Adaptive methods of lines for one-dimensional reaction-diffusion equations

Adaptive and non-adaptive finite difference methods are used to study one-dimensional reaction-diffusion equations whose solutions are characterized by the presence of steep, fast-moving flame fronts. Three non-adaptive techniques based on the methods of lines are described. The first technique uses a finite volume method and yields a system of non-linear, first-order, ordinary differential equations in time. The second technique uses time linearization, discretizes the time derivatives and yields a linear, second-order, ordinary differential equation in space, which is solved by means of three-point, fourth-order accurate, compact differences. The third technique takes advantage of the disparity in the time scales of the reaction and diffusion processes, splits the reaction--diffusion operator into a sequence of reaction and diffusion operators and solves the diffusion operator by means of either a finite volume method or a three-point, fourth-order accurate compact difference expression. The non-adaptive methods of lines presented in this paper may use equaliy or non-equally spaced fixed grids and require a large number of grid points to solve accurately one-dimensional problems characterized by the presence of steep, fast-moving fronts. Three adaptive methods for the solution of reaction-diffusion equations are considered. The first adaptive technique is static and uses a subequidistribution principle to determine the grid points, avoid mesh tangling and node overtaking and obtain smooth grids. The second adaptive technique is dynamic, uses an equidistribution principle with spatial and temporal smoothing and yields a system of first-order, non-linear, ordinary differential equations for the grid point motion. The third adaptive technique is hybrid, combines some features of static and dynamic methods, and uses a predictor-corrector strategy to predict the grid and solve for the dependent variables, respectively. The three adaptive techniques presented in this paper use physical co-ordinates and may employ finite volume or three-point, compact methods. The adaptive and non-adaptive finite difference methods presented in the paper are used to study a decomposition chemical reaction characterized by a scalar, one-dimensional reaction-diffusion equation, the propagation of a one-dimensional, confined, laminar flame in Cartesian co-ordinates and the Dwyer-Sanders model of one-dimensional flame propagation. It is shown that the adaptive moving method presented in this paper requires a smaller number of grid points than adaptive static, adaptive hybrid and non-adaptive methods. The adaptive hybrid method requires a smaller time step than adaptive static techniques, due to the lag between the grid prediction and the solution of the dependent variables. Non-adaptive methods of lines may yield temperature oscillations in front of and behind the flame front if Crank-Nicolson techniques are used to evaluate the time derivatives. Fourth-order accurate methods of lines in space yield larger temperature oscillations than second-order accurate methods of lines, and the magnitude of these oscillations decreases as the time step is decreased. It is also shown that three-point, fourth-order accurate discretizations of the spatial derivatives require the same number of grid points as second-order accurate, finite volume methods, in order to resolve accurately the structure of steep, fast-moving flame fronts.     

Oscillating annular liquid membranes

Summary The response of annular liquid membranes to sinusoidal mass flow rate fluctuations at the nozzle exit is analyzed as a function of the amplitude and frequency of the axial velocity fluctuations at the nozzle exit and thermodynamic compression of the gas enclosed by the membrane. It is shown that both the pressure of the gases enclosed by the annular membrane and the axial distance at which the annular membrane merges on the symmetry axis are periodic functions of time which have the same period as that of the mass flow rate fluctuations at the nozzle exit. They are also nearly sinusoidal functions of time for small amplitudes of the mass flow rate fluctuations at the nozzle exit, and exhibit delay and lag times with respect to the sinusoidal axial velocity fluctuations at the nozzle exit. Both the delay and the lag times are functions of the amplitude and frequency of the mass flow rate fluctuations at the nozzle exit and the polytropic exponent. The amplitudes of both the pressure of the gases enclosed by the annular liquid membrane and the convergence length increase and decrease, resp., as the amplitude and frequency of the mass flow rate fluctuations at the nozzle exit, resp., are increased. They also increase as the polytropic exponent is increased.

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Oszilierende rotationsschalenförmige Flüssigkeitsmembranen

Übersicht Untersucht wird das Verhalten rotationsschalenförmiger Flüssigkeitsmembranen in Abhängigkeit von der Amplitude und Frequenz der axialen Geschwindigkeitsschwankungen an der Düsenmündung und der thermodynamischen Verdichtung des eingeschlossenen Gases, wenn sich der Massestrom an der Düsenmündung sinusförmig ändert. Es wird gezeigt, daß der Druck des eingeschlossenen Gases und der axiale Mündungsabstand des Scheitels der geschlossenen Membran periodische Zeitfunktionen mit der Frequenz der Masseflußschwankung am Düsenaustritt sind. Für kleine Amplituden des Massestroms ist ihr Zeitverhalten ebenfalls fast-sinusförmig, wobei sie bezüglich der sinusförmigen axialen Geschwindigkeitsschwankungen an der Düsenmündung eine Ansprechzeit und Phasenverschiebung aufweisen. Ansprechzeit und Phasenverschiebung sind Funktionen von Amplitude und Frequenz der Massestromschwankung sowie des polytropen Exponenten. Die Amplitude von Gasdruck und Abstand des Membranscheitels von der Düse wächst bzw. fällt mit wachsender Amplitude und Frequenz des Massestroms. Beide nehmen außerdem mit dem polytropen Exponenten zu.

     

An analysis of laminar boundary layers on liquid curtains

A simplified analysis of the laminar boundary layer along an isothermal liquid curtain falling under gravity is presented. The analysis uses a von Kármán-Pohlhausen integral method and includes the effects of gravity, pressure differences, surface tension and nozzle exit geometry on the convergence length of liquid curtains which have applications as chemical reactors and as protection systems in laser fusion reactors. It is shown that the effects of the surrounding gases on the curtain shape and convergence length are small, and that good approximations to the liquid curtain shape can be obtained by using inviscid flow analyses.     

Mathematical modelling of spark-ignition engines

The flow field, scavenging efficiency, power output, heat transfer losses, and unburned hydrocarbon emissions have been numerically studied by means of a two-equation model of turbulence in a four-stroke, homogeneous-charge, spark-ignition engine. The engine is equipped with an intake valve, an exhaust valve, and a constant rate heat source which simulates the spark plug. Combustion has been modelled by means of a one-step irreversible chemical reaction whose rate is controlled by an Arrhenius-type expression. The numerical results indicate that the intake stroke is characterized by the formation of two eddies which persist in the compression stroke. Turbulence is generated at the shear layers of the air jet drawn into the cylinder, but its level decreases in the compression stroke. Due to the heat released by the spark plug and the chemical reaction, a spherical flame kernel is formed. This kernel evolves into a cylindrical flame when the flame front reaches the piston. Fuel remains unburnt at the corner between the cylinder head and the cylinder wall due to heat transfer losses. The numerical results also indicate that despite uncertainties about the turbulence and heat transfer models, an engine model such as the one studied here can be used to understand the flow field, heat transfer losses, scavenging efficiency, and power output in conventional spark-ignition engines. Such capabilities are very helpful in the development and optimization stages of engines. For example, here the engine model thermal and scavenging efficiencies are 15.69% and 94%, respectively. The peak pressure is 33 atm and occurs at 6° ATDC. The unburnt hydrocarbon emissions are 7.41% of the total fuel admitted into the cylinder.     

Mössbauer measurement of the hyperfine magnetic field AT sp site in Heusler alloys Rh2 Mn Z (Z=In and Sb)

We have investigated the systematics of hyperfine magnetic field on a fixed probe at the Z-site in Heusler alloys Rh₂MnZ as the valence of Z (sp element) is varied. The hmf on¹¹⁹Sn in Rh₂MnIn_.98 ¹¹⁹Sn₀₂ has been measured at 293K and 77K. In Rh₂Mn_1.12Sb_.86 ¹¹⁹Sn_.02 the hmf on¹²¹Sb has been measured at 77 K, and on¹¹⁹Sn at 293 K and 77 K. The results are compared with the hmf on¹¹⁹Sn in Rh₂Mn Ge_.98 ¹¹⁹Sn_.02 Rh₂Mn Sn, and Rh₂Mn Pb_.98 ¹¹⁹Sn_.02.Supported by the University Research Council, University of CincinnatiSupported by the Natural Sciences and Engineering Research Council of Canada     

The Overlay of Minimization Diagrams in a Randomized Incremental Construction

In a randomized incremental construction of the minimization diagram of a collection of n hyperplanes in ℝ ^d , for d≥2, the hyperplanes are inserted one by one, in a random order, and the minimization diagram is updated after each insertion. We show that if we retain all the versions of the diagram, without removing any old feature that is now replaced by new features, the expected combinatorial complexity of the resulting overlay does not grow significantly. Specifically, this complexity is O(n ^⌊d/2⌋log n), for d odd, and O(n ^⌊d/2⌋), for d even. The bound is asymptotically tight in the worst case for d even, and we show that this is also the case for d=3. Several implications of this bound, mainly its relation to approximate halfspace range counting, are also discussed.     

Effect of short chain branching in molecular dimensions and Newtonian viscosity of ethylene/1-hexene copolymers: matching conformational and rheological experimental properties and atomistic simulations

The molecular dimensions in dilute solution and the linear viscoelastic melt properties of a series of linear ethylene/1-hexene random copolymers with variable short chain branching content are reported. The results obtained from size exclusion chromatography and viscosimetry in dilute solution show a molecular contraction as the branching level increases. Additionally, a clear dependence of the Newtonian viscosity with the short chain branching content at T = 463 K is obtained. Both experimental observations are in agreement with previous experimental results found in ethylene/α-olefin copolymers, but more interestingly with recent full atomistic simulations made in our group in this type of macromolecular systems. The dependences observed can be related to the changes observed in the macromolecular conformational features and also in the equilibration entanglement time and the molecular weight between entanglements as the number of short chain branches increases.     

Activation of H-H and H-O bonds at phosphorus with diiron complexes bearing pyramidal phosphinidene ligands

The complex [Fe(2)Cp(2)(μ-PMes*)(μ-CO)(CO)(2)] (Mes* = 2,4,6-C(6)H(2)(t)Bu(3)), which in the solid state displays a pyramidal phosphinidene bridge, reacted at room temperature with H(2) (ca. 4 atm) to give the known phosphine complex [Fe(2)Cp(2)(μ-CO)(2)(CO)(PH(2)Mes*)] as the major product, along with small amounts of other byproducts arising from the thermal degradation of the starting material, such as the phosphindole complex [Fe(2)Cp(2)(μ-CO)(2)(CO){PH(CH(2)CMe(2))C(6)H(2)(t)Bu(2)}], the dimer [Fe(2)Cp(2)(CO)(4)], and free phosphine PH(2)Mes*. During the course of the reaction, trace amounts of the mononuclear phosphide complex [FeCp(CO)(2)(PHMes*)] were also detected, a compound later found to be the major product in the carbonylation of the parent phosphinidene complex, with this reaction also yielding the dimer [Fe(2)Cp(2)(CO)(4)] and the known diphosphene Mes*P═PMes*. The outcome of the carbonylation reactions of the title complex could be rationalized by assuming the formation of an unstable tetracarbonyl intermediate [Fe(2)Cp(2)(μ-PMes*)(CO)(4)] (undetected) that would undergo a fast homolytic cleavage of a Fe-P bond, this being followed by subsequent evolution of the radical species so generated through either dimerization or reaction with trace amounts of water present in the reaction media. A more rational synthetic procedure for the phosphide complex was accomplished through deprotonation of the phosphine compound [FeCp(CO)(2)(PH(2)Mes*)](BF(4)) with Na(OH), the latter in turn being prepared via oxidation of [Fe(2)Cp(2)(CO)(4)] with [FeCp(2)](BF(4)) in the presence of PH(2)Mes*. To account for the hydrogenation of the parent phosphinidene complex it was assumed that, in solution, small amounts of an isomer displaying a terminal phosphinidene ligand would coexist with the more stable bridged form, a proposal supported by density functional theory (DFT) calculations of both isomers, with the latter also revealing that the frontier orbitals of the terminal isomer (only 5.7 kJ mol(-1) above of the bridged isomer, in toluene solution) have the right shapes to interact with the H(2) molecule. In contrast to the above behavior, the cyclohexylphosphinidene complex [Fe(2)Cp(2)(μ-PCy)(μ-CO)(CO)(2)] failed to react with H(2) under conditions comparable to those of its PMes* analogue. Instead, it slowly reacted with HOR (R = H, Et) to give the corresponding phosphinous acid (or ethyl phosphinite) complexes [Fe(2)Cp(2)(μ-CO)(2)(CO){PH(OR)Mes*}], a behavior not observed for the PMes* complex. The presence of BEt(3) increased significantly the rate of the above reaction, thus pointing to a pathway initiated with deprotonation of an O-H bond of the reagent by the basic P center of the phosphinidene complex, this being followed by the nucleophilic attack of the OR(-) anion at the P site of the transient cationic phosphide thus formed. The solid-state structure of the cis isomer of the ethanol derivative was determined through a single crystal X-ray diffraction study (Fe-Fe = 2.5112(8) ?, Fe-P = 2.149(1) ?).     

Looking for the physiological role of anthocyanins in the leaves of Coffea arabica

The aim of this study was to determine which anthocyanins are related to the purple coloration of young leaves in Coffea arabica var. Purpurascens and assess their impact on photosynthesis as compared to C. arabica var. Catuaí, with green leaves. Two delphinidin glicosides were identified and histological cross-sections showed they were located throughout the adaxial epidermis in young leaves, disappearing as the leaves mature. Regardless the irradiance level, the photosynthetic performance of Purpurascens leaves did not differ from that observed in leaves of the Catuaí variety, providing no evidence that anthocyanins improve photosynthetic performance in coffee plants. To analyze the photoprotective action of anthocyanins, we evaluated the isomerization process for chlorogenic acids (CGAs) in coffee leaves exposed to UV-B radiation. No differences were observed in the total concentration of phenolic compounds in either variety before or after the UV treatment; however, we observed less degradation of CGA isomers in the Purpurascens leaves and a relative increase of cis-5-caffeoylquinic acid, a positional isomer of one of the most abundant form of CQA in coffee leaves, trans-5-caffeoylquinic acid, suggesting a possible protective role for anthocyanins in this purple coffee variety.