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

Comment on ``Solidification of a Supercooled Liquid in a Narrow Channel''

Comment on PRL v. 86, p. 5084 (2001) [cond-mat/0101016]. We point out that the authors' simulations are consistent with the known theory of steady-state solutions in this system

Observation of the Inverse Cotton-Mouton Effect

We report the observation of the Inverse Cotton-Mouton Effect (ICME) i.e. a magnetization induced in a medium by non resonant linearly polarized light propagating in the presence of a transverse magnetic field. We present a detailed study of the ICME in a TGG crystal showing the dependence of the measured effect on the light intensity, the optical polarization, and on the external magnetic field. We derive a relation between the Cotton-Mouton and Inverse Cotton-Mouton effects that is roughly in agreement with existing experimental data. Our results open the way to applications of the ICME in optical devices

One-Dimensional Ablation with Pyrolysis Gas Flow Using a Full Newton's Method and Finite Control Volume Procedure

The development and verification of a one-dimensional material thermal response code with ablation is presented. The implicit time integrator, control volume finite element spatial discretization, and Newton's method for nonlinear iteration on the entire system of residual equations have been implemented and verified for the thermochemical ablation of internally decomposing materials. This study is a continuation of the work presented in "One-Dimensional Ablation with Pyrolysis Gas Flow Using a Full Newton's Method and Finite Control Volume Procedure" (AIAA-2006-2910), which described the derivation, implementation, and verification of the constant density solid energy equation terms and boundary conditions. The present study extends the model to decomposing materials including decomposition kinetics, pyrolysis gas flow through the porous char layer, and a mixture (solid and gas) energy equation. Verification results are presented for the thermochemical ablation of a carbon-phenolic ablator which involves the solution of the entire system of governing equations

Traitement d'effluents de tannerie-mégisserie par microfiltration tangentielle

Dans ce travail un procédé de traitement des effluents issus de l'unité de préparation des peaux des animaux au tannage (travail en rivière) en tannerie-mégisserie a été étudié en utilisant la technique de microfiltration tangentielle sur membrane minérale en céramique. Les performances de ce procédé en terme de flux de filtration et de rendement épuratoire dépendent aussi bien des paramètres hydrodynamiques de filtration que de la qualité des effluents (collectés en été et en printemps) issus des différents bains de traitement et de rinçage des peaux dans l'atelier de rivière. Le flux de filtration varie entre 15 l/h.m2 pour l'effluent de printemps et 90 l/h.m2 pour l'effluent d'été. Les paramètres hydrodynamiques optimaux ont également été déterminés: la vitesse de circulation U=3 m/s, la pression transmembranaire Ptm=2 bar et la température T=43°C. L'étude de la microfiltration à concentration variable conduit à des facteurs de concentration volumique (FCV) de 6,5 pour l'effluent de l'été et de 2,4 pour l'effluent de printemps.The leather industry is responsible for the transformation of raw animal skin to a final form as shoes, bags, dresses, etc. This industry was known for centuries as a craft activity, and today with industrial development, environmental regulations and new emerging technologies, it has become necessary to include elaborate processes for its wastewater treatment. These industries consume a great amount of water. In Tunisia, more than 15000 tons of skin are treated per year, and about 600000 m3 per year of effluents are discharged. The waste water contains chemicals, fats, hair and protein, varying in composition depending on the season. Figure 1 represents the preparation of raw skin for the tanning operation and the amount of waste water produced. The amount of water used for the preparation of raw skin is about 70% of the total quantity of water used. This waste water has a significant polluting load (chemicals and organic matter), with 5000 - 7500 mg/l of COD and 100 to 150 mg/l of sulfur. Tunisian legislation and regulations concerning the standards for wastewater disposal are 1000 mg/l for COD, 3 mg/l for sulfur and a pH between 6.5-9. Different techniques for wastewater treatment such as: physico-chemical treatment, treatment by electrochemical oxidation and membrane technology were proposed. Wastewater treatment by microfiltration and ultrafiltration with mineral membranes is advantageous because no chemicals are used and it can be combined easily with other physico-chemical or biological pre-treatments. In this study, we have treated two types of effluents from the leather pre-treatment industry collected in the summer (effluent 1), and the spring (effluent 2) seasons. The physico-chemical characteristics of the two types effluents are given in Table 2. The filtration experiments were made on a test bench (Figure 2) equipped with a feed reservoir, a volumetric pump, a filtration module, flow meter, pressure transducers, a heat exchanger and control valves. Ceramic membranes of tubular geometry (7 channels), 0.08 m2 membrane surface area and of 0.1 µm (mean diameter) pores were used. During the microfiltration experiments, the following physico-chemical parameters were analysed in the permeate and retentate: turbidity, specific conductivity, pH, viscosity, chemical oxygen demand (COD), sulfur (volumetric method), fats (Standard JIS 0102.24.2), protein (using Kjeldahl nitrogen), and organic nitrogen. Hydrodynamic parameters such as temperature (25 < T < 50 °C), transmembrane pressure (1 < Ptm < 2.2 bar) and feed velocity (1 < U < 3 m/s) were fixed for experimentation. The COD concentration in the effluent was adjusted and kept constant at 5000 mg/l. The raw effluent was pre-filtered on a screen filter (150 µm pore size). For experiments with variable concentration, we regularly removed the filtrate and the concentration factor was represented by FCV=Vi / Vr, where Vi was the initial volume and Vr was the volume of the retentate. The performance of the microfiltration (J) was expressed in l/h×m2. The retention rate (TR) was defined by: TR=1 - (Cpermeate) / (Cfeed). The total hydraulic resistance (RT) was defined by Darcy's law: Jf=Ptm / µ RT. After each experiment, the membrane was regenerated following a standard protocol and it was verified by measuring water flux. Figure 3a represents the variation of the filtration flux with time for 4 different temperatures: 25 °C, 43 °C, 45 °C and 50 °C with effluent 1. The flux increased from 90 to 118 l/h×m2 when the temperature increased from 25 °C to 43 °C. After 90 min at 50 °C, the filtration flux was 123 l/h×m2. Table 3 shows that the viscosity of the effluent decreased with temperature, while the turbidity of the filtrate increased from 0.63 NTU at T=25 °C to 1.6 NTU for T=50 °C. The retention rate of COD was always superior to 50 %. On the basis of these results, we chose the optimum temperature of 43 °C for other experiments. Figure 4 summarises the variation of flux with transmembrane pressure at flow velocities of 1 m/s, 2 m/s and 3 m/s. The stabilized fluxes were practically the same for the flow velocities of 1 and 2 m/s (of the order of 80 l/h×m2), but were higher at 3 m/s (110 - 115 l/h×m2 at 2 bar). The physico-chemical characteristics of the raw effluent and the permeate obtained after 90 minutes of filtration are summarised in Table 4. Figure 7a shows the variation of filtration flux for 2 types of effluents. The filtration flux for the same conditions of experimentation and at stabilized conditions (at 90 min) was 118 l/h×m2 for effluent 1 and 20 l/h×m2 for effluent 2. The lower filtration flux for effluent 2 can be explained by high deposits of rejected matter on the membrane and in the pores. Table 5 gives a comparison of the characteristics of effluents 1 and 2 before and after microfiltration. At variable feed concentrations, FCV=6.5 for effluent 1 and FCV=2.4 for the effluent 2 and the stabilized flux was about 90 l/h×m2 for the effluent 1 and 15 l/h×m2 for the effluent 2. The time needed for treatment of effluent 1 was about 6 hours, while more that 16 hours was necessary for effluent 2. Table 6 provides physico-chemical characteristics for the two types of effluents. The contents of fat, protein, nitrogen and sulfur in the effluent were important factors for variation. These results indicate that microfiltration is very sensitive to the quantity of polluting matter present in the effluents, particularly sulfur and fat. Increased polluting matter in effluent 2 could be responsible for the membrane polarization and blocking of pores. The resistance model was used to verify this hypothesis. The irreversible resistance values for effluent 2 were greater, thus confirming the hypothesis that the increased adsorption on the membrane surface and passage of pores by the presence of sulfur and organic polluting matter. These experimental results confirm that the best performance can be obtained at the hydrodynamic conditions of: a temperature of 43 °C; a transmembrane pressure of 2 bar; and a flow velocity of 3 m/s. Seasonal variation changed the quality of effluents, which considerably affects the performances of the microfiltration. Effluent 2, which was obtained from the treatment of sheep skin during the spring season, led to more membrane pore blocking than effluent 1 for the same initial concentration in COD. The interactions of fats and sulfur with the membrane layer appear to play an important role in the formation of a cake layer

Conical defects in growing sheets

A growing or shrinking disc will adopt a conical shape, its intrinsic geometry characterized by a surplus angle $se$ at the apex. If growth is slow, the cone will find its equilibrium. Whereas this is trivial if $se <= 0$, the disc can fold into one of a discrete infinite number of states if $se$ is positive. We construct these states in the regime where bending dominates, determine their energies and how stress is distributed in them. For each state a critical value of $se$ is identified beyond which the cone touches itself. Before this occurs, all states are stable; the ground state has two-fold symmetry.Comment: 4 pages, 4 figures, LaTeX, RevTeX style. New version corresponds to the one published in PR

Roughness of moving elastic lines - crack and wetting fronts

We investigate propagating fronts in disordered media that belong to the universality class of wetting contact lines and planar tensile crack fronts. We derive from first principles their nonlinear equations of motion, using the generalized Griffith criterion for crack fronts and three standard mobility laws for contact lines. Then we study their roughness using the self-consistent expansion. When neglecting the irreversibility of fracture and wetting processes, we find a possible dynamic rough phase with a roughness exponent of $\zeta=1/2$ and a dynamic exponent of z=2. When including the irreversibility, we conclude that the front propagation can become history dependent, and thus we consider the value $\zeta=1/2$ as a lower bound for the roughness exponent. Interestingly, for propagating contact line in wetting, where irreversibility is weaker than in fracture, the experimental results are close to 0.5, while for fracture the reported values of 0.55--0.65 are higher.Comment: 15 pages, 6 figure

Elastic theory of unconstrained non-Euclidean plates

Non-Euclidean plates are a subset of the class of elastic bodies having no stress-free configuration. Such bodies exhibit residual stress when relaxed from all external constraints, and may assume complicated equilibrium shapes even in the absence of external forces. In this work we present a mathematical framework for such bodies in terms of a covariant theory of linear elasticity, valid for large displacements. We propose the concept of non-Euclidean plates to approximate many naturally formed thin elastic structures. We derive a thin plate theory, which is a generalization of existing linear plate theories, valid for large displacements but small strains, and arbitrary intrinsic geometry. We study a particular example of a hemispherical plate. We show the occurrence of a spontaneous buckling transition from a stretching dominated configuration to bending dominated configurations, under variation of the plate thickness

Streamer Propagation as a Pattern Formation Problem: Planar Fronts

Streamers often constitute the first stage of dielectric breakdown in strong electric fields: a nonlinear ionization wave transforms a non-ionized medium into a weakly ionized nonequilibrium plasma. New understanding of this old phenomenon can be gained through modern concepts of (interfacial) pattern formation. As a first step towards an effective interface description, we determine the front width, solve the selection problem for planar fronts and calculate their properties. Our results are in good agreement with many features of recent three-dimensional numerical simulations.Comment: 4 pages, revtex, 3 ps file