Pattern Selection: Determined by Symmetry and Modifiable by Distant Effects

We consider Saffman–Taylor channel flow without surface tension on a high-pressure driven interface, but modify the usual infinite-fluid in infinite-channel configuration. Here we include the treatment of efflux by considering a finite connected body of fluid in an arbitrarily long channel, with its second free interface the efflux of this configuration. We show that there is a uniquely determined translating solution for the driven interface, which is exactly the 1/2 width S–T solution, following from correct symmetry for a finite channel flow. We establish that there exist no perturbations about this solution corresponding to a finger propagating with any other width: Selection is locally unique and isolated. The stability of this solution is anomalous, in that all freely impressible perturbations are stabilities, while unstable modes request power proportional to their strength from the external agencies that drive the flow, and so, in principle, are experimentally controllable. This is very different from the behavior of the usual infinite fluid. We conjecture that surface tension on the efflux interface modifies channel-width according to 1–2=/v (i.e., (2)² B of the literature) with v the velocity of the high-pressure tip, but the surface tension of the efflux. That is, is decreased below 1/2 by the effect of smoothing the distant efflux. The perturbation theory created here to deal with transport between two free boundaries is novel and dependent upon a symmetry implied by the equations of motion.     

Asymptotic expansions of the distributions of the latent roots and the latent vector of the Wishart and multivariate F matrices

Asymptotic expansions of the joint distributions of the latent roots of the Wishart matrix and multivariate F matrix are obtained for large degrees of freedom when the population latent roots have arbitrary multiplicity. Asymptotic expansions of the distributions of the latent vectors of the above matrices are also derived when the corresponding population root is simple. The effect of normalizations of the vector is examined.     

The “Error” in the Indian “Taylor Series Approximation” to the Sine

It has been repeatedly noted, but not discussed in detail, that certain so-called “third-order Taylor series approximations” found in the school of the medieval Keralese mathematician M dhava are inaccurate. That is, these formulas, unlike the other series expansions brilliantly developed by M dhava and his followers, do not correspond exactly to the terms of the power series subsequently discovered in Europe, by whose name they are generally known. We discuss a Sanskrit commentary on these rules that suggests a possible derivation explaining this discrepancy, and in the process re-emphasize that the Keralese work on such series was rooted in geometric approximation rather than in analysis per se. © 2001 Elsevier Science (USA).Es ist mehrfach festgestellt bisher aber nicht ausführlich diskutiert worden, daß einige sogenannte Taylor-reihennäherungswerte dritter Ordnung, die in der mittelalterlichen Schule keralesischen M dhava gefunden werden, ungenau sind. Das heißt, diesc Formeln sind den Termen der Potenzreihe, die später in Europa entwickelt wurde und unter dem Namen Taylorreihe bekannt ist, nicht äquivalent, im Gegensatz zu den anderen Entwicklungen von Reihen, die glänzend von M dhava und seinen Nachfolgern entwickelt werden. Wir behandeln einen Sanskritkommentar zu den Regeln, der eine mögliche Herleitung suggeriert, die diese Diskrepanz erklärt. Dabei betonen wir nochmals, daß die keralesische Arbeit über solche Reihen eher in geometrischen Näherungen als in der Analysis an sich ihre Wurzeln hat. © 2001 Elsevier Science (USA).MSC subject classification: 01A32.     

Deformation and Failure Modes of Soft Steel Projectiles Impacting Harder Steel Targets at Increasing Velocity

The present paper describes an experimental and numerical study concerning the impact of blunt steel projectiles against harder steel plates, at impact velocities between 200 and 800 m/s. In contrast with previously published observations, three modes of deformation and failure of the soft steel projectiles were observed in the present study. These included: Taylor cylinder mushrooming, sunflower-like petalling and plugging perforation. Individual velocity ranges and the transitions between the deformation/failure modes are identified by both experiments and numerical simulations. Complex material failure mechanisms of projectile and target play conflicting roles in the various penetration stages. Johnson–Cook models of strength and accumulative damage failure are employed in 3D numerical simulation to describe material behavior of both projectile and target. Computational evolutions of each scenario are offered in detail to understand the deformation and failure of projectile and target plate.     

Modified Taylor approximation of functions with periodic behaviour

We approximate a function with periodic behaviour by means of a small modification of its Taylor polynomial. This modification is based on the work of Scheifele and will simplify the construction of special numerical methods for differential equations with near periodic solutions.     

TAYLOR SERIES AND ORTHOGONALITY OF THE OCTONION ANALYTIC FUNCTIONS

1 IntroductionThe only finite dimensional aIternative di.ili.,1 algebras over R are ffeal algebra R, Com-plex algebra C, Quaternion algebra H, Octonidn algebra O with the embedding relations:R G C C H g O. R and C are commutative and associative, H is associative but notcomlnutative, while O is neither commutative nor a.s..i.ti.e[1].Much earlier the great Swiss mathematician FUeter (a student of Hilbert) and hi8 studentsdeveloped Quaternion analysis up to l950.[2'3] and it was a great…     

Effect of monovalent salts on morphology of calcium carbonate crystallized in Couette‐Taylor reactor

Monovalent ionic additives, Na⁺, K⁺ and NHequation/tex2gif-stack-1.gif impact on the morphology and agglomeration of CaCO₃ crystals. As increasing the additive concentration, the regular shaped crystals such as rhombohedron and spindle are changed to irregular one due to the inclusion of Na⁺ and K⁺ into the crystal structure. The inclusion of Na⁺ and K⁺ is detected using ICP‐AES. The partition of coefficients of Na⁺ and K⁺ are estimated as 9.74 × 10^–4, 9.73×10^–4, respectively and the amount of inclusion in the crystals is about 2×10³ ppm. However, the inclusion of ions does not modify a crystal structure of calcite. Since NH₄⁺ is large in radius, it is not included in crystal but shifts the spindle shape of crystal to the rhombohedral one. It is interesting to find that such modification of crystal morphology begins to appear at high additive concentration (0.05 M). In addition, the crystal agglomeration is promoted because the electric repulsive charge is reduced as increasing the additive concentration. (© 2005 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

JOB-9003的动态性能实验

对JOB-9003进行了SHPB压杆实验和逆向Taylor柱实验,研究其在冲击载荷下的动态特性,为进行JOB-9003的本构模型研究提供实验数据基础。通过对SHPB实验获得的应力应变曲线分析得出:在中低应变率范围,JOB-9003应变率对应力的影响成线性关系。通过对逆向Taylor柱实验获得的变形照片分析,发现破坏损伤对材料的力学性能影响显著,不可忽略。利用Taylor实验的结果对SHPB实验中获得的本构模型进行校核,发现该本构模型并不能准确描述处于高应变率下的材料压缩变形。     

The Fejér-Riesz inequality for the Besov spaces

A Fejér-Riesz inequality for the weighted Besov spaces B _p,q is given. Some characterizations of functions in B _p,q in terms of their Taylor coefficients are obtained.     

Two-equation and multi-fluid turbulence models for Rayleigh–Taylor mixing

This paper presents a new, improved version of the K–L model, as well as a detailed investigation of K–L and multi-fluid models with reference to high-resolution implicit large eddy simulations of compressible Rayleigh–Taylor mixing. The accuracy of the models is examined for different interface pressures and specific heat ratios for Rayleigh–Taylor flows at initial density ratios 3:1 and 20:1. It is shown that the original version of the K–L model requires modifications in order to provide comparable results to the multi-fluid model. The modifications concern the addition of an enthalpy diffusion term to the energy equation; the formulation of the turbulent kinetic energy (source) term in the K equation; and the calculation of the local Atwood number. The proposed modifications significantly improve the results of the K–L model, which are found in good agreement with the multi-fluid model and implicit large eddy simulations with respect to the self-similar mixing width; peak turbulent kinetic energy growth rate, as well as volume fraction and turbulent kinetic energy profiles. However, a key advantage of the two-fluid model is that it can represent the degree of molecular mixing in a direct way, by transferring mass between the two phases. The limitations of the single-fluid K–L model as well as the merits of more advanced Reynolds-averaged Navier–Stokes models are also discussed throughout the paper.