Jeffery Model Research Articles

Effect of shear-thinning rheology on the dynamics and pressure distribution of a single rigid ellipsoidal particle in viscous fluid flow

This paper evaluates the behavior of a single rigid ellipsoidal particle suspended in homogeneous viscous flow with a power-law generalized Newtonian fluid rheology using a custom-built finite element analysis (FEA) simulation. The combined effects of the shear-thinning fluid rheology, the particle aspect ratio, the initial particle orientation, and the shear-extensional rate factor in various homogeneous flow regimes on the particles dynamics and surface pressure evolution are investigated. The shear-thinning fluid behavior was found to modify the particle's trajectory and alter the particle's kinematic response. Moreover, the pressure distribution over the particle's surface is significantly reduced by the shear-thinning fluid rheology. The FEA model is validated by comparing results of the Newtonian case with results obtained from the well-known Jeffery's analytical model. Furthermore, Jeffery's model is extended to define the particle's trajectory in a special class of homogeneous Newtonian flows with combined extension and shear rate components typically found in axisymmetric nozzle flow contractions. The findings provide an improved understanding of key transport phenomenon related to physical processes involving fluid–structure interaction such as that which occurs within the flow field developed during material extrusion–deposition additive manufacturing of fiber-reinforced polymeric composites. These results provide insight into important microstructural formations within the print beads.

Non-isothermal direct bundle simulation of SMC compression molding with a non-Newtonian compressible matrix

Compression molding of Sheet Molding Compounds (SMC) is a manufacturing process in which a stack of discontinuous fiber-reinforced thermoset sheets is formed in a hot mold. The reorientation of fibers during this molding process can be either described by macroscale models based on Jeffery's equation or by direct mesoscale simulations of individual fiber bundles. In complex geometries and for long fibers, direct bundle simulations outperform the accuracy of state-of-the-art macroscale approaches in terms of fiber orientation and fiber volume fraction. However, it remains to be shown that they are able to predict the necessary compression forces considering non-isothermal, non-Newtonian and compaction behavior. In this contribution, both approaches are applied to the elongational flow in a press rheometer and compared to experiments with 23% glass fiber volume fraction. The results show that both models predict contributions to the total compression force and orientation reasonably well for short flow paths. For long flow paths and thick stacks, complex deformation mechanisms arise and potential origins for deviation between simulations models and experimental observations are discussed. Furthermore, Jeffery's basic model is able to predict orientations similar to the high-fidelity mesoscale model. For planar SMC flow, this basic model appears to be even better suited than the more advanced orientation models with diffusion terms developed for injection molding.

Thermal Behaviour of Nanocomposite Phase Change Material for Solar Thermal Applications

The use of solar thermal technologies has shown great prospects towards solar energy conversion into more useful forms of energy and has increasingly expanded solar thermal technology applications. However, the inability to properly store the excess solar energies during peak hours and demands have limited many of their applications. The integration of thermal energy storage (TES) systems with thermal technologies have increased the solar thermal technology performance but the poor thermal characteristics exhibited by phase change materials (PCM) limited the system overall performance. The enhancement of PCM properties by nonadditive have shown increased material performance in TES application and thereby extending the use of solar thermal technology application. Given this narrative and identifies literature gaps, the present study investigated experimentally the enhancement of paraffin PCM using nonadditive metallic of different types and concentrations and analysed their thermal behaviours. The results showed that Cu/paraffin PCM nanocomposites had good thermal reliability in proposed applications even after 150 thermal cycles under different temperatures. Moreover, thermal conductivity was improved significantly as an enhancement of 39% was reported when adding 2.5% of Cu nanoparticles. While specific heat and thermal diffusivity has been enhanced by 16% and 9%, respectively compared to pure paraffin. The obtained results were compared with different theoretical models such as Maxwell mode, Hamilton Crossover model, Jeffery model, and Bruggeman model. The calculated values show a good agreement with experimental ones. As a result, the prepared Cu/Paraffin nanocomposite PCM shows significant promise in thermal energy storage application due to its favourable phase change temperature, comparatively large latent heat, enhanced thermal conductivity and high thermal reliability and conversion. The proposed nanocomposite PCM can be used in many applications such as building materials to reduce interior temperature swings, enhance thermal comfort, and conserve electricity consumption.

Generalized Unsteady MHD Natural Convective Flow of Jeffery Model with ramped wall velocity and Newtonian heating; A Caputo-Fabrizio Approach

This theoretical investigation aims to highlight the unsteady freely convective fractional motion of a Jeffery fluid near an infinite vertical plate. The additional effects of ramped velocity condition, Newtonian heating, magnetohydrodynamics (MHD), and nonlinear radiative heat flux are also examined. A system of fractional order partial differential equations is established by choosing Caputo-Fabrizio fractional derivative as a foundation. Laplace transformation followed by an adequate choice of unit-less parameters is executed to solve the subsequent ordinary differential equations. Stehfest’s and Zakian’s numerical algorithms are invoked to find and justify the inverse Laplace transform of velocity and shear stress. Temperature and velocity gradients are evaluated at the wall to effectively probe the rate of heat transfer and shear stress. In this regard, numerical computations of Nusselt number and shear stress for several inputs of connected parameters are tabulated. Furthermore, graphical elucidations of velocity and temperature profiles are provided to observe the rise and fall subjected to variation in several parameters. Additionally, the velocity profile for both ramped boundary condition and constant boundary condition is analyzed to get a deep insight into the physical phenomenon of the considered problem. Finally, a comparative analysis between Jeffery fluid and second grade fluid is carried out for both factional and ordinary cases, and it is determined that Jeffery fluids exhibit rapid motion in both cases.

Research on viscoelastic fluid unsteady flow model based on torque loss correction

To address the complexity of unsteady shear flow in the initial start-up stage of rheometers, a model considering torque losses in motors and fixtures was developed based on the true torque balance relationship at the rotor boundary of the rheometer and the constitutive equation of the viscoelastic rheological model. The viscoelastic fluid transient flow model was established, and the real wall stress expression of unsteady shear flow of the viscoelastic fluid in the initial stage was obtained. The reliability of the model was verified by comparing the result from the analytical model with numerical calculation. The impact of the geometric dimensions of the test system and the material viscoelastic characteristics on the transient test was analyzed. The results show that the non-uniform velocity distribution has little effect on the initial strain response in the gap of the actual test system of the rheometer. The oscillation curves of strain and stress obtained by numerical and analytical methods coincide completely, proving the reliability of the model in characterizing the transient flow. For a Jeffery fluid, the elastic modulus G mainly affects the oscillation period and amplitude, the parallel viscosity pots η1 mainly impact the amplitude and oscillation attenuation speed, and the series clay pots η2 mainly influence the slope of the strain curve. The greater the system inertia, the more significant the creep oscillation in the unsteady flow process. The coupling between the test system inertia and the viscoelasticity fluid is the essential cause of the creep oscillation. Combined with the mechanical response characteristics of rheometer, a Jeffery model based on short term correction and torque loss was established. The model can accurately describe the strain time curve of gelled crude oil in the initial unsteady flow stage. The results of this study can help understand the rheological properties of materials in the transient start-up process.

Influence of Joule heating and wall slip in electroosmotic flow via peristalsis: second law analysis

Electrokinetic peristaltic slip transport in an asymmetric porous microchannel is studied to explore the entropy production in steady magnetohydrodynamic Jeffery fluid under simulation of Debye and long-wavelength approximations. The emerging two-dimensional bounded problem with electrokinetic body forces is solved numerically. Appropriate combination of heat and momentum equations with Jeffery model, after non-dimensionalization, generated controlling parameters in order to determine velocity, pressure gradient, temperature, entropy production and Bejan number. The trapping mechanism is also visualized by drawing streamlines against governing parameters. The zeta potential signifies the flow and heat response of the system. Former parameters like Brinkman number and Joule heating are compatibly liable for the increase in thermal irreversibilities. The outcomes of the present analysis are applicable in designing the thermofluidic micropumps and biomicrofluidic devices for separation processes and diagnosis.

Learning the Macroscopic Flow Model of Short Fiber Suspensions from Fine-Scale Simulated Data.

Fiber–fiber interaction plays an important role in the evolution of fiber orientation in semi-concentrated suspensions. Flow induced orientation in short-fiber reinforced composites determines the anisotropic properties of manufactured parts and consequently their performances. In the case of dilute suspensions, the orientation evolution can be accurately described by using the Jeffery model; however, as soon as the fiber concentration increases, fiber–fiber interactions cannot be ignored anymore and the final orientation state strongly depends on the modeling of those interactions. First modeling frameworks described these interactions from a diffusion mechanism; however, it was necessary to consider richer descriptions (anisotropic diffusion, etc.) to address experimental observations. Even if different proposals were considered, none of them seem general and accurate enough. In this paper we do not address a new proposal of a fiber interaction model, but a data-driven methodology able to enrich existing models from data, that in our case comes from a direct numerical simulation of well resolved microscopic physics.

Fibre kinematics in dilute non-Newtonian fibre suspensions during confined and lubricated squeeze flow: Direct numerical simulation and analytical modelling

The properties of short fibre-reinforced polymer composites depend on the distribution and the orientation of fibres which drastically changes during the forming of composites. During this stage, these materials behave as fibre suspensions and usually flow in confined geometries. To analyse their flow-induced fibrous microstructures, we previously conducted 3D real-time in situ observations of the compression of non-Newtonian dilute fibre suspensions using fast X-ray microtomography [T. Laurencin et al., Compos Sci Technol134 (2016)]. Here, we successfully simulated these experiments with a multi-domain Finite Element code and compared them with the predictions of Jeffery's model. Often, the Jeffery's equations agree with the experimental and numerical data. However, for fibres closed to compression platens, important deviations were observed with faster simulated and experimental fibre rotation. Adopting the dumbbell approach and revisiting the recent work of Perez et al. [J non-Newtonian Fluid Mech233 (2016)], an extension of the Jeffery's model is proposed to account both for the non-Newtonian rheology of the suspending fluid and confinement effects. Despite its simplicity, the new model allows a good description of simulation and experimental results.

Analytical and numerical assessment of the effect of highly conductive inclusions distribution on the thermal conductivity of particulate composites

Highly conductive composites have found applications in thermal management, and the effective thermal conductivity plays a vital role in understanding the thermo-mechanical behavior of advanced composites. Experimental studies show that when highly conductive inclusions embedded in a polymeric matrix the particle forms conductive chain that drastically increase the effective thermal conductivity of two-phase particulate composites. In this study, we introduce a random network three dimensional (3D) percolation model which closely represent the experimentally observed scenario of the formation of the conductive chain by spherical particles. The prediction of the effective thermal conductivity obtained from percolation models is compared with the conventional micromechanical models of particulate composites having the cubical arrangement, the hexagonal arrangement and the random distribution of the spheres. In addition to that, the capabilities of predicting the effective thermal conductivity of a composite by different analytical models, micromechanical models, and, numerical models are also discussed and compared with the experimental data available in the literature. The results showed that random network percolation models give reasonable estimates of the effective thermal conductivity of the highly conductive particulate composites only in some cases. It is found that the developed percolation models perfectly represent the case of conduction through a composite containing randomly suspended interacting spheres and yield effective thermal conductivity results close to Jeffery's model. It is concluded that a more refined random network percolation model with the directional conductive chain of spheres should be developed to predict the effective thermal conductivity of advanced composites containing highly conductive inclusions.

Electro-osmotic Flow of Non-Newtonian Biofluids Through Wavy Micro-Concentric Tubes

Most biofluids are non-Newtonian fluids with complex flow behaviors; for instance, human blood and DNA samples are shear thinning fluids (Jeffery model). The electro-kinetic transport of such fluids by microperistaltic pumping has been a topic of interest in biomedical engineering and other medical fields. This type of fluid transport requires sophisticated mathematical models and numerical simulations. Motivated by these developments, this study analyzed the simultaneous effects of an electric double layer (EDL) and a transverse magnetic field on the peristaltic flow and transport of blood. However, electro-osmotic flow is most significant when in micro channels/tubes. A constitutive relation is proposed to describe the electro-osmotic flow in the gap between two micro-coaxial horizontal pipes. Under low Reynolds number, long wavelength and Debye linearization approximations, the Poisson-Boltzmann equation, and governing equations are derived with the appropriate boundary conditions. The expressions for the electric potential, stream function, axial velocity, shear wall stress, and axial pressure gradient are obtained. The pressure rise and frictional force per wavelength are numerically evaluated and briefly discussed. The computational evidence suggests that the electric potential is an increasing function of the EDL thickness. Additionally, axial flow is accelerated by a positive electric field and decelerated by a negative electric field. Finally, the bolus size is reduced as the axial external electric field changes from the positive to negative direction.

Mixed convection analysis of variable heat source/sink on MHD Maxwell, Jeffrey, and Oldroyd-B nanofluids over a cone with convective conditions using Buongiorno’s model

This paper investigates the mixed convection of MHD Maxwell, Jeffery, and Oldroyd-B nanofluid models with heat source/sink over a cone geometry. The nonlinear ordinary differential equations are solved numerically by using Runge–Kutta-based shooting technique for transformed systems. Moreover, the non-homogenous Buongiorno’s model is employed, which accounts for both the impact of thermophoresis and Brownian motion of the nanofluids. The accuracy of the numerical data is obtained by comparison against the existed results in the literature. Furthermore, the effects of important physical parameters such as thermophoresis, Biot number, Brownian motion, and magnetic field parameters on the concentration, temperature, and velocity profiles are evaluated in this research. As the vital outcome of this research, it is revealed that the heat and mass transfer rates are more significant in Jeffery model than other two models Maxwell and Oldroyd-B nanofluid models.

Radially varying magnetic field effect on peristaltic motion with heat and mass transfer of a non-Newtonian fluid between two co-axial tubes

The present analysis discusses the effects of thermal-diffusion with thermal radiation, Joule heating and internal heat generation on peristaltic flow of a non-Newtonian fluid obeying Jeffery model. Heat and mass transfer are also taken into consideration, the flow is between two co-axial tubes under the effect of radially varying magnetic field. The inner tube is uniform and at rest, while the outer tube is flexible with sinusoidal wave traveling. The problem is modulated mathematically by a system of partial differential equations which describes the equations of momentum, heat, and mass transfer. These equations are solved analytically under the assumptions of long wave length and low-Reynolds number in non-dimensional form. The solutions are obtained as a functions of physical parameters of the problem. The radially varying magnetic field effect on the temperature and concentration distributions is analyzed and it is shown that the increase of Hartman number tends to reduce the temperature, while it increases the concentration.

The effect of fiber concentration on fiber orientation in injection molded film gated rectangular plates

Fiber orientation in film gated injection‐molded rectangular plates was characterized using micro computed tomography. Polyamide 12 materials with 15, 30, and 50 wt% glass fibers were used in this study. For all three materials, the typical core‐shell layer structure of fiber orientation was observed, but several features of the fiber orientation were highly dependent on the fiber content. Increasing fiber content resulted in increasing core layer thickness and a higher degree of orientation in the shell layer. These effects may be associated with plug flow induced by extra stress from the fibers, which changes the melt rheology compared to nonfilled material. The Dinh–Armstrong model for the fiber‐induced extra stress could not explain the fiber content dependencies. This appears to be attributed to the uncertainty in determining the particle number Np for the high fiber concentrations present in the samples. Analyses of fiber orientation as a function of fiber length revealed a dependence on fiber length in qualitative agreement with Jeffery's model for fiber dynamics. POLYM. COMPOS., 40:615–629, 2019. © 2017 Society of Plastics Engineers

From dilute to entangled fibre suspensions involved in the flow of reinforced polymers: A unified framework

Most suspension descriptions nowadays employed are based on Jeffery model and some of its phenomenological adaptations that do not take into account the possible existence of a relative velocity between the fibres and the suspending fluid when the fibre interactions increase. It is expected that at very low density of contacts, as predicted by standard suspension models, fibres move with the suspending fluid velocity. When the density of fibre interactions becomes extremely high and a percolated network of fibre contacts is established within the suspension, fibres cannot move anymore and then the fluid flows throughout the rigid or moderately deformable entangled fibre skeleton, like a fluid flowing through a porous medium. In between these two limit cases, one could expect that fibres move but with a velocity lower than the one of the suspending fluid. Thus, two contributions are expected, one coming from standard suspension theory in which fibres and fluid move with the same velocity, and the other resulting in a Darcy contribution consisting of the relative fibre/fluid velocity. In this paper, we elaborate a general model able to adapt continuously to all these flow regimes.

Analysis of the Folgar & Tucker model for concentrated fibre suspensions in unconfined and confined shear flows via direct numerical simulation

The classical Jeffery model allows for the prediction of the flow-induced orientation in dilute fibre suspensions. In most industrial applications, however, fibre suspensions are concentrated and fibre-fibre interactions cannot be ignored any longer. These interactions have been traditionally modelled at the mesoscopic and macroscopic scales by introducing a phenomenological diffusion term inducing a randomizing effect. In the so-called Folgar & Tucker (F&T) model, widely used in applications, the diffusion coefficient is assumed to scale linearly with the flow intensity, the latter being described by the second invariant of the rate of strain tensor. Modifications and alternatives to the F&T model have been proposed in view of the difficulty for the F&T model to explain an apparent orientation delay observed experimentally in injection-moulded parts. The noticed deviations were attributed to the intense fibre-fibre interactions, thus pointing to the limitations of a phenomenological diffusion term for describing them. In the present work, we revisit the F&T model and compare its predictions with those obtained bysimplified yet state-of-the-art direct numerical simulation (DNS) in unconfined and confined simple shear flows for a range of shear rates and concentrations, the latter ensuring intense fibre-fibre interactions. In unconfined flows, we find that the F&T model agrees quantitatively with the DNS results once an adequate closure relation is considered for approximating the fourth-order orientation tensor involved in the F&T model. Thus, the results seem to confirm, at least in simple shear flows, the F&T assumption for the form of the isotropic rotary diffusion function scaling linearly with the magnitude of the scalar rate of deformation. Also, a linear scalingof the diffusivity with the fibre concentration is observed. This conclusion remains unexpectedly valid under moderately-confined flow conditions as soon as an advanced fitted closure, like the IBOF, is considered within the F&T model. Other simpler closures (e.g. quadratic or hybrid), however, definitively fail to address confinement issues as also reported in our former work for the dilute regime. Obviously, these conclusions rest on the validity of the considered state-of-the-art DNS, which remains at present an open question.

3D real-time and in situ characterisation of fibre kinematics in dilute non-Newtonian fibre suspensions during confined and lubricated compression flow

The physical and mechanical properties of short fibre-reinforced polymer composites depend on the geometry, content, distribution and orientation of fibres within the polymer matrix. These microstructure features are mainly induced during the forming stage, i.e., when composites usually flow in moulds and behave as non-Newtonian fibre suspensions. Their flow-induced microstructures still cannot be well predicted by current rheological models. To better understand them, non-Newtonian dilute fibre suspensions were prepared and subjected to lubricated compression experiments using a micro-rheometer mounted in a synchrotron X-ray microtomograph. These experiments enabled, for the first time, fast and in situ 3D imaging of the translation and rotation of fibres in the suspending fluid. Fibre motions were compared with the prediction of the Jeffery's model. Despite the use of a non-Newtonian suspending fluid and confined flow conditions, i.e., with a gap between compression platens of the same order of magnitude than the fibre length, we showed that Jeffery's prediction was satisfactory if the fibres were sufficiently far from the compression platens (approximately at a distance of once to twice their diameter). Otherwise, the experimental average orientation rates were higher than the Jeffery's prediction.

Orientation kinematics of short fibres in a second-order viscoelastic fluid

Most theoretical fibre suspension models currently used for predicting the flow-induced evolution of microstructure in the processing of reinforced thermoplastics are based on the Jeffery model of dilute suspensions in a Newtonian suspending fluid or phenomenological adaptations of it that account for fibre-fibre interactions. An important assumption of all these models is the Newtonian character of the fluid in which the fibres are suspended. In industrial practice, the considered fluids are in general molten thermoplastics that exhibit a viscoelastic behaviour. Even though few counterparts of the Jeffery theory exist for second-order fluids, they have been rarely considered and, to our knowledge, never taken into account at the macroscopic scale. In this paper, we address the modelling of short fibre suspensions in second-order fluids throughout the different description scales, from microscopic to macroscopic. We propose a simplified modelling framework that allows one to extend to viscoelastic suspending fluids the standard Folgar and Tucker model widely used in industrial simulation software.

Lorentz covariance of heat conduction laws and a Lorentz-covariant heat conduction model

Lorentz covariance is one of the two basic assumptions in relativity and it has considerable universality in nature. Lorentz covariance has been investigated widely in various fields including thermodynamics, but studies of heat conduction remain very limited. In this study, we demonstrate that several typical heat conduction laws, i.e. Fourier's law, the Cattaneo–Vernotte (CV) model, and Jeffery model, are not Lorentz-covariant. Thus, we propose a new heat conduction model that satisfies Lorentz covariance. Compared with the existing heat conduction laws, which are not Lorentz-covariant, this new Lorentz-covariant model can ensure that singularity and violation of the second law of thermodynamics do not apply in any inertial reference system. The CV model and the new model both predict thermal wave phenomena, but the CV model predicts a pure wave whereas the new model predicts a composite of translational motion and a wave. Therefore, the new Lorentz-covariant model makes the features of heat conduction more coherent.

Fluid-Long Fiber Interaction Based on a Second Gradient Theory

Most suspension descriptions nowadays employed are based on the Jeffery's model andsome phenomenological adaptations of it that do not take into account size effects, that is, the kinematicsand stresses do not introduce a micro-mechanical characteristic length and thus, the rheologicalproperties become independent of the rod length. New models able to enrich first gradient kinematicsas well as to activate rod-bending mechanisms (needed for explaining the mild elasticity experimentallynoticed) are needed. In this paper we propose a second gradient description able to activate rods bending.

Two-dimensional long-flexible fiber simulation in simple shear flow

Long-flexible fiber orientation under the influence of a flow field is an important engineering problem. One can encounter this problem in many fields. For example in fiber reinforced thermoplastics produced in both injection and compression molding, fiber orientation affect final part properties. Fiber orientation models are constructed for short fibers in a simple shear flow case and though this case is important it is not the general case. In this work we extract rotational friction coefficients from Jeffery's model, create a general case long-flexible fiber orientation model, and apply it in a simple shear flow. POLYM. COMPOS., 2015. © 2015 Society of Plastics Engineers