Daniel Lhuillier

Migration of rigid particles in non-Brownian viscous suspensions

There is an obvious need for obtaining a closed set of equations describing multiphase flows and the complexity of their configurations. Despite their “frozen” interfaces, suspensions of rigid particles are also concerned by this issue. The description of the relative motion between the two phases is still controversial, in particular, concerning the forces acting on the particles and involving their concentration gradient or a shear rate gradient. It is our purpose here (a) to develop a two-fluid model especially designed for particles dispersed in a viscous liquid and (b) to close the model for rigid particles with the help of low-Reynolds-number hydrodynamics. Besides the obvious role of gravity forces, the migration of particles relative to the fluid is shown to result from two different physical phenomena: (a) the inhomogeneity of the stress resulting from direct interparticle forces and (b) Fick-like terms in the hydrodynamic force acting on the particles.

Stationary shear flows of dense granular materials: a tentative continuum modelling

Abstract.We propose a simple continuum model to interpret the shearing motion of dense, dry and cohesion-less granular media. Compressibility, dilatancy and Coulomb-like friction are the three basic ingredients. The granular stress is split into a rate-dependent part representing the rebound-less impacts between grains and a rate-independent part associated with long-lived contacts. Because we consider stationary flows only, the grain compaction and the grain velocity are the two main variables. The predicted velocity and compaction profiles are in apparent qualitative agreement with most of the experimental or numerical results concerning free-surface shear flows as well as confined shear flows.

Particle velocity fluctuations and correlation lengths in dilute sedimenting suspensions

We measure the instantaneous velocity of particles sedimenting in a three-dimensional container at low particulate Reynolds numbers. We aim at characterizing the main specificities of the particle velocity fluctuations. We obtain the local and instantaneous Eulerian velocity field from particle image velocimetry: a thin Yag laser light sheet (about two diameters thick) illuminates the particles from one side of the cell to the other. Our measurements are therefore spatially localized and, together with the squared cross sections of the cells, these are the two main originalities of our instrumentation. Four different cells and three different particle sizes give access to aspect ratios (cell width W over particle radius a) ranging from about 50 up to 800. We confirm the existence of eddy-like structures for the velocity fluctuations. The structure size is found to be almost independent of the volume fraction Φ for 6.25×10−4

Internal variables and the non-equilibrium thermodynamics of colloidal suspensions ☆

Abstract Classical thermodynamics of irreversible processes is a valuable tool for the study of suspensions, provided due care is devoted to the selection of internal variables. As a first example, the flow-induced anisotropic microstructure appearing in many a suspension is usually depicted by a vector, the orientation and length of which are submitted to a statistical distribution. We show how irreversible thermodynamics imposes restrictions on the time evolution of that structural internal variable (and consequently on the time evolution of its statistical distribution), and how this evolution is related to the elastic stress of the suspension. As a second example, we consider the possibility of a mean relative motion between the particles and the suspending fluid. Upgrading that relative velocity to the status of an internal variable, we analyse the consequences concerning the transport of mass, momentum and energy, providing a link with the two-fluid model of suspensions, as well as with the well-known description of molecular mixtures.

Flow and jamming of a two-dimensional granular bed: Toward a nonlocal rheology?

In order to test the rheology of granular flows, we performed series of numerical simulations of nearly monodisperse stationary chute flows from rapid to slow and very slow flow regime, namely, close to the jamming transition. We check how existing rheological models (i.e., Bagnold’s model and the I-model) capture the behavior of the numerical flows, and perform an acute characterization of the structure of the flow in terms of grains velocity fluctuations close to the jamming transition. The simulations show that both Bagnold’s and the I-model fail to describe the data points in the slow regime, namely, when I≤2×10−2. Turning to the analysis of grains velocity fluctuations, we compute the associated correlation length λ and show its dependence on the inertial number: λ/d∝I−0.32. The amplitude of the grains velocity fluctuations, namely, the granular temperature, exhibits a power-law dependence on the shear rate and allows for an efficient prediction of the shape of the velocity profiles. The main result consists of a scaling merging all data points for all flow regimes onto the same master curve, and relating granular temperature, shear rate, and the variation of stress between the considered depth and the bottom wall. This scaling can be written as a relation between local stress, local shear rate, and local temperature, provided the introduction of a characteristic length scale ξ=d(H−z)/z where both the distance to the surface and the distance to the bottom wall are involved. This scaling strongly suggests a nonlocal behavior, valid in the flow regime and extending close to the jamming transition, and hints at granular temperature as the variable at the origin of the nonlocality.In order to test the rheology of granular flows, we performed series of numerical simulations of nearly monodisperse stationary chute flows from rapid to slow and very slow flow regime, namely, close to the jamming transition. We check how existing rheological models (i.e., Bagnold’s model and the I-model) capture the behavior of the numerical flows, and perform an acute characterization of the structure of the flow in terms of grains velocity fluctuations close to the jamming transition. The simulations show that both Bagnold’s and the I-model fail to describe the data points in the slow regime, namely, when I≤2×10−2. Turning to the analysis of grains velocity fluctuations, we compute the associated correlation length λ and show its dependence on the inertial number: λ/d∝I−0.32. The amplitude of the grains velocity fluctuations, namely, the granular temperature, exhibits a power-law dependence on the shear rate and allows for an efficient prediction of the shape of the velocity profiles. The main result ...

A mean-field description of two-phase flows with phase changes

Abstract A new version of the two-fluid model is developed, specially devoted to liquid–vapour two-phase mixtures, but also relevant to liquid-gas and liquid–liquid mixtures. It is well-known that, over a large range of volume fractions, liquid–vapour mixtures behave as dispersions of particles in a carrier fluid. But the “particles” belong to one phase at the beginning of the phase change, and to the second phase at the end. Within the present model, the dispersed phase is not prescribed at the outset but is merely the one with the lower volume fraction. To simplify the issue, surface tension and interfacial properties are neglected. However, the differences of pressure, temperature and velocity between the two phases are taken into account. The exchanges of mass, momentum and energy between phases are split into a “mean-field” part corresponding to the average conditions imposed by the whole mixture on the dispersed phase, and a part specifically due to the disturbances created by the particles. Constraints on constitutive relations are obtained from the overall dissipation rate, and result in a closed set of seven equations for seven state variables including one volume fraction. We insist on the general form of the equations but not on the details of the closure relations. The limits of this simple model are clearly stated, and we discuss possible improvements, including a better account of small-scale kinetic phenomena, as well as an eighth equation for the density of interfaces.

Phenomenology of inertia effects in a dispersed solid-fluid mixture

Abstract Combining single particle results, average equations and thermodynamic considerations, we propose a way to build the equations describing a suspension of rigid spherical particles in a carrier fluid, with emphasis on inertia effects including virtual mass. The spatial fluctuations of the fluid velocity field are depicted by two phenomenological functions ƒ(α s ) and g(α s ) of the particle volume fraction, and a third function h(αs) is necessary to describe the intensity of the particles internal stress. It is shown that all inertia effects occurring in the relative translational motion can be derived from the two functions ƒ and g–h only. The conditions under which the above system of equations is hyperbolic are determined and comparison is made with what is presently known about ƒ, g and h in the dilute limit.

Volume averaging of slightly non-homogeneous suspensions

Abstract We consider the problem of volume averaging in a two-phase dispersed system with non-uniform bulk fields, for instance suspensions with gradients of the particle concentration. We limit ourselves to slightly non-homogeneous suspensions for which the scale of macroscopic variation is much larger than the particle distance and we look for the average of quantities which vanish everywhere except inside the particles or at the interfaces. The results appear as a multipolar expansion reminiscent of the one developed by Lorentz for electromagnetic media. We check these results in some simple situations and use them to define the average stress and average interphase force in a suspension with small non-uniformities.

A possible alternative to the FENE dumbbell model of dilute polymer solutions

The dumbbell model of dilute polymer solutions is simple and successful, and its FENE version has progressively become a paradigm. In some transient extensional flows however, an increased dissipation was observed which could not be understood within the FENE model. This prompted us to look for a new dumbbell-like model in which the finite extensibility of the polymer is taken more thoroughly into account, i.e. not only through the Warner potential. The main lines of this alternative model are presented.

The spreading of a granular column from a Bingham point of view

The collapse and spreading of granular columns has been the subject of sustained interest in the last years from both mechanical and geophysical communities. Yet, in spite of this intensive research, the adequate rheology allowing for a reliable continuum modeling of the dynamics of granular column collapse is still open to discussion. Essentially, continuum models rely on shallow‐water approximation for which dissipation and sedimentation processes are taken into account through the introduction of ad hoc laws. However, the rheological origin of the experimental scaling laws exhibited by the granular columns when spreading remains unclear. On these grounds, we adopt an alternative approach consisting of studying the collapse of columns of material obeying a Bingham rheology. Therefore we carried out series of numerical simulations using the Gerris Flow Solver solving the time dependent incompressible Navier‐Stokes equation in two dimensions for the specified rheology. We first check that the mass exhibit similar scaling laws as those shown by granular columns. Then we investigate in which extent rheological parameters do reflect on these scaling laws. A comparative analysis of Bingham and granular flow characteristics ensues.The collapse and spreading of granular columns has been the subject of sustained interest in the last years from both mechanical and geophysical communities. Yet, in spite of this intensive research, the adequate rheology allowing for a reliable continuum modeling of the dynamics of granular column collapse is still open to discussion. Essentially, continuum models rely on shallow‐water approximation for which dissipation and sedimentation processes are taken into account through the introduction of ad hoc laws. However, the rheological origin of the experimental scaling laws exhibited by the granular columns when spreading remains unclear. On these grounds, we adopt an alternative approach consisting of studying the collapse of columns of material obeying a Bingham rheology. Therefore we carried out series of numerical simulations using the Gerris Flow Solver solving the time dependent incompressible Navier‐Stokes equation in two dimensions for the specified rheology. We first check that the mass exhibit...