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Solid–liquid separation is a key unit operation in the wastewater treatment, generally consisting of coagulation and flocculation steps to promote aggregation and increase the particle size, followed by sedimentation, where the particles settle due to the effect of gravity. The sedimentation efficiency is related to the hydrodynamic behavior of the suspended particles that, in turn, depends on the aggregate morphology. In addition, the non-Newtonian rheology of sludges strongly affects the drag coefficient of the suspended particles, leading to deviations from the known settling behavior in Newtonian fluids. In this work, we use direct numerical simulations to study the hydrodynamic drag of fractal-shaped particles suspended in a shear-thinning fluid modeled by the power-law constitutive equation. The fluid dynamics governing equations are solved for an applied force with different orientations uniformly distributed over the unit sphere. The resulting particle velocities are interpolated to compute the aggregate dynamics and the drag correction coefficient. A remarkable effect of the detailed microstructure of the aggregate on the sedimentation process is observed. The orientational dynamics shows a rich behavior characterized by steady-state, bistable, and periodic regimes. In qualitative agreement with spherical particles, shear-thinning increases the drag correction coefficient. Elongated aggregates sediment more slowly than sphere-like particles, with a lower terminal velocity as the aspect ratio increases.

The formation of a line of equally spaced particles at the centerline of a microchannel, referred as "particle ordering," is desired in several microfluidic applications. Recent experiments and simulations highlighted the capability of viscoelastic fluids to form a row of particles characterized by a preferential spacing. When dealing with non-Newtonian fluids in microfluidics, the adherence condition of the liquid at the channel wall may be violated and the liquid can slip over the surface, possibly affecting the ordering efficiency. In this work, we investigate the effect of wall slip on the ordering of particles suspended in a viscoelastic liquid by numerical simulations. The dynamics of a triplet of particles in an infinite cylindrical channel is first addressed by solving the fluid and particle governing equations. The relative velocities computed for the three-particle system are used to predict the dynamics of a train of particles flowing in a long microchannel. The distributions of the interparticle spacing evaluated at different slip coefficients, linear particle concentrations, and distances from the channel inlet show that wall slip slows down the self-assembly mechanism. For strong slipping surfaces, no significant change of the initial microstructure is observed at low particle concentrations, whereas strings of particles in contact form at higher concentrations. The detrimental effect of wall slip on viscoelastic ordering suggests care when designing microdevices, especially in case of hydrophobic surfaces that may enhance the slipping phenomenon.

In this article, we present the mobilities of prolate ellipsoidal micrometric particles close to an air–water interface measured by dual wave reflection interference microscopy. Particle’s position and orientation with respect to the interface are simultaneously measured as a function of time. From the measured mean square displacement, five particle mobilities (3 translational and 2 rotational) and two translational–rotational cross-correlations are extracted. The fluid dynamics governing equations are solved by the finite element method to numerically evaluate the same mobilities, imposing either slip and no-slip boundary conditions to the flow at the air–water interface. The comparison between experiments and simulations reveals an agreement with no-slip boundary conditions prediction for the translation normal to the interface and the out-of-plane rotation, and with slip ones for parallel translations and in-plane rotation. We rationalize these evidences in the framework of surface incompressibility at the interface.

Fused deposition modelling is one of the most widely used additive manufacturing techniques and the diffusion of 3D printers has increased in popularity even further in recent times. Since high precision is required in 3D printing, a good control over the extrusion process is necessary. In this regard, a crucial phenomenon to be accounted for is the die or extrudate swell, i.e., the enlargement of the cross-section of the strand when coming out of the printer nozzle. While this phenomenon has been studied in large scale extruders, it has not yet been investigated in depth for 3D printing processes. In this work, the die swell phenomenon observed in a printed PLA filament is studied by experiments and fluid dynamic simulations. A novel, easy-to-use, accurate and fast procedure for measuring the value of the die swell ratio during the printing process is developed, accounting for typical errors related to a non-constant strand diameter and possible oscillations of the filament with respect to the extrusion direction. As the printing velocity is increased, a linearly increasing swelling ratio is observed at low printing speeds. The trend flattens at moderate speed values. A sudden increase is found at high printing velocities. The swelling ratio measured with the proposed technique is compared with the results of multi-mode viscoelastic simulations at different temperatures. A fair agreement between the experimental measurements and the numerical predictions is found for printing velocities that are typically employed in commercial 3D printers, supporting the reliability of the developed procedure.

The settling dynamics of a rigid ellipsoid in an unbounded viscoelastic fluid under inertialess conditions is studied through direct numerical simulations. The governing equations are solved by the finite element method with an Arbitrary Lagrangian–Eulerian formulation to handle the particle motion. The viscoelastic fluid is modeled through the Giesekus constitutive equation. Simulations are carried out up to a value of Deborah number of 5. The settling of prolate and oblate spheroidal particles is first addressed. The sedimentation, lift, and angular velocities are computed as a function of the orientation angle and for aspect ratios from 1/8 to 8. Regardless of the particle shape, initial orientation, and Deborah number, the particle rotates to align its longest axis along the force direction. A high extensional stress region behind the particle is observed at high aspect ratios due to the large curvature of the tip, leading to a fast decay of the axial fluid velocity downstream and to the appearance of a negative wake. Similarly, a triaxial ellipsoid reaches a final orientation with major axis parallel to the falling direction. The particle shape affects the orientational dynamics, both in terms of the orbits followed by the orientation vectors and the time needed to reach the equilibrium orientation, and the steady-state settling velocity. The fastest sedimentation rate is observed for a prolate spheroid with aspect ratio of about 2 whereas the slowest one for a high aspect ratio oblate spheroid. Triaxial ellipsoids settle with rates in between these two limiting behaviors.

The dynamics of a cell suspended in a Newtonian liquid subjected to a pressure-driven flow at non-negligible inertia in cylindrical and square cross section microfluidic channels is studied through three-dimensional arbitrary Lagrangian–Eulerian finite-element numerical simulations. The cell is modeled through the neo-Hookean hyper-elastic constitutive equation, which can describe biological particles undergoing moderate deformations. The cell-to-channel relative dimension is fixed to 0.2, whereas the Reynolds number Re, measuring the relative importance of liquid inertial and viscous forces, and the elastic capillary number [Formula: see text], measuring the relative importance of liquid viscous stress and solid elastic stress, are varied by several orders of magnitude. In a cylindrical tube, the cell migrates transversally to the flow direction until reaching a radial equilibrium position depending on Re and [Formula: see text]. Given Re, the softer the cell (i.e., the larger [Formula: see text]) the closer its equilibrium position to the tube axis, thus allowing for the separation of healthy and diseased cells which have similar dimensions but different mechanical properties. In a channel with a square cross section, a much more complex dynamics is found. Depending on Re and [Formula: see text], the cell can either migrate to the channel centerline, to the closest median of the channel cross section (thus, four equilibrium positions can be identified due to symmetry), to the closest diagonal (again, four equilibrium positions), or to an intermediate position in between the median and the diagonal (eight equilibrium positions).

Hypothesis Thin liquid films are important in many scientific fields. In particular, films with both the surface layers exposed to a different fluid phase, known as freestanding films, are relevant in the ambit of foams and emulsions. Hence, there is a great interest in developing novel techniques allowing to form large and stable freestanding liquid films and to follow their dynamics. Experiments We develop a novel opto-mechanical tool allowing to perform and study the preparation and the capillary leveling flow of axisymmetric bare freestanding liquid films. The tool is composed by a customized motorized iris diaphragm and by an innovative joint imaging setup combining digital holography and white light color interferometry that enables real-time measurement of film thickness over a large field of view. The dynamics of films made of a model Newtonian fluid, i.e., high-viscosity silicone oil, is studied. Direct numerical simulations and a hydrodynamic model based on the lubrication theory are used to support the experimental results. Findings Iris opening induces the formation of large circular freestanding films with a stepped profile. Once iris opening is stopped, the films undergo a capillary leveling flow tending to flatten their profile. The leveling flow follows the theoretical scaling given by Ilton et al. [1]. We prove through numerical simulations that an equi-biaxial extensional flow occurs at the film center. Furthermore, we observe the formation and dynamics of dimples in bare freestanding films for the first time.

One of the critical issues encountered when particle suspensions are made to flow in microfluidic devices is the adhesion of the suspended particles on the channel surfaces. This process, known as fouling, may lead to a progressive growth of clusters attached to the walls and, possibly, to a complete clogging of the microchannel. In this work, we employ Computational Fluid Dynamics combined with Discrete Element Method to study the initial growth of a cluster at the wall of a slit microchannel. We consider a suspension of ‘soft’ microparticles in a Newtonian liquid under laminar flow conditions, with a well-known simple model to describe particle-particle adhesion. The cluster growth dynamics is quantified in terms of morphology and projected area onto the slit wall. A comparison with some experimental data is carried out, showing that the model captures several qualitative experimental features of the fouling process, e.g., cluster morphology, possibility of break-up and resuspension, and linear growth rate of the cluster projected area onto the channel wall.

In this paper, we investigate the retraction of a circular viscoelastic liquid film with a hole initially present in its center by means of finite element numerical simulations. We study the whole retraction process, aiming at understanding the hole opening dynamics both when the hole does not feel any confinement and when it interacts with the solid wall bounding the film. The retraction behavior is also interpreted through a simple toy model, that highlights the physical mechanism underlying the process.We consider three different viscoelastic constitutive equations, namely, Oldroyd-B, Giesekus (Gsk), and Phan Thien-Tanner (PTT) models, and several system geometries, in terms of the film initial radius and thickness. For each given geometry, we investigate the effects of liquid inertia, elasticity, and flow-dependent viscosity on the dynamics of the hole opening. Depending on the relative strength of such parameters, qualitatively different features can appear in the retracting film shape and dynamics.When inertia is relevant, as far as the opening hole does not interact withthe wall bounding the film, the influence of liquid elasticity is very moderate,and the retraction dynamics tends to the one of Newtonian sheets; whenthe hole starts to interact with the solid wall, hole radius/opening velocityoscillations are detected. Such oscillations enhance at increasing elasticity.From the morphological point of view, the formation of a rim at the edge ofthe retracting film is observed. If inertial forces become less relevant withrespect to viscous forces, R-oscillations disappear, the hole opening velocitygoes through a maximum and then monotonically decays to zero, and norim forms during the film retraction. Geometrical changes have the effect ofenlarging or reducing the portion of the retraction dynamics not influencedby the presence of the solid wall with respect to the one governed by thehole-wall interactions.

Controlling the fate of particles and cells in microfluidic devices is critical in many biomedical applications, such as particle and cell alignment and separation. Recently, viscoelastic polymer solutions have been successfully used to promote transversal migration of particles and cells toward fixed positions in straight microchannels. When inertia is negligible, numerical simulations have shown that strongly shear-thinning polymer solutions (fluids with a shear viscosity that decreases with increasing flow rates) promote transversal migration of particles and cells toward the corners or toward the centerline in a straight microchannel with a square cross section, as a function of particle size, cell deformability, and channel height. However, no experimental evidence of such shifting in the positions for particles or cells suspended in strongly shear-thinning liquids has been presented so far. In this work, we demonstrate that particle positions over the channel cross section can be shifted "from the edge to the center" in a strongly shear-thinning liquid. We investigate the viscoelasticity-induced migration of both rigid particles and living cells (Jurkat cells and NIH 3T3 fibroblasts) in an aqueous 0.8 wt % hyaluronic acid solution. The combined effect of fluid elasticity, shear-thinning, geometric confinement, and cell deformability on the distribution of the particle/cell positions over the channel cross section is presented and discussed. In the same shear-thinning liquid, separation of 10 and 20 μm particles is also achieved in a straight microchannel with an abrupt expansion. Our results envisage further applications in viscoelasticity-based microfluidics, such as deformability-based cell separation and viscoelastic spacing of particles/cells.

The fast growth of microfluidic applications based on complex fluids is a result of the unique fluid dynamics of these systems, enabling the creation of devices for health care or biological and chemical analysis. Microchannels designed to focus, concentrate, or separate particles suspended in viscoelastic liquids are becoming common. The key fluid dynamical issue on which such devices work is viscoelasticity-induced lateral migration. This phenomenon was discovered in the 1960s in macroscopic channels and has received great attention within the microfluidic community in the past decade. This review presents the current understanding, both from experiments and theoretical analysis, of viscoelasticity-driven cross-flow migration. Examples of promising microfluidic applications show the unprecedented capabilities offered by such technology based on geometrically simple microchannels and rheologically complex liquids.

The aggregation/fragmentation dynamics of hard-gel microparticles suspended in a Newtonian liquid flowing through a straight channel is studied by numerical simulations. A one-way Discrete Element Method is employed to simulate the motion and the adhesion of the particles. The formation and fragmentation of aggregates, and their deposition at the channel walls are investigated by varying the Reynolds number and the strength of the adhesive force. A non-periodic channel is considered to simulate the start-up phase of aggregate formation till a ‘pseudo steady-state’ condition. Results in terms of microstructures, particle velocity profiles, and spatial and temporal evolution of aggregates in 2D and cylindrical (3D) channels are presented and discussed.

We address the effect of wall slip on the dynamics of a spherical particle suspended in an inertialess Newtonian or viscoelastic shear-thinning fluid under shear or Poiseuille flow. The study is performed through 3D direct finite element simulations by employing an Arbitrary Lagrangian-Eulerian method for the particle motion.In both shear and Poiseuille flows, wall slip reduces the difference between the particle translational velocity along the flow direction and the velocity of the unperturbed fluid, and slows down the particle rotational velocity. Remarkably, in a viscoelastic fluid, the presence of wall slip reverses the migration direction as compared to the no-slip case. Hence, for sufficiently high slip coefficients, all the particles migrate toward the channel midplane in shear flow and toward the channel centerline in Poiseuille flow, regardless of their initial position through the channel.

In this work a design tool to control cell-opening in gas foaming of thermoplastic polymers is developed. The sequence of events following bubble nucleation, namely, bubble growth and impingement, is modeled to gain a comprehensive, perspective view on the mechanisms of bubble wall rupture and on the conditions for achieving a fully open-cell morphology. In particular, unlike the previously published literature, the polymer elastic recovery is recognized as an important factor for wall retraction, which is typically considered as solely driven by surface tension. The new approach is experimentally validated on poly(e-caprolactone) (PCL), foamed with CO 2 , as a model polymer/gas system.

This thesis work arises from a collaboration between University of Naples Federico II and the Institute of Applied Sciences and Intelligent Systems belonging to National Council of Research (CNR-ISASI) on a European project, SensApp (grant agreement No 829104), financed by the European Union’s Horizon 2020 research and innovation programme. The project, in which six partners are involved (CNR, Vrije Universiteit Brussel VUB, Johannes Kepler University Linz JKU, Technical Research Centre of Finland VTT, Centro Neurolesi Pulejo Messina, Ginolis GIN), has the aim to develop a super-sensor device able to give an early diagnosis of the Alzheimer’s disease by a simple blood test.The contribution of this thesis work is to give reliable dimensions and geometry shape for the device channel, region in which the blood sample will be loaded, by simulating the dynamics of fluid through an orifice in order to choose the size giving thinner drop meniscus shape. Subsequently laboratory tests have been performed to have a comparison between simulated and realistic results and to study the fluid behaviour under the electric field.

In this work, we numerically investigate the dynamics of the growth and impingement of two gas bubbles in a Newtonian liquid in the presence of a rigid spherical particle. The computational analysis is carried out through 3D Arbitrary Lagrangian Eulerian (ALE) Finite Element Method (FEM) simulations. During their growth, as the bubbles start to ‘feel’ each other, they lose their spherical shape, with the side facing the other bubble becoming almost flat. In the liquid layer between the gas inclusions, an essentially biaxial extensional flow takes place. Depending on its initial position with respect to the bubbles, the solid particle can be ‘captured’ by the coalescing bubbles or ‘escape’ them. The effects of the physical and geometrical parameters of the system on such phenomenon are studied.

We study the dynamics of a slippery spherical particle suspended in an inertialess Newtonian or viscoelastic shear-thinning fluid, under shear or Poiseuille flow, by means of 3D direct numerical simulations. In particular, we investigate on the effect of particle slip on the cross-stream migration induced by fluid viscoelasticity. The governing equations are solved through the finite element method, by adopting an Arbitrary Lagrangian-Eulerian (ALE) formulation to handle the particle motion.In shear flow, the migration dynamics is qualitatively unchanged as compared to the no-slip case, i.e. the particle always moves towards the nearest wall regardless of the initial position. For increasing slip, the migration velocity first reaches a maximum, and then decreases to values lower than the no-slip one. Thus, a pronounced particle slip slows down the migration phenomenon.In Poiseuille flow, at variance with the no-slip case for a shear-thinning viscoelastic fluid, the tube wall becomes a `hydrodynamic repulsor' for high slip values, and all the particles migrate towards the channel centerline (`attractor'). In this sense, slippery particles are more easily aligned along the channel centerline than no-slip particles.

In this paper, we present a finite element implementation of fluctuating hydrodynamics with a moving boundary fitted mesh for treating the suspended particles. The thermal fluctuations are incorporated into the continuum equations using the Landau and Lifshitz approach [1]. The proposed implementation fulfills the fluctuation-dissipation theorem exactly at the discrete level. Since we restrict the equations to the creeping flow case, this takes the form of a relation between the diffusion coefficient matrix and friction matrix both at the particle and nodal level of the finite elements. Brownian motion of arbitrarily shaped particles in complex confinements can be considered within the present formulation. A multi-step time integration scheme is developed to correctly capture the drift term required in the stochastic differential equation (SDE) describing the evolution of the positions of the particles.The proposed approach is validated by simulating the Brownian motion of a sphere between two parallel plates and the motion of a spherical particle in a cylindrical cavity. The time integration algorithm and the fluctuating hydrodynamics implementation are then applied to study the diffusion and the equilibrium probability distribution of a confined circle under an external harmonic potential.

Strings of equally spaced particles (particle train) are tremendously important in a variety of microflu- idic applications. By using inertial microfluidics, particle trains can be formed near the channel walls. However, the high particle rotation and large local shear gradient near the microchannel walls can lead to blurred images and cell damage, thus negatively affecting applications related to flow cytometry. To address this challenge, we demonstrate that adding a tiny amount of hyaluronic acid biopolymer to an aqueous suspension drives self-assembly of a particle train on the centerline of a square-shaped straight microchannel, with a throughput up to approximately 2400 particles/s. The fraction of equally spaced par- ticles increases by increasing the volumetric flow rate and the distance from the channel inlet. Numerical simulations corroborate the experimental observations and, together with a simple qualitative argument on the particle train stability, shed insights on the underlying mechanism leading to particle ordering.

Systems made by particles in viscoelastic liquids are ubiquitous in a variety of industrial and biological applications. Much work has been done in the last half-century in understanding the effect of non-Newtonian properties on the dynamics of the suspended particles. Theoretical predictions, experimental observations and numerical simulations highlighted peculiar phenomena induced by fluid elasticity that dramatically affect the particle motion and patterning. In this review, the existing literature on the dynamics of non-Brownian particles in viscoelastic fluids is discussed. The main part is focused on the dynamics of rigid particles passively transported in flowing viscoelastic liquids. The available results are classified by increasing level of complexity in terms of hydrodynamic interactions (single-particle problems, binary interactions, multi-body systems) and according to the flow field. Recent results on soft and active viscoelastic suspensions are also discussed.

Migration and chaining of noncolloidal spheres in a worm-like micellar, viscoelastic solution under shear flow have been studied both experimentally and by numerical simulations. The microstructure dynamics have been experimentally investigated in the flow-gradient and in the flow-vorticity planes. 20 simulations in the flow-gradient plane have been performed for the same geometry, and with a proper selection for the constitutive equation of the suspending liquid. Experimental results show the formation of particle chains in the bulk, along with migration of a considerable fraction of spheres to the walls. At long times, chains in the bulk are stable, and cross-flow migration of individual spheres is suppressed. Numerical simulations with a standard viscoelastic constitutive equation (Giesekus fluid) reproduce the same phenomena observed experimentally, both in terms of fast particle migration to the wall and bulk chain stability. No alignment is, instead, found in simulations with a constant-viscosity, elastic fluid (Oldroyd-B model), in agreement with previous experimental results with Boger fluids. (C) 2013 Elsevier B.V. All rights reserved.

The dynamics of pairs and triplets of particles suspended in a viscoelastic fluid flowing along the centerline of a cylindrical channel is studied by numerical simulations. The governing equations are solved by the finite element method by employing an ALE formulation to handle the particle motion. For a pair of particles, at variance with the Newtonian case, the viscoelastic nature of the suspending medium alters the interparticle distance during the flow. For low and moderate Deborah numbers, the particles can approach or separate depending on the initial distance. For high Deborah numbers, the approaching dynamics disappears. Different fluid rheology and confinement ratio only quantitatively alter such a scenario. The three-particle dynamics is more complex. In a Newtonian liquid, the leftmost particle of the triplet approaches and slows down the middle one. Consequently, the rightmost particle separates, giving rise to a pair and an isolated particle. A similar scenario occurs for a viscoelastic liquid. In this case, however, depending on the initial configuration and the Deborah number, the particles forming the pair can subsequently approach or separate. In the latter case, the final configuration is the formation of three isolated particles.

The rising of spherical bodies in structured fluids with yield stress is studied. The system is a suspension of hydrogenated castor oil colloidal fibers in a surfactant micellar solution. The fiber network confers to the fluid a viscoelastic behavior, with a well-defined yield stress, which increases with increasing fiber concentration. Various fluids with different fiber contents are prepared and rheologically characterized. A home-made time-lapse photography setup is used to monitor the time evolution position of the spherical particles, and the rising motion of both hollow spheres and air bubbles, in the diameter range 65–550 μm, is measured. The experiments last as long as several weeks, corresponding to significantly low measured velocities. Finite element simulations are performed to support the experimental data, assuming both interfacial slip and no slip conditions. The fluid dynamic phenomenon is studied and discussed in terms of dimensionless numbers, such as yield ratio, Bingham number, and Stokes drag coefficient. The results are novel for the system (suspending medium and hollow spheres) and for the covered Bingham number range, which is extended over three orders of magnitude in comparison with already available literature results. Our values provide quantitative data of the mechanical properties (i.e., yield stress value) at very low shear rates, in a prohibitive range for a traditional rheometer, and agree with the macroscopic rheological response. Moreover, the important role of the power law index n of the Herschel-Bulkley model, used to fit the data, has been highlighted. Our results, based on a Bingham-like fluid, are compared with the experimental data already available with Carbopol, treated as a Herschel Bulkley fluid with n = 0.5. The results could have important implications in the fabric and personal care detergency, a technological area where many fluids have composition and show rheological properties similar to those considered in the current work.

The viscoelasticity-induced migration of a sphere in pressure-driven flow in a square-shaped microchannel is investigated under inertialess conditions. The effects of fluid rheology, i.e. of shear thinning and normal stresses, is studied by means of 3D finite element simulations. Two constitutive models are selected, in order to highlight differences due to rheological properties. A strong influence of the suspending fluid rheology on the migration phenomenon is shown, by particle trajectory analysis. When the second normal stress difference is negligible and, as a consequence, no secondary flows appear, the particle migrates towards the channel centerline or the closest corner, depending on its initial position. As shear thinning is increased, the center-attractive region is reduced, and the migration rate is faster. On the other hand, the existence of secondary flows, linked to the existence of a second normal stress difference, alters the migration scenario. The competition between the particle-wall hydrodynamic interactions, promoting the migration mechanism, and the secondary flow velocity components gives rise to further ‘equilibrium’ positions within the channel cross-section. Particles driven towards such positions trace out a spiral trajectory, following the vortex structure of the secondary flows. However, as the particle dimension is increased or the Deborah number is reduced, the cross-streamline migration velocity overcomes the secondary flow velocity. In this case, most of the particles are driven towards the channel centerline, i.e. a strong flow-focusing effect results.

In this work, we study the dynamics of an elastic particle suspended in an inertialess Newtonian fluid flowing through a channel with an orthogonal side branch (asymmetric T-shaped bifurcation) by means of 2D Arbitrary Lagrangian Eulerian (ALE) Finite Element Method (FEM) direct numerical simulations. The simulations show that, when the fluid is equally partitioned in the two downstream branches and the particle starts from the inflow channel centerline, a sufficiently deformable particle is deviated into the ‘side’ branch, at variance with a rigid particle, which remains in the ‘main’ outlet. The effects of the elastic capillary number and the confinement ratio on the particle trajectory and deformation near the bifurcation are investigated. We discuss how this device can be exploited for separating particles based on their elasticity.

A decoupling time-marching algorithm for the transient simulation of viscoelastic fluids with inertia is proposed. The method is an extension of the second-order implicit stress formulation previously reported for inertialess problems [D’Avino G, Hulsen MA. Decoupled second-order transient schemes for the flow of viscoelastic fluids without a solvent contribution. J Non-Newton Fluid Mech 2010;165:1602–12]. At each time step, the momentum and continuity equations are solved by using a time-discretized but space-continuous form of the constitutive equation, based on a first-order, semi-implicit Euler scheme. The advantage of using such a form for the force term in the momentum balance is the possibility to use the decoupled procedure even when the solvent viscosity is small or absent. In the next substep, the stress unknowns are computed by using the calculated velocity field. For the time discretization of the momentum equation, four schemes, i.e. Gear, Crank–Nicolson, Gear mixed explicit–implicit and Gear with velocity predictor, have been compared. A second-order, semi-implicit Gear scheme is adopted for the discretization of the constitutive equation, where the convection term is taken implicitly whereas all the other terms are explicit. Three test problems have been considered to validate the proposed algorithm. The method is demonstrated to be second-order accurate in time. The Crank–Nicolson scheme is found to give numerical oscillations for any time step size. The Gear-based discretizations are stable and the choice of the specific scheme is related to the investigated problem and the set of parameters. New results are also given for a falling sphere in a Giesekus fluid without solvent contribution.

The development of a parallel software for simulating large-scale time-dependent 3D problems involving the flow of multimode viscoelastic fluids is presented. The computing environment relies on PETSc [27] (Portable, Extensible Toolkit for Scientific Computation) components integrated with a finite element solver. A DEVSS-G/SUPG formulation together a log-representation of the conformation tensor are used to stabilize the code. An operator-splitting time-integration scheme is implemented whereby the solution of the continuity and momentum balance equations are decoupled. The large linear system arising from the continuity and momentum discretization is solved by the multifrontal, massively parallel solver (MUMPS) [37]. The time-integration scheme applied to the constitutive equation gives rise to six, decoupled linear systems (one for each stress component) solved by MUMPS as well. Three test cases are selected to show the performances and the limitations of the parallel code. In the first example, the flow of a Giesekus fluid in a pressure-driven square-shaped channel is examined. In the second test, the unsteady drag on a sphere is computed and, in the third example, a solid particle in a sheared viscoelastic fluid is considered, thus involving a moving boundary. The parallel software gives high performances for problems such that the finite element mesh is independent of time. In this case, the coefficients of the Stokes-like system are constant in time and the matrix factorization is computed at the first time step only and kept during the whole simulation, drastically reducing the total computational time. On the other hand, whether the factorization needs to be computed at each time step, the solution time increases by more than one order of magnitude, suggesting to seek for alternative iterative solvers. Finally, for all the test cases, scalability of parallel software is evaluated using the speed-up normalized with respect to the software execution time on a number of processors greater than 1. This was necessary due to the high memory requirements. Experimental results validate the performance gain of the parallel code as both the problem size and the processor number increase. (C) 2011 Elsevier B.V. All rights reserved.

The cross-streamline migration of a spherical particle in a viscoelastic fluid flowing in a wide slit microdevice is here investigated through 3D finite element simulations. The study is performed by neglecting both fluid and particle inertia, which is a common assumption in microfluidic systems. In order to highlight the role of the suspending liquid rheology, two viscoelastic constitutive equations were chosen, i.e. the Giesekus and the Phan Thien–Tanner models. Because of the large cross-section aspect ratio, the influence of the lateral walls can be neglected, and the migration is unidirectional, along the gap (small) height. For small confinement ratios, i.e. for small particle-gap dimensions ratios, a multistable dynamics is found whereby the particle is driven towards the channel centerplane or the closest wall depending on its initial position through the gap, thus leading to the existence of an unstable separatrix. The position of such a separatrix is strongly influenced by the fluid rheology. For the Giesekus suspending liquid, the centerplane attractive region is always found to be wider than the wall attractive zone, independently of the flow rate. On the other hand, for the Phan Thien–Tanner liquid, large flow rates drastically move the separatrix towards the channel center. Beyond a critical particle size, the multistable behavior disappears and the separatrix collapses on the centerplane or the wall depending on the second normal stress difference. Finally, the calculated particle distributions along the channel gap at different distances from the inlet show that the migration is faster for large particles, higher flow rates, and more shear thinning fluids.