spurt and instability in a two-layer johnson-segalman liquid
TRANSCRIPT
Theoret. Comput. Fluid Dynamics (1995) 7:463-475 Theoretical and Computational Fluid Dynamics © Springer-Verlag 1995
Spurt and Instability in a Two-Layer Johnson-Segalman Liquid1
Yuriko Yamamuro Renardy
Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA 24061-0123, U.S.A.
Communicated by Kang Ping Chen and Thomas B. Gatski
Received 6 December 1993 and accepted 20 July 1994
Abstract. The Johnson-Segalman model is an example of a model that exhibits a nonmonotone curve for the shear stress in terms of shear rate. There are many works based on such models for an explanation of the spurt phenomenon but they have concerned the one-dimensional problem. This paper concerns a model problem, taking a one-dimensionally stable "spurted solution," viewed in two dimensions. A two-layer arrangement between walls in parallel shear, with a thin layer in the higher shear rate and the bulk of the fluid in the lower shear rate, is examined for linear stability in two dimensions. The spectrum is computed numerically for normal mode solutions. Instabilities with dominant growth rates for short waves are found. Thus, the one-dimensionally stable solutions of this model are actually two-dimensionally unstable.
This work is ded ica ted to Professor Danie l D. Joseph on his 65th b i r t h d a y in apprec ia t ion of his help t h rough the years.
There once was an engineer Whose ideas were fun and clear. He showed us flows, amazing us. Why, he's our Pythagoras!
There once was a pipe flow. Two viscous fluids on the go. Both would like us to bestow Easy rules for them to know. "Past results we can dispute." Their energy, we do compute.
Dan studies how flows bifurcate. Many theorems he would state Describing flows that rotate. All such flows can find their fate Let they Dan investigate.
This research was sponsored by ONR Grant No. N00014-92-J-1664 and NSF Grant CTS-9307238.
463
464 Y.Y. Renardy
1. Introduction
Some constitutive models, including the Johnson-Segalman model and the Giesekus model (Kolkka et al., 1988; Kolkka and Ierley, 1989) have a nonmonotone shear-stress-shear-rate curve. We focus on the Johnson-Segalman fluid. Johnson and Seglman (1977) begin with a constitutive equation that is derived systematically from both the molecular theory of Gaussian networks, and the molecular bead-spring model with Hookean springs. In order to allow the model to describe such viscoelastic properties as a shear- thinning viscosity, they generalize the model. They do this by introducing a scalar parameter and "by developing a continuum theory that allows two histories of deformation which may be nonaffine. One history is made to give rise to the current state of stress according to the constitutive equation from molecular theory. The second is made the observed smooth continuum deformation. The two motions are connected by an appropriate constitutive equation." They apply their theory to shear flows. A critical discussion of its application to elongational flows is given by Adewale and Leonov (1993). The model contains one relaxation time, which is thought of as the dominant relaxation time of the polymeric melt, one "polymer viscosity" which is identified as the contribution to the total viscosity from the component with the dominant relaxation time, and a second "solvent viscosity" which is identified with the viscosity arising from the shorter relaxation times present in the melt; the apparent viscosity is shear-dependent. Figure l shows the steady shear stress w(¢) against shear rate ~ for the Johnson-Segalman fluid:
¢ w(¢) = 1 + 42(1 - a 2) ~ e{, (1)
where ~ denotes the retardation parameter and a is a model constant with ½ < a < 1. The range of e < ~, a < 1, leads to the nonomonotone behavior, and popular values to model polymer melts have been picked for this figure. It has been proposed (Hunter and Slemrod, 1983; Malkus et al., 1990, 1991) that such differential constitutive laws may explain the spurt phenomenon.
Melt fracture is an instability that is observed in the extrusion of molten polymers. The spurt phenomenon is a stage of melt fracture that occurs in some types of polymer melts. The stages involved are described for linear low-density polyethylene by Denn (1990), Kalika and Denn (1987), and Moynihan et al. (1990). Below a certain critical shear stress, the surface of the extrudate is smooth. At a first critical shear stress, very short wavelength disturbances called sharkskin appear on the extrudate surface. At a second critical shear stress, the surface becomes alternately sharkskin and smooth and the flow is known as stick-slip or spurt. The flow rate increases in this regime. At a third critical shear stress, there is onset of a more severe extrudate distortion, called wavy fracture. Larson (1992) gives a review of experiments and theories, and in particular describes the dependence of the stages of melt fracture on the materials and flow conditions. There are a number of suggested causes for the onset of melt fracture: partial slip at the polymer/die interface, and the role of prestressing conditions upstream of the exit (Moynihan et al., 1990). The slip hypothesis is that for low flow rates, the material adheres to the wall, while at higher flow rates, it goes through alternate sticking and slipping, and at the highest flow rates, the material slips. Models for the wall-stress-slip-velocity dependence have been developed by Leonov (1984, 1990) and Leonov and Srinivasan (1993). Another possible and totally different explanation is based on the role of bulk properties, and has attracted much interest from analysts. This is the nonmonotone constitutive model of the type in (1). Given the shear stress, there are two possible values for the shear rate once a critical value is reached, ignoring the decreasing part of the curve in Figure 1. The sudden increase of the flow rate in the spurt regime
0.9
0.8
0.7
0.6
2.5 5 7.5 10 12.5 15
Figure 1. Graph of the shear stress w(¢) versus shear rate ~ for the Johnson Segalman fluid with e = 0.05, a = 0.8. The increasing parts of the curve along 0.866 < ~ _< 0.9213 and 6.820 _< ~ < 14.694 yield two valnes of ~ for a given w(~). The decreasing part of the curve is unstable (Kolkka et al., 1988; Malkus et al., 1990). The max imum and min imum values of the shear stress in the nonmonotone part are 0.921 and 0.725, respectively.
Spurt and Instability in a Two-Layer Johnson-Segalman Liquid 465
is modeled by the two shear rates coexisting in a flow, with the higher shear rate close to the wall to mimic slip at observable scales.
K01kka et al. (1988) refer to the work of Yerushalmi et al. (1970) on the linear stability of plane Couette flow for nonmonotone shear-thinning fluids, and extend their work to show that the one-dimensional solutions along the increasing portions of the curve in Figure I are linearly stable but along the decreasing portion of the curve they are linearly unstable. Malkus et al. (1990) show that the problem is well posed when the retardation parameter e is strictly positive, unlike the generalized Newtonian model which suffers from the Hadamard instability when the steady shear stress decreases with shear rate.
Malkus et al. (1991) and Nohel and coworkers (1990, 1993) analyze systems of ordinary differential equations that approximate the dynamics of one-dimensional shear flow through a slit die. In particular, they treat highly elastic very viscous flows to model the spurt experiments. The velocity profile through the slit die is calculated as a function of the distance from the wall and time. In a cross section of the die the shear stress has its maximum value at the wall and is zero at the centerline. They determine the global dynamics using a phase.plane analysis. One example is the case of quasi-static loading, in which the pressure gradient is increased in small steps so that a steady state is achieved at each step. Under loading, the shear stress at the wall increases until the first hump in Figure 1 is reached. Past this point, the ve][ocity profile develops a kink at the wall, as the fluid at the wall jumps to the higher shear rate, and the bulk of the fluid remains at the lower shear rate. This jump in the shear rate at the top of the curve is called "top jumping." The layered solution is a "spurted solution." As the driving pressure gradient is increased further, the kink moves into the interior of the flow. Other properties such as latency and hysteresis under loading and unloading are predicted and they suggest experiments that may be performed to verify their model. Kolkka et al. (1988) and the works of Malkus et al. fit the data at the onset of the spurt regime in the experiments of Vinogradov et al. (1972).
The separate case of one-dimensional piston-driven shear flow for prescribed volumetric flow rate is treated by Malkus et al. (1994). They predict oscillations in the extrudate due to a Hopf bifurcation to periodic solutions, as the volumetric flow rate is increased beyond a critical value. These solutions have a layered structure except at very low flow rates. They draw the analogy between this periodicity and the stick-slip regime, where periodic pressure pulsations have been observed by Lim (1988) and Lim and Schowalter (1989).
In this paper we pursue the idea of the one-dimensional spurted solution. We take the following simple model problem (as opposed to a physical model). A two-dimensional spurted Couette flow is driven by the motion of the upper wall. The lower layer of fluid occupies the bulk of the flow and has a shear rate which is just below the critical value. They upper layer has the higher shear rate and occupies a thin region. The two layers are separated by a free surface, but are composed of the same liquids. Admittedly, spurt experiments are pressure-driven or piston-driven flows and would be better modeled by a Poiseuille flow, with a lower shear stress at the center of the velocity profile, and the highest shear attained close to the wall. However, since stick-slip apparently occurs first at the wall and then moves inward, the aim of our Couette-ftow analysis is to model the region close to the wall at the onset of an instability.
Linear stability to perturbations which are periodic in the direction of the flow is examined by computing the eigenvalues numerically. The results are presented in Section 3. It is !round that there are always short-wave instabilities. The most unstable mode is found to be the interracial mode. Depending on the relative depths of the fluids, other modes which may be termed internal modes may also cause instabilities but with lesser growth rates than the interracial mode. In the case of quasi-static loading, the one- dimensional results of Malkus et al. (1990) suggest that a likely situation is top-jumping at the hump of the curve in Figure 1. In such an arrangement, long and order-one waves are found to be stable, and short waves are unstable, the growth rate increases with wave number until it levels offfor the shortest waves. A number of internal or bulk modes may also be unstable for a range of wave numbers. As the kink in the velocity profile moves inward from the wall, it is found that the interracial mode is unstable down to longer waves, with comparable growth rates for order-one waves and shorter. Any instability from the internal modes also shift from short to order-one wave numbers. Near the trough of the curve in Figure 1, spurted solutions are unstable for all wavelengths of perturbation, with dominant growth rates occurring for order-one and short waves.
Since the one-dimensionally stable layered solutions are unstable to two-dimensional perturbations, how would this affect the dynamic evolution predicted previously with the one-dimensional models, and how
466 Y.Y. Renardy
can these steady spurted solutions be achieved during an actual experiment? It is possible that even if a one-dimensional steady solution is unstable, it may be describing what is happening in an average sense. The short-wave interfacial instability is an intriguing feature. The eigenfunction for this instability decays exponentially fast away from the interface position. Its effect is localized in a boundary layer near the interface, not disturbing the bulk of the flow, which in this case is steady. As the disturbance wavelength becomes longer, the corresponding eigenfunction begins to affect the bulk of the flow. Thus, the presence of a short-wave interfacial instability for a just-spurted arrangement is reminiscent of the loss of gloss of the extrudate surface at the first instability of the onset of melt fracture, described by the formation of the spurted solution with the layer at the wall too thin to be observable, together with the short-wave instability which leaves the bulk of the flow steady. In any case, there is uncertainty about spurt models, and the results of this paper raise further questions.
Chen and Joseph (1992) suggest a different "apparent slip" explanation, which motivated them to do a stability analysis of two upper-convected Maxwell liquids in layers. The idea is that at some critical Weissenberg number, high molecular-weight polymers are detached from the wall and migrate to the lower shear-rate region, creating a lubricating layer near the wall. Hence, the layer near the wall is a polymer-depleted layer as opposed to a layer of a different material, and therefore surace tension between the two layers is negligible. In this way, one can start with a single layer and end up with two layers without resorting to the use of nonmonotone constitutive equations. They treat core-annular flow in which the pipe radius is slightly larger than the nominal radius separating the high and low molecular-weight polymers. The stratification in density, viscosity, and relaxation times affect the stability of this two-layer flow. The viscosity of the polymer core is much larger than the average viscosity of the low molecular-weight annulus, and the core is more elastic. Both instability and stability are possible depending on the con- ditions. The instability may be suggestive of sharkskin, with the physical mechanism being the leaching of polymers from the solution at the wall. The stress-induced diffusion of polymer chains has been experimentally observed by a number of research groups for polymer solutions (Tirrell and Malone, 1977).
On the other hand, the situation is different for polymer melts. In the case of polymer melts, the distribution of molecular weights is narrow. It is not clear how a mechanism like leaching would happen in a polymer melt. If the separation were present, there would be a strong dependence of spurt on the molecular-weight distribution, and this has not been reported. The observed critical stress (Vinogradov et al., 1972) is independent of molecular weight. In the absence of a separation mechanism, one choice is the adoption of the nonmonotone model which describes the initial flow of one fluid and subsequent separation into two layers of the identical fluid, and provides an explanation of where the second layer came from with a single constitutive equation across the entire flow.
2. Governing Equations
We model a freshly spurted flow with the following two-layer Couette flow of a Johnson-Segalman liquid. Two layers of a fluid of density p, polymer or shear viscosity/~, Newtonian or solvent viscosity t/, and relaxation time 2 lie in the (x*, z*) plane between infinite parallel plates located at z* = 0, l*. Asterisks are used for dimensional variables. The upper plate moves with velocity (Uv*, 0) and the bottom plate is at rest. In the basic flow, layer 1 occupies 0 _< z* _< 11" and layer 2 occupies l* _< z* _< l*. The velocity, distance, time, and pressure are made dimensionless with respect to l*/2,1", A, and ]~/2. The dimensionless upper plate speed is defined to be the Weissenberg number, representing the average shear rate multiplied by the re- laxation time. The Reynolds number, retardation parameter, and Weissenberg number, respectively, are defined by
The equation of motion is
1"212 rl U * 2 R = (~2,)' ~ ~t' W = Up = I* (2)
) \ & + (u.V)n = V-T - Vp + 8Au. (3)
Spurt and Instability in a Two-Layer Johnson-Segalman Liquid 467
The total stress is T + e(Vu + Vu T) - pI. The constitutive law for the Johnson-Segalman fluid (Johnson and Segalman, 1977) is
DT T + ~ - = Vu + (Vu) r, (4)
where D / D t is the Oldroyd derivative
DT 0T (1 +a)(vu)T ~ ~ (1 +a)T(Vu)T. Dt - & + (u'V)T 2 + _ _ (Vu)TT + T(Vu) 2 (5)
These equations reduce to those of the upper-convected Maxwell liquid when a = 1. For the fluid to be rod-climbing, ½ < a < 1. For the nonmonotone behavior of Figure 1, a < 1 and ~ < -~.
At the kink in the velocity profile, the extra stress has a jump corresponding to the jump in the velocity gradient. This discontinuity is transported along with the fluid particles, as evident from the material derivative terms in (5) of the constitutive law. Hence the kink in the base flow lies on a material surface. At this interface, the velocity and total tangential stress are continuous, the jump in the normal stress is zero, and the kinematic free surface condition holds.
The dimensionless basic velocity (U(z), 0) satisfying no-slip at the walls is
U" , ~ k l z , O<_z<_ll , ~z) ~- ~k2( z _ 1) + Up, l 1 <_ z <_ 1, (6)
where kl is chosen so that it lies in the interval of shear rate where the variation with the shear stress is nonmonotone. At the interface z = l 1, continuity of shear stress yields the value of k2:
kt k2 1 + k](1 - a 2) ~- ekx - 1 + k~(1 - a 2) +- ek2" (7)
Continuity of velocity yields the Weissenberg number or the upper plate speed:
W = Up = k l l 1 + k212, 11 + 12 = 1. (8)
The normal stress condition gives the difference in the constant basic pressures. The basic extra stress tensor in fluid i (i = 1, 2) is
C2 C3/ (9)
ki C 1 =(1 + a ) C a k , , C 2 - 1 + k2(1-aZ) ' C 3 = - ( 1 - a ) C 2 k v
Solutions that are perturbations of the above basic flow are sought in normal modes proportional to exp(i~x + at). The perturbations to the velocity, pressure, and interface position are denoted by (u, v), p, and h, respectively, with u = 34`/~z = 4`', v -= - ic@, where 4 ̀is the streamfunction. We use a prime to denote differentiation with respect to z. The perturbation to the extra stress tensor is denoted by
r l 2
The equations of motion (3) in each fluid yield
R ( ~ + i ~ U ( z ) ) ( 4 ` " - c~24 )̀ = e(4`iv _ 2c~24`, + ~40) + i~T,11 + T£ 2 + ~2Tl 2 _ i~T22. (11)
The constitutive equations (4)-(5) yield
( a + i c ~ U ( z ) + 1)Ttl--(1 + a ) U ' T 1 2 + ( 1 - a ) C 2 c t 2 4 ` - 2 i ~ ( a C 1 + 1 ) 0 ' - (1 + a)C24`" = 0, (12)
_ , (1 + a ) _ _ , ( l - a ) r (a + 1 + i~U(z))r~2 - t ; - - 5 - - - '~i2 + tJ ~ ~ + 4'"((1 - a2)k~C2 - 1 ) - ~24`(k~C2(1 + a 2) + 1) --- 0, (13)
(a + 1 + iag(z ) )T22 + (1 -- a)U'T12 -- 4 c̀~2(1 + a)C 2 + 2ic@'(aC 3 + 1) + 4`"(1 - a)C 2 = 0. (14)
468 Y.Y. Renardy
The boundary conditions of no slip are ~ = ~' = 0 at z = 0, 1. The conditions at the interface are posed at the unknown position z = 11 + h(x, t) where h(x, t) is small, and the conditions are linearized at z = 11. Continuity of velocity yields
h[U'~ + [0'~ = 0 (15) and
[0~ = 0, (16)
where ~x~ denotes x(fluid 1) - x(fluid 2). The total stress is
T t o t a 1 = T + e(Vu + V(u) r) - pI,
where T here denotes the sum of the basic stress (9) and the perturbation (10), u denotes the sum of the basic velocity (6) and the perturbation, and p denotes the basic plus the perturbation pressure. Continuity of shear stress is then
It "Ttota I" n~ ----- 0,
where the tangent to the interface is t= (1 ,h , ) / x /1 +h~, and the normal to the interface is
n = ( -h~, 1)/x/i- + h 2. Thus,
~T12 ~ + ieh[C 3 - C1~ + e[~," + e2~9~ = 0. (17)
Here, ~C 2 + eki~ = 0 and ~P~ = 0 in the basic flow. The balance of normal stress is
[n'Ttota,-n~ = 0,
where again T denotes the basic stress (9) plus the perturbation (10): In terms of the perturbations, this is
~[T22 + 2evz -- p~ = O.
We express this in terms of the streamfunction:
-- c~T22 ~ -1- ig(30~2{¢'~ -- [~"~) + R(ia~[~k'~ -- ctU(I1)[O' ~ + c~tpU'~) + ~ T l l ~ - i[T'12 ~ ----- 0. (18)
The kinematic free surface condition yields
ha + U(ll)icth + ieOl(ll) = 0. (19)
3. Results on the Spectrum
The strips of continuous spectra are found by reducing the governing equations to a single equation for the streamfunction and setting the coefficient of the highest derivative to zero. Results on compact operators (Dunford and Schwartz, 1958) are applied to the ordinary differential equation for the streamfunction defined on the bounded interval. It is known that if the coefficient of the highest derivative is not zero, then the eigenvalues are discrete. If the coefficient of the highest derivative vanishes, then the eigenfunctions may be singular and yield a continuous spectrum. In order to do the reduction, we define
21(Z ) = (7 "~- 1 "~ i~U(z), 0 < z < 1. (20)
This is the term that multiplies the stress component Tll in the constitutive equation (12) which is re-expressed as
r l l , h = r l r l ~ + r2(0),
F 1 = (1 + a)U',
F2(O) = - (1 - a)C2~2~ + 2ietk~(aC 1 + 1) + (1 + a)C2Ozz.
The constitutive equation for the component T22 also involves ~,, 21, and T12;
21 T22 = r3(~,) + F4T12,
F3(O) = C~2(1 q - a)C21 p -- 2i~(1 + aC3)O= -- (1 -- a)C2~b=,
F 4 = --(1 -- a)U'.
(21)
(22)
(23)
(24)
(25)
(26)
Spurt and Instability in a Two-Layer Johnson-Segalman Liquid 469
We substitute into the Navier-Stokes equation (11) and express the stress terms on the right-hand side of it in terms of T12. In order to do this, we require T'11 and T22 in terms of T12. From (21),
2~T'11 + 2'1T11 = r l T i ~ + r l ,
where the prime denotes d/dz, and substitution from (21) for T 11 yields
21Ti ~ _ -2'1 ( r t T12 + r2 ) + r l Ti 2 + r l " 21
We multiply this through by 21:
2~rql = - 4 1 ( r l T l z + r2) + 21(F1Ti2 + r l ) . (27)
From (24), ! t !
21T22 + 4'1T22 = F 3 + F4T12 ,
and substitution from (24) for T22 yields
t 4 t t !
41 T22 = --"~2!1 (F 3 + F4T12 ) -t- F 3 + F4T12, ,tl
so that
412 T22, ..= _ 2,1(F3 q_ F4T12) -t- 21(F 3 -1- F4 T'12 ). (28)
We use (27)-(28) to re-express the stress terms in (11) in terms of 7"12:
ieT'tl + T12 -'}-e2Tlz- ic~T'z2
= 2--~t (-- 24](-- C2~2@ -q- io@z(aC 1 + aC 3 + 2) + C2@zz ) + 221(- C2o~21[tz 71- ic~Ozz(aC 1 + aC 3 + 2) + C2@zzz) )
T 1 2 2 r 2 2 ~2 Y' +--~-1 (2c~ U + e ~1) + ~ ie2U' + T~2. (29)
We next express T12 in terms of ~. We re-express (13) as
41T12 = FsT22 + F 6 T l l + F7(O) , (30)
F s = U,(1 +a ) = _ U , ( 1 - a ) 2 ' F6 2 ' FT= -tP"((1-aZ)kiCz-1)+c~zt~(kiC2(1-t-a2)+l)"
We substitute from (21) and (24) and find
22T12 = FsF 3 + F6F2 + 21F7, 22(z) = 22 + U,2( 1 _ aZ), (31)
which shows that the highest derivative of ~k in T 1 z is ~,". Equations (30)-(31) show that we should multiply the Navier-Stokes equation (11) through by 21222, and find that the highest derivative of ~, Oiv, is multiplied by
g4~42 -- 4az((1 -- a2)U'C2(41 + 1) - 21) = 22(842 -- (1 - a2)U'C2(41 + 1) + 41).
This yields
The first strip is
2 1 = 0 , e(22+U'2(1-a2))-(1-aZ)U'Cz(21+l)+21=O.
Reo- = - 1 , Im a in [0, - s U p ] .
The second equation in (32) yields two constant values of 21 for each fluid given by
1 2 1 - --A-t-(AZ-4eB(e-A))l/22e ' B=U'2(1-a2) ' A - I + B'
(32)
(33)
(34)
470 Y.Y. Renardy
resulting in two strips per fluid at
= - 1 + 21 - ic~U(z). (35)
This set of eigenvalues gives rise to instability on the decreasing portion of the curve in Figure 1 for the following reason. If the square root in (34) is taken over a negative term, then 21 is a complex conjugate with negative real part, and o- is stable. Thus, if 2 t is to cause an instability, it must be real and must be greater than one. If 2 t = 1, then (32) yields C2(k i ) = 0 where C 2 is defined in (10). Thus when 21 is greater than one, this yields the condition that C'z(ki) < 0. This is consistent with the results of Malkus e t al. (1990) where the phase plane analysis of the one-dimensional time-dependent problem showed that if any point on the velocity profile lands on the decreasing part of the shear-stress-shear-rate curve, then it is a saddle point and is not a steady-state solution.
The numerical calculation of the eignvalues was done with the Chebyshev-tau method (Gottlieb and Orszag, 1983; Orszag, 1971). This approximates the eigenvalues for C~-eigenfunctions with infinite-order accuracy. The streamfunction, T t 1, T12, and T22 are discretized in the z-direction in terms of Chebyshev polynomials up to order N, N - 2, N - 2, and N, respectively, in each fluid. Together with h, the total number of unknowns is 8N - 1. The equation of motion is approximated to the (N - 4)th degree, the constitutive equations for Tll and T12 to the ( N - 2)th degree, and for T2z up to the ( N - 1)th degree. Together with the four boundary conditions and five interface conditions, this yields 8N - 1 equations. The eigenvalues are computed in complex quadruple precision with an NAG routine. The numerical results of this paper have been convergence tested.
An additional test for the numerical results is the case c~ = 0, for which the solution to the linearized problem is known in closed form. In particular, for R = 0, (31) which comes from the constitutive equations and the Navier-Stokes equation are
T12((g + 1) 2 + U'2(1 - a2)) + 0"(U'(1 -- a 2 ) C 2 + (1 - a 2 ) U ' C 2 - 1)(a + 1)) = 0 (36)
and e01v + T~ 2 = 0. (37)
These imply that Oi~ = 0. The streamfunctions are therefore cubic polynomials in z:
= a l + a 2 z + a 3 z 2 -~ a4z 3 in layer 1,
= bt + b2(1 - z) + b3(1 - z) 2 + b4(1 - z) 3 in layer 2.
The boundary conditions at z = 0, 1 yield a t = 0, a z -- 0, b 1 = 0, and b 2 = 0. At the interface, the kinematic condition reduces to h = 0. The remaining conditions are the continuity of the two velocity components, the shear-stress balance and the normal-stress balance. This yields a four-by-four determinant equation for the eigenvalues. The eigenvalues computed with the Chebyshev-tau scheme have been checked to satisfy the determinant equation. The stability at ~ = 0 is inferred from the works of Malkus e t al. (1990). An inspection of the governing equations shows that the growth rate of the interfacial eigenvalue a is proportional to cd as
~ 0. All other eigenvalues are real or complex conjugates and are found to be stable. We begin by reminding the reader of results available for the one-layer case. The one-layer flow is
expected to be linearly stable, from previous results on related constitutive models (Renardy and Renardy, 1986). A computation at a = 0.8, ~ = 0.05, and kl = k 2 = Up = 0.8 confirms this for the increasing portions of the shear-stress-shear-rate curve. Due to the reasons following (34), the one-layer flow is unstable along the decreasing part of the curve.
A number of works on the linear stability of two-layer flows of related constitutive models have appeared, many with applications to the production of multilayered materials (Joseph and Renardy, 1992). The upper-convected Maxwell (UCM) model is the case e = 0 and a = 1. This is a singular limit in the Johnson-Segalman model since e multiplies the highest derivative in the governing equations. The short-wave asymptotic analysis for the interfacial mode was done for the UCM model by Renardy (1988) for zero surface tension and equal densities. The eigenfunction decays exponentially fast away from the interface, so that the conditions at the walls do not enter the calculation. This is a boundary-layer-type singular perturbation problem. The growth rate of the interfacial eigenvalue is expanded in powers of l/e". At leading order, the governing equation for the streamfunction is a fourth-order ordinary differential equation with the complication being the variable coefficients. This is solved with a decoupling technique used by Gorodtsov and Leonov (1967). The growth rate is order one, and the competition between viscous
Spurt and Instability in a Two-Layer Johnson-Segalman Liquid 471
and elastic stratifications appears. In Newtonian flows the effect of viscosity stratification appears at growth rates of O(1/cd) (Hooper and Boyd, 1983), so the effect of elastic stratification is stronger than this. The order-one growth rate for short waves is seen also in the numerical results for the Johnson-Segalman model.
Chen (1991b) derives the long-wave expansion (e small) for the interfacial mode for two-layer UCM and Oldroyd-B liquids in Couette flow. The Oldroyd-B model reduces to the UCM model when the characteristic retardation time is zero. He notes that the jump in the first normal stress difference across the interface is not balanced for the basic flow, and this induces an elastic instability. This instability occurs J
when the more elastic fluid occupies less than half of the total volume inside the channel. He computes neutral stability curves for this and for core-annular flow (Chen, 1991a; Chen and Zhang, 1993). Chen and Joseph (1992) analyze the short-wave case for core-annular flow.
S u and Khomami (1991, 1992a, 1992b) treat the stability of two-layer plane Poiseuille flow of power-law fluids, second-order fluids, and Oldroyd-B fluids. They calculate the long-wave formulas and numerically compute the neutral stability curves over wave numbers of order one. For the Oldroyd-B model, the purely elastic instability is again found (Su and Khomami, 1992b). In the absence of other effects such as surface tension, the interface is unstable for long and order one wave numbers if the more-elastic fluid occupies less than half of the channel. If the less-elastic fluid is in the thinner layer, then stability depends on the elasticities, depth ratio, and wave number. The maximum growth rates occur at wave numbers, made dimensionless with respect to the thickness of the more elastic layer, of order one. Growth rates are shown for situations where long waves are unstable and the highest growth rates occur at order-one wave numbers. The opposite situation of the stable long waves and unstable short waves arises in the numerical results below for the Johnson-Segalman model. A direct comparison between the numerical results for the Johnson-Segalman model and the Oldroyd-B model is hampered by the question of how to evaluate the effective elasticities in each layer for the Johnson-Segalman case, since both the viscosities and the elasticities compete to control stability.
The Reynolds number for polymer melts is close to zero. Results are shown for R = 0, a = 0.8 and e = 0.05. A comparison between R = 0 and R = 1 a t k 1 = 1, 11 = 0.9 has shown that the modes which cause instability for low R are captured by looking at R = 0. The inclusion of nonzero R leads to a family of solutions which do not cause instability at low R.
In an experiment which begins at low shear rates, the flow is initially in a one-fluid arrangement and, under quasi-static loading, follows the increasing part of the curve in Figure 1 until it reaches the crest at
= 1.866. At this point, the flow develops spurt with two regions of different shear rates. At the onset of spurt, the region of high shear is expected to be relatively thin. The phase plane analysis of Malkus et al. (1990, 1991) on their one-dimensional model predicts the flow will approach a steady spurt solution in which the jump in the strain rate occurs at the maximum stress ("top-jumping") with the kink in the velocity profile located as close as possible to the wall. This occurs for the case of quasi-static loading where the stress is increased in small steps.
As an example of top-jumping, consider the lower fluid of depth 11 -- 0.99 occupying the bulk of the flow with shear rate k 1 = 1.85, and a thin layer of depth 12 = 0.01 at the wall with shear rate k 2 = 14.69335. The Weissenberg number is Up = 1.978433. Figure 2 illustrates the growth rate versus the wave number of the
Figure 2. Growth rate versus wave number for the unstable mode for a=0.8, e=0.05, 11=0.99, k 1=1.85, k 2=14.69335, w(k0= w(k2)=0.92, and Up= 1.9784. Other eigenvalues are stable.
0.8
0.6
0.4 Re o-
0.2
--0. '0 10 20 30 40 50 60 70
472 Y.Y. Renardy
perturbation. In this range of wave numbers only the interfacial mode was found to be unstable. There is stability for long and order-one waves. Instability sets in for wave numbers greater than about eight and the maximum growth rate is achieved for ~ ~ 42, with growth rates for shorter waves at about the same order of magnitude thereafter. Thus, the wavelength for instabilities is on the scale of the thickness of the thin spurted layer. The short-wave growth rate appears to be at most order one, just as in the UCM case discussed above (Chen and Joseph, 1992; Renardy, 1988). Instabilities may set in for the bulk modes at shorter wavelengths than the ones investigated here. The significant amount of short-wave instability is reminiscent of the loss of gloss observed in the sharkskin regime, with very fine extrudate distortions. Larson (1992) writes that "sharkskin is a surface roughness that usually modulates the extrudate diameter by no more than one percent or so."
In the case of an abrupt loading where the change in shear stress in an experiment is not small, it may be possible for spurt to occur for a shear stress below that of top-jumping. As an example of this, consider the bulk of the fluid of depth l 1 = 0.99 at the shear rate k 1 = 1, with a thin layer of fluid at the wall at the shear rate k 2 = 10.584, w(k~) = w(k2) = 0.79. The Weissenberg number is Up = 1.096. Figure 3 shows the growth rate or Re o- versus the wave number ~ of the disturbance. The interfacial mode is stable for long waves and is unstable for wave numbers larger than 3.8. The maximum growth rate is attained at c~ ~ 40 and thereafter remains at about the same level. A second mode is unstable for short waves in the region 41 < c~ and appears to become stable for wavelengths shorter than the ones recorded here. At ~ ~ 67, a third mode appears to be nearing instability. The instability of the bulk modes is a feature that has not appeared in previous works on two-layer flows. For the Oldroyd-B model, it would be expected that the bulk modes be unstable for sufficiently high flow rates. In the Johnson-Segalman case we have the bulk modes being unstable at shear rates that are less than the ones where they were stable (see also Renardy, 1996).
The effect of an abrupt loading rather than top-jumping is that instabilities have moved down to the longer waves and long waves are less stable. The short waves attain slightly larger growth rates. The predominantly short-wave instability of Figure 2 suggests that it more closely describes sharkskin than the situation in Figure 3.
As an example of a situation where the region of high shear has traveled inward into the fluid, la is chosen to be 0.9. Figure 4 shows the growth rate for the unstable modes for 0 < c~ _< 70 for 11 = 0.9, k~ = 1, k 2 = 10.584, w(k l ) = w(k2) = 0.79, and U v = 1.96. The most unstable mode is the interracial mode which is unstable for the entire range of c~ < 70 which was examined, and at ~ = 70 the growth rate appears to be tapering to a constant. The maximum growth rate is reached at ~ ~ 4 and approximately this level is
1
S p u r t a n d Ins tab i l i ty in a T w o - L a y e r J o h n s o n - S e g a l m a n L i q u i d 473
6
4
2
Imf f 0 ~
- 2
- 4
- -618-16 - 1 4 - 1 2 - 1 0 -'8 -'6 -'4 -'2 Re o" (a)
Im o-
- (
-1( - 2 5 o -2o -io
Re (b)
I - 5 0
4
2
0
- 2
- 4 I m cr
- 6
- 8
- 1 0
- 1 2
- 1 4 - - - - 3 0
k
-25 -2o -10 -s 6 5 Re cr
(c)
Figure 5. T h e s p e c t r u m for a = 0.8, 8 = 0.05, l 1 = 0.9, k 1 = 1, k 2 = 10.584, w ( k l ) = w ( k 2 ) = 0.79, a n d U , = 1.958. (a) ~ = 0, (b) e = 1.7,
a n d (c) c~ = 3.3.
a t t a ined for shor te r waves. There is a second mode which is uns tab le over 4.1 < c~ < 11 and has a m a x i m u m growth ra te a t c~ ~ 5.6. A th i rd mode is uns tab le over 7 < e < 11. The second and th i rd modes are bu lk modes , which p o p ou t of the con t inuous spec t rum and this is i l lus t ra ted next.
F igure 5(a) shows the spec t rum at c~ -- 0, where the con t inuous spectra, by (32)-(35), have d iminished to the po in ts G -- - 1, - 0 . 6 7 1 8 , - 16.034, and - 1.24198 + 4.55536i. A r o u n d the con t inuous spectra, there are a n u m b e r of discrete modes . This is s imilar to the behav io r found for the two- layer U C M liquid, in which the con t inuous spec t rum consists of two strips. In between these strips, four discrete modes were cap tu red by the shor t -wave a sympto t i c expans ion for the interracial m o d e (Renardy, 1988). The next leas t -s table mode for c~ = 0 in F igure 5(a) is o- = - 1.0042 _+ 0.59753/i and is deno ted by M o d e 1 in F igure 4. At e ~ 1.7, this m o d e merges in to a con t inuous spec t rum at Re o- = - 0 . 6 7 1 8 , - 1.53 < Im o- < 0. The c o m p u t e d spec t rum for c~ = 1.7 is shown in F igure 5(b). There are four o ther str ips of con t inuous spect ra at:
(i) R e o - = 1, 3 . 3 2 9 2 < I m o - < 0 . ( i i ) R e o- = - 16.034, - 1.53 < Im cr < 0.
(iii) Re a = - 1.242, 1.226 < Im ~r < 3.025. (iv) Re o- = - 1.242, - 7 . 8 8 4 6 < I m a < --6.085.
There are a n u m b e r of i sola ted discrete modes close to these spec t ra in F igure 5(b). The uns tab le m o d e labe led M o d e 2 in F igure 4 emerges at e ~ 3.3 f rom the con t inuous spec t rum loca ted
at Re ~ = - 1, - 6.46 < Im o- < 0. F igure 5(c) shows the c o m p u t e d spec t rum at c~ = 3.3, where the o ther
474 Y.Y. Renardy
1.6
1.4
1.2
1
0.8 Rea
0.6
0.4
0.2
0
- 0 . 2 L- 0 1'0 20 30 4'0 50 60 70
Figure 6. Growth rate versus wave number for the least stable modes for a=0.8, e= 0.05, 11 =0.99, k t =0.87, k z = 7.391946, w(kl) = w(k2)= 0.727, and Up = 0.9352. The interfacial mode is unstable over all the wave numbers shown.
strips of continuous spectra are at:
(i) Re o- = -0 .6718 and Re a = - 16.034 with -2 .97 < Im a < 0. ( i i ) Re o- = - 1.242 with - 1.907 < Im a < 1.585 and - 11.018 < Im cr < -7.525.
Once a flow has undergone spurt, quasi-static unloading, in which the shear stress is reduced in small steps until a smooth flow is recovered may be applied. At this point, the value of the shear stress is lower than is required for the onset of spurt. This hysteresis effect on unloading has been described in the one- dimensional model equation of Malkus et al. (1991), where a smooth flow is achieved once the shear stress falls below the trough of the curve. This is called bottom-jumping. The trough in Figure 1 occurs at w(~) = 0.725, ¢ = 0.866, 6.82. If the flow is in a two-layer state around the trough, then consider 11 = 0.99, k 1 = 0.87, k 2 = 7.392, and U~ -- 0.935. Figure 6 shows the growth rates versus wave numbers. The interfacial mode is unstable over all wave numbers. The stability of long waves is influenced by both the elasticities and by the viscosities of each of the regions. A number of bulk modes are also unstable. Compared with the situations in Figures 2 and 3, the instabilities have moved down to the longer waves and the long-wave stability is lost. The unstable modes achieve higher growth rates than when the shear stress was higher. The spectrum for the top-jumping situation appears more closely related to sharkskin than the spectrum for bottom-jumping.
The results show that the two-layer arrangement modeling spurt along the nonmonotone part of Figure 1 is unstable to short-wave disturbances. Thus the path described by the arrows in Figure 11 of Malkus et al. (1991) lands on steady states which are two-dimensionally unstable. At the onset of spurt for top-jumping, the largest growth rates are found for short wavelengths, consistent with the observation of the loss of gloss. As the fluid in high shear increases in depth, the largest growth rates move in to the order-one wavelength with the same order of magnitude for shorter waves. The spurted arrangement at the trough of the shear-stress-shear-strain curve is unstable for all wavelengths and the growth rates are larger than for higher shear stresses. The Johnson-Segalman model has been examined for the stability of steady states exhibiting spurt to two-dimensional perturbations.
R e f e r e n c e s
Adewale, K.E.P., and Leonov, A.I. (1993). On modeling spurt flows of polymers, J. Non-Newtonian Fluid Mech., 49, 133-138. Chen, K. (1991a). Interfacial instability due to elastic stratification in concentric coextrusion of two viscoelastic fluids, J. Non-
Newtonian Fluid Mech., 40, 155-175. Chen, K. (199 lb). Elastic instability of the interface in Couette flow of viscoelastic liquids, J. Non-Newtonian Fluid. Mech., 40, 261-267. Chen, K., and Joseph, D.D. (1992). Elastic shortwave instability in extrusion flows of viscoelastic liquids, J. Non-Newtonian Fluid
Mech., 42, 189-211. Chen, K., and Zhang, Y. (1993). Stability of the interface in co-extrusion flow of two viscoelastic fluids through a pipe, J. Fluid Mech.,
247, 489-502. Denn, M.M. (1990). Issues in viscoelastic fluid mechanics, Ann. Rev. Fluid Mech., 22, 13-34. Dunford, N., and Schwartz, J.T. (1958). Linear Operators, vol. 1, Wiley, New York.
Spurt and Instability in a Two-Layer Johnson Segalman Liquid 475
Gorodtsov, V.A., and Leonov, A.I. (1967). On a linear instability of a plane parallel Couette flow of viscoelastic fluid, J. Appl. Math. Mech., 31, 310-319.
Gottlieb, D., and Orszag, S. (1983). Numerical Analysis of Spectral Methods: Theory and Applications, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA.
Hopper, A.P., and Boyd, W.G.C. (1983). Shear flow instability at the interface between two viscous fluids, J. Fluid Mech., 128, 507-528. Hunter, J., and Slemrod, M. (1983). Viscoelastic fluid flow exhibiting hysteretic phase changes, Phys. Fluids, 26, 2345-2351. Johnson, M., and Segalman, D. (1977). A model for viscoelastic fluid behavior which allows non-aNne deformation, J. Non-Newtonian
Fluid Mech., 2, 255-270. Joseph, D.D., and Renardy, Y. (1992). Fundamentals of Two-Fluid Dynamics, Parts 1 and 2, Springer-Verlag, New York. Kalika, D.S., and Denn, M.M. (1987). Wall slip and extrudate distortion in linear low-density polyethylene, J. Rheol., 31,815 834. Kolkka, R., and Ierley, G. (1989). Spurt phenomena for the Giesekus viscoelastic liquid model, J. Non-Newtonian Fluid. Mech.,
33, 305-323. Kolkka, R., Malkus, D., Hansen, M., Ierley, G., and Worthing, R. (1988). Spurt phenomena of the Johnson Segalman fluid and related
models, J. Non-Newtonian Fluid Mech., 29, 303-335. Larson, R.G. (1992). Instabilities in viscoelastic flows, Rheol. Acta, 31,213-263. Leonov, A.I. (1984). A linear model of the stick-slip phenomena in polymer flow in rheometers, Rheol. Acta, 23, 591-600. Leonov, A.I. (1990). On the dependence of friction force on sliding velocity in the theory of adhesive friction of elastomers, Wear, 141,
131-145. Leonov, A.I., and Srinivasan, A. (1993). Self-oscillations of an elastic plate sliding over a smooth surface, Internat. J. Engrg. Sci., 31,
453-473. Lira, FJ. (1988). Wall slip in narrow molecular weight distribution polybutadienes, Ph.D. Thesis, Princeton University. Lim, F.J., and Schowalter, W.R. (1989). Wall slip of narrow molecular weight distribution polybutadienes, J. Rheol., 33, 1359 1382. Malkus, D.S., Nobel, J.A., and Plohr, B.J., (1990). Dynamics of shear flow of a non-Newtonian fluid, J. Comput. Phys., 87(2), 464-487. Malkus, D.S., Nohel, J.A., and Plohr, B.J. (1991). Analysis of new phenomena in shear flow of non-Newtonian fluids, SlAM J. Appl.
Math., 51(4), 899-929. Malkus, D.S., Nohel, J.A., and Plohr, B.J., (1994). Oscillations in piston-driven shear flow of a non-Newtonian fluid, Forschungsin-
stitut fiir Mathematik, ETH Ziirich, preprint. Moynihan, R.H., Barid, D.G., and Ramanathan, R. (1990). Additional observations on the surface melt fracture behavior of linear
low-density polyethylene, J. Non-Newtonian Fluid Mech., 36, 255-263. Nohel, J.A., Pego, R.L., and Tzavaras, A.E. (1990). Stability of discontinuous steady states in shearing motions of a non-Newtonian
fluid, Proc. Roy. Soe. Edinburoh Sect. A, 115, 39-59. Nohel, J.A., and Pego, R.L. (1993). Nonlinear stability and asymptotic behavior of shearing motions of a non-Newtonian fluid,
preprint. Orszag, S. (1971). Accurate solutions of the Orr-Sommerfeld stability equation, J. Fluid Mech., 50, 689-703. Renardy, Y. (1988). Stability of the interface in two-layer Couette flow of upper convected Maxwell liquids, J. Non-Newtonian Fluid
Mech., 28, 99-115. Renardy, Y. (1996). An instability of plane Couette flow of the Johnson-Segalman liquid, in Advances in Multi-Fluid Flows,
Y. Renardy, D. Papageorgiou, and S.M. Sun, eds., SIAM, Philadelphia, PA, to appear. Renardy, M., and Renardy, Y. (1986). Linear stability of plane Couette flow of an upper convected Maxwell fluid, J. Non-Newtonian
Fluid Mech., 22, 23-33. Su, Y.-Y., and Khomami, B. (1991). Stability of multilayer power law and second-order fluids in plane Poiseuille flow, Chem. Engr 9.
Comm., 109, 209-223. Su, Y.-Y., and Khomami, B. (1992a). Interracial stability of multilayer viscoelastic fluids in slit and converging channel die geometries,
J. Rheol., 36, 357-387. Su, Y.-Y., and Khomami, B. (1992b). Purely elastic interfacial instabilities in superposed flow of polymeric fluids, Rheol. Acta, 31,
413-420. Tirrell, M., and Malone, M.F. (1977). Stress-induced diffusion of macromolecules, J. Polym. Sci., 15, 1569-1583. Yerushalmi, J., Katz, S., and Shinnar, R. (1970). The stability of steady shear flows of some viscoelastic fluids, Chem. Engrg. Sci., 25,
1891-1902.