Journal of Non-Newtonian Fluid Mechanics

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1 J. Non-Newtonian Fluid Mech Contents lists available at ScienceDirect Journal of Non-Newtonian Fluid Mechanics journal homepage: Robustness of pulsating jet-like layers in sheared nano-rod dispersions M. Gregory Forest a, Sebastian Heidenreich b,, Siegfried Hess b, Xiaofeng Yang c, Ruhai Zhou d a Departments of Mathematics & Biomedical Engineering, Institute for Advanced Materials, University of North Carolina at Chapel Hill, Chapel Hill, NC , United States b Institute for Theoretical Physics, Technische Universität Berlin, Hardenbergstrasse 36, D Berlin, Germany c Department of Mathematics, University of North Carolina at Chapel Hill, Chapel Hill, NC , United States d Department of Mathematics and Statistics, Old Dominion University, Norfolk, VA 23529, United States article info abstract Article history: Received 16 February 2008 Received in revised form 2 May 2008 Accepted 2 June 2008 Keywords: Shear flow Nematic polymers Numerical simulation Structure formation PACS: v Vx Fd Fa Nano-rod dispersions in steady shear exhibit persistent transient responses both in experiments and simulations. The rotational contribution from shear flow couples with orientational diffusion, excludedvolume interactions, and distortional elasticity to yield complex dynamics and gradient morphology of the rod ensemble. The classification of sheared responses has mostly focused on nematodynamics of the collective particle response known as tumbling, wagging and kayaking; in heterogeneous simulations, one monitors the variability in nematodynamics across the domain. In this paper, we focus on flow coupling and non-newtonian feedback in transient heterogeneous simulations, and in particular on a remarkable effect: the formation of localized, pulsating jet layers in the shear gap. We solve the Navier Stokes momentum equations coupled through an orientation-dependent stress to three different orientational models a kinetic Smoluchowski equation and two tensor models, one from kinetic closure and another from irreversible thermodynamics. A similar spurt phenomenon was reported in 1D simulations of a model for planar nematic liquids by Kupferman et al. [R. Kupferman, M. Kawaguchi, M.M. Denn, Emergence of structure in models of liquid crystalline polymers with elasticity, J. Non-Newt. Fluid Mech ], which we extend to full orientational configuration space. We show: the pulsating jet layers correlate, in space and time, with the formation of a non-topological oblate defect phase in which the principal axis of orientation spreads from a unique direction to a circle; the jet-defect layers form where the local nematodynamics transitions from finite oscillation wagging to continuous rotation tumbling, and when neighboring directors lose phase coherence; and, a negative first normal stress difference develops in the pulsating jet-defect layers. Finally, we extend one model algorithm to two space dimensions and show numerical stability of the jet-defect phenomenon to 2D perturbations Elsevier B.V. All rights reserved. 1. Introduction The nano-rod dispersion is a general and simple model for a wide range of anisotropic, non-newtonian fluids that consist of small-to-large molecules or Brownian structures with properties similar to rigid rods or platelets. Representative examples are liquid crystals, liquid crystal polymers, worm-like micelles, and tobacco mosaic virus suspensions. In each of these model systems, orientational degrees of freedom and the possibility to form different mesoscopic phases isotropic and nematic lead to surprising orientational behavior and flow feedback in shear-dominated rotational flows. The large literature on shear banding [2] in sheared wormlike micelles [3] is an example of the remarkable non-newtonian flow feedback that is possible in such systems. Corresponding author. Tel.: ; fax: address: sebastian@itp.physik.tu-berlin.de S. Heidenreich. In this article we present numerical results for a dramatic shear banding phenomenon, the occurrence of pulsating jet layers, in conditions that mimic a parallel plate shear cell with imposed steady plate speeds. This behavior was first reported for a model twodimensional 2D nematic liquid the rods are 2D, whereas the flow is one-dimensional 1D with variations across the plate gap by Kupferman et al. [1]. Here, we present numerical results for nematic liquids with a three-dimensional 3D rod configuration space, for three completely different models, each solved by different numerical algorithms at our respective institutions. Numerical benchmarks are provided for each code to confirm these pulsating jet layers are model and not numerical phenomena, and one code is extended to two physical space dimensions to show the behavior is stable even to 2D perturbations in physical space. We use the rest of the Introduction to give some background and motivation for the special phenomena reported here. In steady shear driving conditions, nano-rod dispersions the reader may think of the particular example of liquid crystal /$ see front matter 2008 Elsevier B.V. All rights reserved. doi: /j.jnnfm

2 M.G. Forest et al. / J. Non-Newtonian Fluid Mech polymers as the most studied of these fluid systems exhibit both stationary and transient orientational responses. This behavior has been modeled primarily in the monodomain limit, where one posits simple shear and analyzes the resulting orientational dynamics of the rod phase. This analysis decouples the hydrodynamics and suppresses spatial dependence of the orientational distribution. This dimensional reduction of physical space, leaving a purely time-dependent model in orientational configuration space, has worked remarkably well in interpreting liquid crystal polymer experiments dating back to Kiss and Porter [4]. For example, the novel observation of negative first normal stress differences in steady shear were subsequently duplicated by the monodomain models of Doi Hess type [16 19] that are special limits of the models presented in this paper. Most notably, sign changes in the first normal stress difference are associated in each of these models with transitions to or among periodic orientational responses in steady shear. This model reduction has worked either because the flow response functions are well approximated by socalled viscometric flow in the experimental operating conditions, or because any strongly nonlinear shear flow does not qualitatively alter the measurable orientation and stress response functions. As noted just above, nematic liquids in steady shear exhibit several distinct modes of transient behavior, which have picturesque names in the liquid crystal and nematic polymer literature. Tumbling refers to continuous rotation of the peak axis of orientation of the rod ensemble in the plane of shear normal to the vorticity axis. Wagging refers to a finite oscillation of the major axis of orientation, again in the shear plane. More limit cycles were computed numerically by Larson and Ottinger [5] in which the major director either rotates around the vorticity axis called kayaking or rotates around some axis midway between the shear plane and vorticity axis tilted kayaking. These long-time transient responses have been identified in experiments [4,6 8] and confirmed through theoretical descriptions [9 15]. More complex, apparently chaotic behavior is also found in diverse models [12,14,20 22], arising through a period-doubling bifurcation route, which Berry [6] associated with the rapid development of turbidity strong heterogeneity in experiments. In this flow regime, the two assumptions of a homogeneous orientational distribution and of a simple, steady linear shear become especially suspect, motivating several research groups to undertake spatiotemporal numerical studies. To do so, one relaxes the above two assumptions, proceeding first to models and simulations of 1D heterogeneity. For spatially inhomogeneous systems, it becomes possible to explore the impact of physical boundary conditions on the orientational distribution at the moving or stationary plates. The standard approach has been to impose quiescent equilibrium boundary conditions, i.e., a uniform nematic phase with the principal axis fixed experimentally, this is done by mechanical or chemical preparations. The fundamental issue is whether plate anchoring on the orientational distribution is felt only in a wall boundary layer or the anchoring at solid walls penetrates into the shear gap. Diverse models show that limit cycles found for the monodomain assumptions persist in the presence of heterogeneity for example, if one monitors the orientational distribution at a specific spatial location. Indeed, all limit cycles and complex chaotic behavior found in monodomain models have been observed, and phase transitions continue to occur as one varies gap dimensions, plate speeds, and distortional elasticity potentials [23 26]. In some studies, the flow is imposed as simple linear shear and only orientational gradients are allowed, while other, more resolved simulations perform a self-consistent computation of the flow. A particularly interesting flow feedback phenomenon where there was compelling experimental evidence is the formation of steady roll cells at very low shear rates. These structures were reported experimentally by Larson and Mead [27,28], and successfully modeled by Feng et al. [29] with a liquid crystal director theory, and more recently by Klein et al. [30] with a Doi Hess Marrucci Greco tensor model applicable to nematic polymers and nano-rod dispersions. The flow feedback behavior of interest in this paper is in the strongly nonlinear regime, where both anisotropy and focusingdefocusing of the orientational distribution are important. Yet, the phenomenon generation of a nonlinear and non-monotone shear profile across the plate gap is predominantly 1D in physical space. Thus, a full orientation tensor, or a full kinetic model for the orientational distribution function, is necessary, but a one space-dimensional flow-orientation solver is sufficient to capture the phenomenon. We return at the end to explore stability to 2D perturbations. The pulsating flow phenomena turn out to arise when there is a significant orientational mismatch in the local monodomain response functions across the gap: the interior has a tumbling response function, whereas near the plates the local response function is wagging. These two monodomain responses are spatially incompatible for the following reason. The principal axis of a wagging orbit cannot continuously reside next to a tumbling orbit, and thus the major director of neighboring wagging and tumbling orbits will become significantly out of phase. Either the principal axis will develop enormous local gradients which the distortional potential penalizes, or the orientational distribution must defocus through a degeneracy in the principal values. This latter scenario costs much less energy, creating what we call an order-parameter or non-topological defect. This happens locally: in space, at the transition site from tumbling to wagging, and in time, during the part of the tumbling cycle when the principal axis rapidly rotates before resetting in phase with the neighboring wagging orbit. This picture explains intuitively why the flow feedback effect, whatever it is, occurs locally in space and in periodic pulses. Since this behavior generates gradients in the principal values of the orientational distribution, the extra stress from the rod distribution will generate flow feedback. The intriguing nature of this flow structure is that it leads to local pulsating layers where the flow accelerates, and furthermore we can tune parameters to amplify the effect so that jet-layers form that speed ahead of the flow above and below. Since our primary numerical studies are confined to one-space dimensional heterogeneity, the issue still remains as to stability with respect to higher space-dimensional perturbations. We find evidence in our comprehensive studies [32] of 1D orientational distributions that are stable in two space dimensions. The flow was imposed in that study. We proceed for this paper to perform an analogous numerical stability study of the full coupled system in two space dimensions. One might also surmise that these jet layer oscillations are exclusive to special anchoring conditions at the plates. We present one simulation below where the flow-orientation behavior, persists when the wall anchoring conditions are shifted out of plane. In these conditions, starting from the same parameter specifications as with tangential anchoring, the tumbling-wagging composite dynamics across the gap is replaced by their out-of-plane monodomain analogs: kayaking in the center of the gap, and tilted kayaking near each plate. Again, the jet layer forms at the transition layer where the local monodomain responses get completely out of phase, and order parameter defects form during the jet phase. Given all of these studies, the hydrodynamic feedback mechanism leading to a pulsating jet layer in each half of the shear gap appears to be considerably more robust that previously anticipated.

3 132 M.G. Forest et al. / J. Non-Newtonian Fluid Mech We turn now to the modeling of this paper and the order of presentation. We consider a dispersion of nano-rods between oppositely translating, parallel plates at velocities v 0 and v 0, respectively. We present three different models for the orientational distribution and the associated stress constitutive law that couples to the Navier Stokes momentum equations. We then perform one-space dimensional simulations of the initialboundary-value problem for flow between the translating plates, starting with an equilibrium, nematic phase across the shear gap. Numerical convergence studies are presented in an Appendix B to confirm these results are not numerical artifacts. In each model, we impose consistent parameter values for the strength of the excluded-volume potential, Frank elasticity constants, Reynolds number, and Deborah number. The spatio-temporal attractor of each system is periodic, and the remarkable non-monotone, jetlike, primary shear profile emerges and disappears. Thus our terminology of an oscillating or pulsating jet layer. When the jet layers reside sufficiently near the plates, as observed in the Kupferman et al. model [1], they move faster than the translating plates that drive the experiment! We proceed to show the jet layers can be translated toward the interior or shifted closer to the plates, and the strength of the jet can be controlled, by varying the Ericksen and Deborah numbers. The jet layers are then correlated with the orientational distribution and stress tensor, with a consistent picture for each model. The jet layers coincide with an order parameter degeneracy in the orientational distribution, a non-topological defect structure whose existence is independent of physical space dimensions. In these defect phases, the principal axis of orientation often called the major director coincides with the secondary principal axis to form a circle of maximum likelihood. This is a strong defocusing event, with a broadening of the peak of the orientational distribution, which always arises when local anisotropic monodomain responses get strongly out of phase. This order parameter defect structure is the tensorial and kinetic theory mechanism to avoid discontinuities in the principal axis of orientation, providing a continuous transition between wagging and tumbling at neighboring spatial sites. We note that this phenomenon is not captured by a Leslie Ericksen Frank theory nor by the model 2D liquid of Kupferman et al. In a planar liquid, the only order parameter degeneracy is the isotropic phase; we never observe isotropic phases in our simulations, which involve all three eigenvalues of the second-moment tensor to collide. This is an interesting observation, suggesting that the kinetic and second-moment models select this oblate defect phase degeneracy over higher order degeneracy in the distribution such as complete isotropy. We develop some new graphics for our simulation results to amplify the formation of these oblate defect phases during the pulsating jet layer events. Finally, we explore and confirm robustness of these phenomena to two-space dimensional perturbations in both flow and orientation, using a spectral-galerkin code for the so-called Doi Marrucci Greco tensor-flow model. 2. The model equations for a spatially inhomogeneous anisotropic fluid In this section, we give the descriptions of three models implemented for numerical simulations. The models are the kinetic model, the Doi Marrucci Greco model and the alignment tensor model. All models have the same physical ingredients to describe the orientational dynamics: orientational diffusion, the orienting torque caused by the symmetric traceless part of the velocity gradient, the local molecular rotation induced by the vorticity flow, and, elastic and diffusional terms associated with spatial inhomogeneities of the local orientation. The stress tensor of the alignment tensor model differs in some respects from that used in the kinetic and Doi Marrucci Greco model The Doi Hess kinetic model, Marrucci Greco gradient elasticity, and flow coupling We consider plane shear flow between two plates located at y = ±h, in Cartesian coordinatesx = x, y, z, and moving with corresponding velocity v = ±v 0, 0, 0, respectively. There are two apparent length scales in this problem: the gap width 2h, an external length scale, and the finite range l of molecular interaction, an internal length scale, set by the distortional elasticity in the Doi Marrucci Greco DMG model. When the plates move relative to each other at a constant speed, it sets a bulk flow time scale t 0 = h/v 0 ; the nematic average rotary diffusivity D r sets another internal time scale t n = 1/D r, and the ratio t n /t 0 defines the Deborah number De: De = t n t 0 = v 0 hd r. 1 There are also scales associated with solvent viscosity and the three nematic viscosities, and with elastic distortion, which due to the flow-nematic-plate interaction are not a priori known. We nondimensionalize the DMG model using the length scale h, the time scale t n and the characteristic stress 0 = h 2 /t 2 n. The dimensionless flow and stress variables become: ṽ = t n h v, x = 1 t x, t =, =, p = p. 2 h t n 0 0 The following seven dimensionless parameters arise: Re = 0 t n /,, Er, i and. Here Re is the solvent Reynolds number; measures the strength of entropic relative to kinetic energy; Er is the Ericksen number which measures short-range nematic potential strength relative to distortional elasticity strength depicted by the persistence length l; i,i= 1, 2 and 3 are three nematic Reynolds numbers; and is a fraction between 0 and 1 that corresponds to equal = 0 or distinct /= 0 elasticity constants [33]. For other parameters, we refer to [33]. We drop the tilde on all variables; all figures correspond to normalized variables and length, time scales. We neglect the effect of translational diffusion and employ an approximate rotary diffusivity. The dimensionless Smoluchowski or generalized Fokker Planck equation [18,19] for the probability distribution function PDF f m, x,t becomes: Df = R [Rf + f RV] R [m ṁf ], Dt ṁ = m + a[d m D : mmm], where D/Dt = / t + v, R = m / m is the rotational gradient operator, D and are the dimensionless rate-of-strain and vorticity tensors, respectively, and a = r 2 1/r 2 + 1, where r is the rod aspect ratio. The coupled Maier Saupe and Marrucci Greco potential is V = 3N 2 M : mm 1 [ M + M] :mm, 4 2Er where N is a dimensionless rod volume fraction which governs the strength of short-range excluded volume interactions. The rank-2 tensor M is the second moment of f: M = Mf = m =1 3 mmf m, x, tdm. 5

4 M.G. Forest et al. / J. Non-Newtonian Fluid Mech The dimensionless forms of the balance of linear momentum, the continuity equation, and the stress constitutive equation are [33]: dv = pi +, v = 0, dt 2 = Re + 3 D + 1 D M + M D + 2 D : M 4 +a M 1 3 I N M M + N M : M 4 a 6Er M M + M M 2 M : M 4 12Er [2 M M M M + M c] a 12Er [M M d + M d M 4 M :M 4 ] 12Er [M d M M M d M e ], where M c = M : M M :M, 7 M d = M + M T, 8 M e = M M Mˇj, M ij,i, 9 M 4 = m =1 6 mmmm f m, x,tdm. 10 In the simulations presented below, we fix the following parameter values: Er = 500, De = 1, N = 6, = 0.5, = 20, 1 = 0.004, 2 = 1.5, 3 = 0.1. We also impose tangential director is parallel to the flow direction anchoring conditions on the equilibrium nematic phase at the plates cf. [34]. We consider 1D physical space the interval between the two parallel plates. The boundary conditions on the velocity v = v x, 0, 0 are given by the Deborah number: v x y =±1,t =±De. 11 We assume homogeneous tangential anchoring at the plates, given by the quiescent nematic equilibrium: f m,y=±1,t = f e m, 12 where f e m is an equilibrium solution of the Smoluchowski equation corresponding to the tangential anchoring of particles at the plates when v = Doi Marrucci Greco second-moment orientation model In Landau de Gennes models, the probability distribution function f m, x, t of Doi Hess kinetic theory is projected onto the second moment tensor, M, a symmetric, trace 1, positive semidefinite tensor of rank 2. The orientation tensor Q is the trace zero form of M, Q = M I/3. The distinctions among models can be framed in terms of closure rules necessary to reduce the above coupled flow-kinetic orientation model and stress formula to a system that closes on the second-moment orientation tensor Q and the flow variables v and p. The dimensionless forms of the stress constitutive equation and orientation tensor equation that are often called the Doi Marrucci Greco DMG model [45,46] are given as follows [35]: 2 = Re + 3 D + a FQ + a Q : Q Q + I 12 3Er 3 QQ + Q Q 13 Q + 1 3Er 2 Q Q QQ 1 13 Q : Q Q : Q Q + I D + D Q + I + 2 D : Q Q + I, where Er is the Ericksen number, and i are the three nematic Reynolds numbers normalized viscosities all defined earlier, and the short-range excluded volume effects are captured by FQ = 1 N 3 Q NQ 2 + NQ : Q Q + I 3, 14 where N is a dimensionless concentration of nematic polymers, which controls the strength of the mesoscopic approximation, FQ, of the gradient of the Landau de Gennes type potential. The orientation tensor equation with the DMG closure rule is DQ Dt = Q Q + adq + QD + 2a Q 3 D 2aD : Q + I 3 FQ + 1 Q : Q Q + I 3Er 3 12 QQ + Q Q 13 Q. 15 The boundary condition for the scaled velocity is the same as in the kinetic simulations 11. We assume homogeneous tangential anchoring at the plates, given by the quiescent nematic equilibrium: Q eq = s e x e x 13 I, 16 where e x = 1, 0, 0 and s = Alignment tensor model In this section we introduce a different model for the second rank alignment tensor that was first derived in the framework of irreversible thermodynamics, but also inferred from the kinetic equation with a closure relation slightly different from the one of DMG. We present the model equations in the notation of the DMG model for a more transparent comparison. This notation therefore differs from the original literature cited below; the correspondence and a discussion of its relationship to the DMG model are given in Appendix A. The equation for the alignment tensor Q [36,39] in the Doi Marrucci Greco model notation reads see Appendix 44: D Dt Q = Q Q + DQ + QD 2 3 D : QI + D a FQ De Ẽr 2 Q 1 De FQ + 1 Ẽr Q 3 2 KD. 17 The parameters De, Ẽr, and K are related to the DMG model parameters see Appendix A. The parameter D a does not have an analog in the DMG model: D a is a flow-alignment diffusion parameter In our study we set = 0 and D a = 0. This means we presents results for a minimal model, which is as simple as possible, but not too simple. The function F denotes the derivative of the spatially homogeneous part of the amended Landau de Gennes potential: Q AM Q = FQ = ϑq 3 6 QQ 13 Q : QI Q : QQ Q : Q 2 /Qmax, 4 which is a modification of the corresponding Landau de Gennes potential: LDG = 1 2 ϑq : Q 6QQ :Q Q : Q2. 18 The parameter ϑ in 18 has been used with ϑn = A 0 1 N/N /1 N K /N [17]. The pseudocritical concentration N and

5 134 M.G. Forest et al. / J. Non-Newtonian Fluid Mech Table 1 Characteristic scales Scale DH and DMG model AT model Time 1/D r 1/DeD r Space h h Velocity h D r De h D r Stress h D r 2 /mk BT D r denotes the nematic average rotary diffusivity and De the Deborah number De = v 0/hD r. 2h is the gap height. is the density. the concentration at phase coexistence N K are also model parameters. The value of A 0 depends on the proportionality coefficient chosen between Q and uu. The choice made above implies A 0 = 1, cf. [36]. The coefficients, on the one hand, are linked with measurable quantities and, on the other hand, can be related to molecular quantities within the framework of a mesoscopic theory based on a kinetic equation for the orientational pdf [18,17,16]. In previous studies it was demonstrated that the Landau de Gennes potential does not restrict the order parameter to physically admissible values, in particular the order parameter blows up at high extensional flow rates. To cure this problem a modified potential amended potential was introduced and discussed in [40]. From [40], the value 2.5 is implemented for the cut-off parameter Q max. This is a plausible value, e.g. liquid crystals, where the Maier Saupe order parameter S = P 2 is about 0.4 at the transition temperature. Thus, the maximum value of unity for S with the Maier Saupe potential is larger by a factor 2.5. In the Landau de Gennes case, one has Q = a K = 1 at the transition temperature ϑ = ϑ K = 1. For the amended potential with Q max = 2.5 one has the transition at ϑ = ϑ K with a K Because of the small difference between these values, it is convenient to maintain the Landau de Gennes scaling for the physical variables. In the Table 1 a summary of the different scalings is given. Note the alignment tensor Q in 17 differs from the DMG tensorial order parameter by the constant factor a K 15/2, i.e. Q AT = a K 15/2 Q DMG. Upon the assumption that the alignment tensor is uniaxial and that its magnitude is constant, the spatial derivative Q just acts on the director n. In this limiting case the Frank elasticity is recovered [37,38]. For simplicity, the isotropic case is considered which implies that, in the nematic phase, all three Frank elasticity coefficients are equal. This is consistent with the choice made in the DMG tensor model, whereas the kinetic model allows for two distinct elasticity constants. The phenomena we report are insensitive to one versus two elasticity constants. The orientation-dependent stress constitutive equation is given by 3 K = iso D 2 ˇ FQ 3 K + 2 ˇẼr Q + ˇ Q FQ + FQQ 2 3 FQ :QI 19 1 Q Q + QQ 23 Ẽr ˇ Q :QI. The parameter ˇ is inversely proportional to : 3v 0 2 ˇ = L 2 Dr 0 a K A, 20 K where D r is the rotational diffusion constant, as in the other two models. Again, we assume homogeneous tangential anchoring at the plates, given by the quiescent nematic equilibrium: 3 Q h, t = Qh, t = 2 q eq e x e x 13 I, 21 Table 2 Parameter values used in three models Parameter DH model DMG model Parameter AT model Er Ẽr 100 De De 1.0 N 6 6 N ˇ 1.0 a K iso where q eq = 5a K s, here s = 0.61 with a K = 1. For the velocity, no-slip boundary conditions are used: u h, t = v 0, uh, t = v In Table 2 we compare the values used in the three models. Fig. 1. The space time primary velocity profile across the gap. Top: DH model. Middle: DMG model. Bottom: AT model.

6 M.G. Forest et al. / J. Non-Newtonian Fluid Mech Model robustness of pulsating jet layers We now present results of numerical simulations of each of the 3 models presented above, at parameter specifications given in the Tables. The spatio-temporal attractors for each model are remarkably consistent: We focus exclusively on the longtime attractor properties which are independent of initial conditions and shortlived transients. We begin with graphics of the flow structure, then the orientational structure, and finally the stress. We present and compare each feature from all three model results The pulsating jet layer phenomenon We begin with surface plots of v x y, t analogous to those in Kupferman et al. [1]. Fig. 1 top is from the Doi Hess kineticflow DH model, Fig. 1middle is from the Doi Marrucci Greco tensor-flow DMG model, and Fig. 1bottom is from the alignment tensor-flow AT model. Each surface reveals the periodic appearance and disappearance of a localized layer on each side of the shear gap within which the flow accelerates to speeds that surpass the flow above or below it. We coin this phenomenon a pulsating jet layer. Unlike the reported nonlinear shear profiles in most studies of flow-nematic coupling, these jet layers are distinguished by non-monotone shear profiles. The three models, at these parameter values, differ in regard to the location of the jet layers within the gap, the temporal period between successive jet pulses, and in the nonlinear flow profile across the remainder of the gap Fig. 4. Snapshots of the primary velocity profile across the gap in the AT model. The top snapshots are during the jet formation, while the lower plots are during the quiet phase. Fig. 2. Snapshots of the primary velocity profile across the shear gap from the kinetic model. Fig. 3. Snapshots of the primary flow profile, v xy, t on the interval 184 <t<186, from the DMG tensor model. Fig. 5. The primary velocity time series, v xy,t, at the gap height y where the peak velocity of the jet occurs. Top: DH model at y = 0.77; middle: DMG model at y = 0.96; bottom: AT model at y = 0.2.

7 136 M.G. Forest et al. / J. Non-Newtonian Fluid Mech during and in between the jet pulses. These quantitative features are all parameter-dependent within each model, so that a comprehensive parameter study could be undertaken to minimize the differences among the models. Instead, we prefer to highlight the similarities that occur without trying to force them; later we give some results on jet pulse locations versus Deborah and Ericksen number. We focus next on snapshots of the primary flow profile during, before, and after the jet pulses form. The kinetic model snapshots are given in Fig. 2. During the relatively long quiet phase, the flow profile is nearly linear. Then, prior to the jet formation, a strongly nonlinear, but still monotone shear profile develops; these snapshots reveal a structure consisting of layers with plug-like flow sandwiched between strong shear bands with local shear rates much greater than the bulk shear rate. Similar monotone nonlinear shear band structures have been reported in several studies of the DMG tensor-flow model [35,41]. Next, the flow gradients concentrate, forming the jet layer, with nearly linear flow in the remainder of the gap. The corresponding DMG model snapshots are given in Fig. 3. The snapshot with the jet layers reveals jets close to each plate as in the kinetic model. The remainder of the gap has strongly nonlinear and non-monotone behavior with large regions of flow reversal in the top and bottom of the gap. As the jet disappears, the snapshots reveal a composite shear band and plug flow profile, with a mid-gap nearly stationary flow layer, and two strong shear bands emanating from each plate. The quiet phase with a nearly linear profile across the gap is shown in the t = 187snapshot. The AT model snapshots, Fig. 4, also show the formation of jet layers, which at these parameter values are much closer to the center of the gap, and an intermediate quiet phase where the flow is nearly linear across the gap as in the other model simulations. To foreshadow the correlations of these flow features with the orientational distribution, we show in subsequent figures that the quiet phase in the flow correlates with a phase synchrony of the major director across the shear gap. The jets correlate with major director asynchrony, in particular as one layer oscillates wagging orbits while the neighboring layer continues to monotonically rotate tumbling orbits. In the DMG model, there is no quiet phase with nearly linear flow across the entire gap, and the flow profile is a composite of nearly plug flow layers separated by strong shear bands. Next, we show the time series Fig. 5 of the primary velocity stationed at the gap height in each model where the peak velocity of the jet occurs. The normalized velocity at the center of the jet fluctuates: between 0.60 and 1.25 of the plate speed at y = 0.77 in the kinetic model, and between 0.68 and 1.36 of the plate speed at y = 0.96 in Fig. 6. The space time surface plot of the order parameter s = d 1 d 2. Top: DH model. Middle: DMG model. Bottom: AT model. Fig. 7. A snapshot of the spatial variation of the order parameter s = d 1 d 2 taken when the jet is fully formed in each model. Top: DH model. Middle: DMG model. Bottom: AT model.

8 M.G. Forest et al. / J. Non-Newtonian Fluid Mech the DMG model. The analogous time series for the AT model shows a fluctuation between 0.1 and 1.25 of the plate speed at y = 0.2. The time series at the pulsating jet center has a consistent signature between the kinetic and DMG closure models, and one can see a similar time series from the AT model surface plot, even though the locations of the jet layers are quite different at the parameter values presented here Orientational correlations with the pulsating jet layer The hydrodynamic feedback phenomenon shown above arises from orientational gradients in the stress constitutive law, whose divergence couples to the Navier Stokes momentum balance. Thus, there must be an orientational space time signature that accompanies the pulsating jet layer. It is reasonable to expect, based on the long history of liquid crystal theory in support of experimental phenomena, that this anisotropic non-newtonian flow behavior is associated with defects. However, one-space dimensional behavior cannot be associated with topological defects, which correspond to non-zero winding numbers of the unique major director around a closed curve: there are no closed loops in one space dimension. The tensor order parameter is needed to resolve the equilibrium and flow phase diagrams, precisely because of behavior associated with the degrees of order. These degrees of freedom are captured by the scalar order parameters defined earlier from the second moment tensor of the probability distribution function. Before giving the simulation results, recall our earlier discussion of the second moment tensor M, which affords a geometric representation of the orientational distribution in terms of triax- ial ellipsoids. Non-degenerate nematic phases are characterized by s>0, for which there is a unique principal axis n 1. Recall s = d 1 d 2, the difference in the two largest principal values of M. Degenerate phases are defined by the condition s = 0. The greatest degeneracy arises in the isotropic phase, where ˇ = 0 as well, so that all directions on the sphere are equally probable, and the corresponding ellipsoid is a sphere. There remains an intermediate degeneracy between the nematic and isotropic phases, where s = 0, yet ˇ>0. This oblate defect phase corresponds geometrically to an oblate spheroid, in which there is a circle of principal axes. Recall the plate boundary conditions are uniaxial equilibrium phases s >0,ˇ= 0 with the major director aligned with the primary flow axis, corresponding to a prolate spheroid aligned with the flow. For the parameter values chosen in Table 2, the equilibrium value of s is 0.77 for the kinetic model, 0.86 for the DMG model, and q eq = 1.37 for the AT model. In Fig. 6, we show the respective surface plots of sy, t during the same evolution of the primary flow in Fig. 1, from the kinetic, DMG and AT models. We do not show ˇ, which remains bounded away from zero so the phases are not isotropic; this will be confirmed in the graphics below of the orientation ellipsoids. Clearly, all figures show a pulsating defect layer characterized by a precipitous drop of s close to zero, corresponding to an oblate defect phase, which correlates precisely with the location and timing of the pulsating jet layer in each model! Next in Fig. 7, we present Fig. 8. The time series of the order parameter s = d 1 d 2 at the gap height y where the peak velocity of the jet occurs. Top: DH model at y = Middle: DMG model at y = Bottom: AT model at y = Fig. 9. The space time plot of orientation ellipses from the kinetic-flow model, constructed from the shear plane x y projection of the principal axes associated with the two largest eigenvalues d 1,d 2 of the second moment tensor. Near the plate, the major director exhibits finite, in-plane oscillations wagging, whereas in the middle of the gap the major director continuously rotates tumbles. A thin ellipse corresponds to strong focusing of the PDF, whereas a circle corresponds to the oblate defect phase.

9 138 M.G. Forest et al. / J. Non-Newtonian Fluid Mech Fig. 11. The ellipsoids of the AT model simulation, analogous to Fig. 10. Fig. 10. The second-moment tensor ellipsoids of the DMG model simulation: a space time sequence capturing the jet-defect phase formation around t = 185, and a spatial blow-up of the orientational ellipsoids from the t = 185 snapshot, capturing the gradient morphology around the defect phase. a snapshot of the scalar order parameter sy, t from each model simulation, coinciding with the peak jet layer snapshots shown earlier for each model. Since the snapshots are taken at the same time, clearly there is a temporal correlation, and from the figures, clearly there is a spatial correlation with the minimum of s and the maximum of v x.nextinfig. 8, we show the time series of sy,tatthe location y of the center of the jet layer, which is to be compared with the velocity time series in Fig. 5 at the same gap height for each model. One finds further confirmation of the temporal correlation between the oblate defect formation and the jet pulses in each model. That is, the pulsating jet layer correlates in space and time with a pulsating non-topological defect structure. Next, we add information captured by the principal axes of M by presenting the orientation ellipsoids in Figs. 9 11, first in space time for comparison with the surface plots of v x and s, and then a snapshot of the spatial profile of the ellipsoids during the peak of the jet. The ellipsoids are presented in the coordinate frame of the x, y plane, meaning the horizontal axis is the flow direction and the vertical axis is y, the flow gradient axis, and the vorticity axis is normal to the paper. Notice that the ellipsoids never tilt out of the plane. This means that the principal axis is always in-plane, and equally notable, even during the oblate phase degeneracy, the circle of principal orientation remains in the shear plane. The minor axis of orientation associated with d 3 is always aligned with the vorticity axis. The space time correlations between oblate defect phases and jet layers are amplified through the ellipsoid graphics. This remarkable in plane symmetry of the orientation tensor and probability distribution for each model implies the dynamics of the attractor at each gap height involves an in-plane limit cycle. The principal axis n 1, when defined, is always orthogonal to the vorticity axis, with polar coordinate angle in the x,y plane. Fig. 12 show the time series of t, y i at several gap heights from the plates where = 0 is the anchoring condition to the center y of the pulsating jet, to the mid-gap y = 0. In each model, we find the orbit of the major director is wagging between the plates and the center of the jet layer, and tumbling in the remainder of the gap! The wagging orbits continuously grow in amplitude from the plates to the center of the jet-defect layer, and then the major director tumbles in phase in the interior of the gap. This strong phase coherence of the major director across the middle of the gap is consistent with a weak gradient morphology, since the jet-defect layers are close to the plates and well-separated. In the AT model, for the parameters chosen here, the jet-defect layers are close to one another, and therefore the interior of the gap has significantly greater gradient morphology, and less heterogeneity near the plates. We note that the formation of composite tumbling-wagging 1D heterogeneous attractors have been observed previously, including studies of Tsuji and Rey [13] and the authors [31], where in-plane symmetry was imposed and pure shear was also imposed. The present simulations do not enforce in-plane symmetry, and solve for the fully coupled flow, yet we find the space time attractor is in-plane; the out-of-plane degrees of freedom in the tensor and full kinetic distribution function simply decay to zero. The conditions on De and Er simply have to be tuned to amplify the flow feedback jet phenomenon where the flow profile becomes nonmonotone.

10 M.G. Forest et al. / J. Non-Newtonian Fluid Mech Fig. 12. The time series of the in-plane Leslie angle of the major director at different gap locations. Top: DH model. Middle: DMG model. Bottom: AT model. Left column: the wagging states. Gap heights: y = DH model, DMG model, y = AT model. Right column: the tumbling states. Gap heights: y = DH model, DMG model, y = AT model Shear and normal stress correlations with the pulsating jet layer A major achievement of the Landau de Gennes tensor and Doi Hess kinetic model lies in their prediction of sign changes in the first normal stress difference associated with the tumbling transition, in agreement with classical experiments of Kiss and Porter. Here, we of course have heterogeneous and dynamical responses in the flow and orientational distribution, so it is of interest to see what normal and shear stress features are associated with these flow-orientation responses. Fig. 13 gives the surface plots of N 1 y, t associated with the earlier plots of the primary flow v x y, t and scalar order parameter sy, t. Pulsating layers of negative N 1 coincide with the jet and oblate defect pulsating layers! 3.4. Scaling behavior of the pulsating jet-oblate defect layers So far, each model has been simulated at a special parameter set, and the features of the pulsating jet-oblate defect layer have been described. In particular, the locations and strengths of the jet layer vary from model to model, and now we give results from the AT model regarding the scaling behavior of the center of the jet with respect to Deborah and Ericksen numbers. In the AT model simulation presented, the jet center is close to the middle of the gap, at y = 0.2. In the kinetic and DMG tensor model simulations, the jet center was near the plates. Fig. 14 and then Fig. 15 show the location of the jet-layer depending on the parameter Ẽr and De, respectively. This indicates model robustness of the jet phenomenon, and shows how the jet center can be moved relative to the plates in each half of the shear cell. We omit similar parameter studies of the other two models, which also reflect a similar scaling behavior Persistence of the pulsating jet-oblate defect layer to out-of-plane anchoring The tangential anchoring condition has led to an in-plane orientational distribution for all three models, where the jet layer pulsates and oblate defect pulses coincide with the maximum dephasing of the wagging orbits near the plates and the tumbling orbits in the center of the gap. One might question whether these flow-orientation phenomena are specific to the special in-plane anchoring conditions. We choose the kinetic model to illustrate the persistence of the pulsating jet-oblate defect layer phenomenon

11 140 M.G. Forest et al. / J. Non-Newtonian Fluid Mech Fig. 15. Scaling of the mid-point of the jet-defect layer with Deborah number. The scaled distance = y/2h between the wall and the middle of the nearest hydrodynamical jet in values of the gap width 2h is plotted vs. the Deborah number De for the AT model. The points are the averages and the lines are the error bars. Fig. 16. The space time surface plot of the primary velocity with an out-of-plane plate anchoring condition from the kinetic-flow model. Fig. 13. Space time surface plots of the first normal stress difference N 1 = xx yy. Top: DH model. Middle: DMG model. Bottom: AT model. with out-of-plane anchoring, and thus an out-of-plane orientational response across the shear gap. Fig. 16 shows the velocity profile from out-of-plane anchoring with major director n = sin 0 cos 0, sin 0 sin 0, cos 0, where 0 = 85, 0 = 5, which is close to the tangential anchoring. It is clearly seen that the jet phenomena persists. Fig. 17 shows the polar and the in-plane Leslie angle from the same simulation. The attractor exhibits tilted kayaking in layers near the plates, then kayaking in the interior of the gap. The pulsating jet arises at the transition from kayaking to tilted kayaking during the times when nearby principal axes become strongly out of phase. 4. Numerical stability of the pulsating jet-oblate defect layers to 2D spatial perturbations Fig. 14. Scaling of the mid-point of the jet-defect layer with Ericksen number. The scaled distance = y/2h between the wall and the middle of the nearest hydrodynamical jet in values of the gap width 2h is plotted versus Ericksen number Ẽr for the AT model. The points are the averages and the lines are the error bars. The curve is the best fit of the function / Ẽr 3.9. Since the simulations reported here, for all three models, are confined to one space dimension, it is natural to question whether the phenomena are stable in higher space dimensions. For the DMG tensor-flow model, we have a spectral-galerkin code with which we can perform a numerical stability analysis. Following a recent study of the authors [32], we use the attractor above to populate initial data. The snapshots shown above are combined with noisy perturbations in full 2D spatial modes, which obey the boundary conditions on flow and orientation at the plates, and which are periodic in the z direction. The results of the simulations reveal a convergence back to the 1D attractors, shown in Fig. 18. A snapshot of the secondary flow,

12 M.G. Forest et al. / J. Non-Newtonian Fluid Mech Fig. 17. The orbit of the polar and the in-plane Leslie angle at different gap heights for the out-of-plane anchoring condition in the kinetic-flow model. The right column shows kayaking orbits the out of plane analog of tumbling at locations in the middle of the gap, while the left column shows tilted kayaking orbits the out of plane analog of wagging near the plate. Fig. 18. Two-space y, z dimensional simulations of the DMG tensor-flow model with initial data consisting of the 1D attractor above superimposed y, z-dependent perturbations in flow and orientation. Snapshots of a the primary velocity v xy, z, b the y velocity component v yy, z, c the velocity component v zy, z, d the shear stress, and e first normal stress difference N 1, at a fixed time after transients have passed.

13 142 M.G. Forest et al. / J. Non-Newtonian Fluid Mech Table 3 Error of the numerical solutions versus the number of mesh points, compared with the numerical solution using 801 mesh points Mesh points Kinetic model error AT model error Fig. 19. Two-space y, z dimensional simulations of the DMG tensor-flow model with initial data consisting of the 1D attractor above superimposed y, z-dependent perturbations in flow and orientation. The snapshot of the secondary flow v y, v zy,z is displayed at a fixed time after transients have passed. local axes of orientation regain phase coherence, the flow returns to monotone shear profiles. The pulsating jet layer-oblate defect phase phenomenon was then shown to be robust as two constraints were released. First, the tangential anchoring of the orientational distribution at the plates was shifted to an out-of-plane boundary condition. The result is a composite spatio-temporal attractor which exhibits kayaking in the middle of the shear gap and tilted kayaking in layers buffering each plate. Again, the pulsating jet layers form precisely at the transition from monotone major director rotation kayaking to finite oscillations tilted kayaking, and furthermore, oblate defect phases form, correlated in space and time with the pulsating jet, when the local orbits lose phase coherence. Finally, we use one model to establish the numerical stability of these one-space dimensional attractors to 2D perturbations. As pointed out by one of the referees the strong tangential anchoring boundary condition is a first step to model nano-rod dispersions or liquid crystals in Couette flow cells. A more realistic approach would impose an orienting field associated with an anchoring energy at the wall, which could be driven out of equilibrium by stresses propagating from the interior. While an intriguing project, this treatment of non-equilibrium anchoring conditions is deferred to the future. In summary, three separate models and codes predict model and dimensional robustness of the non-newtonian effect of pulsating jet layers in a sheared nano-rod dispersion, first discovered and reported by Kupferman et al. [1] in a reduced planar flow-nematic model. Acknowledgement Fig. 20. The mesh convergence rate of the velocity for the DMG model. v y,v z y, z, is also presented, showing extremely weak roll cells where the values of these flow components are O10 11 orders of magnitude below the primary flow and is therefore negligible Fig Concluding remarks We have presented results of simulations for three different flow-nematic models for nano-rod dispersions in a steady parallelplate Couette experiment. Each model is simulated with one-space dimensional heterogeneity and physical flow-orientation conditions at the plates, and with full orientational space resolution. Pulsating jet layers are shown to arise in each model, where the flow profile across the shear gap is non-monotone. The jet layer forms precisely at the location and time when neighboring orbits of the orientational distribution transition from tumbling to wagging, i.e., monotone rotation versus oscillation in the principal axis. The orientational distribution forms an oblate defect phase that coincides with the jet layer pulse, which occurs when nearby tumbling and wagging orbits develop director phase incoherence. When the This research has been performed under the auspices of the Sonderforschungsbereich 448 Mesoskopisch strukturierte Verbundsysteme financially supported by the Deutsche Forschungsgemeinschaft DFG. This research is also sponsored by AFOSR grant FA , NSF grants DMS , DMS , DMS and DMS , NASA URETI BIMat award No. NCC , and the Army Research Office. Appendix A. The relationship between the AT tensor model and the DMG tensor model The alignment of the effectively uniaxial particles with a molecular axis parallel to the unit vector u is characterized by an orientational distribution function f u,t. In the tensor model the order parameter is the second rank alignment tensor: Q = 15 2 uu f u,t 15 2 uu d2 u, 23 which is the anisotropic second moment characterizing the distribution. The symbol x indicates the symmetric traceless part of a tensor x, i.e. with Cartesian components denoted by Greek subscripts, one has x = 1/2x + x 1/3x ı. The alignment tensor is also often referred to the S-tensor or