Steady,oblique, detonation waves

Normal and oblique, steady planar detonation waves have been theoretically and computationally examined using the Zeldovich, von Neumann, Döring model. Combustion is between a methane/hydrogen mixture and dry air assuming, first, complete combustion, then an equilibrium solution. Prescribed parameters are the upstream values for the pressure, temperature, and Mach number, the fuel/air equivalence ratio, a hydrogen/methane ratio, and the detonation wave angle. For a given upstream state, the angle varies from its normal wave value in increments of 10 ^o to non-integer wave angles that correspond to the Chapman-Jouguet state for complete combustion and for an equilibrium solution. For each solution, detailed results are provided for the upstream state, the state just downstream of the shock, and the two downstream states. Over 340 solutions in a report (Emanuel and Tuckness 2002) are provided, thereby establishing, for the first time, comprehensive tables that can be used to provide quick estimates, establish trends, and check CFD results. This paper describes the basis for the model, briefly outlines the analytical and numerical method, and discusses several insights.     

Relaxation patterns of long,linear, flexible,monodisperse polymers: BSW spectrum revisited

Theoretical predictions for the dynamic moduli of long, linear, flexible, monodisperse polymers are summarized and compared with experimental observations. Surprisingly, the predicted 1/2 power scaling of the long-time modes of the relaxation spectrum is not found in the experiments. Instead, scaling with a power of about 1/4 extends all the way up to the longest relaxation times near τ/τ _max = 1. This is expressed in the empirical relaxation time spectrum of Baumgaertel-Schausberger-Winter, denoted as “BSW spectrum,” and justifies a closer look at the properties of the BSW spectrum. Working with the BSW spectrum, however, is made difficult by the fact that hypergeometric functions occur naturally in BSW-based rheological material functions. BSW provides no explicit solutions for the dynamic moduli, G ^′(ω), G ^″(ω), or the relaxation modulus G(t). To overcome this problem, close approximations of simple analytical form are shown for these moduli. With these approximations, analysis of linear viscoelastic data allows the direct determination of BSW parameters.     

Symbolic Image, Hyperbolicity, and Structural Stability

The aim of the paper is substantiation of a constructive method for verification of hyperbolicity and structural stability of discrete dynamical systems. The main tool here is a symbolic image which is a directed graph constructed by a finite covering of the projective bundle. Hyperbolicity is tested by calculation of the Morse spectrum (the limit set of Lyapunov exponents of pseudo trajectories) which can be found for a given accuracy by the symbolic image [24]. If the Morse spectrum does not contain 0, then the chain recurrent set is hyperbolic and the system is Ω-stable. Thus, the symbolic image gives an opportunity to verify these properties. A diffeomorphism f is shown to be structurally stable if and only if the Morse spectrum does not contain 0 and for the complementary differential there is no connection CR ⁺→CR ^? on the protective bundle. These conditions are verified by an algorithm based on the symbolic image of the complementary differential.     

An automated high-pressure,high-temperature,low-frequency viscometer

A computerized apparatus for the measurement of low shear rate viscosity on polymer solutions at elevated pressure and temperature is described. The pressure is variable from 1 bar to 400 bar with an accuracy of ±5 bar. The temperature is variable from 0° to 100°C with an accuracy of ±0.3 K. The instrument is operated in a free-relaxation mode in which the decay of oscillation of the torsion pendulum after an initial displacement is recorded and used to compute the viscosity of the sample. The measurements are performed according to the contents of a user-specified control file, and the oscillation data are stored digitally and later analyzed for parameter estimation. The instrument operates in the very-low frequency range (0.03 to 0.25 Hz) and the accuracy of the measured viscosity is ±0.03 x 10^–3 Ns/m2.     

PM2.5 in China: Measurements,sources, visibility and health effects,and mitigation

Concern over the health effects of fine particles in the ambient environment led the U.S. Environmental Protection Agency to develop the first standard for PM_2.5 (particulate matter less than 2.5 μm) in 1997. The Particle Technology Laboratory at the University of Minnesota has helped to establish the PM_2.5 standard by developing many instruments and samplers to perform atmospheric measurements. In this paper, we review various aspects of PM_2.5, including its measurement, source apportionment, visibility and health effects, and mitigation. We focus on PM_2.5 studies in China and where appropriate, compare them with those obtained in the U.S. Based on accurate PM_2.5 sampling, chemical analysis, and source apportionment models, the major PM_2.5 sources in China have been identified to be coal combustion, motor vehicle emissions, and industrial sources. Atmospheric visibility has been found to correlate well with PM_2.5 concentration. Sulfate, ammonium, and nitrate carried by PM_2.5, commonly found in coal burning and vehicle emissions, are the dominant contributors to regional haze in China. Short-term exposure to PM_2.5 is strongly associated with the increased risk of morbidity and mortality from cardiovascular and respiratory diseases in China. The strategy for PM_2.5 mitigation must be based on reducing the pollutants from the two primary sources of coal-fired power plants and vehicle emissions. Although conventional Particulate Emission Control Devices (PECD) such as electrostatic precipitators in Chinese coal-fired power plants are generally effective for large particles, most of them may not have high collection efficiency of PM_2.5. Baghouse filtration is gradually incorporated into the PECD to increase the PM_2.5 collection efficiency. By adopting stringent vehicle emissions standard such as Euro 5 and 6, the emissions from vehicles can be gradually reduced over the years. An integrative approach, from collaboration among academia, government, and industries, can effectively manage and mitigate the PM_2.5 pollution in China.     

Crown incident radiant heat flux measurements in an industrial,regenerative, gas-fired,flat-glass furnace

Crown incident radiant heat flux measurements performed during both firing and non-firing cycles are reported, for the first time, in the combustion space of a regenerative, side-port, 455 metric ton/day, gas-fired, flat-glass furnace. Measurements were acquired through six crown access holes along the furnace axial centerline. Video and visual observations of the glass surface were also made through access ports in the furnace. A three-dimensional numerical model of the turbulent mixing, reaction, and heat transfer processes is also used to predict radiant heat flux to the crown. The measured crown incident radiant heat flux profile during firing cycles rises from 425 kW/m² close to the batch feeder to a peak of 710 kW/m² near the center of the combustion space, followed by a drop to approximately 575 kW/m² near the furnace working end. Numerical model results are in relatively good agreement with measured results. During non-firing reversal cycles, measured flux levels at the crown rise from 320 kW/m² near the batch feeder, to a maximum of 565 kW/m² closest to the spring zone. Increases in crown incident radiant heat flux due to combustion are quantified, with nominal increases of 105 kW/m² in regions closest to the batch feeder and approximately 155 kW/m² in the center of the combustion space. Lower increases from combustion (85 and 12 kW/m²) are exhibited in locations closest to the furnace working end. During the 20–25 s non-firing reversal period, the incident heat flux to the crown typically decreased between 20 and 50 kW/m² at each measured location. Variation of heat flux to the crown during 15-min firing cycles is typically 3–6% of the total incident heat flux, with a maximum typically occurring one-third of the way into the cycle (5–6 min) and declining during the remaining two-thirds of the period.     

Damping Properties of Sequoia, Birch, Pine, and Aspen under Shock Loading

Results of an experimental study of strength—deformation properties of sequoia, aspen, pine, and birch wood for various loading rates, temperatures, humidities, and orientation angles of wood fibers with respect to the loading direction are described. Stress—strain diagrams and an analytical dependence of strength on humidity, temperature, and loading rate are presented.     

Stability,buckling, and postbuckling of elastic structures

Summary This paper is the continuation of a research on the stability and postbuckling behaviour of elastic structures; in the first Part [n. 4, Vol. X, 1975 of this Journal], the potential energy of elastic tridimensional bodies has been defined in a suitable configuration space, and its properties of regularity have been studied. According to the program given in the previous Paper, we examine now the potential energy functional for some monodimensional and bidimensional structural models. Then the Part II of this research is developed so that the Liapunov stability of the equilibrium configuration of continuous elastic structures is analyzed by means of suitable energy criteria.

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Sommario Questo lavoro costituisce il proseguimento di uno studio sulle stabilità e comportamento postcritico delle strutture elastiche; nel precedente lavoro, pubblicato sul n. 4, vol. X, 1975 di questa rivista, è stato esaminato il problema della definizione dell'energia potenziale di un corpo elastico tridimensionale in un opportuno spazio funzionale delle configurazioni, e sono state esaminate le proprietà di regolarità di questo funzionale. Proseguendo il programma formulato nel lavoro precedente, viene ora innanzitutto completato lo studio del funzionale energia potenziale con riferimento a modelli strutturali mono e bidimensionali. Viene poi sviluppata la Parte II e cioè l'analisi della stabilità secondo Liapunov di una configurazione di equilibrio attraverso opportune formulazioni dei criteri dell'energia.

     

Asymptotic analysis of an end-loaded, transversely isotropic, elastic, semi-infinite strip weak in shear

We use linear elasticity to study a transversely isotropic (or specially orthotropic), semiinfinite slab in plane strain, free of traction on its faces and at infinity and subject to edge loads or displacements that produce stresses and displacements that decay in the axial direction. The governing equations (which are identical to those for a strip in plane stress, free of traction on its long sides and at infinity, and subject to tractions or displacements on its short side) are reduced, in the standard way. to a fourth-order partial differential equation with boundary conditions for a dimensionless Airy stress function ƒ. We study the asymptotic solutions to this equation for four sets of end conditions—traction, mixed (two), displacement—as g3, the ratio of the shear modulus to the geometric mean of the axial and transverse extensional moduli, approaches zero. In all cases, the solutions for ƒ consist of a “wide” boundary layer that decays slowly in the axial direction (over a distance that is long compared to the width of the strip) plus a “narrow” boundary layer that decays rapidly in the axial direction (over a distance that is short compared to the width of the strip). Moreover, we find that the narrow boundary layer has a “sinuous” part that varies rapidly in the transverse direction, but which, to lowest order, does not enter the boundary conditions nor affect the transverse normal stress or the displacements. Because the exact biorthogonality condition for the cigenfunctions associated with ƒ can be replaced by simpler orthogonality conditions in the limit as →b 0, we are able to obtain, to lowest order, explicit formulae for the coeflicients in the eigenfunction expansions of ƒ for the four different end conditions.     

Use specification of multiscale materials for life spanned over macro-, micro-, nano-, and pico-scale

Multiscale material intends to enhance the strength and life of mechanical systems by matching the transmitted spatiotemporal energy distribution to the constituents at the different scale, say—macro, micro, nano, and pico,—, depending on the needs. Lower scale entities are, particularly, critical to small size systems. Large structures are less sensitive to microscopic effects. Scale shifting laws will be developed for relating test data from nano-, micro-, and macro-specimens. The benefit of reinforcement at the lower scale constituents needs to be justified at the macroscopic scale. Filling the void and space in regions of high energy density is considered.Material inhomogeneity interacts with specimen size. Their combined effect is non-equilibrium. Energy exchange between the environment and specimen becomes increasingly more significant as the specimen size is reduced. Perturbation of the operational conditions can further aggravate the situation. Scale transitional functions and/or f_j/j+1 are introduced to quantify these characteristics. They are represented, respectively, by , and (f_mi/ma,f_na/mi,f_pi/na). The abbreviations pi, na, mi, and ma refer to pico, nano, micro and macro.Local damage is assumed to initiate at a small scale, grows to a larger scale, and terminate at an even larger scale. The mechanism of energy absorption and dissipation will be introduced to develop a consistent book keeping system. Compaction of mass density for constituents of size 10⁻¹², 10⁻⁹, 10⁻⁶, 10⁻³ m, will be considered. Energy dissipation at all scales must be accounted for. Dissipations at the smaller scale must not only be included but they must abide by the same physical and mathematical interpretation, in order to avoid inconsistencies when making connections with those at the larger scale where dissipations are eminent.Three fundamental Problems I, II, and III are stated. They correspond to the commonly used service conditions. Reference is made to a Representative Tip (RT), the location where energy absorption and dissipation takes place. The RT can be a crack tip or a particle. At the larger size scales, RT can refer to a region. Scale shifting of results from the very small to the very large is needed to identify the benefit of using multiscale materials.     

Chain recurrence,semiflows, and gradients

This paper is a study of chain recurrence and attractors for maps and semiflows on arbitrary metric spaces. The main results are as follows. (i) C. Conley's characterization of chain recurrence in terms of attractors holds for maps and semiflows on any metric space. (ii) An alternative definition of chain recurrence for semiflows is given and is shown to be equivalent to the usual definition. The alternative definition uses chains formed of orbit segments whose lengths are at least 1, while in the usual definition these lengths are required to be arbitrarily long. (iii) The chain recurrent set of a continuous semiflow is the same as the chain recurrent set of its time-one map. (iv) Conditions on a real-valued function are given that ensure that the semiflow generated by its gradient has only equilibria in its chain recurrent set. An example is given (onR ³) showing that a gradient flow may have nonequilibrium chain recurrent points if these conditions are violated.     

Singular Perturbations, Transversality, and Sil'nikov Saddle-Focus Homoclinic Orbits

We consider the singularly perturbed system $\dot x$ =εf(x,y,ε,λ), $\dot y$ =g(x,y,ε,λ). We assume that for small (ε,λ), (0,0) is a hyperbolic equilibrium on the normally hyperbolic centre manifold y=0 and that y ₀(t) is a homoclinic solution of $\dot y$ =g(0,y,0,0). Under an additional condition, we show that there is a curve in the (ε,λ) parameter space on which the perturbed system has a homoclinic orbit also. We investigate the transversality properties of this orbit and use our results to give examples of 4 dimensional systems with Sil'nikov saddle-focus homoclinic orbits.     

Tensor Products, Positive Linear Operators, and Delay-Differential Equations

We develop the theory of compound functional differential equations, which are tensor and exterior products of linear functional differential equations. Of particular interest is the equation $$\begin{aligned} \dot{x}(t)=-\alpha (t)x(t)-\beta (t)x(t-1) \end{aligned}$$ with a single delay, where the delay coefficient is of one sign, say $\delta \beta (t)\ge 0$ with $\delta \in \{-1,1\}$ . Positivity properties are studied, with the result that if $(-1)^k=\delta $ then the $k$ -fold exterior product of the above system generates a linear process which is positive with respect to a certain cone in the phase space. Additionally, if the coefficients $\alpha (t)$ and $\beta (t)$ are periodic of the same period, and $\beta (t)$ satisfies a uniform sign condition, then there is an infinite set of Floquet multipliers which are complete with respect to an associated lap number. Finally, the concept of $u_0$ -positivity of the exterior product is investigated when $\beta (t)$ satisfies a uniform sign condition.     

Onset of Cavitation in Compressible,Isotropic, Hyperelastic Solids

In this work, we derive a closed-form criterion for the onset of cavitation in compressible, isotropic, hyperelastic solids subjected to non-symmetric loading conditions. The criterion is based on the solution of a boundary value problem where a hyperelastic solid, which is infinite in extent and contains a single vacuous inhomogeneity, is subjected to uniform displacement boundary conditions. By making use of the “linear-comparison” variational procedure of Lopez-Pamies and Ponte Castañeda (J. Mech. Phys. Solids 54:807–830, 2006), we solve this problem approximately and generate variational estimates for the critical stretches applied on the boundary at which the cavity suddenly starts growing. The accuracy of the proposed analytical result is assessed by comparisons with exact solutions available from the literature for radially symmetric cavitation, as well as with finite element simulations. In addition, applications are presented for a variety of materials of practical and theoretical interest, including the harmonic, Blatz-Ko, and compressible Neo-Hookean materials.     

Physiology,mechanics, and rheology

Biological systems possess rather specialized mechanical properties, acquired as part and parcel of the evolutionary development of the system as a whole. Their optimization permits the system to function physiologically in the context of a biologically essential, but mechanically often widely varying environment with adequate efficiency. The system's environment is its source of food and shelter; it represents the space in which it forages or preys on other creatures and in which it has to defend itself against still others. Thus, the system has to develop an adequately pliant, rheologically matched, energy-use efficient, mechanical interface between it and its surroundings. This must be an interface that both effectively excludes, but also effectively admits, the external. Internally, as well, it has to adapt the mechanical properties of cell and connective tissue to physiological function and the efficient performance of useful work. This will be illustrated by way of examples. Blood rheology is briefly discussed and put into the context of clinical hemorheology and epithelial protection; and function, by way of a mucus coating or a mucociliary clearance system, is reviewed in some detail. The importance of all aspects of rheological matching is demonstrated.Delivered as a Keynote Lecture at the Golden Jubilee Conference of the British Society of Rheology and Third European Rheology Conference, Edinburgh, 3–7 September, 1990.     

Elastic silicon-film-based nanoshells: Formation, properties, and applications

Controllable formation and properties of solid single-crystal micro-and nanoshells of various shapes (tubes and spirals, vertically positioned rings and cylinders, and bent and trough-shaped cantilevers) are briefly reviewed, and new results are given. The shells and complicated structures of prescribed size and shape are formed with the use of elastic energy of initial strained SiGe/Si films of nanometer thickness and methods of highly selective and directed detachment of the films from the silicon substrates. It is experimentally demonstrated that the diameters of the fabricated SiGe/Si nanotubes are several times smaller than the values predicted by the continuum elasticity theory. The properties of the shells made of semiconductor and hybrid (metal-semiconductor and metal-dielectric-semiconductor) films and their applications in micro-and nanoscale electrical engineering are discussed. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 114–128, November–December, 2006.     

Filter Feeding, Chaotic Filtration, and a Blinking Stokeslet

The filtering mechanisms in bivalve molluscs, such as the mussel Mytilus edulis, and in sessile organisms, such as Vorticella or Stentor, involve complex fluid mechanical phenomena. In the former example, three different sets of cilia serving different functions are involved in the process whereas in the sessile organisms the flexibility and contractile nature of the stalk may play an important role in increasing the filtering efficiency of the organisms. In both cases, beating microscopic cilia are the “engines” driving the fluid motion, so the fluid mechanics will be dominated entirely by viscous forces. A fluid mechanical model is developed for the filtering mechanism in mussels that enables estimates to be made of the pressure drop through the gill filaments due to (i) latero-frontal filtering cilia, (ii) the lateral (pumping) cilia, and (iii) through the non-ciliated zone of the ventral end of the filament. The velocity profile across the filaments indicates that a backflow can occur in the centre of the channel leading to the formation of two “standing” eddies which may drive particles towards the mucus-laden short cilia, the third set of cilia. Filter feeding in the sessile organisms is modelled by a point force above a rigid boundary. The point force periodically changes its point of application according to a given protocol (a blinking stokeslet). The resulting fluid field is illustrated via Poincaré sections and particle dispersion—showing the potential for a much improved filtering efficiency. Returning to filter feeding in bivalve molluscs, this concept is extended to a pair of blinking stokeslets above a rigid boundary to give insight into possible mechanisms for movement of food particles onto the short mucus-bearing cilia. The appendix contains a Latin and English version of an “Ode of Achievement” in celebration of Sir James Lighthill's contributions to mathematics and fluid mechanics. Received 20 December 1996 and accepted 22 April 1997     

Propagation of very long water waves, with vorticity, over variable depth, with applications to tsunamis

We present a theory of very long waves propagating on the surface of water. The waves evolve slowly, both on the scale ε (weak nonlinearity), and on the scale, σ, of the depth variation. In our model, dispersion does not affect the evolution of the wave even over the large distances that tsunamis may travel. We allow a distribution of vorticity, in addition to variable depth. Our solution is not valid for depth=O(ε^4/5); the equations here are expressed in terms of the single parameter ε^2/5σ and matched to the solution in deep water. For a slow depth variation of the background state (consistent with our model), we prove that a constant-vorticity solution exists, from deep water to shoreline, and that regions of isolated vorticity can also exist, for appropriate bottom profiles. We describe how the wave properties are modified by the presence of vorticity. Some graphical examples of our various solutions are presented.     

Energy release,friction, and supplemental relations at phase interfaces

An Eulerian expression for the dissipation due to a moving surface of discontinuity in mass density, velocity, stress, energy, and heat flux is obtained. This leads to Eulerian measures forenergy release andfriction — measures that are work conjugate tomass flux andvelocity slip, respectively. Constitutive equations involving these quantities are proposed as a means to determine transition kinetics in materials that change phase.     

Stability, Instability, and Error of the Force-based Quasicontinuum Approximation

Due to their algorithmic simplicity and high accuracy, force-based model coupling techniques are popular tools in computational physics. For example, the force-based quasicontinuum (QCF) approximation is the only known pointwise consistent quasicontinuum approximation for coupling a general atomistic model with a finite element continuum model. In this paper, we present a detailed stability and error analysis of this method. Our optimal order error estimates provide a theoretical justification for the high accuracy of the QCF approximation: they clearly demonstrate that the computational efficiency of continuum modeling can be utilized without a significant loss of accuracy if defects are captured in the atomistic region. The main challenge we need to overcome is the fact that the linearized QCF operator is typically not positive definite. Moreover, we prove that no uniform inf-sup stability condition holds for discrete versions of the W ^1,p -W ^1,q “duality pairing” with 1/p + 1/q = 1, if 1 ≤ p < ∞. However, we were able to establish an inf-sup stability condition for a discrete version of the W ^1,∞-W ^1,1 “duality pairing” which leads to optimal order error estimates in a discrete W ^1,∞-norm.