An overview of the proper generalized decomposition with applications in computational rheology

We review the foundations and applications of the proper generalized decomposition (PGD), a powerful model reduction technique that computes a priori by means of successive enrichment a separated representation of the unknown field. The computational complexity of the PGD scales linearly with the dimension of the space wherein the model is defined, which is in marked contrast with the exponential scaling of standard grid-based methods. First introduced in the context of computational rheology by Ammar et al. [3] and [4], the PGD has since been further developed and applied in a variety of applications ranging from the solution of the Schrödinger equation of quantum mechanics to the analysis of laminate composites. In this paper, we illustrate the use of the PGD in four problem categories related to computational rheology: (i) the direct solution of the Fokker-Planck equation for complex fluids in configuration spaces of high dimension, (ii) the development of very efficient non-incremental algorithms for transient problems, (iii) the fully three-dimensional solution of problems defined in degenerate plate or shell-like domains often encountered in polymer processing or composites manufacturing, and finally (iv) the solution of multidimensional parametric models obtained by introducing various sources of problem variability as additional coordinates

Modelling the effect of particle inertia on the orientation kinematics of fibres and spheroids immersed in a simple shear flow

Simulations of flows containing non-spherical particles (fibres or ellipsoids) rely on the knowledge of the equation governing the particle motion in the flow. Most models used nowadays are based on the pioneering work of Jeffery (1922), who obtained an equation for the motion of an ellipsoidal particle immersed in a Newtonian fluid, despite the fact that this model relies on strong assumptions: negligible inertia, unconfined flow, dilute regime, flow unperturbed by the presence of the suspended particle, etc. In this work, we propose a dumbbell-based model aimed to describe the motion of an inertial fibre or ellipsoid suspended in a Newtonian fluid. We then use this model to study the orientation kinematics of such particle in a linear shear flow and compare it to the inertialess case. In the case of fibres, we observe the appearance of periodic orbits (whereas inertialess fibres just align in the flow field). For spheroids, our model predicts an orbit drift towards the flow-gradient plane, either gradually (slight inertia) or by first rotating around a moving oblique axis (heavy particles). Multi-Particle Collision Dynamics (MPCD) simulations were carried out to assess the model predictions in the case of inertial fibres and revealed similar behaviours

A simple microstructural viscoelastic model for flowing foams

The numerical modelling of forming processes involving the flow of foams requires taking into account the different problem scales. Thus, in industrial applications a macroscopic approach is suitable, whereas the macroscopic flow parameters depend on the cellular structure: cell size, shape, orientation, etc. Moreover, the shape and orientation of the cells are induced by the flow. A fully microscopic description remains useful to understand the foam behaviour and the topological changes induced by the cell elongation or distortion, however, from an industrial point of view, microscopic simulations remain challenging to address practical applications involving flows in complex 3D geometries. In this paper, we propose a viscoelastic flow model where the foam microstructure is represented from suitable microstructure descriptors whose evolution is governed by the macroscopic flow kinematics. This is a post-peer-review, pre-copyedit version of an article published in International journal of material forming

A Physically-Based Fractional Diffusion Model for Semi-Dilute Suspensions of Rods in a Newtonian Fluid

[EN] The rheological behaviour of suspensions involving interacting (functionalized) rods remains nowadays incompletely understood, in particular with regard to the evolution of the elastic modulus with the applied frequency in small-amplitude oscillatory flows. In a previous work, we addressed this issue by assuming a fractional diffusion mechanism, however the approach followed was purely phenomenological. The present work revisits the topic from a phys ical viewpoint, with the aim of justifying the fractional nature of diffusion. After accomplishing this first objective, we explore by means of numerical ex periments the consequences of the proposed fractional modelling approach in linear and non-linear rheology.Nadal, E.; Aguado-López, JV.; Abisset-Chavanne, E.; Chinesta Soria, FJ.; Keunings, R.; Cueto, E. (2017). A Physically-Based Fractional Diffusion Model for Semi-Dilute Suspensions of Rods in a Newtonian Fluid. Applied Mathematical Modelling. 51:58-67. https://doi.org/10.1016/j.apm.2017.06.009S58675

Microscopic modelling of orientation kinematics of non-spherical particles suspended in confined flows using unilateral mechanics

The properties of reinforced polymers strongly depend on the microstructural state, that is, the orientation state of the fibres suspended in the polymeric matrix, induced by the forming process. Understanding flow-induced anisotropy is thus a key element to optimize both materials and process. Despite the important progresses accomplished in the modelling and simulation of suspensions, few works addressed the fact that usual processing flows evolve in confined configurations, where particles characteristic lengths may be greater than the thickness of the narrow gaps in which the flow takes place. In those circumstances, orientation kinematics models proposed for unconfined flows must be extended to the confined case. In this short communication, we propose an alternative modelling framework based on the use of unilateral mechanics, consequently exhibiting a clear analogy with plasticity and contact mechanics. This framework allows us to revisit the motion of confined particles in Newtonian and non-Newtonian matrices. We also prove that the confined kinematics provided by this model are identical to those derived from microstructural approache

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

Enhanced pressure drop, planar contraction flows and continuous spectrum models

This study addresses a rheological problem that has been outstanding now for the past few decades, raised by the experimental findings of Binding and Walters [1]. There, it was established experimentally that planar contraction flows for some Boger fluids could display enhanced pressure-drops above Newtonian flows, as was the case for their tubular counterparts. Nevertheless, flow-structures to achieve this result were reported to be markedly different, planar to circular. In this article, it is shown how predictive differential-viscoelastic solutions with continuum models can replicate these observations. Key to this success has been the derivation of a new definition for the third-invariant of the rate-of-deformation tensor in planar flows, mimicking that of the circular case [2], [3]. This provides a mechanism to successfully incorporate dissipation within planar flows, as performed earlier for tubular flows. Still, to reach the necessary large deformation-rates to achieve planar enhanced pressure-drops, and whilst maintaining steady flow-conditions, it has been found crucial to invoke a continuous-spectrum relaxation-time model [3]. The rheological power and flexibility of such a model is clearly demonstrated, over its counterpart Maxwellian single-averaged relaxation-time approximation; the latter transcending the boundaries of steady-to-unsteady flow to manifest equivalent levels of enhanced pressure-drops. Then, the role of extensional viscosity and first normal-stress difference, each play their part to achieve such planar enhanced pressure-drops. As a by-product, the distinctive planar ‘bulb-flow’ structures discovered by Binding and Walters [1], absent in tubular flows, are also predicted under the associated regime of high deformation-rates where enhanced pressure-drop arise

Subduction Initiation With Vertical Lithospheric Heterogeneities and New Fault Formation

How subduction initiates with mechanically unfavorable lithospheric heterogeneities is important and rarely studied. We investigate this with a geodynamic model for the Puysegur Incipient Subduction Zone (PISZ) south of New Zealand. The model incorporates a true free surface, elasto-visco-plastic rheology and phase changes. Our predictions fit the morphology of the Puysegur Trench and Ridge and the deformation history on the overriding plate. We show how a new thrust fault forms and evolves into a smooth subduction interface, and how a preexisting weak zone can become a vertical fault inboard of the thrust fault during subduction initiation, consistent with two-fault system at PISZ. The model suggests that the PISZ may not yet be self-sustaining. We propose that the Snares Zone (or Snares Trough) is caused by plate coupling differences between shallower and deeper parts, the tectonic sliver between two faults experiences strong rotation, and low-density material accumulates beneath the Snares Zone