to (Ynot small, that are not predicted by this work. Before discussing the implications of the results given above we shall first review some of the results of previous authors mentioned in the introduction. In all cases and in the present work we shall follow Yih [l] and only give consideration to the case when the liquids have equal densities (y = 1). Two Newtonian liquids
In Fig. 1 we encapsulate the results of Yih [l] for two Newtonian liquids (obtained from (27)-(29) by putting S, = S, in both layers) onto one diagram in which the sign of Im cr and hence the stability or instability of the flow is indicated in the (log,,& log,,/3) plane. Obviously there is no contribution from II/J2 in this case. It is easily observed that the interface is unstable for the viscosity ratio j3 > 1 and the layer thickness ratio S < 1 or /I < 1 and 6 > 1. This is consistent with the stability of the interface being unchanged if the layers are completely interchanged, whereby (30) as noted by Yih [l].
Fig. 1. Regions of stability in the (log,,, 6, log,,, #I) plane for two Newtonian
liquids.
170 Two Power-law liquids For two power-law liquids with consistency constants K~, ~~~ and powerlaw indices n, and n,, respectively Waters [lo] found that eqn. (27) reduces (31)
2.0 b)
UNSTABLE 1.0
0.0
I ‘1 I
-1.0
UNSTABLE
Fig, 2. Regions of stability in the (log,, 8, log,, .$) plane for two power-law liquids N = (i) 0.1 - - - - - -; (ii) 0.5 -- -- --; (iii) 1.0 (b) N=(i) 1.0 -; --_; (i;i) 5.0 ___.
when (a) (ii) 2.0
171 where R is a suitably modified Reynolds number N = n,,/n,
and .$= (U/d,)N-‘(~,,/~,)““‘.
5”’ represents the ratio of the reference viscosities of the two liquids and 5 is equivalent to /?, the viscosity ratio in the Newtonian case, when n I = n II = 1. In Figs. 2(a) and 2(b) we see the regions of interfacial stability and instability in the (log,, 8, log,, 4) plane for two such liquids for different values of the power-law index ratio N. They illustrate the dramatic effect that the power-law parameters can have on the regions of interfacial stability and can obviously be chosen to stabilize or destabilize a given flow. For two Newtonian liquids there is a dividing contour at j? = 1 which corresponds to the case of equal viscosities and shear rates. The distortion in this contour for two power-law liquids can easily be predicted by putting c1i= a,, which leads to the relationship 5 = (1 + 8)N-1 which can be confirmed by comparison with Figs. 2(a) and (b) (a, = an * c; = 0 * Im c1 = 0 by eqns. (26)-(29)). In Couette flow the shear rates in both layers is constant and thus shear-thinning liquids exhibit constant apparent viscosities different from their reference viscosities. It is interesting to ascertain whether the changes in the regions of interfacial stability for the two power-law liquids are due solely to the variation in the viscosities with the shear rate. In Fig. 3 we plot
STABLE
UNSTABLE
Fig. 3. Regions of stability in the (log,, 6, log,, &) plane showing the changes due to using the actual viscosity ratio & for two power-law liquids when N = (i) 0.1 - - - - --; (ii) 1.0 - - --; (iii) 5.0 -.
172
the regions of interfacial stability in the (log,, 6, log,, /3,) plane where & is the ratio of the apparent viscosities. It can readily be shown that J1 = J1( N, /I,, 8, y) and we observe that although the regions are now much more like the Newtonian case they are still dependent on N, the index ratio. The shape of the dividing contour at j3, = 1 re-appears and is unaltered as N changes but the other contour changes shape as well as being shifted, commensurate with a scaling of 8.
(a)
UNSTABLE
STABLE
l.O-
l.O-
UNSTABLE
UNSTABLE
173
to-
UNSTABLE
STABLE
I?
UNSTABLE
Fig. 4. Regions of stability in the (log,, 6, log,, /?) plane for two constant viscosity Oldroyd liquids when (a) k = 0.1 and W/R = (i) 0.001 -; (ii) 0.5 - - -; (iii) 10.0 -----. (b) k =l.O and W/R = (i) 0.001 -; (ii) 0.5 - - --; (iii) 10.0 -----. (c) k-10.0 (ii) 0.05 ----; (iii) 1.0 -----. and W/R = (i) 0.001 -;
Two elastico-viscous
liquids with constant viscosity
On putting Mr = i&f,,= 0 in eqns. (27)-(29) we obtain the expression obtained by Li[4] for two Oldroyd B liquids, which exhibit no variation in viscosity with rate of shear. In this case WJ, is not zero and eqns. (27)-(29) become Re ci =O,
Im ci =RI,(S,
/3, y) + WJ,(S, p, k).
(32)
A relationship similar to eqn. (30) can be derived when the layers are completely reversed. In figs. 4(a), (b) and (c) we illustrate the regions of interfacial stability in the (log,, 8, log,, p) plane for small, moderate and large values of W/R when the elasticity ratio k = 0.1, 1.0 and 10 respectively. (W/R = [ 7j( A, -A2)/pd2]I represents the ratio of the time scale associated with the elasticity with that of viscosity in layer I, and can take any value without being inconsistent with aR and aW both being small compared to unity). The small and large values of W/R correspond to the contours due to Ji and J2 respectively and these figures present the results given by Li-[4] in an economical fashion. Complete agreement is found with the results of Li. We conclude that elasticity can stabilize or destabilize the interface but that it has no effect on the stability, for the range of the viscosity ratio j? shown, if either layer is much wider than the other. This is
174 illustrated by the distortions to the regions of stability being confined to a fairly narrow band of values of 6. Two elastico-viscous Equations (27)-(29)
liquids with shear-dependent viscosities are of the form
Re c,=O,
j = I, II.
(33)
Unfortunately the relative algebraic simplicity of the previous cases has now disappeared; the products MS, and MS, from both layers occuring independently in Ji and J,. However, they only contribute to J1 through the shear rates a, and a,, and m = /?Qrr/Qr, JI being independent of k, the elasticity ratio. We find that the presence of shear-thinning in one layer usually only affects the stability of the interface if that layer is the less (or potentially the less) viscous one. Thus, for example, if layer II is shear-thinning the stability contours in the (log,, 6, log,, p) plane are mainly affected in the region j3 < 1. If both layers are shear-thinning only the shear-thinning in the less-viscous layer has much effect on the stability contours. Thus we
-I+,
-VE
+VE
Fig. 5. Regions of sign of J1 in the (log 1” 8, logIcj /3) plane for two Oldroyd liquids when (ii) 10.0 ------; Mu = 0, (s, - S,),, = 0.5, (S,/&),, = 0.125 and MU = (i) 0.1 -; (iii) 1.0 - - -.
175 confine our investigation to the case when only one layer is shear-thinning and we choose that layer to be layer II (MI = 0, Mu # 0). The parameters (MS,),, and (MS,),, only distort the regions of sign predicted by J1 from those for the viscous case when we approach the maximum shear-thinning possible for an Oldroyd liquid (0 < l/9 < AZ/X, < 1). In Fig. 5 we illustrate this for (S/S,),, = (h,/X,),, = 0.125, remem-
2.0 (b)
0.0_-
-VE
+VE
l.O-
-
-
-
-VE
C-N \
+VE .\ \ \
-l.O-
\ \ \ \ \
-24
- *o
I 3.0
dil
I.0
AI
176
Fig. 6. Regions of sign of J2 in the (log,, 8, log,, /3) plane for two Oldroyd liquids when M,=O, (S,-S,),,-0.5 and (i) MII=O.l and (S2/S1)II=0.125 ---, (ii) MII=O (constant viscosity in both layers) -, for k = (a) 0.1, (b) 1.0 and (c) 10.0.
bering that this represents the factor by which the viscosity in layer II is reduced at high shear rates. We see that the effect changes dramatically with increasing Mu and is destabilizing in that the interface is stable for a smaller range of 6. The apparently stabilizing effect which we observe is the distortion of the contour corresponding to a, = a,, from the line /3 = 1. This is due of course to the change in the actual viscosity in layer II by shear-thinning. In order to interpret the more complicated diagrams which follow it may help to remember that, for any contour or contours drawn for particular values of the relevant parameters, crossing a contour changes the sign or the stability behaviour from that already indicated. It is impractical to indicate on the diagrams the sign or stability of every region. In Figs. 6(a), (b) and (c) we show how the regions of sign of J2 vary for elasticity ratios k = 0.1, 1.0 and 10.0 respectively. In each case we take (S, - S,) II = 0.5, M, = 0, Mu = 0.1 and (S/S,) = 0.125 and show for comparison the case with no shear-thinning (M, = M,, = 0). We observe that shear-thinning can affect the regions of sign of J2 quite dramatically in the region p c 1, S < 10. Even more dramatic effects can be achieved by increasing (S, - S,) rI to the much larger value 5.0 and taking M,, = 1.0 for the same values of the other parameters. This is illustrated in Figs. 7(a), (b) and (c) where we observe that the contour which is at p = 1 for the case MI = M,, = 0 is now noticeably deformed. This also occurs for J1 with these
177 values of the parameters, and the shape of this deformed contour can be predicted by putting u1 = a,, in eqns. (10) and (11). These larger values of the parameters are included for illustration but we doubt whether they are consistent with the smaII parameter expansion used earlier. It is of interest to note that when we redraw Figs. 5 and 7 with p replaced by & the apparent viscosity ratio, the central dividing contour & = 1 is
178
Fig. 7. Regions of sign of Jz in the (loglo 8, log,, /3) plane for two Oldroyd liquids when M,=O, (S,-S,)II-5.0 and (i) M,,=l.O and (S~/S1)II=0.125 ---, (ii) MII=O (constant viscosity in both layers) -, for k = (a) 0.1 (b) 1.0 and (c) 10.0.
re-established in a similar fashion to when we carried out the same exercise with two power-law liquids. For example Figs. 5 and 7(a) become Figs. 8(a) and (b) and we note that the other contour distortions do not disappear but only change shape somewhat. This implies again that the effect of shearthinning on the stability is not solely due to the variation of the viscosities with shear rate. In order to assess the combined effect of the variation of J1 and J2 with shear-thinning we consider the regions of sign of J1 + (W/R) J2 given by eqn. (33) using (28) and (29) for the typical case M, = 0, M,, = 0.1, (S, S,) II = 0.5 and (SJS’,) II = 0.125 for specified values of W/R when k = 0.1 and 10.0 in Figs. 9(a) and (b) respectively. Decreasing W/R gives contours which tend to those given by J1 alone (cf. Fig. 5(a)(i)) and increasing W/R tend to those given by J, (cf. Fig. 6(a)(i) and (c)(i)). The contours in the region j3 > 1 are affected only by the elasticity ratio, in line with the behaviour predicted in fig. 4. This is consistent with our previous statement that shear-thinning usually affects the stability only when it is present in the less viscous layer. In part, this must be due to the fact that the shear rate is usually higher in the less-viscous layer. Conclusions (1) In Fig. 1 we have encapsulated the results of Yih [l], showing the regions of interfacial stability for two Newtonian liquids.
179
1.0-
f
VE
-VE I
.*' <
0.0
-1.0-
-2.0 -24
/
,A
!
- VE
+VE
-1.0
o!o
l!O
z’“‘“’
-2.0
-1.o
0.0
1.0
2.0
Fig. 8. Regions of sign in the (log,, 6, log,, 8,) plane showing the changes due to using the actual viscosity ratio 8, for two Oldroyd liquids: (a) J1 corresponding to Fig. 5, (b) J2 corresponding to Fig. 7(a).
(2) We have presented the results of Waters [8] for two power-law liquids in a similar fashion in Fig. 2 indicating the possible dramatic effect of the power-law parameters. (3) The results of Li [4] for two constant viscosity Oldroyd liquids are similarly presented in Fig. 4, and we have shown that the presence of
180 2.0-l
(a)
la-
UNSTABLE
STABLE
STABLE
UNSTABLE
-l.O-
2.0
lb)
1.0
UNSTABLE
STABLE
0.0 STABLE
UNSTABLE -1.0
-2.0+ 1 -2.0
-1.0
0.0
1.0
0
Fig. 9. Regions of stability of the (loglo 6, logI, 8) plane for two Oldroyd liquids when -0.125, (a) k-0.1 and W/R -0.1, (b) MI = 0, MIX= 0.1, (S, - WII = 0.5, (s2/&)11 k = 10.0 and W/R = 0.01.
elasticity can stabilize or destabilize the flow but appears not to effect the stability if one layer is very much wider than the other. (4) We find that for two shear-thinning Oldroyd liquids the presence of shear-thinning in layer I mainly affects the stability for /3 > 1 and similarly shear-thinning in layer II affects it mainly for j3 -C1. We thus have the
181 interesting conclusion that shear-thinning usually affects the interfacial stability only if it occurs in the less-viscous layer. This allows us to restrict our investigation to the case when shear-thinning is present in only one layer. (5) For liquids with shear-dependent viscosities the true (or apparent) viscosity ratio & is obviously different from p which is calculated from the reference viscosities. When we use & instead of /3 to draw our stability regions we might expect them to reduce to those for the constant viscosity case. Although this does occur for one of the dividing contours (this is the one that corresponds to Im ci = 0 in the case of equal viscosities) other distortions still remain. This indicates that the stability is not only influenced by the fact that the viscosities change but also by the mode by which they change. Acknowledgement The authors would like to thank one of the referees for spotting an algebraic error and for making helpful comments and suggestions. References 1 2 3 4 5 6 7 8 9 10 11 12 13
C-S. Yih, J. Fluid Mech., 27(2) (1967) 337. C.E. Hickox, Phys. Fluids., 14 (1971) 251. A.P. Hooper and W.G.C. Boyd, J. Fluid Mech., 128 (1983) 507. C-H. Li, Phys. Fluids, 12 (13) (1969) 531. J.G. Oldroyd, Proc. Roy. Sot., A245 (1958) 278. A.A. Khan and C.D. Han, Trans. Sot. Rheol., 20 (4) ((1976) 595. A.A. Khan and C.D. Han, Trans. Sot. Rheol., 21(l) (1976) 101. B. Coleman and W. Noll, Rev. Mod. Phys., 33 (1961) 239. P.R. Williams and R.W. Williams, J.N.N.F.M., 19 (1985) 53. N.D. Waters, J.N.N.F.M., 12 (1983) 85. J.G. Oldroyd, Proc. Roy. Sot., A200 (1950) 523. H.B. Squire, Proc. Roy. Sot., Al42 (1933) 621. C-S. Yih, Quart. Appl. Mech., 12 (1955) 434.