On the measurement of normal stress differences in steady shear flow

Summary Compared to the similar pressure-distribution cone-and-plate apparatus of Adams and Lodge (4), the new apparatus' improvements include: temperature control of the cone (as well as the plate); increased stiffening of the frame; four (not three) pressuremeasuring holes in the cone/plate region; inclusion of a pressure-measuring hole on the axis under the cone truncation; exclusive use of a vertical free liquid boundary at the cone rim (without a sea of liquid). Temperature control of the rotating cone and of the fixed plate leads to acceptable temperature uniformity in the test liquid for working temperatures within 10°C or 20°C of ambient; the corresponding interval is about 1°C if the cone temperature control is abandoned. Pressure gradients measured using a Newtonian liquid agree with those calculated using Walters' eq. (3). For a viscoelastic liquid, after subtracting inertial contributions, pressure distributions measured at a given shear rate in the cone/plate region do not change when the gap angle is changed from 2° to 3°, showing that the effects of secondary flow are negligible. Values ofN ₃ =N ₁ + 2N ₂ obtained from the gradients of these distributions are believed to be in error by not more than ±1 Pa, in favorable cases. The present most useful ranges are: 10 to 5000 Pa forN ₃; 0.1 to 200 sec^–1 for shear rate; up to 5 Pa s for viscosity; and 5 to 40°C for temperature. As an application, it is shown that adding 0.1% of a high molecular weight polyisobutylene to a 2% polyisobutylene solution doublesN ₃ and has no detectable effect on the viscosity measured at low shear rates with a Ferranti-Shirley viscometer.

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Zusammenfassung Im Vergleich zu dem ähnlichen Kegel-Platte-Gerät von Adams und Lodge (4) zur Messung der Druckverteilung wurden an dem neuen Gerät die folgenden Verbesserungen vorgenommen: Temperaturregelung an Kegel und Platte, Versteifung des Rahmens, vier (anstatt drei) Druckmeßlöcher im Kegel-Platte-Bereich, ein zusätzliches Druckmeßloch auf der Achse unter der Kegelstumpf-Deckfläche, ausschließliche Verwendung einer vertikalen freien Grenzfläche der Flüssigkeit am Kegelrand (ohne umgebenden Flüssigkeitssee). Die Temperaturregelung des rotierenden Kegels und der festen Platte führt zu einer ausreichenden Temperaturgleichförmigkeit in der Testflüssigkeit für Betriebstemperaturen, die höchstens um 10–20°C von der Umgebungstemperatur abweichen. Dieses Intervall beträgt dagegen nur etwa 1°C, wenn auf die Temperaturregelung am Kegel verzichtet wird. Für newtonsche Flüssigkeiten entsprechen die gemessenen Druckgradienten den mittels der Gleichung von Walters (3) berechneten. Für viskoelastische Flüssigkeiten zeigen sich bei der Änderung des Spaltwinkels von 2° auf 3° nach Abzug der Trägheitsbeiträge keine Änderungen der bei einer bestimmten Schergeschwindigkeit gemessenen Druckverteilung. Dies zeigt, daß Sekundärströmungseffekte vernachlässigbar sind. Es darf angenommen werden, daß die Werte vonN ₃ =N ₁ + 2N ₂, die man aus den Gradienten dieser Verteilungen erhält, unter günstigen Umständen mit einem Fehler von nicht mehr als ±1 Pa behaftet sind. Gegenwärtig liegen die günstigsten Bereiche bei 10 bis 5000 Pa fürN ₃, 0,1 bis 200 s^–1 für die Schergeschwindigkeit, unterhalb von 5 Pa s für die Viskosität und 5 bis 40°C für die Temperatur. Als Anwendung wird gezeigt, daß ein Zusatz von 0,1% hochmolekularen Polyisobutylens zu einer 2%igen Polyisobutylenlösung den Wert vonN ₃ verdoppelt, aber keinen erkennbaren Einfluß auf die (bei geringen Schergeschwindigkeiten mit einem Ferranti-Shirley-Viskosimeter gemessen) Viskosität hat.

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udsf unidirectional shear flow - TCP truncated-cone and plate - N ₁,N ₂ 1st and 2nd normal stress differences in udsf - N ₃ N ₁ + 2N ₂ - : = A is defined by the equationA := B - P ^* hole pressurePw – Pm; Pw, Pm = pressures measured by flush transducer and by hole-mounted transducer - t time - , strain rate, shear rate - (P,t) covariant body metric tensor at particleP and timet - ⁱ , _i covariant and contravariant udsf body base vectors (i = 1, 2, 3) - ^–1 inverse of - R, plate radius, cone/plate gap angle - r ₀,h ₀ radius and height of cone truncation - r,, spherical polar coordinates; cone axis = 0; plate surface = /2 - physical components of stress; for a tensile component - cone angular velocity - p on the plate = /2 - ,T, density, absolute temperature, viscosity - P 0.15 ²(r ² –R ²) (inertial contribution) [2.7] - P _ve contribution [2.8] from flow perturbations of viscoelastic origin - r _i i = 1,2,3,4; values ofr at centers of holes in cone/plate region - P _i () pressure change recorded by transducerTi when cone angular velocity goes from zero to - 1/2 {P _i ()+ P _i (–)} (average for 2 senses of rotation) - rim pressure, from least-squares line through four points - Re Reynolds' number:R ²/ - (P,t)/t With 11 figures and 2 tables     

High temperature normal phase liquid chromatography of aromatic hydrocarbons on bare zirconia

The normal phase HPLC behavior of a bare zirconia column was studied at temperatures up to 200 °C using a hexane mobile phase. The use of elevated column temperatures significantly decreased the retention of twenty five aromatic model compounds according to the van't Hoff equation (>30-fold decrease for some compounds). Large improvements in peak shape, efficiency (>2.2-fold), aromatic group-type selectivity, and column re-equilibration times (>5-fold) were obtained at elevated temperatures. The thermal decomposition of two polar nitrogen compounds (indole and carbazole) was observed in a hexane/dichloromethane mobile phase at temperatures greater than 100 °C. The first order decomposition of carbazole was studied in further detail.     

Chromatographic characteristics of poly(1-vinylimidazole)-grafted silica in normal-phase HPLC

A stationary phase based on poly(1-vinylimidazole)-grafted silica has been prepared by the surface radical chain-transfer reaction. The stationary phase was characterized by infrared spectra, X-ray photoelectron spectroscopy and elemental analysis. Chromatographic characteristics of the stationary phase were investigated in normal-phase HPLC. The results showed that both weak polar compounds (polycyclic aromatic hydrocarbons, dialkyl phthalates) and polar compounds (anilines, phenols) could be successfully separated on this stationary phase, implying better separation performance than blank silica and conventional aminopropyl-bonded silica under the same conditions. The excellent performance can be attributed to multiple interactions between surface modifier and the analytes that might include dipole, hydrogen bonding, H-π, electrostatic and inductive interactions.     

Self-assembled highly ordered ethane-bridged periodic mesoporous organosilica and its application in HPLC

Monodisperse spherical periodic mesoporous organosilicas (PMOs) with ethane integrated in the framework were synthesized and their application as stationary phase for chromatographic separation is demonstrated. The ethane-PMOs were prepared by condensation of 1,2-bis(triethoxysilyl)ethane (BTSE) in basic condition using octadecyltrimethylammonium chloride (C(18)TMACl) as template and ethanol as co-solvent. The morphology and mesoporous structure of ethane-PMOs were controlled under different concentrations of sodium hydroxide (NaOH) and EtOH. The results of scanning electron microscopy (SEM), transmission electron microscopy (TEM), powder X-ray diffraction (XRD), nitrogen sorption measurement, Fourier transform infrared spectroscopy (FT-IR) and elemental analysis showed that ethane-PMOs have spherical morphology, uniform particle distribution, highly ordered pore structure, high surface area and narrow pore-size distribution. The column packed with these materials exhibits good permeability, high chemical stability and good selectivity of mixtures of aromatic hydrocarbons in normal phase high-performance liquid chromatography (HPLC).     

Spectral decomposability of rank-one perturbations of normal operators

This paper is a continuation of the study by Foias, Jung, Ko, and Pearcy (2007) [4] and Foias, Jung, Ko, and Pearcy (2008) [5] of rank-one perturbations of diagonalizable normal operators. In Foias, Jung, Ko, and Pearcy (2007) [4] we showed that there is a large class of such operators each of which has a nontrivial hyperinvariant subspace, and in Foias, Jung, Ko, and Pearcy (2008) [5] we proved that the commutant of each of these rank-one perturbations is abelian. In this paper we show that the operators considered in Foias, Jung, Ko, and Pearcy (2007) [4] have more structure - namely, that they are decomposable operators in the sense of Colojoar? and Foias (1968) [1].     

Maximal admissible faces and asymptotic bounds for the normal surface solution space

The enumeration of normal surfaces is a key bottleneck in computational three-dimensional topology. The underlying procedure is the enumeration of admissible vertices of a high-dimensional polytope, where admissibility is a powerful but non-linear and non-convex constraint. The main results of this paper are significant improvements upon the best known asymptotic bounds on the number of admissible vertices, using polytopes in both the standard normal surface coordinate system and the streamlined quadrilateral coordinate system.To achieve these results we examine the layout of admissible points within these polytopes. We show that these points correspond to well-behaved substructures of the face lattice, and we study properties of the corresponding “admissible faces”. Key lemmata include upper bounds on the number of maximal admissible faces of each dimension, and a bijection between the maximal admissible faces in the two coordinate systems mentioned above.     

Sensitivity of risk measures with respect to the normal approximation of total claim distributions

A simple and commonly used method to approximate the total claim distribution of a (possibly weakly dependent) insurance collective is the normal approximation. In this article, we investigate the error made when the normal approximation is plugged in a fairly general distribution-invariant risk measure. We focus on the rate of convergence of the error relative to the number of clients, we specify the relative error’s asymptotic distribution, and we illustrate our results by means of a numerical example. Regarding the risk measure, we take into account distortion risk measures as well as distribution-invariant coherent risk measures.     

Fractional normal inverse Gaussian diffusion

A fractional normal inverse Gaussian (FNIG) process is a fractional Brownian motion subordinated to an inverse Gaussian process. This paper shows how the FNIG process emerges naturally as the limit of a random walk with correlated jumps separated by i.i.d. waiting times. Similarly, we show that the NIG process, a Brownian motion subordinated to an inverse Gaussian process, is the limit of a random walk with uncorrelated jumps separated by i.i.d. waiting times. The FNIG process is also derived as the limit of a fractional ARIMA processes. Finally, the NIG densities are shown to solve the relativistic diffusion equation from statistical physics.     

Only the first term of some series counts

Let X,X₁,X₂,… be i.i.d. random variables, and set S_n=X₁+?+X_n. We prove that for three important distributions of X, namely normal, exponential and geometric, series of the type ∑_n≥1a_nP(|S_n|≥xb_n) or ∑_n≥1a_nP(S_n≥xb_n) behave like their first term as x→∞.     

A Criterion of Normality Concerning Holomorphic Functions Whose Derivative Omits a Function

The authors discuss the normality concerning holomorphic functions and get the following result. Let F be a family of holomorphic functions on a domain D ⊂ ℂ, all of whose zeros have multiplicity at least k, where k ≥ 2 is an integer. And let h(z) ≢ 0 be a holomorphic function on D. Assume also that the following two conditions hold for every f ∈ F: (a) f(z) = 0 ⇒ |f ^(k)(z)| < |h(z)|; (b) f ^(k)(z) ≠ h(z). Then F is normal on D.