Mixed Kinematic Research Articles

Chaotic mixing in a single-screw extruder with a moving internal baffle

A geometry in which the mixing of a single-screw extruder was enhanced by a reciprocating baffle is proposed in this article. The effect of the baffle's amplitude on the mixing kinematics of the screw channel was investigated. A model with the baffle lower than the screw channel and the corresponding mathematical model were developed. The periodic flow and mixing performance of Newtonian fluid in such an extruder were numerically simulated. The finite volume method was used, and the flow domain was meshed by staggered grids with the periodic boundary conditions of the barrier motion being imposed by the mesh supposition technique. Fluid particle tracking was performed by a fourth-order Runge–Kutta scheme. Distributive mixing was visualized by the evolution of passive tracers initially located at different positions. The growth of the interface stretch of tracers with time and the cumulative residence time distribution were also obtained. Poincaré sections were applied to reveal the geometrical scale of chaotic mixing patterns and the regions with embedded regular laminar flows. For comparison, the mixing performance in a conventional single extruder with fixed baffle was also investigated. POLYM. ENG. SCI., 54:198–207, 2014. © 2013 Society of Plastics Engineers

A mixed formulation of mortar-based frictionless contact

A class of mortar-based frictionless contact formulations is derived based on a classical three-field mixed variational framework. Within a penalty regularization complemented by Uzawa augmentations, discrete mortar constraints are naturally induced by the variational setting. Major aspects of earlier mortar approaches are obtained through constrained, lumped or unconstrained recovery procedures for the mixed kinematic and kinetic mortar quantities from their projected counterparts. Two- and three-dimensional examples at the infinitesimal and finite deformation regimes highlight the local and global quality of the contact interactions.

From streamline jumping to strange eigenmodes: Bridging the Lagrangian and Eulerian pictures of the kinematics of mixing in granular flows

Through a combined computational–experimental study of flow in a slowly rotating quasi-two-dimensional container, we show several new aspects related to the kinematics of granular mixing. In the Lagrangian frame, for small numbers of revolutions, the mixing pattern is captured by a model termed “streamline jumping.” This minimal model, arising at the limit of a vanishingly thin surface flowing layer, possesses no intrinsic stretching or streamline crossing in the usual sense, yet it can lead to complex particle trajectories. Meanwhile, for intermediate numbers of revolutions, we show the presence of naturally persistent granular mixing patterns, i.e., “strange” eigenmodes of the advection-diffusion operator governing the mixing process in Eulerian frame. Through a comparative analysis of the structure of eigenmodes and the corresponding Poincaré section and finite-time Lyapunov exponent field of the flow, the relationship between the Eulerian and Lagrangian descriptions of mixing is highlighted. Finally, we show how the mapping method for scalar transport can be modified to include diffusion. This allows us to examine (for the first time in a granular flow) the change in shape, lifespan, and eventual decay of eigenmodes due to diffusive effects at larger numbers of revolutions.

Continuum and multi-scale simulation of mixed kinematics polymeric flows with stagnation points: Closure approximation and the high Weissenberg number problem

Abstract It is a well known fact that the upper We limit encountered in continuum level viscoelastic flow simulations with typical constitutive equations for dilute polymeric solutions that predict bounded extensional viscosities in geometries with internal stagnation points on solid surfaces is a strong function of the strain hardening nature of the fluid. Specifically, the upper We limit decreases as the level of strain hardening of the fluid is increased. To provide further insight into the high We limitations in continuum level computations of this class of flows, we have performed extensive continuum and multiscale flow simulations in two benchmark flow problems, namely sedimentation of sphere in a tube and flow past a cylinder in a channel, utilizing the FENE-P (continuum and Brownian Configuration Fields (BCF)) and Giesekus (continuum) constitutive equations as well as the FENE (BCF) dumbbell micromechanical model. Extremely large stress gradients in the axial normal stress along the plane of symmetry in the wake of the cylinder and sphere are observed in FENE-P and Giesekus predictions using both multiscale and continuum numerical techniques at We of O(1) where the numerical simulations begin to breakdown for significantly strain hardening fluids, i.e., b > 300, α < 0.005. The existence of very large localized polymeric stresses and stress gradients is shown to be the consequence of significant over prediction of macromolecular extension by the closed form constitutive equations in the extensionally dominated region of the flow. The inability of the aforementioned constitutive equations to accurately describe the flow microstructure coupling downstream of stagnation points in complex kinematics flows should motivate use of regularization techniques for the polymeric stress or utilization of more sophisticated constitutive equations in flows with strong straining components.

Prediction of serrated chip formation in orthogonal metal cutting by advanced adaptive 2D numerical methodology

In this work a complete numerical methodology combining ‘advanced’ thermo-elasto-viscoplastic constitutive equations accounting for mixed (isotropic and kinematic) non-linear hardening, thermal effects, isotropic ductile damage and contact with friction is proposed to simulate the 2D orthogonal cutting process of AISI4340 steel. First, the fully coupled constitutive equations are presented and their specificities highlighted. The relevant numerical aspects concerning both the local integration scheme as well as the global resolution strategy together with the 2D adaptive remeshing facility are discussed. This model is implemented into ABAQUS/EXPLICIT using the Vumat user subroutine and connected with an adaptive 2D meshing program. Application is made to the 2D orthogonal metal cutting by chip formation. A special care is put in the prediction of the primary shear band where the temperature, the strain and the damage are highly localised giving the serrated shape and possible segmentation of the ship.

Micro-damage propagation in ultra-high vacuum seals

The paper addresses a fundamental problem of tightness of ultra-high vacuum systems (UHV) at cryogenic temperatures in the light of continuum damage mechanics (CDM). The problem of indentation of a rigid punch into an elastic–plastic half-space is investigated based on rate independent plasticity with mixed kinematic and isotropic hardening. The micro-damage fields are modeled by using an anisotropic approach with a kinetic law of damage evolution suitable for ductile materials and cryogenic temperatures. The model has been experimentally validated and the results are used to predict the onset of macro-cracking (loss of tightness) and the corresponding load (contact pressure). The algorithm is applied in the design of UHV systems for particle accelerators.

Mixed kinematic and dynamic sideslip angle observer for accurate control of fast off‐road mobile robots

AbstractAutomation in outdoor applications (farming, surveillance, military activities, etc.) requires highly accurate control of mobile robots, at high speed, although they are moving on low‐grip terrain. To meet such expectations, advanced control laws accounting for natural ground specificities (mainly sliding effects) must be derived. In previous work, adaptive and predictive control algorithms, based on an extended kinematic representation, have been proposed. Satisfactory experimental results have been reported (accurate to within ±10 cm, whatever the grip conditions), but at limited velocity (below 3 m·s−1). Nevertheless, simulations reveal that control accuracy is decreased when vehicle speed is increased (up to 10 m·s−1). In particular, oscillations are observed at curvature transition. This drawback is due to delays in sideslip angle estimation, unavoidable at high speed because only an extended kinematic representation was used. In this paper, a mixed backstepping kinematic and dynamic observer is designed to improve observation of these variables: the slow‐varying data are still estimated from a kinematic representation, which is then injected into a dynamic observer to supply reactive and reliable sliding variable (namely sideslip angle) estimation, without increasing the noise level. The algorithm is evaluated via advanced simulations (coupling Adams and MatLab software) investigating high‐speed capabilities. Actual experiments at lower speed (experimental platform maximum velocity) demonstrate the benefits of the proposed approach. © 2009 Wiley Periodicals, Inc.

Anomalous pressure drop behaviour of mixed kinematics flows of viscoelastic polymer solutions: a multiscale simulation approach

A long-standing unresolved problem in non-Newtonian fluid mechanics, namely, the relationship between friction drag and flow rate in inertialess complex kinematics flows of dilute polymeric solutions is investigated via self-consistent multiscale flow simulations. Specifically, flow of a highly elastic dilute polymeric solution, described by first principles micromechanical models, through a 4:1:4 axisymmetric contraction and expansion geometry is examined utilizing our recently developed highly efficient multiscale flow simulation algorithm (Koppol, Sureshkumar &amp; Khomami, J. Non-Newtonian Fluid Mech., vol. 141, 2007, p. 180). Comparison with experimental measurements (Rothstein &amp; McKinley, J. Non-Newtonian Fluid Mech., vol. 86, 1999, p. 61) shows that the pressure drop evolution as a function of flow rate can be accurately predicted when the chain dynamics is described by multi-segment bead-spring micromechanical models that closely capture the transient extensional viscosity of the experimental fluid. Specifically, for the first time the experimentally observed doubling of the dimensionless excess pressure drop at intermediate flow rates is predicted. Moreover, based on an energy dissipation analysis it has been shown that the variation of the excess pressure drop with the flow rate is controlled by the flow-microstructure coupling in the extensional flow dominated region of the flow. Finally, the influence of the macromolecular chain extensibility on the vortex dynamics, i.e. growth of the upstream corner vortex at low chain extensibility or the shrinkage of the upstream corner vortex coupled with the formation of a lip vortex that eventually merges with the upstream corner vortex at high chain extensibility is elucidated.

Low Reynolds number scalar transport enhancement in viscous and non-Newtonian fluids

Enhancement of heat and/or mass transfer via turbulence is often not feasible for highly viscous, non-Newtonian or shear sensitive fluids. One alternative to improve transport within such materials is chaotic advection, whereby Lagrangian chaos occurs within regular (non-turbulent) flows [J.M. Ottino, The Kinematics of Mixing: Stretching, Chaos and Transport, Cambridge University Press, Cambridge, 1989]. Complex interactions between chaotic advection and diffusion yields enhanced dispersion, and the topology of the Lagrangian dynamics is governed by the set of control parameters for the flow device. What parameter set maximises scalar dispersion for a given fluid rheology and diffusivity? Most studies to date have only considered a handful of points in the parameter space Q , but as this space may be large and the solution distribution complex (fractal), robust optimisation requires detailed global resolution of Q . By utilising a novel spectral method [D.R. Lester, G. Metcalfe, M. Rudman, H. Blackburn, Global parametric solutions of scalar transport, J. Comput. Phys. (2007). doi: doi:10.1016/j.jcp.2007.10.015] which exploits the symmetries often present in chaotic flows, we can resolve the asymptotic transport dynamics over Q , facilitating the identification of optima and elucidating the global structure of transport. We employ this method to optimize scalar transport for both Newtonian and non-Newtonian fluids in a chaotic mixing device, the Rotated Arc Mixer (RAM). Significant (up to sixfold) acceleration of scalar transfer is observed at Peclét number Pe = 10 3, which furthermore increases with Pe.

Dynamics of size segregation and mixing of granular materials in a 3D-blender by NMR imaging investigation

Magnetic Resonance Imaging (MRI) was used to characterize the kinematics of mixing and size segregation of dry binary mixtures (diameters d min and d max) in a Turbula® shaker-mixer. It was found that the filling level F of the cylindrical container should not exceed 80%; otherwise, a dead zone appears in the centre of the cell. When F=66% and the two different species ( d min≠ d max) are in equivalent proportion, segregation is observed when R= d max/ d min is greater than 1.1. Furthermore, the slower the rotation, the larger the segregation. Moreover, it is demonstrated that this blender is very efficient for dry materials within the three following conditions: (i) the rotation speed is fast enough, (ii) one tries to mix a little amount of small particles in a sea of large ones, and (iii) the concentration of the smaller particles does not exceeds 10%. Otherwise, large particles segregate quite fast towards the container walls. A segregation index S, based on density fluctuation, has been defined. When studied as a function of the number of rotations, S allows to define a characteristic time that is much shorter (1.4 rotations) for segregation than for mixing (10.7 rotations). However, it is demonstrated that this index S is not sufficient to study the real segregation mechanism and the flow pattern. It is also shown that segregation in Turbula® results from both surface and bulk segregation mechanisms. The surface effect is related to shear percolation during flow close to the free surface and it is observed whatever the rotation speed is. On the other hand, the bulk effect disappears when sample rotation is large enough since “it averages the gravity force at zero”. At last, it is proved that a low concentration system can be understood via a self-consistent approach with a single small particle in a “sea” of large ones.

The kinematics of bend-induced mixing in micro-conduits

The absence of turbulence and the difficulty associated with introducing moving components into micro-fluidic systems make the mixing problem in micro-devices challenging. We studied steady, laminar, incompressible flow through a sequence of conduits with rectangular cross-sections aligned to form 90° with each other. The feasibility of taking advantage of bend-induced vortices to stir the fluid and enhance the mixing process was evaluated theoretically and experimentally. Since at very low Reynolds numbers the bend-induced vortices decay rapidly, it was necessary to utilize a large number of bends to achieve the desired effects. Since it is not practical to directly simulate the flow through a large number of bends, we borrowed Jones et al.’s [J. Fluid Mech. 209 (1989) 335–357] idea of constructing a two-dimensional map to project fluid particles from a cross-section upstream of the bend to a cross-section downstream of the bend. This map was then applied repetitively to trace particle trajectories in various bend arrangements. Under certain conditions, chaotic advection was predicted. A prototype of a stirrer was fabricated with low temperature co-fired ceramic tapes.

Prediction of forming limit diagram with mixed anisotropic kinematic–isotropic hardening plastic constitutive model based on stress criteria

This paper presents an investigation of predicting the forming limit diagram (FLD) based on a stress criterion. The prediction is achieved by employing an anisotropic plastic model with a mixed kinematic–isotropic hardening formulation recently developed by the authors. The loading conditions considered include both proportional and non-proportional loading. The applicability of the stress FLD (SFLD) under the loading conditions and different rolling directions is examined and discussed for AL2008-T4.

An analytical model for material line kinematics in steady and unsteady buoyancy driven flows in 2-d channels

In this note, we use both developing and fully developed buoyant flow in a 2-d channel as a simple flow model useful for approximating buoyancy driven interpenetrating mixing phenomenon, such as the Rayleigh–Taylor instability. Our approach towards understanding and quantifying mixing within this flow follows the development by Ottino (The kinematics of mixing: stretching, chaos and transport, Cambridge University Press, Cambridge, 1989) that mixing is the kinematically efficient stretching and folding of material lines. Like Ottino, we use simple flows as a prototype to understand more complex problems. Here we derive the stretch length i.e. the deformation of a unit filament subjected to both the fully developed and developing buoyant channel flow. We use the simpler fully developed flow to compute the specific rate of stretching for the fully developed flow. An early time contraction–expansion effect is noted for stretching. Asymptotically, the stretching rate decays by the expected t −1 behavior. This note indicates that early time buoyancy induced inter-penetrative mixing is a typical 2-d, shear driven phenomenon, exhibiting neither strong re-orientation (with attendant enhance mixing) nor negligible filament deformation (indicating minimal disruption of material surfaces).

Mixing efficiency in a pin mixing section for single‐screw extruders

AbstractNon‐Newtonian, non‐isothermal, 3D finite‐element simulation of mixing performance in a pin mixing section with different axial gaps in the pins has been carried out according to their realistic configurations. The quantitative evaluation of mixing ability was based on the theory of kinematics of fluid mixing. To learn and to compare the local mixing performance in a standard screw and a pin mixing section, the local mixing efficiency distribution proposed by Ottino was calculated. Also, the RTDs of these mixers were calculated in an attemt to measure mixing. The integration of the two, namely, the integrating local mixing efficiency along a number of particle pathlines from entrance to exit, together with statistical treatment, which was referred as integral mixing efficiency, then gives a quantitative judgment of the total mixing ability of a continuous mixer. The calculated results showed a nonlinear dependence of the mixing ability of a pin mixing section on the axial gap of the pins. Finally, the calculation results were compared with the experimental ones obtained in our previous study.

Computing reduced equations for robotic systems with constraints and symmetries

Develops easily computable methods for deriving the reduced equations for mechanical systems with Lie group symmetries. These types of systems occur frequently in robotics, and are found generically in robotic locomotion, wheeled mobile robots, and satellites or underwater vehicles with robotic arms. Results are presented for two important cases: (1) the unconstrained case, for both body and spatial representations; and (2) the constrained (mixed kinematic and dynamic) case. In each case, the dynamic equations for these nonholonomic mechanical systems are given, and illustrated by the appropriate calculations for an example system. A primary result of the paper is to show that the spectrum of possible constraints-ranging from no constraints to fully constrained systems-can be expressed within a single unifying principle for calculating the reduced equations. In this process, the structure of the reduced Lagrangian directly reveals two useful components in the reduction process, namely the local forms of the locked inertia tensor and the mechanical connection. Finally, it is shown that the reduced dynamics decouple from any explicit dependence on the group configuration variables.

An application of the theory of kinematics of mixing to the study of tropospheric dispersion

The most common technique used for numerical simulations of tracer mixing is that of the numerical solution of the advection–diffusion equation with the unresolved fluxes parameterized using the similarity theory. Despite correct predictions of the overall directions of transport, models based on a numerical solution of the advection–diffusion equation lack sufficient accuracy to correctly reproduce the coupling of mixing with small scale processes which are sensitive to the microstructure of the tracer distribution. The objective of this paper is to revisit the basic formalism employed in numerical models used to investigate atmospheric tracers. The main mathematical method proposed here is the theory of kinematics of mixing which could be applied effectively for simulations of atmospheric transport processes. At the beginning of the paper, we introduce simple mathematical transformations in order to demonstrate how complex topological structures are created by mixing processes. These idealistic flow systems are essential to explain transport properties of much more complex three-dimensional geophysical flows. An example of the application of the kinematics of mixing to the analysis of tracer transport on a planetary scale is presented in the following sections. The complex filamentary structures simulated in the numerical experiment are evaluated using some commonly applied statistical measures in order to compare the results with the data published in the literature. The results of the experiment are also analysed with the help of simple conceptual models of fluid filaments. The microstructure of the tracer distribution introduced in the paper is essential to increase our understanding of atmospheric transport and to develop more realistic parameterizations of small-scale mixing. The presented results could also be used to improve calculations of the coupling between microphysical processes and tracer mixing.

Reduced equations for nonholonomic mechanical systems with dissipative forces

This paper addresses the problem of computing reduced equations for mechanical systems with nonholonomic constraints, Lie group symmetries, and dissipative forces. Results are presented for two important cases: the unconstrained case, for both body and spatial representations, and the constrained (mixed kinematic and dynamic) case. For unconstrained systems, we show that the structure of the reduced Lagrangian almost transparently reveals two useful components in the reduction process, namely the local forms of the locked inertia tensor and the mechanical connection. For the case when nonholonomic constraints are present, we develop an extension to the nonholonomic momentum equation of Bloch, Krishnaprasad, Marsden, and Murray [5] that includes general forcing functions. We then provide an alternate proof to give additional intuitive insight into the origin and structure of these equations. Finally, we present a method for modeling (invariant) damping forces using a Rayleigh dissipation function. These techniques are illustrated with examples from robotics, including the snakeboard example.

Chaotic Advection in Large-Scale Convection

We study the kinematics of mixing and transport of "passive" particles in two-dimensional thermal convection using a low-order spectral model proposed by Howard and Krishnamurti [1986]. This model allows for a large-scale flow spanning the layer width. Mixing by chaos in the oscillatory flow is demonstrated with the help of numerical Poincaré maps. The chaos induced transport process is characterized from a relation of the form ΔX2(t)~ tm, for large t, where ΔX2(t) is the mean square distance traveled by a cloud of particles. It is shown that the transport process can be either shear-flow dominated (m=2) or Brownian type (m=1) or an intermediate type characterized by fractional exponents (1&lt;m&lt;2), depending on the external parameters in the problem. The intermediate process has been found to be intermittent in nature. The present results are compared with earlier studies of chaotic advection and also some experimental observations. Directions for future work are also pointed out.

Two-Dimensional Dynamic Analysis of Distributive Mixing in Pin Mixing Section of Screw Extruder.

In a previous study (Yao et al, 1997), a direct eveluation of distributive mixing performance in a new type of pin mixing section based on the kinematics of mixing, i.e., local and integral mixing efficiency, was carried out. The calculated local mixing efficiency can quantitatively identify the regions where mixing is good and those where it is not. Furthermore, the calculation of integral mixing efficiency and its distribution in a mixer based on the results of local mixing efficiency allowed a quantitative comparison of a mixer with various configurations. In this study, to justify the results obtained above, dynamic analysis to quantify distributive mixing in the pin mixing section, i.e., tracking the evolution of trajectories of particles and calculating the standard deviation of particle concentration distribution, is carried out.

Pattern formation during mixing and segregation of flowing granular materials

Powder mixing plays an important role in a number of industries ranging from pharmaceuticals and food to ceramics and mining. Avalanches provide a mechanism for the stretching and folding needed to mix granular solids. However, unlike fluids, when particles dissimilar in size, density, or shape flow, they can spontaneously demix or segregate. Using magnetic resonance imaging, we track the transport of granular solids in a slowly rotating tube both with and without segregation effects. Compared with experiments in a 2-dimensional rotating disk partially filled with colored particles, the mixing kinematics and the granular pattern formation in a tube are changed by an axial flow instability. From simple physical principles we argue how size and density segregation mechanisms can be made to cancel, allowing good mixing of dissimilar particles, and we show experiments verifying this. Further experiments isolate the axial transport in the slowly rotating tube. Axial transport can appear faster with segregation than without.