Liquid–crystalline nematic polymers revisited

A new model for nematic polymers is proposed, based on the probability ψ(u,u,t) for a macromolecule to be oriented along direction u while embedded in a u environment created by its neighbours. The potential of the internal forces is written Φ(u,u) accordingly. The free energy contains a contribution ν Φ + k_BT ln ψ where the brackets mean an average over the probability distribution, while ν is the (uniform) polymer number density. An equation is derived for the time-evolution of the order parameter S = uu − I/3, together with an expression for the stress tensor. These two results offer a generalization of the Doi Model in so far as they include a distortional energy, analogue to the Frank elastic energy for low molecular mass nematics. Extending the Maier–Saupe variational procedure, we specify the way that the internal potential Φ(u,u) must be written for it to favour non-zero values of the order parameter, while giving a penalty to situations with gradients of the order parameter. The result is quite different from the potential proposed a decade ago by Marrucci and Greco (their Φ depends on u only), while it has a clear connection with the so-called Landau-de Gennes (LdG) tensor models, which are based on a free-energy depending on the order parameter and its gradients.     

Duality in constitutive formulation of finite-strain elastoplasticity based on F=FeFp and F=FF decompositions

A constitutive theory for large elastic–plastic deformations is presented by employing F=F^pF^e decomposition of the total deformation gradient. A duality in constitutive formulation based on this and the well-known Lee's decomposition F=F_eF_p is established for isotropic polycrystalline and single crystal plasticity.     

Explicit expressions of S(v), H(v) and L(v) for anisotropic elastic materials

The three matrices L(v), S(v) and H(v), appearing frequently in the investigations of the two-dimensional steady state motions of elastic solids, are expressed explicitly in terms of the elastic stiffness for general anisotropic materials. The special cases of monoclinic materials with a plane of symmetry at x₃ = 0, x₁ = 0, and x₂ = 0 are all deduced. Results for orthotropic materials appearing in the literature may be recovered from the present explicit expressions.     

Quantitative Impact Testing of Energy Dissipation at Surfaces

Impact testing with nanoscale spatial, force, and temporal resolution has been developed to address quantitatively the response of surfaces to impingement of local contact at elevated velocities. Here, an impact is generated by imparting energy to a pendulum carrying an indenter, which then swings towards a specimen surface. The pendulum displacement as a function of time x(t) is recorded, from which one can extract the maximum material penetration x _max , residual deformation x _r , and indentation durations t _in and t _out. In an inverse application one can use the x(t) response to extract material constants characterizing the impact deformation and extent of energy absorption, including material specific resistance coefficient C_in, coefficient of restitution e, and dynamic hardness H _imp . This approach also enables direct access to the ratio H/E, or resilience of the deformed material volume, at impact velocities of interest. The impact response of aluminum was studied for different contact velocities, and the mechanical response was found to correlate well with our one-dimensional contact model. Further experiments on annealed and work hardened gold showed that dynamic hardness H _imp scales with contact velocity and highlighted the importance of rate-dependent energy absorption mechanisms that can be captured by the proposed experimental approach.

A study of a 3-D double backward-facing step

An investigation of the flow over a three-dimensional (3-D) double backward-facing step is presented using a combination of both quantitative measurements from a particle image velocimetry (PIV) system and qualitative oil-flow visualizations. The arrangement of the PIV instrument allows for snap-shots of the (x, y) and (y, z) planes at various axial and spanwise positions. The measurements illustrate characteristics that are found in both two-dimensional (2-D) backward-facing steps and 3-D flows around wall mounted cubes. In particular, the development of a horseshoe vortex is found after each step alongside other vortical motions introduced by the geometry of the model. Large turbulence levels are found to be confined to a region in the center of the backstep; their mean square levels being much larger than what has been observed in 2-D backward-facing steps. The large turbulent fluctuations are attributed to a quasi-periodic shedding of the horseshoe vortex as it continuously draws energy from the spiral nodes of separation, which form to create the base of the horseshoe vortex. A combination of effects including the shedding of the first horseshoe vortex, the horizontal entrainment of air and the presence of two counter rotating vortices initiated at reattachment, are shown to cause the steering vector of the flow to jettison away from the surface in the first redeveloping region and along the center at z/h = 0. Oil-flow visualizations confirm these observations.

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Dependence of the zero shear-rate viscosity and the viscosity function of linear high-density polyethylenes on the mass-average molar mass and polydispersity

Linear high-density polyethylenes with molar masses M _w between 240 and 1,000,000 g/mol, obtained by metallocene catalysts, were characterized in shear using oscillatory and creep tests. The polydispersities of the molar mass distributions (MMDs) lay between 1 and 16. The resulting zero shear-rate viscosities η₀ covered a range from 2.5×10⁻³ to around 10⁸ Pas. Above a critical molar mass of M _c≈2,900 g/mol, the experimental results can be described by the relation η₀ ∼ M _w^3.6, independently of the MMD. The oscillatory data were fitted with a Carreau–Yasuda equation. The resulting parameters were correlated to molecular structure. The parameter a, being a quantity for the width of the transition between the Newtonian and the non-Newtonian regime, showed a dependence on the molar mass M _w but not on M _w/M _n. The parameter λ of the Carreau-Yasuda equation was found to be the reciprocal crossover frequency for all samples with a log-Gaussian MMD. λ depends on the molar mass M _w and also on M _w/M _n.

Explicit expression of derivatives of elastic Green’s functions for general anisotropic materials

The analytical expressions of Green’s function and their derivatives for three-dimensional anisotropic materials are presented here. By following the Fourier integral solutions developed by Barnett [Phys. Stat. Sol. (b) 49 (1972) 741], we characterize the contour integral formulations for the derivatives into three types of integrals H, M, and N. With Cauchy’s residues theorem and the roots of a sextic equation from Stroh eigenrelation, these integrals can be solved explicitly in terms of the Stroh eigenvalues P_i (i=1,2,3) on the oblique plane whose normal is the position vector. The results of Green’s functions and stress distributions for a transversely isotropic material are discussed in this paper.     

Label-free visualization of microfluidic mixture concentration fields using a surface plasmon resonance (spr) reflectance imaging

A label-free visualization technique based on surface plasmon resonance (SPR) illumination sensing is applied for the nonintrusive and real-time mapping of microscale mixture concentration fields. The key idea is that the SPR reflectance sensitively varies with the refractive index of the near-wall region of the test mixture fluid contacting the metal (Au) layer, of tuned 47.5 nm thickness. The Fresnel equation, based on Kretschmann’s theory, correlates the SPR reflectance with the refractive index, or dielectric constant, of the test medium, and then, the measured refractive index correlates with the mixture concentration. An example application is presented for the case of ethanol penetrating into water contained in a micro-channel with a rectangular cross-section of 91 μm wide and 50 μm high. The measurement sensitivity, uncertainties and detection limitations of the implemented SPR imaging sensor are carefully examined for its potential as a nonintrusive means of microscale concentration field mapping.

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Experimental investigation of the flow around a radially vibrating circular cylinder

This work aims to develop a process for controlling a cylinder wake, especially the von Karman vortex street, in such way so as to drastically reduce the drag coefficient. A new technique for influencing the cylinder wake is proposed in the present experimental study. The flow around a circular cylinder is perturbed by temporarily changing the cylinder diameter. Experiments have been performed for Reynolds numbers in the range Re=9,500 to Re=31,500. Three values of the controlling frequencies are considered: fs₁=0.41, fs₂=0.54 and fs₃=0.73, in addition to the stationary case corresponding to a non-deformable cylinder, fs₀=0. The visualisation flow shows that the pulsing motion of the cylinder walls greatly influences both the near and far wake dynamics. A decrease of the drag is expected.

Local mass transfer measurements in an inclined enclosure at high Rayleigh number

An electrochemical technique is used to study local mass transfer coefficients on surfaces of inclined enclosures over the range 1.1×10⁴ < Ra_H < 1.4×10¹⁰ for a nominal Schmidt number of 2280. Scaling with gcos instead of g in the Rayleigh number correlates the data well at low angles of inclination; however, as either the aspect ratio or the angle of inclination increase, the longitudinal density stratification causes the data to deviate from a power law scaling.

Dynamic Crack Growth Past a Stiff Inclusion: Optical Investigation of Inclusion Eccentricity and Inclusion-matrix Adhesion Strength

Interactions between a dynamically growing matrix crack and a stationary stiff cylindrical inclusion are studied optically. Test specimens with two different bond strengths (weak and strong) and three crack-inclusion eccentricities (e = 0, d/2 and 3d/4, d being inclusion diameter) are studied using reflection mode Coherent Gradient Sensing (CGS) and high-speed photography. These variants produce distinct dynamic crack trajectories and failure behaviors. A weaker inclusion-matrix interface attracts a propagating crack while a stronger one deflects the crack away. The former results in a propagating crack lodging (‘key-hole’) into the inclusion-matrix interface whereas in the latter the crack tends to circumvent the inclusion. When the inclusion is in the prospective crack path, the maximum attained crack speed is much higher in the weakly bonded inclusion cases relative to the strongly bonded counterparts. For a crack propagating towards a weakly bonded inclusion, the effective stress intensity factor (K _e) value remains constant for each inclusion eccentricity considered. But these constant K _e values increase with increasing eccentricity. A distinct drop in K _e occurs when the crack is near the inclusion. In strongly bonded inclusion cases, on the other hand, monotonically increasing K _e before the crack reaches the inclusion is observed. A drop in K _e is seen just before the crack reaches the inclusion. The mode-mixity estimates are of opposite signs for weakly and strongly bonded inclusions in case of the largest eccentricity studied, confirming the observed crack attraction and deflection mechanisms.

Identification of vortex pairs in aircraft wakes from sectional velocity data

The dynamics of multiple-vortex wake systems behind aircraft endangering air traffic can be assessed also from physical modelling. Large-scale laboratory investigations of multiple-vortex systems have been performed in a free-flight laboratory and in a water towing tank. Specialized PIV measurements provide time-resolved flow velocity fields normal to the wake axis. The applicability of various ∇u-based vortex identification schemes to planar velocity data is addressed and demonstrated for unequal-strength co- and counter-rotating vortex pairs. Large vortices shed off the wing tips and flaps are identified employing a ∇u-based criterion. Their cooperative mechanisms of generation and decay are evidenced from iso-surfaces of squared swirling strength and from further characteristic vortex parameters.

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A large deformation theory for rate-dependent elastic–plastic materials with combined isotropic and kinematic hardening

We have developed a large deformation viscoplasticity theory with combined isotropic and kinematic hardening based on the dual decompositions F=F^eF^p [Kröner, E., 1960. Allgemeine kontinuumstheorie der versetzungen und eigenspannungen. Archive for Rational Mechanics and Analysis 4, 273–334] and [Lion, A., 2000. Constitutive modelling in finite thermoviscoplasticity: a physical approach based on nonlinear rheological models. International Journal of Plasticity 16, 469–494]. The elastic distortion F^e contributes to a standard elastic free-energy ψ^(e), while , the energetic part of F^p, contributes to a defect energy ψ^(p) – these two additive contributions to the total free energy in turn lead to the standard Cauchy stress and a back-stress. Since F^e=FF^p-1 and , the evolution of the Cauchy stress and the back-stress in a deformation-driven problem is governed by evolution equations for F^p and – the two flow rules of the theory.We have also developed a simple, stable, semi-implicit time-integration procedure for the constitutive theory for implementation in displacement-based finite element programs. The procedure that we develop is “simple” in the sense that it only involves the solution of one non-linear equation, rather than a system of non-linear equations. We show that our time-integration procedure is stable for relatively large time steps, is first-order accurate, and is objective.     

Axial annular flow of plastic fluids: dead zones and plug-free flow

In axial annular flow, the shear stress decreases from its value τ(κR) at the inner cylinder to 0 at r = λR and increases from then on to τ(R) at the outer cylinder. For plastic fluids with a yield stress τ _c, λ will be such that flow commences when τ(κR) = τ(R) = τ _c. For fluids with position-dependent yield stresses (electro- and magnetorheological fluids are examples), the situation is more complex. While it is possible that yielding and flow occur everywhere, it is also possible that flow occurs only in parts of the fluid-filled space, and a dead zone (region in which the fluid is at rest) close to one of the walls exists. In that case, the fluid will flow no matter how small the applied pressure difference is. If P is large enough, the dead zone ceases to exist and flow without any plug is possible. The fluid flows as if no yield stress exists.

A comparison of three different methods for measuring both normal stress differences of viscoelastic liquids in torsional rheometers

A novel pressure sensor plate (normal stress sensor (NSS) from RheoSense, Inc.) was adapted to an Advanced Rheometrics Expansion System rheometer in order to measure the radial pressure profile for a standard viscoelastic fluid, a poly(isobutylene) solution, during cone–plate and parallel-plate shearing flows at room temperature. We observed in our previous experimental work that use of the NSS in cone-and-plate shearing flow is suitable for determining the first and second normal stress differences N ₁ and N ₂ of various complex fluids. This is true, in part, because the uniformity of the shear rate at small cone angles ensures the existence of a simple linear relationship between the pressure [i.e., the vertical diagonal component of the total stress tensor (Π₂₂)] and the logarithm of the radial position r (Christiansen and coworkers, Magda et al.). However, both normal stress differences can also be calculated from the radial pressure distribution measured in parallel-plate torsional flows. This approach has rarely been attempted, perhaps because of the additional complication that the shear rate value increases linearly with radial position. In this work, three different methods are used to investigate N ₁ and N ₂ as a function of shear rate in steady shear flow. These methods are: (1) pressure distribution cone–plate (PDCP) method, (2) pressure distribution parallel-plate (PDPP) method, and (3) total force cone–plate parallel-plate (TFCPPP) method. Good agreement was obtained between N ₁ and N ₂ values obtained from the PDCP and PDPP methods. However, the measured N ₁ values were 10–15% below the certified values for the standard poly(isobutylene) solution at higher shear rates. The TFCPPP method yielded N ₁ values that were in better agreement with the certified values but gave positive N ₂ values at most shear rates, in striking disagreement with published results for the standard poly(isobutylene) solution.

Multilayer nano-particle image velocimetry

Nano-particle image velocimetry (nPIV), based on evanescent-wave illumination of fluorescent colloidal tracers, measures the two velocity components parallel to the wall averaged over the first few hundred nanometers next to the wall. The intensity of the evanescent wave decays exponentially with z, or the distance normal to the wall. Illuminated tracers closer to the wall therefore have images that are brighter than those farther from the wall. This nonuniform illumination presents the possibility to extend the technique to “multilayer nPIV,” where the two velocity components parallel to the wall can be estimated at different z-locations within the illuminated region. In this paper, the variation of tracer image intensity with distance from the wall was predicted using diffraction optics-based approaches. The predictions, which were validated by calibration experiments, show that particle image intensity decays exponentially with distance normal to the wall. The feasibility of multilayer nPIV was evaluated using artificial images of plane Couette flow that incorporate evanescent-wave illumination, hindered Brownian diffusion and image noise. Each image was divided into three sub-images based on tracer image intensity, and standard techniques were then used to extract temporally and spatially averaged velocities at three different z-locations. In these simulations, velocity data were obtained within 80 nm of the wall, a threefold improvement over previous measurements. The results demonstrate that multilayer nPIV is feasible if appropriate classification techniques are developed and used to separate tracer images into different layers.

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The entrainment behavior of a turbulent axisymmetric jet in a viscous host fluid

Although turbulent jets have been studied extensively, one configuration that has not received much attention is the viscosity-stratified jet, wherein a turbulent jet of lower viscosity issues into a density-matched host liquid of higher viscosity. We present experimental data for scalar dispersion and two-dimensional velocity measurements in the axial plane of a turbulent axisymmetric jet with a Reynolds number (Re) of 2,000 issuing into a viscous host liquid at viscosity ratios (m) ranging from 1 to 55. The presence of a strong viscosity discontinuity across the jet edge results in a significant decrease in the scalar spread rate. We attribute this to the rapid reduction in turbulence intensity and the suppression of large engulfing eddies at the jet edge. The velocity profile, on the other hand, indicates that the velocity width and mass flux reduce with increasing m up to about 20, but then increase for higher values of m. This non-monotonic variation is explained by the growing influence of viscous stress for m>20. The scalar spread rate, the velocity spread rate, the centerline velocity decay rate, and the jet mass flux are all minimized for m20 for Re=2,000.

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Enhanced boiling heat transfer using radial fins

A numerical bifurcation analysis is carried out in order to determine the solution structure of radial fins subjected to multi-boiling heat transfer mode. One-dimensional conduction is employed throughout the thermal analysis. The fluid heat transfer coefficient is temperature dependent on the three regimes of phase-change of the fluid. Six fin profiles, defined in the text, are considered. Multiplicity structure is obtained to determine different types of bifurcation diagrams, which describe the dependence of a state variable of the system like the temperature or the heat dissipation on the fin design parameters, conduction–convection parameter (CCP) or base temperature difference (ΔT). Specifically, the effects of ΔT, CCP and Biot number are analyzed. The results are presented graphically, showing the significant behavioral features of the heat rejection mechanism.

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Two- and three-dimensional flows in nearly rectangular cavities driven by collinear motion of two facing walls

Two- and three-dimensional flows in nearly cuboidal cavities are investigated experimentally. A tight cavity is formed in the gap between two long and parallel cylinders of large radii by adding rigid top, bottom, and end walls. The cross-section perpendicular to the axes of the cylinders is nearly rectangular with aspect ratio Γ. The axial aspect ratio Λ > 10 is large to suppress end-wall effects. The fluid motion is driven by independent and steady rotation of the cylinders about their axes which defines two Reynolds numbers Re _1,2. Stability boundaries of the nearly two-dimensional steady flow have been determined as functions of Re _1,2 for Γ = 0.76 and Γ = 1. Up to six different three-dimensional supercritical modes have been identified. The critical thresholds for the onset of most of the three-dimensional modes, three of which have been observed for the first time, agree well with corresponding linear-stability calculations. Particular attention is paid to the flow for Γ = 1 under symmetric and parallel wall motion. In that case the basic flow consists of two mirror symmetric counter-rotating parallel vortices. They become modulated in span-wise direction as the driving increases. Detailed LDV measurements of the supercritical three-dimensional velocity field and the bifurcation show an excellent agreement with numerical simulations.

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Statistics of the surface temperature field of an air/water interface under air flow

Experimental results are presented that reveal the relationship between the root mean square of the surface temperature field of an air/water interface (σ) and the heat flux (q′′) emanating from that interface, over a range of wind speeds. Experiments were conducted for wind speeds ranging from 1.0 to 4.0 m/s to determine if and how the σ versus q′′ relationship was affected by wind speed. Consistent surfactant coverage conditions were maintained for wind speeds ranging from 1.0 to 2.6 m/s, and these are the focus of the results presented herein. For wind speeds above 2.6 m/s the surfactant was consistently pushed downstream, resulting in an inhomogeneous surface condition for the air/water interface. For wind speeds less than 2.6 m/s the relationship between σ and q′′ is approximately linear and is weakly dependent on wind speed. The surface temperature field was obtained by infrared (IR) imaging. Sample IR images are presented in addition to the σ versus q′′ data. IR images are presented for surfaces covered with insoluble surfactants (liquid phase and solid phase), a soluble surfactant, and a clean water surface.

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