Bivariate averaging functions,translation and scale autocorrelations,Fourier and Mellin transforms,the Wiener-Khinchine theorem and their inter-relationships

We first draw attention to the fact that the position operator, , its translation generator, , and its scale generator, , form an important group of triplet of operators that appear in the Heisenberg uncertainty relation stated in its most general form. The pair forms the phase-space and they have led to Fourier transform pair, the autocorrelation function, the Wiener-Khinchine theorem, and the Wigner function with many different applications to wave phenomena. The importance of the pairs and has been pointed out by Moses and Quesada (1972, 1973, 1974) who showed that we must then consider a Mellin transform pair, a scale autocorrelation function, and a corresponding Wiener-Khinchine theorem. In the present paper, we define and explore properties of a bivariate averaging function defined in a new “phase-space” involving the Mellin transform variable and its partner which can either be the position or momentum, analogous to the Wigner function. The not-necessarily positive feature of the bivariate averaging functions is traced to the general Heisenberg uncertainty mentioned above. The properties and their inter-relationships among the averaging functions are given. We hope this will be of use in discussing physical phenomena involving fractals, turbulence, and near phase transitions where the scaling properties are of importance.     

Thermal analysis of alumina precursors prepared by ‘PFHS’ methods

Thermal analysis of alumina precursors prepared by two different PFHS (precipitation from homogeneous solution) methods and a conventional method is described. All three precursors exhibit distinct thermal behaviour patterns, marked by multiple phase transformations, to yield-Al₂O₃ ultimately. Thermal analysis studies, coupled with XRD, IR and elemental analysis data, indicate that the precursors obtained by the PFHS methods are monophasic in nature, and hence yield relatively small, uniform microspheroidal alumina in comparison with the alumina obtained by the conventional method.

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Zusammenfassung Die Thermoanalyse von Aluminiumoxidpräkursoren, hergestellt durch zwei verschiedene PFHS (Precipitation From Homogeneous Solution) und eine herkömmliche Methode, wurde beschrieben. Alle drei Präkursoren zeigen ausgeprägte thermische Verhaltensformen, gekennzeichnet durch Mehrfachphasenübergange zur letztendlichen Bildung von-Al₂O₃. Die Thermoanalysen, verbunden mit Röntgendiffraktions-, Infrarot- und Elementaranalysen zeigen, dass die durch PFHS-Methoden erhaltenen Präkursoren über Einphasen verfügen und verglichen zu herkömmlich hergestelltem Aluminiumoxid relativ kleines, einheitliches Kugelaluminiumoxid liefern.

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« » , - , . «» , -l₂₃. , , , « », , , .

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One of us BSM thanks the University Grants Commission, India, for the award of a Research Fellowship.     

Phenomenological theory of the kinetics of ultrasonic emulsification

Summary A working model is given for the rate of ultrasonic emulsification, considering the dispersion at the interface (areaA) and the coagulations in the volumeV of the emulsion. A bimolecular coagulation leads to the equationc=c _∞tanhbt;c _∞=(Aα/Vβ)^1/2;b=(Aαβ/V)^1/2 while a monomolecular coagulation givesc=c _∞{1−exp (−at)};c _∞=Aα/Vβ;a=β. The experiments on the dependence of c_∞,a andb uponA andV favour the bimolecular coagulation. The results are satisfactorily explained on general theoretical grounds.

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Zusammenfassung Ein Arbeitsmodell für die Geschwindigkeit der Ultraschallemulgierung wird entwickelt, das Dispersion in der Grenzfl?che (Fl?cheA) und Koagulation im Volumen (V) der Emulsion annimmt. Eine bimolekulare Koagulation führt zu der Gleichung:c=c _∞ tanhbt;c _∞=(Aα/Vβ)^1/2;b=(Aαβ/V)^1/2, eine monomolekulare dagegen zu:c=c _∞{1−exp (at−)};c _∞=Aα/Vβ;α=β. Die Versuche über die Abh?ngigkeit vonc _∞,a undb vonA undV scheinen für bimolekulare Koagulation zu sprechen. Die Ergebnisse werden auf der Basis dieser einfachen theoretischen Grundlagen befriedigend erkl?rt.

     

On the Deformation of the Circumflex Coronary Artery During Inflation Tests at Constant Length

Here we investigate whether the deformation observed in an experiment in which the porcine circumflex coronary artery is subjected to inflation at constant length included in the class, , , . We find that this is not the case and discuss its implications in the study of the mechanics of this artery. Moreover, we identify and quantify the uncertainty in the value of the invariants of the left Cauchy–Green tensor inferred from the 2D motion of markers affixed to the surface of the test specimen, and suggest that 3D tracking of markers is needed due to inherent bending and twisting induced by pressurization in vitro.     

Stabilized mixed three‐field formulation for a generalized incompressible Oldroyd‐B model

This paper presents a generalization of the incompressible Oldroyd‐B model based on a thermodynamic framework within which the fluid can be viewed to exist in multiple natural configurations. The response of the fluid is viewed as a combination of an elastic component and a dissipative component. The dissipative component leads to the evolution of the underlying natural configurations, while the response from the natural configuration to the current configuration is considered elastic and therefore non‐dissipative. For an incompressible fluid, it is necessary that both the elastic behavior as well as the dissipative behavior is isochoric. This is achieved by ensuring that the determinant of the stretch tensor associated with the elastic response meets the constraint that its determinant is unity. A new stabilized mixed method is developed for the velocity, pressure and the kinematic tensor fields. Analytical models for fine scale fields are derived via the solution of the fine‐scale equations facilitated by the Variational Multiscale framework that are then variationally embedded in the coarse‐scale variational equations. The resulting method inherits the attributes of the classical SUPG and GLS methods, while a significant new contribution is that the form of the stabilization tensors is explicitly derived. A family of linear and quadratic tetrahedral and hexahedral elements is developed with equal‐order interpolations for the various fields. Numerical tests are presented that validate the incompressibility of the elastic stretch tensor, show optimal L₂ convergence for the conformation tensor field, and present stable response for high Weissenberg number flows. Copyright © 2016 John Wiley & Sons, Ltd.     

A note on viscoelastic materials that can age

Materials can stress relax for a variety of reasons, for example stress relaxation in classical viscoelastic bodies and that due to aging or damage. Here, within the context of a specific model we show that the stress relaxation that is characteristic of viscoelasticity is distinctly different from that of aging.