Orienting properties of discotic nematic liquid crystals in Jeffrey-Hamel flows

The orienting properties of incompressible discotic nematic liquid crystals for creeping flows between converging and diverging planar walls (Jeffrey-Hamel) are analyzed using the Leslie-Ericksen theory. The dependence of director orientation on the reactive parameter and the flow kinematics is presented. Closed form stationary solutions for the director orientation are found when elastic effects are neglected. Stationary numerical solutions for the velocity and director fields using the full Leslie-Ericksen theory are presented. The director field in converging flow is characterized by azimuthal (radial) centerline orientation, by being asymmetric with respect to the azimuthal (radial) direction, and by having an allowed orientation range that spans two half-quadrants (full quadrants). In the limiting case of perfectly flat disk ( –) the flow-induced director orientation in converging flow is the azimuthal direction, while in diverging flow the director rotates by a full n radians. By reducing the vertex angle between the walls to vanishingly small values, converging flow solutions properly reduce to those of flow between parallel plates, but diverging flows are expected to lead to a new instability.     

The end correction for power-law fluids in the capillary rheometer

The finite element scheme developed by Nickell, Tanner and Caswell is used to compute the entry and exit losses for creeping flow of power-law fluids in a capillary rheometer. The predicted entry losses for a Newtonian fluid agree well with available experimental and theoretical results. The entry losses for inelastic power-law fluids increased with decreasing flow behaviour index and show an increasing deviation from available upper bound results as the flow behaviour index in the power-law decreases.The exit losses are found to be finite for inelastic power-law fluids and increase as the flow behaviour index decreases. The predicted die swell for Newtonian fluids agrees well with the available experimental data while the influence of shear thinning is to reduce the die swell.The end correction which is the sum of the entry and exit losses relative to twice the viscometric wall shear stress varies from 0.834 for n = 1 to 2.917 for n = 1/6. This figure reaches a very high value as n tends to zero. The experimental variation in the Couette correction factor in capillary rheometry is explained in terms of the shear thinning characteristics of the fluid. It is concluded that the exit flow is not viscometric, contrary to a common assumption.     

Frottements et pertes de pression des fluides non newtoniens dans des conduites non circulaires

The study of fluid flow in a duct requires characteristic parameters of the flow and dimensionless numbers to correlate and compare experimental results. For Newtonian fluids in simple configurations, the definition of the Reynolds number is quite standard, but for non-Newtonian fluid flows in ducts with arbitrary shape of cross section, the dependence of the apparent viscosity with the shear rate requires a generalization of this dimensionless number. This note proposes a general method valid for a large class of non-Newtonian fluids and for all duct shapes. An application is developed for a viscoelastic flow through a rectangular duct. Results obtained in the present investigation are in a good agreement with available correlations. To cite this article: M. Mahfoud et al., C. R. Mecanique 333 (2005).     

Conoscopic observations of shear-induced rotations in nematic liquid crystals

Orientational changes in monodomains of flow-aligning liquid crystals, 4-n-pentyl-4′-cyanobiphenyl and N-(4-methoxybenzylidene)-4-butylaniline, were studied during shear and recovery in a linear shearing device fitted to an optical microscope. Planar alignment (director in the shear plane) allows the study of twist effects and was generated by strong planar anchoring at the walls with orientations in a range of 0–90° with the shear direction. While being held back by the anchoring walls, shear caused the bulk director to rotate towards a steady-state alignment angle in the shear direction (Leslie angle θ_L). The transient director rotation was observed with conoscopy. It was found that increasing the initial alignment towards the vorticity direction increased the measured θ_L. Upon stopping the flow, the bulk director returned to its initial state. With initial alignment orientation changing from parallel to perpendicular to the flow direction, the rate of the twist-driven recovery process increases. This rate increase is not seen in the splay-driven recovery which is constant and consistently faster than twist-driven recovery at all orientations studied. Received: 10 December 1998/Accepted: 7 June 1999     

Study of a simple oscillatory flow in polar fluids

Summary In the present paper, we have undertaken a comparative study of the flow behaviour of two types of fluids —Eringen's mioropolar fluid andStokes' couple stress fluid — in a simple oscillatory flow. This study was undertaken with a view to see if the close resemblance of the flow behaviour of these two fluids in steady shearing flows was maintained even in time dependent flows. We find that the flow behaviour of these two fluids are widely different in oscillatory motion.     

Simultaneous measurements of velocity and stress distributions in polyisobutylenes using laser-Doppler velocimetry and flow induced birefringence

An experimental device was set up for the synchronous measurement of velocities and stresses in polyisobutylenes using laser-Doppler velocimetry (LDV) and the two-colour flow-induced birefringence method (FIB). The materials investigated are three low molecular polyisobutylenes. Velocity (LDV) and stress (FIB) measurements are performed in the flow entrance region and inside a slit die with a contraction ratio of 1:10. The behaviour of the polyisobutylenes is Newtonian under the flow conditions applied. Therefore, the stresses inside the fluids can be calculated and compared to the stresses experimentally determined. A good agreement in shear and elongational flows was found between the calculated (LDV) and directly measured stresses (FIB). This result demonstrates the applicability of the experimental setup as an optical rheometer that can preferentially be used to measure elongational properties of low viscous fluids.

Entry flow behaviour of viscoelastic fluids in an annulus

Developing and fully developed velocity profiles in the entrance region of an abrupt 2-to-1 annular contraction were measured for a number of visco-elastic polymer solutions. Experimental results were obtained for Reynolds number and flow behaviour index in the range 9.8 ? Re ? 355 and 0.372 ? n ? 0.55 respectively. A momentum-energy integral technique was employed in the boundary layer analysis. The deviation from inelastic behaviour was indicated by the ratio of elastic to inertial forces, Ws/Re. Within the limits of confidence of the experimental results, good agreement with theoretical predictions was obtained and very little deviation from inelastic behaviour was observed for Ws/Re < 0.08. For the test fluids investigated, the entrance length was found to be longer than that predicted for the corresponding inelastic fluids of the same n.     

Modelling of pH-dependent electro-osmotic flowsModélisation des variations de flux électro-osmotiques avec le pH.

NaCl-water saturated kaolinite is observed to show pH-dependent electro-osmotic flow. This behaviour is modelled by computing the electro-osmotic flow between parallel planes as a function of their electrical surface charge density. The latter can be related to pH through the physico-chemical properties of the material. In case of kaolinite particles, isomorphic substitution and two surface dissociation reactions allow to predict the pH-dependence of the electrical charge density and thus the variations of electro-osmotic flows with pH. To cite this article: P. Dangla et al., C. R. Mecanique 332 (2004).     

Skewed Poiseuille-Couette flows of sPTT fluids in concentric annuli and channels

Analytical solutions have been derived for the helical flow of PTT fluids in concentric annuli, due to inner cylinder rotation, as well as for Poiseuille flow in a channel skewed by the movement of one plate in the spanwise direction, which constitutes a simpler solution for helical flow in the limit of very thin annuli. Since the constitutive equation is a non-linear differential equation, the axial and tangential/spanwise flows are coupled in a complex way. Expressions are derived for the radial variation of the axial and tangential velocities, as well as for the three shear stresses and the two normal stresses. For engineering purposes expressions are given relating the friction factor and the torque coefficient to the Reynolds number, the Taylor number, a nondimensional number quantifying elastic effects (εDe²) and the radius ratio. For axial dominated flows fRe and C_M are found to depend only on εDe² and the radius ratio, but as the strength of rotation increases both coefficients become dependent on the velocity ratio (ξ) which efficiently compacts the effects of Reynolds and Taylor numbers. Similar expressions are derived for the simpler planar case flow using adequate non-dimensional numbers.     

Lattice Boltzmann simulations of viscoplastic fluid flows through complex flow channels

We present the results of lattice Boltzmann (LB) simulations for the planar-flow of viscoplastic fluids through complex flow channels. In this study, the Bingham and Casson model fluids are covered as viscoplastic fluid. The Papanastasiou (modified Bingham) model and the modified Casson model are employed in our LB simulations. The Bingham number is an essential physical parameter when considering viscoplastic fluid flows and the modified Bingham number is proposed for modified viscoplastic models. When the value of the modified Bingham number agrees with that of the “normal” Bingham number, viscoplastic fluid flows formulated by modified viscoplastic models strictly reproduce the flow behavior of the ideal viscoplastic fluids. LB simulations are extensively performed for viscoplastic fluid flows through complex flow channels with rectangular and circular obstacles. It is shown that the LB method (LBM) allows us to successfully compute the flow behavior of viscoplastic fluids in various complicated-flow channels with rectangular and circular obstacles. For even low Re and high Bn numbers corresponding to plastic-property dominant condition, it is clearly manifested that the viscosity for both the viscoplastic fluids is largely decreased around solid obstacles. Also, it is shown that the viscosity profile is quite different between both the viscoplastic fluids due to the inherent nature of the models. The viscosity of the Bingham fluid sharply drops down close to the plastic viscosity, whereas the viscosity of the Casson fluid does not rapidly fall. From this study, it is demonstrated that the LBM can be also an effective methodology for computing viscoplastic fluid flows through complex channels including circular obstacles.     

Structure and properties of sheared fiber suspensions with mechanical contacts

The properties of fiber suspensions are highly sensitive to the suspension microstructure. In dilute or semi-dilute suspensions, nL²d≪1, the fibers' orientation distribution is controlled by hydrodynamic interactions among the fibers. However, direct mechanical contacts among the fibers play an important role in semi-concentrated suspensions, nL²d=O(1). Here, n is the number of fibers per unit volume, L is the fiber length and d is the fiber diameter. We have performed dynamic simulations of fiber suspensions including contact forces that prevent any two fibers from passing through one another. Collisions between the fibers cause them to flip more frequently in the shear flow, leading to a spread of the orientation distribution away from the flow direction. Both this increased orientational dispersion and the direct stress transmitted through the contacts enhance the shear viscosity of the suspension significantly. The contacts also give rise to normal stress differences. The results of the simulation are compared with experiments and the relative importance of contacts and hydrodynamic interactions is discussed.     

The effect of inextensibility on elastic surface waves

It is shown that when the complications associated with material anisotropy are absent a simple exact analysis can be given of the effect of unidirectional inextensibility on the propagation of surface waves in a semi-infinite elastic body. Provided that the direction of inextensibility e is not orthogonal to either m or m Λ n (m being the outward unit normal to the traction-free boundary of the body and n the wave normal), a unique surface wave exists with displacement everywhere orthogonal to e. The surface-wave solution is assembled from inhomogeneous plane waves in the usual manner, but a novel feature is the presence of a degenerate wave producing no displacement yet perturbing sinusoidally the tension in the inextensible fibres. When the aforementioned provisos are not met the surface wave either degenerates continuously into a shear wave (when $\text{(}\text{m}\text{Λ}\text{n}\text{)·}\text{e}\text{= 0,}\text{m}\text{·}\text{e}\text{≠ 0)}$, ceases to exist (when $\text{m}\text{·}\text{e}\text{= 0,}\text{n}\text{·}\text{e}\text{≠ 0)}$, or merges smoothly into a Rayleigh wave (when $\text{(}\text{e}\text{=±}\text{m}\text{Λ}\text{n}$, the inextensibility constraint then being inoperative).     

Characteristics of developing vertical bubbly flow under normal and microgravity conditions

The axial development of the void fraction profile, interfacial area concentration and Sauter mean bubble diameter of adiabatic nitrogen-water bubbly flows in a 9 mm-diameter pipe were measured using stereo image processing under normal and microgravity conditions. The flow measurements were performed at four axial locations (axial distance from the inlet, z normalized by the pipe diameter, D, z/D = 5, 20, 40 and 60) and with various flows: superficial gas velocity of 0.00840-0.0298 m/s, and superficial liquid velocity of 0.138-0.914 m/s. The effect of gravity on radial distribution of bubbles and the axial development of two-phase flow parameters is discussed in detail based on the obtained database and visual observation. Following Serizawa-Kataoka’s phase distribution pattern criteria under normal gravity conditions, the phase distribution pattern map was developed. Similar to normal gravity two-phase flows, wall, core and intermediate void peak patterns are observed under microgravity conditions but a transition void distribution pattern is not observed in the current experimental conditions. The data obtained in the current experiment are expected to contribute to the benchmarking of CFD simulation of phase distribution pattern and interfacial area concentration in forced convective pipe flow under microgravity conditions.     

Turbulent transport of particles in a straight square duct

Turbulent flow through a duct of square cross-section gives rise to off-axis secondary flows, which are known to transfer momentum between fluid layers thereby flattening the velocity profile. The aim of this study is to investigate the role of the secondary flows in the transport and dispersion of particles suspended in a turbulent square duct flow. We have numerically simulated a flow through a square duct having a Reynolds number of Re_τ = 300 through discretization of the Navier–Stokes equations, and followed the trajectories of a large number of passive tracers and finite-inertia particles under a one-way coupling assumption. Snapshots of particle locations and statistics of single-particle and particle pair dispersion were analyzed. It was found that lateral mixing is enhanced for passive tracers and low-inertia particles due to the lateral advective transport that is absent in straight pipe and channels flows. Higher inertia particles accumulate close to the wall, and thus tend to mix more efficiently in the streamwise direction since a large number of the particles spend more time in a region where the mean fluid velocity is small compared to the bulk. Passive tracers tend to remain within the secondary swirling flows, circulating between the core and boundary of the duct.     

Extrema of Young’s modulus for cubic and transversely isotropic solids

For a homogeneous anisotropic and linearly elastic solid, the general expression of Young’s modulus E(n), embracing all classes that characterize the anisotropy, is given. A constrained extremum problem is then formulated for the evaluation of those directions n at which E(n) attains stationary values. Cubic and transversely isotropic symmetry classes are dealt with, and explicit solutions for such directions n are provided. For each case, relevant properties of these directions and corresponding values of the modulus are discussed as well. Results are shown in terms of suitable combinations of elements of the elastic tensor that embody the discrepancy from isotropy. On the basis of such material parameters, for cubic symmetry two classes of behavior can be distinguished and, in the case of transversely isotropic solids, the classes are found to be four. For both symmetries and for each class of behavior, some examples for real materials are shown and graphical representations of the dependence of Young’s modulus on direction n are given as well.     

A classical problem revisited: rheology of nematic polymer monodomains in small amplitude oscillatory shear

We revisit the classical problem of the viscoelastic response of nematic (liquid crystal) polymers to small amplitude oscillatory shear. A multiple time scale perturbation analysis is applied to the Doi–Hess mesoscopic orientation tensor model to describe key features observed of longtime experiments, both physical (Moldenaers and Mewis, J Rheol, 30:567–584, 1986; Larson and Mead, J Rheol, 33:1251–1281, 1989b) and numerical (herein). First, there is a very slow time scale drift in the envelope of oscillations of the major director; we characterize the mean director angle and the envelope of oscillation. Second, there are bistable asymptotic orientational states, distinguished in that they are precisely the zero-stress orientational distributions noted in Larson and Mead (J Rheol, 33:185–206, 1989a). Third, the drift dynamics and asymptotic mean director angle are determined by the initial orientation of the director, not by material properties; we characterize the domain of attraction of each bistable state. Finally, the director drift leads to a predicted longtime decrease in the storage and loss moduli, consistent with experimental observations.

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Orientation dynamics in commercial thermotropic liquid crystalline polymers in transient shear flows

In situ X-ray scattering measurements of molecular orientation under shear are reported for two commercial thermotropic liquid crystalline polymers (TLCPs), Vectra A950® and Vectra B950®. Transient shear flow protocols (reversals, step changes, and flow cessation) are used to investigate the underlying director dynamics. Synchrotron X-ray scattering in conjunction with a high-speed area detector provides sufficient time resolution to limit the total time spent in the melt during testing, whereas a redesigned X-ray capable shear cell provides a more robust platform for working with TLCP melts at high temperatures. The transient orientation response upon flow inception or flow reversal does not provide definitive signatures of either tumbling or shear alignment. However, the observation of clear transient responses to step increases or step decreases in shear rate contrasts with expectations and experience with shear-aligning nematics and suggests that these polymers are of the tumbling class. Finally, these two polymers show opposite trends in orientation following flow cessation, which appears to correlate with the evolution of dynamic modulus during relaxation. Specifically, Vectra B shows an increase in orientation upon flow cessation, an observation that can only be rationalized by the assumption of tumbling dynamics in shear. Together with prior observations of commercial LCP melts in channel flows, these results suggest that this class of materials, as a rule, exhibits director tumbling.     

Smoothed particle hydrodynamics techniques for the solution of kinetic theory problems: Part 2. The effect of flow perturbations on the simple shear behavior of LCPs

The effect of slight perturbations to simple shear flow of liquid-crystalline polymers (LCPs) is explored by using the SPH technique to solve the unapproximated orientation distribution function equation arising from the Doi Theory. First, the case of simple shear flow is outlined, and it is shown that skewed distributions play an important role in the transition from periodic to steady behavior as the shear rate is increased. Next, we consider perturbations to flows that are slightly more extensional than simple shear, parametrized by the flow type parameter α. They are shown to eliminate all periodic director behavior (tumbling and wagging), even when the relative increment in flow type is small. At lower shear rates (or more properly, lower Peclet number Pe based upon the rotational diffusivity), the elimination occurs through a homoclinic bifurcation, the transition being rather abrupt as the flow type is changed. At higher Pe, periodic behavior is suppressed more gradually through a Hopf bifurcation, with tumbling being replaced by wagging and negative θ flow-aligning, where θ is the angle of the director in the shear plane. The effect of these perturbations on rheological behavior is also explored. As the flow is made slightly more extensional, the zero-shear rate limiting value of the generalized viscosity η decreases dramatically, due to the slowing down of tumbling as the system approaches a homoclinic orbit; as Pe is increased, the viscosity rises again before falling, due to the induction of wagging behavior where tumbling would normally prevail in simple shear. Finally, it is found when the flow type is changed sufficiently, the interesting, non-monotonic behavior of rheological functions seen in simple shear of LCPs is replaced by monotonic behavior, even though the flow is still relatively close to simple shear.     

Highly elastic solutions for Oldroyd-B and Phan-Thien/Tanner fluids with a finite volume/element method: planar contraction flows

A hybrid finite volume/element method is analysed through the computation of creeping flows of viscoelastic fluids in plane 4:1 sharp and rounded-corner contraction geometries. Simulations are presented for three models: a constant viscosity Oldroyd-B fluid, and Phan-Thien/Tanner (PTT) shear thinning fluids of exponential and linear approximation form. A Taylor–Galerkin/pressure-correction scheme is implemented as the base time-stepping framework. The momentum equations are solved by a finite element method, whilst the constitutive equations are solved by a finite volume approach. Mesh convergence is analysed via refinement around the contraction to capture boundary layers and flow structure. Pressure drop is shown to increase with flow rate for a fixed fluid. For the Oldroyd-B model, singular behaviour is reported in the main stress component as one approaches the corner in the rounded, as with the sharp geometry. Velocity components display an asymptotic trend with a positive slope. Higher values of Weissenberg numbers (We) are reached with these finite volume schemes compared to their finite element counterparts, attributing this to superior accuracy properties.