Finger behavior of a shear thinning fluid in a Hele-Shaw cell

We make a theoretical study of the behavior of a simple fluid displacing a shear thinning fluid confined in a Hele-Shaw cell. To study the Saffman-Taylor instability when the displaced fluid is non Newtonian we face the problem of having a field which is non laplacian. By means of an hodographic transformation we are able to solve the problem in the case of weak shear thinning while taking into account the non laplacian character of the equation. Our results predict that the finger width decreases towards zero for small values of the surface tension parameter which is inversely proportional to the finger velocity.Comment: 4 pages, 1 figure, RevTeX, to appear in Phys. Rev. Let

Creases and cusps in growing soft matter

The buckling of a soft elastic sample under growth or swelling has highlighted a new interest in materials science, morphogenesis, and biology or physiology. Indeed, the change of mass or volume is a common fact of any living species, and on a scale larger than the cell size, a macroscopic view can help to explain many features of common observation. Many morphologies of soft materials result from the accumulation of elastic compressive stress due to growth, and thus from the minimization of a nonlinear elastic energy. The similarity between growth and compression of a piece of rubber has revived the instability formalism of nonlinear elastic samples under compression, and in particular Biot's instability. Here we present a modern treatment of this instability in the light of complex analysis and demonstrate the richness of possible profiles that an interface can present under buckling, even if one restricts oneself to the two spatial dimensions. Special attention is given to wrinkles, folds and cusps, a surprising observation in swelling gels or clays. The standard techniques of complex analysis, nonlinear bifurcation theory and path-independent integrals are revisited to highlight the role of physical parameters at the origin of the observed patterns below and above the Biot threshold.Comment: 40 pages. 8 figures 179 reference

Evaporation of a thin film: diffusion of the vapour and Marangoni instabilities

The stability of an evaporating thin liquid film on a solid substrate is investigated within lubrication theory. The heat flux due to evaporation induces thermal gradients; the generated Marangoni stresses are accounted for. Assuming the gas phase at rest, the dynamics of the vapour reduces to diffusion. The boundary condition at the interface couples transfer from the liquid to its vapour and diffusion flux. A non-local lubrication equation is obtained; this non-local nature comes from the Laplace equation associated with quasi-static diffusion. The linear stability of a flat film is studied in this general framework. The subsequent analysis is restricted to moderately thick films for which it is shown that evaporation is diffusion limited and that the gas phase is saturated in vapour in the vicinity of the interface. The stability depends only on two control parameters, the capillary and Marangoni numbers. The Marangoni effect is destabilising whereas capillarity and evaporation are stabilising processes. The results of the linear stability analysis are compared with the experiments of Poulard et al (2003) performed in a different geometry. In order to study the resulting patterns, the amplitude equation is obtained through a systematic multiple-scale expansion. The evaporation rate is needed and is computed perturbatively by solving the Laplace problem for the diffusion of vapour. The bifurcation from the flat state is found to be a supercritical transition. Moreover, it appears that the non-local nature of the diffusion problem unusually affects the amplitude equation

Morphology of melanocytic lesions in situ

International audienceMelanoma is a solid tumour with its own specificity from the biological and morphological viewpoint. On one hand, numerous mutations are already known affecting different pathways. They usually concern proliferation rate, apoptosis, cell senescence and cell behaviour. On the other hand, several visual criteria at the tissue level are used by physicians in order to diagnose skin lesions. Nevertheless, the mechanisms between the changes from the mutations at the cell level to the morphology exhibited at the tissue level are still not fully understood. Using physical tools, we develop a simple model. We demonstrate analytically that it contains the necessary ingredients to understand several specificities of melanoma such as the presence of microstructures inside a skin lesion or the absence of a necrotic core. We also explain the importance of senescence for growth arrest in benign skin lesions. Thanks to numerical simulations, we successfully compare this model to biological data

Onset of nonlinearity in a stochastic model for auto-chemotactic advancing epithelia

International audienceWe investigate the role of auto-chemotaxis in the growth and motility of an epithelium advancing on a solid substrate. In this process, cells create their own chemoattractant allowing communications among neighbours, thus leading to a signaling pathway. As known, chemotaxis provokes the onset of cellular density gradients and spatial inhomogeneities mostly at the front, a phenomenon able to predict some features revealed in in vitro experiments. A continuous model is proposed where the coupling between the cellular proliferation, the friction on the substrate and chemotaxis is investigated. According to our results, the friction and proliferation stabilize the front whereas auto-chemotaxis is a factor of destabilization. This antagonist role induces a fingering pattern with a selected wavenumber k0. However, in the planar front case, the translational invariance of the experimental set-up gives also a mode at k = 0 and the coupling between these two modes in the nonlinear regime is responsible for the onset of a Hopf-bifurcation. The time-dependent oscillations of patterns observed experimentally can be predicted simply in this continuous non-linear approach. Finally the effects of noise are also investigated below the instability threshold

Exact results with the J-integral applied to free-boundary flows

We apply the J-integral to free-boundary flows in a channel geometry such as viscous fingering or blob injection in Hele-Shaw cells, void propagation in electromigration, and injection of air bubbles into inviscid liquids. The theory of that and related conservation integrals, developed in elasticity, is outlined in a way that is applicable to fluid mechanics problems. Depending on the boundary conditions, for infinite bubbles in Laplacian fields we are able to use the J-integral to predict finger width if such solutions exist or to predict that there are no solutions. For finite sized bubbles, bounds can sometimes be derived. In the case of Hele-Shaw flows, in which solutions appear as a continuum, finger width cannot be constrained, but we do obtain a new derivation and generalization of Richardson moment conservation. Applications to vortex motion are also outlined briefly.Earth and Planetary SciencesEngineering and Applied Science

A theoretical treatment of void electromigration in the strip geometry

The void electromigration process in the strip geometry is investigated analytically and numerically. The void is assumed to travel either along the axis of symmetry of the metal strip or at the boundary. In each case, the shape, the velocity of the void and the characteristic electrical current are predicte

Models of void electromigration

We study the motion of voids in conductors subject to intense electrical current densities. We use a free-boundary model in which the flow of current around the insulating void is coupled to a law of motion (kinematic condition) for the void boundary. In the first part of the paper, we apply a new complex variable formulation of the model to an infinite domain and use this to (i) consider the stability of circular and flat front travelling waves, (ii) show that, in the unbounded metal domain, the only travelling waves of finite void area are circular, and (iii) consider possible static solutions. In the second part of the paper, we look at a conducting strip (which can be used to model interconnects in electronic devices) and use asymptotic methods to investigate the motion of long wavelength voids on the conductor boundary. In this case we derive a nonlinear parabolic PDE describing the evolution of the free boundary and, using this simpler model, are able to make some predictions about the evolution of the void over long times.<br/

Morphogenesis in space offers challenges and opportunities for soft matter and biophysics

Abstract The effects of microgravity on soft matter morphogenesis have been documented in countless experiments, but physical understanding is still lacking in many cases. Here we review how gravity affects shape emergence and pattern formation for both inert matter and living systems of different biological complexities. We highlight the importance of building physical models for understanding the experimental results available. Answering these fundamental questions will not only solve basic scientific problems, but will also enable several industrial applications relevant to space exploration