Linear instability of planar shear banded flow

0.2 MB·

by  Suzanne M Fielding

Linear instability of planar shear banded flow
S. M. Fielding∗
Polymer IRC and School of Physics & Astronomy,
University of Leeds, Leeds LS2 9JT, United Kingdom
(Dated: February 2, 2008)
We study the linear stability of planar shear banded flow with respect to perturbations with
5 wavevectorintheplaneofthebandinginterface,withinthenonlocalJohnsonSegalmanmodel. We
0 find that perturbations grow in time, over a range of wavevectors, rendering the interface linearly
0 unstable. Results for the unstable eigenfunction are used to discuss the nature of the instability.
2 Wealso comment on the stability of phase separated domains to shear flow in model H.

n
PACS numbers: 47.50.+d Non-Newtonian fluid flows– 83.60.Wc Flow instabilities – 47.55.Kf Multi-phase
a
andparticleladenflows–61.25.HqMacromolecularandpolymersolutions;polymermelts;swelling
J

1
2 Complex fluids such as wormlike micellar surfac- The viscoelastic stress evolves with dJS dynamics [8, 9]

tants [1], lamellar onion phases [2], polymer solutions [3]
ft] and soft glasses [4] commonly undergo flow instabilities Σ♦=2GD− Σ + l2∇2Σ, (3)
o and flow-induced transitions that result in spatially het- τ τ
s erogeneous“shearbanded”states. Thiseffectiscaptured
withplateaumodulus Gandrelaxationtimeτ. The non
t. byseveralnotablerheologicalmodels[5]inwhichtheun- local diffusive term accounts for spatial gradients across
a
derlying constitutive curve of shear stress vs. shear rate,
m the interface between the bands. It arises naturally in
T (γ˙), is non-monotonic (Fig. 1), allowing the coexis-
xy models of liquid crystals, and diffusion of strained poly-
- tence of bands of differing shear rate at common shear
d mer molecules [12]. The time derivative
stress, Fig. 2. However, most theoretical studies have
n
o consideredonlyonespatialdimension(1D)[6,7],normal ♦
Σ=(∂ +V·∇)Σ−a(D·Σ+Σ·D)−(Σ·Ω−Ω·Σ),
c to the interface between the bands (the flow gradientdi- t
[ rection, y). The stability of 1D bandedprofiles in higher
in which D and Ω are the symmetric and antisymmetric
1 dimensions has been implicitly assumed, but is in fact parts of the velocity gradient tensor, (∇V) ≡ ∂ v .
αβ α β
v anopen question. In this Letter, therefore,we study nu- The “slip parameter” a measures the non-affinity of de-
8 merically the linear stability of 1D planar shear banded
formation of the viscoelastic component [8]. Slip occurs
1 profiles with respect to perturbations with wavevectors
for |a| < 1. The underlying constitutive curve T (γ˙) is
5 in the interfacial plane (x,z)=(flow,vorticity). xy
1 then capable of the non-monotonic behaviour of Fig. 1.
We workwithinthe JohnsonSegalman(JS)model[8],
0 Within this model we consider planar shear between
modified to include non local diffusive terms [9]. These
5 infinite, flat parallel plates at y = 0,L. We use units in
0 account for gradients in the order parameters across the whichG=1,τ =1 andL=1; andboundary conditions
t/ banding interface, conferring a surface tension. This aty =0,1of∂yΣαβ =0∀α,β fortheviscoelasticstress,
a “dJS”modelisoftentakenasaparadigmofshearband-
with no slip and no penetration for the velocity.
m ingsystems. Ourmainresultwillbethatinterfacialfluc- Foranimposedshearrateγ¯˙ intheregionofdecreasing
- tuationstypicallygrowintime,renderingthe1Dbanded stress, dT /dγ˙ < 0, homogeneous flow is unstable [13].
d profile linearly unstable. This potentially opens the way xy
A 1D analysis in the flow gradient dimension then pre-
n
tonontrivialinterfacialdynamicsandcouldformastart-
o dicts a separation into two bands of differing shear rates
ing point for understanding an emerging body of data
c γ˙1,γ˙2 at common shear stress, Tb, separated by an in-
: revealing erratic fluctuations of shear banded flows [10]. terface of width O(l). As the applied shear rate γ¯˙ is
v
This work is a timely counterpart to new techniques for
i tracked across the banding regime, the relative width-
X measuring interfacial dynamics [11]. It is also relevant
fraction of the bands adjusts to maintain the constraint
r industrially, to processing instability and oil extraction. R dyγ˙(y)=γ¯˙, while γ˙1,γ˙2 and Tb stay constant, leading
a The model is defined as follows. The generalised
to a plateau in the steady state flow curve (Fig. 1).
Navier Stokes equation for a viscoelastic material in a
We verified this 1D scenario by numerically evolving
Newtonian solvent of viscosity η and density ρ is:
Eqns.1 to 3, allowingspatialvariationsonly in the flow-
gradient direction y. We used a Crank Nicholson algo-
ρ(∂ +V.∇)V =∇.(Σ+η∇V−PI), (1)
t rithm [14] within a finite difference scheme on a uniform
whereV(R)is the velocityfieldandΣ(R)the viscoelas- mesh of “full” points y0,y1...yNbase for Σ and staggered
tic part of the stress. For homogeneous planar shear, “half” points y21,y23...yNbase−21 for V. We evolved with
V=yγ˙xˆ, the total shear stress T =Σ (γ˙)+ηγ˙. The time-stepDtforatimetmax tosteadystate,checkingfor
xy xy
pressure P is determined by incompressibility, convergenceto the limit Nbase →∞,Dt→0,tmax →∞.
The resulting flow curve is shown in Fig. 1. A typical
∇.V=0. (2) steady state shear banded profile V(y), Σ(y) is given
2
0.6 l = 0.0025
0.6
l = 0.005
l = 0.01 (adapted mesh)
T
b 0.4 l = 0.01 (uniform mesh)
0.4
T Re ω
xy max 0.2
0.2
underlying constitutive curve 0
steady state flow curve
0 -0.2
0 γ.1 2 4 γ. 6 γ.2 8 10 0 5 10 15 20
q
x
FIG. 1: Underlying constitutive curve; steady state flow
FIG.3: Realpartoftheeigenvalueofthemostunstablemode.
curve. a = 0.3, η = 0.05. Banding occurs on the plateau.
a = 0.3, η = 0.05, γ¯˙ = 2.0, Reynolds number ρ/η = 0. The
2 data for l = 0.01 correspond to the base profile in Fig. 2.
Symbols: data. Solid lines: cubic splines.
1
Σ
plane of perturbation wavevectors(q ,q ).
yy x z
Σ Welinearisedthemodelequations1to3forsmallper-
xy 0 turbations (lower case) about the (upper case) base pro-
Σ
xx Σ file, Φ˜(x,y,z,t) = Φ(y)+φq(y)exp(ωqt+iqxx+iqzz).
Vx Σyy The vector Φ comprises all components Φ = (Σαβ,Vα),
-1 xy the pressure being eliminated by incompressibility. This
Σ
xx linearisationresultsinaneigenvalueequationwithanop-
V
x erator L, which acts linearly on the perturbation φq(y):
-2
0 0.2 0.4 0.6 0.8 1
y
ω φ (y)=L(Φ(y),q,∂ ,∂2...)φ (y). (4)
q q y y q
FIG. 2: 1D banded profile, with spatial gradients restricted For numerical study, we discretized this equation on a
to the flow gradient direction, y. γ¯˙ =2.0, towards the left of staggered mesh. The 1D base profile Φ(y) was read in
theplateau in Fig. 1. l=0.01, Nbase =800. fromthecalculationalreadydescribed. Fornarrowinter-
faces,itsuniformmeshhadtoomanynodesforuseinthe
eigenvalue problem, so we adapted it to put most atten-
in Fig. 2. The velocity normal to the interface Vy = 0 tionneartheinterface. WethenusedaNAGroutine[18]
in this 1D profile. The smooth variation of the order to find the eigenmodes of this discretized problem.
parametersacrossthe interface results fromthe diffusive The results, discussed below, were checked as follows:
term in Eqn. 3, which confers an interface width O(l). (i) for convergence with respect to mesh structure; (ii)
This is in contrast to local models (l = 0) in which the thatforahomogeneousbasestateontheunderlyingcon-
interface is a sharp discontinuity. In fact, local models stitutive curve our results match those of Ref. [19]; (iii)
arepathologicalinthesensethatthe bandedstateisnot that for a = 0, l = 0 (the local Oldroyd B model), our
uniquely selected, but depends on flow history [7, 9]. method gives results consistent with Fig. 3 of Ref. [20];
The linear stability of the sharply banded profiles (iv)thatlinearisationaboutasemi-evolved(non-steady)
of local models was studied by previous authors. Re- banded state using the analytically derived Eqn. 4 gives
nardy [15] found instability with respect to interfacial the same results in the limit q = 0,q → 0 as a partic-
x z
fluctuations of high wavevector,qx →∞, in the local JS ular direct numerical linearisation performed about the
model restricted to the case of a thin high shear band. same profile in the code that evolves the 1D base state;
McLeish [16] studied capillary flow, for general band (v) for robustness with respect to first evolving the base
thickness. He demonstrated a long wavelength (qx →0) state on either a uniform or adapted grid, using either a
instability due to the jump in normal stresses across the semi-implicit or explicit algorithm; (vi) that two differ-
interface. ThismechanismwasalsodiscussedinRef.[17]. entmethodsofeliminatingthepressure(usingtheOseen
Herewestudynumericallythenon localcase,inwhich tensor, and the curl operator) agree.
the 1D banded profile is uniquely selected [7, 9]. The For any base profile Φ(y) and wavevectorq, the num-
non zero interfacial width, l ≪ L, now confers a surface ber of eigenmodes is equal to the number of order pa-
tension, which was absent fromthe localcase. We study rameters summed over all mesh points. In this Letter,
generalbandthicknessesandthe full (velocity,vorticity) we onlyconsider the eigenvalue ωmax(q) with the largest
3
1
0.5
0.8
0
ω 0.6
peak y
0.4
-0.5 l = 0.0025
l = 0.005
l = 0.01
0.2
0 2 4 6 8
_γ_. 0
0.3
FIG. 4: Peak of the dispersion relation, i.e. ℜωmax at
dℜωmax/dqx = 0. Parameters as for Fig. 3. Limits of the
bandingregimeshownbyverticallines. Symbols: data. Dot-
0.25
ted lines: cubicsplines, as a guide to theeye.
y
realpart,ℜωmax(q). Inparticular,weaskifthismodeis 0.2
stable, ℜωmax < 0, or unstable, ℜωmax > 0. All results
given are for a low solvent viscosity η =0.05≪Gτ ≡1,
consistentwithexperiment. Weseta=0.3,althoughour
0.15
findings are qualitatively robust to variations in a. This 0 0.25 0.5 0.75 1
leavestheappliedshearrateγ¯˙ asthetunable parameter. q x/2π
x
The dispersion relation ℜωmax(qx,qz = 0) for fluctu-
ations with wavevector confined to the direction of the FIG. 5: Perturbation to flow field s1ℜv(y)eiqxx (arrows),
unperturbed flow is shown in Fig. 3 for γ¯˙ = 2.0. At and contour lines of perturbed normal stress Σ˜xx(x,y) =
any qx, ℜωmax increases with decreasing l, and for small Σxx(y)+s2ℜσxx(y)eiqxx (dottedlines), corresponding tothe
enough l the dispersion relation is positive over a range eigenvalue of Fig. 3 with l=0.01, qx =2.0. Contours down-
of wavevectors, rendering the 1D profile unstable. For wards: 0.45, 0.60, 0.75, 0.90, 1.05, 1.20, 1.35 (middle value
small l this applies to shear rates right across the stress shownthicker). Arbitraryscalefactorss1 =1.5ands2 =0.3.
plateau of Fig. 1, as shown in Fig. 4. Because the l val-
ues accessed here – l=O(1−10µm) for a 1mm rheome-
ter gap – are even larger than those expected physically,
l = O(100nm), our results suggest that, experimentally, stressjump∆Σxxacrosstheinterface(recallFig.2). This
the entire stress plateau will be unstable. triggersahorizontalperturbationtotheflowfieldℑvx in
theseregions,whichrecirculates,givinganO(q2)vertical
In the limit l → 0, q → 0, the corresponding x
x
velocityℜv atq x/2π =0.0,0.5,1.0. This enhances the
eigenfunction {∂ v ,v = 0,σ (y)} tends to the spa- y x
y x y αβ
original displacement and so causes instability. Stability
tial derivative of the base state, ∂ {∂ V ,V = 0,Σ },
y y x y αβ
is restoredfor higher q (Fig. 3), a feature that is absent
representing a simple displacement of the interface in x
in the local case.
the flow-gradient direction, with small corrections in
the bulk phases to maintain γ¯˙ = constant. As qx in- The eigenvalue ℜωmax(q) over the (qx,qz) plane is
creases from zero, this displacement is modulated by a showninFig.6. Modeswithwavevectoralongtheqxaxis
wave of wavevector qxxˆ with an eigenvalue ωmax(qx) = are much more prone to instability than those along the
ω0 + iqxω1 + qx2ω2 with ω2 > 0, signifying instability. qz axis. Nonetheless, for smaller values of l (not shown),
A natural question is whether this instability has the modes along the line qx =0 can go unstable as well.
same origin as that described by McLeish for the local We note finally an important bound on the validity of
model [16]. It is not obvious, a priori, that this should our calculation. The expansion used to obtain Eqn. 4
be true because, for the base state at least, the limit is valid for perturbations that are small at any point in
l→0is singular[7]. Indeed, adetailedanalysis(workin space. Forexample,forthestresscomponentswerequire
progress) is more complicated in this case, and deferred σαβ ≪ 1. Displacement of the interface by a distance
toalongerpublication. However,thenumericalresultsof ∆ gives σαβ = ∆dΣαβ/dy, which is O(∆/l), because
Fig.5arequalitativelyconsistentwiththe mechanismof the base profile Σαβ changes by O(1) overthe interfacial
McLeish, as follows. A wavelike interfacial displacement width O(l). We are thus restricted to small displace-
with extrema at q x/2π = 0.0,0.5,1.0 causes an inter- ments, ∆≪l. In future work, we will consider ∆≫l.
x
facial tilt near q x/2π =0.25,0.75, exposing the normal We comment briefly on the stability of a sheared in-
x
4
tor in the plane of the banding interface, within the non
6 local Johnson Segalman model. This applies to shear
rates right across the stress plateau, suggesting that the
5 instability is ubiquitous and therefore that the existing
theoretical picture of two stable shear bands separated
4
by a steady interface needs further thought. Indeed, our
finding is consistent with accumulating evidence for er-
q 3
z ratic fluctuations [10] and band breakup [23] in several
systems. Future work will study the fate of the interface
2
inthenonlinearregime,beyondthevalidityofthislinear
study. One possibility is that the instability is self limit-
1
ing beyond a criticalamplitude set by l (e.g., l1/2). This
0 would be consistent with a narrowly localized but still
0 1 2 3 4 5 6
unsteadyinterface,whichmightbe interpretedassteady
q
x in experiments that did not have high spatialresolution.
This might evenreconcileearly data showingapparently
FIG. 6: Real part of the most unstable eigenvalue. a = 0.3, steadyinterfaceswithrecentworkrevealingfluctuations.
η =0.05, γ¯˙ = 2.0, Reynolds number ρ/η =0.01 (negligible),
Bycontrast,ifthe instability werefoundnottobe self
l = 0.01. Contours are −0.45,−0.40... (dotted line), 0.00
limiting, and yet ubiquitous in existing banding models
(dashed) and ...0.25,0.30 (solid).
(work in progress), one would then need a new theoreti-
calpictureof(reasonably)steadyshearbandsthatcould
stillaccommodatetherequirednormalstressjumpacross
terface between two phases of a binary fluid in “model
the interface. Other open questions include the status of
H” [21]. Although this was studied in Ref. [22], that
the instabilityincurvedCouette geometry;andthe rela-
work integrated over space to get a simple equation for
tive importance of instabilities at non-zero q (as studied
the position of the interface. Such an approach neglects
here)tothosefoundatzeroqinrecentmodelsofspatio-
changes in the interface’s profile, and any fluid flow nor-
temporal rheochaos [24].
mal to the interface, so is not guaranteed to agree with
ours. Nonetheless,wefoundtheinterfacetobestable,as The author thanks Paul Callaghan, Mike Cates, Tan-
in Ref. [22]. This supports the idea that normal stresses niemolaLiverpool,TomMcLeish,PeterOlmstedandHe-
(absentinmodelH)causetheinstabilitydescribedabove. len Wilson for useful discussions and feedback; and the
To conclude, we have found 1D planar shear banded EPSRC GR/S29560/01for funding.
flowtobelinearlyunstabletofluctuationswithwavevec-
∗
Electronic address: s.m.fi[email protected] [10] A. S. Wunenburger,A. Colin, J. Leng, A. Arneodo, and
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