Well-posed and ill-posed behaviour of the μ(I)-rheology for granular flow

In light of the successes of the Navier–Stokes equations in the study of fluid flows, similar continuum treatment of granular materials is a long-standing ambition. This is due to their wide-ranging applications in the pharmaceutical and engineering industries as well as to geophysical phenomena such as avalanches and landslides. Historically this has been attempted through modification of the dissipation terms in the momentum balance equations, effectively introducing pressure and strain-rate dependence into the viscosity. Originally, a popular model for this granular viscosity, the Coulomb rheology, proposed rate-independent plastic behaviour scaled by a constant friction coefficient μ . Unfortunately, the resultant equations are always ill-posed. Mathematically ill-posed problems suffer from unbounded growth of short-wavelength perturbations, which necessarily leads to grid-dependent numerical results that do not converge as the spatial resolution is enhanced. This is unrealistic as all physical systems are subject to noise and do not blow up catastrophically. It is therefore vital to seek well-posed equations to make realistic predictions. The recent μ(I) -rheology is a major step forward, which allows granular flows in chutes and shear cells to be predicted. This is achieved by introducing a dependence on the non-dimensional inertial number I in the friction coefficient μ . In this paper it is shown that the μ(I) -rheology is well-posed for intermediate values of I , but that it is ill-posed for both high and low inertial numbers. This result is not obvious from casual inspection of the equations, and suggests that additional physics, such as enduring force chains and binary collisions, becomes important in these limits. The theoretical results are validated numerically using two implicit schemes for non-Newtonian flows. In particular, it is shown explicitly that at a given resolution a standard numerical scheme used to compute steady-uniform Bagnold flow is stable in the well-posed region of parameter space, but is unstable to small perturbations, which grow exponentially quickly, in the ill-posed domain

Shear band dynamics from a mesoscopic modeling of plasticity

The ubiquitous appearance of regions of localized deformation (shear bands) in different kinds of disordered materials under shear is studied in the context of a mesoscopic model of plasticity. The model may or may not include relaxational (aging) effects. In the absence of relaxational effects the model displays a monotonously increasing dependence of stress on strain-rate, and stationary shear bands do not occur. However, in start up experiments transient (although long lived) shear bands occur, that widen without bound in time. I investigate this transient effect in detail, reproducing and explaining a t^1/2 law for the thickness increase of the shear band that has been obtained in atomistic numerical simulations. Relaxation produces a negative sloped region in the stress vs. strain-rate curve that stabilizes the formation of shear bands of a well defined width, which is a function of strain-rate. Simulations at very low strain-rates reveal a non-trivial stick-slip dynamics of very thin shear bands that has relevance in the study of seismic phenomena. In addition, other non-stationary processes, such as stop-and-go, or strain-rate inversion situations display a phenomenology that matches very well the results of recent experimental studies.Comment: 10 pages, 10 figure

Tunable diffusive lateral inhibition in chemical cells

The Belousov-Zhabotinsky (BZ) reaction has become the prototype of nonlinear chemical dynamics. Microfluidic techniques provide a convenient method for emulsifying BZ solutions into monodispersed drops with diameters of tens to hundreds of microns, providing a unique system in which reaction-diffusion theory can be quantitatively tested. In this work, we investigate monolayers of microfluidically generated BZ drops confined in close-packed two-dimensional (2D) arrays through experiments and finite element simulations. We describe the transition from oscillatory to stationary chemical states with increasing coupling strength, controlled by independently varying the reaction chemistry within a drop and diffusive flux between drops. For stationary drops, we studied how the ratio of stationary oxidized to stationary reduced drops varies with coupling strength. In addition, using simulation, we quantified the chemical heterogeneity sufficient to induce mixed stationary and oscillatory patterns

Approaches to molecular communication between synthetic compartments based on encapsulated chemical oscillators

The use of confined micro-oscillators as paradigmatic model for studying communication and information exchange among network of synthetic cells is rapidly growing in the last years. In this paper we report the first steps of an ongoing investigation on the encapsulation of the Belousov-Zhabotinsky oscillating reaction inside phospholipid vesicles (liposomes). The preparation of liposomes encapsulating water soluble molecules can be efficiently carried out in two steps: 1. confining the solute inside water-in-oil droplet, 2. transformation of droplets into liposomes. Here we have started the investigation of chemical oscillation emerging behavior in these two types of compartments. We firstly show interesting dynamical behavior within emulsion droplets. Next, we assess the influence of additives (sugars in particular, which are necessary for the liposomes production) on the oscillation pattern. Future studies will be devoted to the encapsulation of Belousov-Zhabotinsky reaction within liposomes. The potentiality of our systems for modeling intercellular communication pathways and its applications in the Bio-Chem-ITs context are shortly discussed. © Springer International Publishing Switzerland 2014

Approaches to molecular communication between synthetic compartments based on encapsulated chemical oscillators.

The use of confined micro-oscillators as paradigmatic model for studying communication and information exchange among network of synthetic cells is rapidly growing in the last years. In this paper we report the first steps of an ongoing investigation on the encapsulation of the Belousov-Zhabotinsky oscillating reaction inside phospholipid vesicles (liposomes). The preparation of liposomes encapsulating water soluble molecules can be efficiently carried out in two steps: 1. confining the solute inside water-in-oil droplet, 2. transformation of droplets into liposomes. Here we have started the investigation of chemical oscillation emerging behavior in these two types of compartments. We firstly show interesting dynamical behavior within emulsion droplets. Next, we assess the influence of additives (sugars in particular, which are necessary for the liposomes production) on the oscillation pattern. Future studies will be devoted to the encapsulation of Belousov-Zhabotinsky reaction within liposomes. The potentiality of our systems for modeling intercellular communication pathways and its applications in the Bio-Chem-ITs context are shortly discussed

Modelling Approach to Enzymatic pH Oscillators in Giant Lipid Vesicles

The urease-catalyzed hydrolysis of urea can display feedback driven by base production (NH 3 ) resulting in a switch from acidic to basic pH under non-buffered conditions. Thus, this enzymatic reaction is a good candidate for investigation of chemical oscillations or bistability. In order to determine the best conditions for oscillations, a two-variable model was initially derived in which acid and urea were supplied at rates k H and k S from an external medium to an enzyme-containing compartment. Oscillations were theoretically observed providing the necessary condition that k H > k S was met. To apply this model, we devised an experimental system able to ensure the fast transport of acid compared to that of urea. In particular, by means of the droplet transfer method, we encapsulated the enzyme, together with a proper pH probe, in 1-palmitoyl-2-oleoyl-sn-glycero-3-phosphatidylcholine (POPC) based liposomes, where differential diffusion of H + and urea is ensured by the different permeability (P m ) of the membrane to the two species. Here we present an improved theoretical model that accounts for the products transport and for the probe hydrolysis, to obtain a better guidance for the experiments