Uniqueness and the maximum principle for quasilinear elliptic equations with quadratic growth conditions

In this paper, we show that the maximum principle holds for quasilinear elliptic equations with quadratic growth under general structure conditions.Two typical particular cases of our results are the following. On one hand, we prove that the equation (1) {ie77-01} where {ie77-02} and {ie77-03} satisfies the maximum principle for solutions in H ¹()L(), i.e., that two solutions u ₁, u ₂H¹() L() of (1) such that u ₁u₂ on , satisfy u ₁u₂ in . This implies in particular the uniqueness of the solution of (1) in H ₀ ¹ ()L().On the other hand, we prove that the equation (2) {ie77-04} where fH^–1() and g(u)>0, g(0)=0, satisfies the maximum principle for solutions uH¹() such that g(u)¦Du|{²L¹(). Again this implies the uniqueness of the solution of (2) in the class uH ₀ ¹ () with g(u)¦Du|{²L¹().In both cases, the method of proof consists in making a certain change of function u=(v) in equation (1) or (2), and in proving that the transformed equation, which is of the form (3) {ie77-05}satisfies a certain structure condition, which using ((v₁ -v ₂)⁺)ⁿ for some n>0 as a test function, allows us to prove the maximum principle.     

Non-linear normal modes,invariance, and modal dynamics approximations of non-linear systems

Non-linear systems are here tackled in a manner directly inherited from linear ones, that is, by using proper normal modes of motion. These are defined in terms of invariant manifolds in the system's phase space, on which the uncoupled system dynamics can be studied. Two different methodologies which were previously developed to derive the non-linear normal modes of continuous systems — one based on a purely continuous approach, and one based on a discretized approach to which the theory developed for discrete systems can be applied-are simultaneously applied to the same study case-an Euler-Bernoulli beam constrained by a non-linear spring-and compared as regards accuracy and reliability. Numerical simulations of pure non-linear modal motions are performed using these approaches, and compared to simulations of equations obtained by a classical projection onto the linear modes. The invariance properties of the non-linear normal modes are demonstrated, and it is also found that, for a pure non-linear modal motion, the invariant manifold approach achieves the same accuracy as that obtained using several linear normal modes, but with significantly reduced computational cost. This is mainly due to the possibility of obtaining high-order accuracy in the dynamics by solving only one non-linear ordinary differential equation.     

Two-phase flow distributors for fuel cell flow channels

All existing proton exchange membrane （PEM） fuel cell gas flow fields have been designed on the basis of single-phase gas flow distribution. The presence of liquid water in the flow causes non-uniform gas distribution, leading to poor cell performance. This paper demonstrates that a gas flow restrictor/distributor, as is commonly used in two-phase flow to stabilize multiphase transport lines and multiphase reactors, can improve the gas flow distribution by significantly reducing gas real-distribution caused by either non-uniform water formation in parallel flow channels or flow instability associated with negative-slope pressure drop characteristic of two-phase horizontal flow systems.     

Experimental characterization of the 3D dynamics of a laminar shallow vortex dipole

Experimental results on the dynamics of a vortex dipole evolving in a shallow fluid layer are presented. In particular, the generation of a spanwise vortex at the front of the dipole is observed in agreement with previous experiments at larger Reynolds numbers. The results show that this secondary vortex is of comparable strength to the dipole. The present physical analysis suggests that the origin of this structure involves the stretching induced by the dipole of the boundary-layer vorticity generated by the dipole’s advection over the no-slip bottom.     

Rheological characterization of a solder paste for surface mount applications

Solder pastes used in surface mount soldering techniques (SMT) are very complex suspensions containing high volumes of metallic powder in a carrier fluid. The rheological complexity results largely from the carrier fluid itself, which is a suspension of colloidal particles. In this work, we have characterized the rheological properties of a typical carrier fluid and its solder paste containing 64 vol.% metallic powder. A six-blade vane geometry was used to avoid wall slip and sample fracture. All measurements were carried out following pre-shearing and rest time in order to obtain reproducible results. Steady shear experiments showed that the solder paste was highly shear-thinning and thixotropic. In oscillatory shear, the linear viscoelastic domain was found to be very narrow for both the suspending fluid and the paste. Frequency sweep tests in the linear domain revealed a gel-like structure with a nearly constant G′ for the suspending fluid and a slightly increasing G′ for the solder paste. From creep experiments, a yield stress of about 40 Pa was determined for the suspending fluid at temperatures between 25 and 40°C, and of 100 Pa at 4°C. A much larger yield stress, 480 Pa, was determined for the solder paste at 25°C.     

Self-assembled nanoribbons and nanotubes in water: energetic vs entropic networks

We present a comparative investigation of two opposite classes of self-assembled fibrillar networks. Ribbons and tubes having cross-sectional dimensions in the nanoscale can be formed in aqueous solutions of steroids derived, respectively, from deoxycholic (DC) and lithocholic (LC) acids. Rheological features distinguish energetic networks of DC ribbons rigidly fixed in cylindrical bundles and entropic transient networks of LC tubes weakly interacting in shear-sensitive suspensions. The two classes are characterized by their frequency sweep profiles, viscoelastic linear domains, scaling laws of the elastic shear modulus vs concentration, kinetics of formation of the networks, and their optical birefringence aspects. A theoretical context for networks of rigid fibers is used to account for the scaling exponents α in the G’ (and σ*) ∝C ^α laws (α=2.0 and 1.0, respectively, for DC and LC). The evolution observed in DC gels from ribbons to cylindrical fibers with monodisperse sections made up with four ribbons is an indication of an equilibrated balance between face-to-face attractions and untwisting elastic processes of the constitutive ribbons.Paper presented at the Annual Meeting of the European Society of Rheology, Grenoble, April 2005     

Effects of timber skidding on chemical characteristics of herbaceous cover, forest floor and topsoil on skidroad in an oak (Quercus petrea L.) forest

This paper examines the effects of timber harvesting by skidding on some soil properties (sand, silt, clay, pH, organic carbon, bulk density and compaction), herbaceous cover (unit mass) and forest floor (unit weight) properties. Also N (%), P, K, Na, Ca, Mg, Fe, Zn, Cu and Mn (ppm) were determined in all herbaceous cover, forest floor and two soil depth (0–5 cm and 5–10 cm) on skidroad of an oak (Quercus petrea L.) stand in Istanbul Belgrad Forest – Turkey. In this study, obtained results are; the forest floor and the herbaceous cover amount on the skidroad have been found considerably lower than undisturbed area. There were some crucial changes in the characteristics of the soil which has been investigated down to 10 cm depth. Soil bulk density was found quite high in the samples taken from the skidroad subject to compaction compared to the ones on the undisturbed area. Nevertheless, no important difference had been detected between the skidroad and the undisturbed area at both soil depths in terms of organic carbon contents. Moreover, the soil acidity (pH) values showed noteworthy differences in the analysis of soil samples taken from both soil depths on the skidroad and on the undisturbed area. Fe and Cu contents of herbaceous samples on skidroad were significantly higher than undisturbed area. Forest floor on skidroad had significantly higher K content, and significantly lower Zn, Mn and N content compared to undisturbed area. P, Fe, Zn and Mn contents were found significantly lower in 0–5 cm soil depth on skidroad than undisturbed area. In 5–10 cm soil depth, concentrations of N, P, Fe, Zn and Mn were significantly lower, while Mg and Cu contents were significantly higher than undisturbed area. Results indicate that long-term harvest using skidding techniques on these sites had adversely affected soil cation concentrations, physical soil conditions and mass of herbaceous cover and forest floor.     

Effect of flow history on the structure of a non-polar polymer/clay nanocomposite model system

The effect of flow history on the linear and non-linear viscoelastic properties of non-polar polymer nanocomposites (PNCs) has been investigated by means of a suitable model system based on a Newtonian matrix. The structural recovery of this model suspension after cessation of different pre-shear rates was monitored by measuring its linear viscoelastic properties while its structural evolution under shear flow was followed by using stepwise changes in shear rate including flow reversal measurements. To assess the kinetics of the structural evolution at rest and under flow, empirical relations of stretched exponential form were used. It is shown that for different pre-shear rates, different equilibrium structures were reached at rest but with a similar kinetics of recovery. As a result, the low frequency behaviour was typical of solid-like or weak gel material, strongly dependent on the flow history. After any given shear rate under steady state, only one reversible equilibrium structure was reached after a kinetics that was dependent on the pre-shear history. Finally, typical flow reversal responses as observed for PNCs are reported and interpreted in light of the microstructure evolution under flow. This paper was presented at the Annual Meeting of the European Society of Rheology, Hersonisos, Greece April 2006.     

Near-wall velocity measurements by particle-shadow tracking

We report a new method to measure the velocity of a fluid in the vicinity of a wall. The method, that we call particle-shadow tracking (PST), simply consists in seeding the fluid with a small number of fine tracer particles of density close to that of the fluid. The position of each particle and of its shadow on the wall are then tracked simultaneously, allowing one to accurately determine the distance separating tracers from the wall and therefore to extract the velocity field. We present an application of the method to the determination of the velocity profile inside a laminar density current flowing along an inclined plane.     

Virtual testing applied to transverse multiple cracking of tows in woven ceramic composites

The present paper proposes a method of virtual testing with a view to investigating the local response of tows within textile ceramic matrix composite (CMC) under various loading conditions. The method was developed on 2D woven SiC/SiC composites. It capitalizes on knowledge on mechanical damage phenomenology and data established in previous works. It is applied to isolated transverse tows subjected to uniaxial loading by parallel longitudinal tows. The transverse tows contain heterogeneities like matrix voids, fibres and interphases. Mesh for finite element analysis is constructed from micrographs of composite cross section. Cracks were introduced into the mesh for simulation of multiple cracking. Transverse tow tensile behavior and data on distributions of flaw populations were derived from finite element computations of stress-state. Results were compared to experimental observations.