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.     

On the eddy viscosity model of periodic turbulent shear flows

Physical argument shows that eddy viscosity is essentially different from molecular viscosity. By direct numerical simulation, it was shown that for periodic turbulent flows, there is phase difference between Reynolds stress and rate of strain. This finding posed great challenge to turbulence modeling, because most turbulence modeling, which use the idea of eddy viscosity, do not take this effect into account. The project supported by the National Natural Science Foundation of China (19732005) and Liu Hui Center for Applied Mathematics of Nankai & Tianjin University     

LES prediction of space-time correlations in turbulent shear flows

We compare the space-time correlations calculated from direct numerical simulation(DNS) and large-eddy simulation(LES) of turbulent channel flows.It is found from the comparisons that the LES with an eddy-viscosity subgrid scale(SGS) model over-predicts the space-time correlations than the DNS.The overpredictions are further quantified by the integral scales of directional correlations and convection velocities.A physical argument for the overprediction is provided that the eddy-viscosity SGS model alone does not includes the backscatter effects although it correctly represents the energy dissipations of SGS motions.This argument is confirmed by the recently developed elliptic model for space-time correlations in turbulent shear flows.It suggests that enstrophy is crucial to the LES prediction of spacetime correlations.The random forcing models and stochastic SGS models are proposed to overcome the overpredictions on space-time correlations.     

Subcritical and supercritical shear flows above an uneven bottom

The concepts of subcritical and supercritical flows are introduced for the long-wave approximation model describing stationary free-boundary rotational flows of an ideal incompressible fluid. Shear flows of a fluid layer above an uneven bottom are analyzed. Exact solutions describing different flow regimes are constructed, and the flow properties are studied as a function of the flow regime. Flows with backward streamlines are considered. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 4, pp. 26–38, July–August, 2006. An erratum to this article is available at .     

Interaction of shear flows of an ideal incompressible fluid in a channel

The problem of the decay of an arbitrary discontinuity for the equations describing plane-parallel shear flows of an ideal fluid in a narrow channel is considered. The class of particular solutions corresponding to fluid flows with piecewise constant vorticity is studied. In this class, the existence of self-similar solutions describing all possible unsteady wave configurations resulting from the nonlinear interaction of the specified shear flows is established. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 34–47, November–December, 2006.     

Brownian dynamics simulation of rodlike polymers under shear flow

The highly nonlinear behaviors of rodlike polymers in nematic phase under shear flow are studied with Brownian dynamics simulation. The LebwohlLasher nematogen model is taken as the prototype of the simulation and the mean-field approximation is avoided. By considering the nearest-neighbor intermolecular interaction, the spatial orientational correlation is introduced and therefore the spatial inhomogeneity such as the multiple-domain effect can automatically be incorporated. The transient order parameters, birefringence axes, shear stresses and first normal stress differences are calculated. The important finding of this work is that the director wagging and damped oscillation share the same molecular origin as director tumbling. The only difference is that the system is split into micro-domains which tumble with different phase angles in the wagging and damped oscillation regimes. The tumbling of the director of the whole system is suppressed due to the spatial inhomogeneity of director fields and then the damped oscillation of macroscopic stresses becomes predominant. The negative first normal stress difference exists at moderate shear rates, where both elasticity and viscosity play important role. Our simulation results including some dimensionless scaling parameters find good agreement with experimental observations in literature.     

Analyzing the influence of compressibility on the rapid pressure–strain rate correlation in turbulent shear flows

The influence of compressibility on the rapid pressure–strain rate tensor is investigated using the Green’s function for the wave equation governing pressure fluctuations in compressible homogeneous shear flow. The solution for the Green’s function is obtained as a combination of parabolic cylinder functions; it is oscillatory with monotonically increasing frequency and decreasing amplitude at large times, and anisotropic in wave-vector space. The Green’s function depends explicitly on the turbulent Mach number M _t , given by the root mean square turbulent velocity fluctuations divided by the speed of sound, and the gradient Mach number M _g , which is the mean shear rate times the transverse integral scale of the turbulence divided by the speed of sound. Assuming a form for the temporal decorrelation of velocity fluctuations brought about by the turbulence, the rapid pressure–strain rate tensor is expressed exactly in terms of the energy (or Reynolds stress) spectrum tensor and the time integral of the Green’s function times a decaying exponential. A model for the energy spectrum tensor linear in Reynolds stress anisotropies and in mean shear is assumed for closure. The expression for the rapid pressure–strain correlation is evaluated using parameters applicable to a mixing layer and a boundary layer. It is found that for the same range of M _t there is a large reduction of the pressure–strain correlation in the mixing layer but not in the boundary layer. Implications for compressible turbulence modeling are also explored.      

Rheology of rodlike polymers in the nematic phase with tumbling or shear orientation

The theory of rodlike polymers in the nematic phase is now sufficiently well developed as to allow predictions of the rheological behaviour that qualitatively compare well with the experimental observations. One of the main results of the theory is the prediction that rodlike polymers are nematics of the tumbling type at low shear rates, whereas they become shear oriented at high rates: a nonlinear effect, which is absent in low molecular weight nematics. This aspect is here reviewed in an effort to highlight the intuitive aspects of the matter. First, the low shear-rate situation is discussed in order to investigate the conditions that determine the existence of a stationary solution as opposed to a periodic one (tumbling). Then, the high shear-rate range is considered, where the shear-oriented situation prevails under all conditions. The intermediate range of shear rates is the most interesting one for its peculiar rheological behavior.Delivered as a Keynote Lecture at the Golden Jubilee Conference of the British Society of Rheology and Third European Rheology Conference, Edinburgh, 3–7 September, 1990.     

Theory of stability of rotating shear flows

The stability of rotating horizontal-shear flows is investigated within the framework of the linear approximation. The shear flow perturbations are divided into three classes (symmetric and two- and three-dimensional) and sufficient conditions of stability are obtained for each class. The perturbation dynamics in a flow with constant horizontal shear are described and the algebraic instability of the flow with respect to three-dimensional perturbations is detected. It is shown that the symmetric perturbations may be localized (trapped) inside the shear layer. The problem of finding the growth rates and frequencies of the trapped waves is reduced to a quantum-mechanical Schrödinger equation. Exact solutions are obtained for a “triangular” jet and hyperbolic shear.     

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.     

Newtonian drop in a Newtonian matrix subjected to large amplitude oscillatory shear flows

The dynamics of a single Newtonian drop immersed in a Newtonian matrix subjected to large-amplitude oscillatory shear flow is investigated. The ratio of the drop and matrix viscosity is above criticality, and thus break-up is absent under constant shear flow. At small forcing amplitudes the drop shape follows a regular oscillation. As the forcing amplitude increases, multipeaked oscillations of drop shape and orientation are observed. Experimental results are compared with predictions obtained with a phenomenological model. Model predictions are in qualitative good agreement with experimental data. The model suggests that the appearance of higher harmonics in the drop response is mainly due to flow nonaffinity.     

Stability of some shear flows for concentrated suspensions

In this paper we investigate the stability of some viscometric flows for a concentrated suspension model which allows for the effects of shear-induced migration, including plane and circular Couette and Poiseulle flows, and unbounded and bounded torsional flows. In the bounded torsional flow, where its radial outer boundary is assumed frictionless, an exact closeform solution is given. With the exception of torsional flows, we find that a limit point for all the steady-state solutions can exist for certain range in the parameter values. In all cases, disturbances can persist for a long time, O (H ²/a ²), where H is a dimension of the flow field, and a is the particles' radius.     

Swirl effects on variable-density jet mixing in confined flows

An experimental investigation on swirl effects on inhomogeneous confined jet mixing in a combustor configuration is reported. The confined swirling flow was simulated by a swirler with a central jet mounted in a cyclindrical tube. Helium and air jets set at different velocities were injected into the confined swirling air flow. The resulting flow fields due to two vane swirlers with constant vane angles of 35° and 66° were compared. Results show that the 35° vane swirler produces a solid-body rotation core with a slope about twice that created by the 66° vane swirler. It is the behavior of this solid-body rotation core that determines jet mixing rather than the swirler vane angle. Consequently, the coaxial jet decays much faster, the mixing is more intense, and the turbulence intensities are higher for the 35° vane swirler. In view of these results, combustor designers should be more concerned with behavior of the solid-body rotation core produced by the swirler, instead of the swirler vane angle.     

Modeling spurt and stress oscillations in flows of molten polymers

The paper presents an approach for modeling polymer flows with non-slip, slip and changing non-slip — slip boundary conditions at the wall. The model consists of a viscoelastic constitutive equation for polymer flows in the bulk, prediction of the transition from non-slip to sliding boundary conditions, a wall slip model, and a model for the compressibility effects in capillary polymer flows. The bulk viscoelastic constitutive equation contains a hardening parameter which is solely determined by the polymer molecular characteristics. It delimits the conditions for the onset of solid, rubber-like behavior. The non-monotone wall slip model introduced for polymer melts, modifies a slip model derived from a simple stochastic model of interface molecular dynamics for cross-linked elastomers. The predictions for the onset of spurt, as well as the numerical simulations of hysteresis, spurt, and stress oscillations are demonstrated. They are also compared with available data for a high molecular weight, narrow distributed polyisoprene. By using this model beyond the critical conditions, many of the qualitative features of the spurt and oscillations observed in capillary and Couette flows of molten polymers, are described.Notations upper convected derivative of elastic strain tensor - f, f_m, f_min dimensionless (sliding) shear friction characteristics, and its maximum and minimum - G Hookean elastic modulus - G_p plateau modulus - G, G storage and loss moduli - I₁, I₂ first and second invariant of strain tensor - I₁, I₀ capillary and barrel lengths - M non-dimensional mass flow rate - M_C critical molecular weight - M^*, M_e molecular weights of a statistical segment, and of polymer chain between entanglements - M_n, M_W number average and weight average molecular weights - m, k two fitting parameters of slip model - _s , _s ^o nominal and characteristic sliding velocities - u non-dimensional sliding velocity - u _sc initial (infinitesimal) slip velocity - u ₁ upper limit of u on the lower branch - u ₂ lower limit of u on the upper branch - u _max value of u corresponding to f_min - u _min value of u corresponding to f_max - U piston speed - Q nominal volumetric flow rate - q non-dimensional volumetric flow rate - R, R_o capillary and barrel radii - M non-dimensional mass flow rate     

Finitely extensible bead spring chains in time-dependent shear flows

The configurational and rheological properties of bead spring chains in time-dependent shear flows are calculated. The finite extensibility is incorporated through the constraint of constant contour length of the chain. Start-up of shear flow yields a stress overshoot, whereas oscillatory shear flow yields the same frequency dependence of the dynamic moduli as the simple bead spring model. The results show that finite extensibility can lead to non-linear rheological behavior of dilute polymer solutions. The influence of preaveraged hydrodynamic interaction on the obtained results is discussed.     

Likelihood and expected-time statistics of monodomain attractors in sheared discotic and rod-like nematic polymers

Employing a mesoscopic Doi tensor model, we develop transient statistical properties of sheared nematic polymer monodomains consistent with typical experimental protocols. Our goal is to convey to the experimentalist a list of expected outcomes, based not only on properties of the nematic liquid and imposed flow rate, but also on the timescale of the experiment and variability in the initial conditions. Step 1 is deterministic: we solve the model equations completely, then compile the flow-phase diagram of all monodomain attractors and phase transitions versus nematic concentration and Peclet number (shear rate normalized by molecular relaxation rate). Step 2 is to overlay on the phase diagram a statistical diagnostic of the expected time, t_A, to reach a small neighborhood of every attractor A. The statistics are taken over the arbitrary quiescent director angle on the sphere, modeling experiments that begin from rest. Step 3 is to explore parameter regimes with multiple attractors, where we statistically determine the likelihood of convergence to each attractor. These statistical properties are critical for any application of theoretical models to the interpretation of experimental data. If t_A is longer than the timescale of the experiment, attractor A is never fully resonated and the relevant stress and scattering predictions are those of the transients, not the attractor. In bi-stable and tri-stable parameter regimes, which are typical of nematic polymers, a distribution of monodomains of each type will populate the sample, so experimental data must be compared with weighted averages based on the likelihood of each attractor (see Grosso et al (2003) Phys Rev Lett 90:098304). The final step is to give statistics of shear stress and normal stress differences during the approach to each attractor type, as well as typical paths of the major director that are contrasted with the results of Van Horn et al (Rheol Acta (2003) 42(6):585–589) with Leslie-Ericksen theory.     

Numerical investigation of off-design regimes of flow past waveriders on the basis of axisymmetric conical flows

Off-design flow past waveriders is studied on the M_∞ = 4 to 12 freestream Mach number range. Configurations based on the flows behind axisymmetric conical shocks are considered. The waverider geometry is determined by preassigning the transverse contour of the undersurfaces in the base section using power-law functions. The results are obtained by finite-volume solution of the Euler equations using higher-order Runge-Kutta TVD schemes.     

Poiseuille flow of Leslie–Ericksen discotic liquid crystals: solution multiplicity, multistability, and non-Newtonian rheology

Computational modeling of the steady capillary Poiseuille flow of flow-aligning discotic nematic liquid crystals (DNLCs) using the Leslie–Ericksen (LE) equations predicts solution multiplicity and multistability. The phenomena are independent of boundary conditions. The steady state solutions are classified into: (a) primary, (b) secondary, and (c) hybrid. Primary solutions exist for all orientation boundary conditions and all flow rates, and are characterized by a flow-alignment angle that is closest to the anchoring angle at the bounding surface. Secondary solutions exist for all orientation boundary conditions and flow rates above a certain critical value. The secondary solutions are characterized by a flow-alignment angle which can be either the nearest neighbor below the primary solution or any multiple of π above. Hybrid solutions interpolate between the primary and the nearest secondary solutions, and hence exhibit two alignment angles. All solutions are stable to planar finite amplitude perturbations. Hybrid solutions are unstable to front propagation and lead to primary or secondary solutions. The non-Newtonian rheology of the primary and secondary solutions is characterized by non-classical shear thinning and thickening apparent viscosity behavior. Well-aligned monodomains can lead to shear thickening, thinning, or a sequence of both. The degree of rheological uncertainty is present for planar and homeotropic anchoring conditions. The non-Newtonian rheology of non-aligned samples leads to shear thinning and lack the uncertainty of well-aligned samples, since the apparent viscosity becomes insensitive to orientation.     

Stability analysis of unbounded uniform shear flows of dense, slightly inelastic spheres based on a frictional-kinetic theory

Asymptotic and transient stability analyses of unbounded uniform shear flows of dense, slightly inelastic, spherical particles were carried out using a frictional-kinetic theory. This model proposed for describing dense flows is based on a critical state plasticity theory and a simplified kinetic theory. In this model, the bulk and shear viscosities, the “thermal” conductivity, and the energy dissipation rate are proportional to a “mean pressure” which is composed of a quasistatic-frictional-contribution pressure considered for slow, plasticity deformations and a granular-kinetic-theorycollisional-contribution pressure. We studied two-dimensional stability analyses of layering disturbances (i.e., the perturbations whose wave number vectors are aligned only in the gradient) as well as nonlayering disturbances (the wave number vectors have nonzero streamwise components). Although this model has a simpler framework, it predicted similar results to those obtained using a more elaborate frictional-kinetic model. For instance, nonlayering disturbances are asymptotically stable at large time; the maximum transient growth of disturbances increases as the solids fraction or the friction coefficient is increased; and transient growths of disturbances can be significant due to the non-normality of the system. However, the prediction of the asymptotic stability of layering disturbances may be questionable because the collisional-contribution terms of the present model were over-simplified.