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At least 289 records · Page 16

Monodisperse Single-Material Granular Tribocharging Modeling and Experimental Validation

Charge transfer between insulating grains has been a topic of interest for many years as this phenomenon is extremely important to many areas in industry. Of particular interest to NASA is the behavior of electrically insulative dust grains such as those found on the lunar surface. Whether poured from a scoop during sample collection activities, agitated inside a drum via mining robots, or fluidized by a gas plume, particle-particle interactions between similar granular materials will be widespread on the Moon. Designers hope to have a predictive model for how dust grains are charged, transported, and deposited on flight systems to better understand issues electrostatics may cause in future missions. While difference in work function is often seen as a driver for charge transfer between two materials, this is an incomplete representation for insulative particles. Other asymmetries in the system can also lead to charge being transferred: one such being the difference between static vs dynamic particles. Experiments performed under vacuum in the Electrostatics and Surface Physics Laboratory (ESPL) clearly show that particles having more contacts with other particles tend to charge positively and particles having fewer contacts tend to charge negatively, despite being made from the same material. During these experiments, monodisperse (population of single diameter) spherical particles were held in a reservoir which was then elevated at one end to slope into a Faraday cup used to measure bulk charge of the grains. Care was taken to limit the number of spheres bouncing out of the cup, to isolate the beads from the container itself so that only grain-to-grain interactions occurred, and to assure neutrality of the beads before the experiment began. A discrete element method (DEM) modeling package already incorporating many granular mechanics interactions was augmented to include the electrostatic behaviors seen in experiments. The model can reproduce the experimental results above by tracking the number of contacts between pairs of individual grains and applying a charge transfer condition related to this contact parameter. The experimental results as well as their incorporation into the improved modeling suite will be discussed. Future work includes adding additional asymmetries into the experiments/model such as polydisperse populations of spherical particles and irregularly shaped grains.

electrostatics↗

Macrosegregation and nucleation in undercooled Pb-Sn alloys

A technique resulting in large undercoolings in bulk samples (23g) of lead-tin alloys was developed. Samples of Pb-12.5 wt percent Sn, Pb-61 wt percent Sn, and Pb-77 wt percent Sn were processed with undercoolings ranging from 4 to 34 K and with cooling rates varying between 0.04 and 4 K/sec. The nucleation behavior of the Pb-Sn system was found to be nonreciprocal. The solid Sn phase effectively nucleated the Pb phase of the eutectic; however, large undercoolings developed in Sn-rich eutectic liquid in the presence of the solid Pb phase. This phenomenon is believed to be mainly the result of differences in interfacial energies between solid Sn-eutectic liquid, and solid Pb-eutectic liquid rather than lattice misfit between Pb and Sn. Large amounts of segregation developed in the highly undercooled eutectic ingots. This macrosegregation was found to increase as undercooling increases. Macrosegregation in these undercooled eutectic alloys was found to be primarily due to a sink/float mechanism and the nucleation behavior of the alloy. Lead-rich dendrites are the primary phase in the undercooled eutectic system. These dendrites grow rapidly into the undercooled bath and soon break apart due to recalescence and Sn enrichment of the liquid. These fragmented Pb dendrites are then free to settle to the bottom portion of the ingot causing the macrosegregation observed in this study. A eutectic Pb-Sn alloy undercooled 20 K and cooled at 4 K/sec had a composition of about Pb-72 wt percent Sn at the top and 55 percent Sn at the bottom.

Degroh, Henry C., III↗

Lunar and Planetary Science XXXV: From Ancient Mists: Presolar and Nebular Processes

The session "From Ancient Mists: Presolar and Nebular Processes" included the following reports: Interpretation of the Meteoritic Extinct Radioactivity - Mean Life Relation; On the Issue of Molybdenum Isotopic Anomalies in Meteorites: Is It Still "FUN"?; (26Al/27Al)o of the Solar Nebula Inferred from Al-Mg Systematic in Bulk CAIs from CV3 Chondrites; Magnesium Isotopic Compositions of Igneous CAIs in the CR Carbonaceous Chondrites: Evidence for an Early and Late-stage Melting of CAIs; The 26Al-26Mg Chronology of a Type C CAI and POIs in Ningqiang Carbonaceous Chondrite; Bulk Compositions of CAIs and Al-rich Chondrules: Implications of the Reversal of the Anorthite / Forsterite Condensation Sequence at Low Nebular Pressures; Synthesis of Refractory Minerals by High-Temperature Condensation of a Gas of Solar Composition; Elemental and Isotopic Fractionation by Diffusion-limited Evaporation; "Nonideal" Isotopic Fractionation Behavior of Magnesium in Evaporation Residues; Determination of Primordial Refractory Inclusion Compositions; Zoning Patterns in Spinel from Type B Ca-Al-rich Inclusions: Constraints on Sub-Solidus Thermal History; Radial Migration of Materials from Inner to Outer Solar Nebula: Evidence from Meteorite Matrix; and Refractory Forsterites in Chondritic Meteorites, a Link Between CAIs and Chondrules.

Source record↗

Nonequilibrium synthesis of NbAl3 and Nb-Al-V alloys by laser cladding. II - Oxidation behavior

Isothermal oxidation behaviors of NbAl3 alloy synthesized by laser cladding were investigated at temperatures between 800 and 1400 C, and the effect of vanadium microalloying on the oxidation of the laser-clad alloy was examined. The oxidation kinetics of the two alloys were monitored using thermal gravimetric weight gain data, and the bulk and surface chemistries were analyzed using XRD and XPS, respectively. It was found that NbAl3 did not form an exclusive layer of protective Al2O3. The oxidation products at 800 C were found to be a mixture of Nb2O5 and Al2O3. At 1200 C, a mixture of NbAlO4, Nb2O5, and Al2O3 formed; and at 1400 C, a mixture of NbAlO4, Al2O3, NbO2, NbO(2.432), and Nb2O5 formed. The addition of V led to a dramatic increase of the oxidation rate, which may be related to the formation of (Nb, V)2O5 and VO2, which grows in preference to protective Al2O3.

Haasch, R. T.↗

Ag Isotopic Evolution of the Mantle During Accretion: New Constraints from Pd and Ag Metal-Silicate Partitioning

Decay of (sup 107) Pd to (sup 107) Ag has a half-life of 6.5 times 10 (sup 6) mega-annums. Because these elements are siderophile but also volatile, they offer potential constraints on the timing of core formation as well as volatile addition. Initial modelling has shown that the Ag isotopic composition of the bulk silicate Earth (BSE) can be explained if accretion occurs with late volatile addition. These arguments were tested for sensitivity for pre-cursor Pd/Ag contents, and for a fixed Pd/Ag ratio of the BSE of 0.1. New Ag and Pd partitioning data has allowed a better understanding of the partitioning behavior of Pd and Ag during core formation. The effects of S, C and Si, and the effect of high temperature and pressure has been evaluated. We can now calculate D(Ag) and D(Pd) over the wide range of PT conditions and variable metallic liquid compositions that are known during accretion. We then use this new partitioning information to revisit the Ag isotopic composition of the BSE during accretion.

Righter, K.↗

Crystallization behavior and properties of BaO-Al2O3-2SiO2 glass matrices

Glass of stoichiometric celsian composition, BaO-Al2O3-SiO2, has a density of 3.39 g/cu cm, a thermal expansion coefficient of 6.6 x 10 to the -6th/C, a glass-transition temperature of 910 C, and a dilatometric softening point of 925 C. On heat treatment, only hexacelsian crystallized out on the surface, but both celsian and hexacelsian were present in the bulk. Effects of cold isostatic pressing (CIP), sintering, and hot-pressing, in the presence and absence of an additive, on the formation of the celsian phase in the glass have been studied. CIP'd samples, after appropriate heat treatments, always crystallized out as celsian, whereas presence of 5-10 wt pct of an additive was necessary for formation of celsian in sintered as well as hot-pressed specimens. Green density increased with CIP'ing pressure but had no effect on sintered density. Hot-pressing resulted in fully dense samples.

Drummond, Charles H., III↗

Effective Forces Between Colloidal Particles

Colloidal suspensions have proven to be excellent model systems for the study of condensed matter and its phase behavior. Many of the properties of colloidal suspensions can be investigated with a systematic variation of the characteristics of the systems and, in addition, the energy, length and time scales associated with them allow for experimental probing of otherwise inaccessible regimes. The latter property also makes colloidal systems vulnerable to external influences such as gravity. Experiments performed in micro-ravity by Chaikin and Russell have been invaluable in extracting the true behavior of the systems without an external field. Weitz and Pusey intend to use mixtures of colloidal particles with additives such as polymers to induce aggregation and form weak, tenuous, highly disordered fractal structures that would be stable in the absence of gravitational forces. When dispersed in a polarizable medium, colloidal particles can ionize, emitting counterions into the solution. The standard interaction potential in these charged colloidal suspensions was first obtained by Derjaguin, Landau, Verwey and Overbeek. The DLVO potential is obtained in the mean-field linearized Poisson-Boltzmann approximation and thus has limited applicability. For more precise calculations, we have used ab initio density functional theory. In our model, colloidal particles are charged hard spheres, the counterions are described by a continuum density field and the solvent is treated as a homogeneous medium with a specified dielectric constant. We calculate the effective forces between charged colloidal particles by integrating over the solvent and counterion degrees of freedom, taking into account the direct interactions between the particles as well as particle-counterion, counterion-counterion Coulomb, counterion entropic and correlation contributions. We obtain the effective interaction potential between charged colloidal particles in different configurations. We evaluate two- and three-body forces in the bulk as well as study the influence of soft walls. We qualitatively explain the effects of the walls on the forces and demonstrate that many-body effects are negligible in our system. With adjustments in the parameters, the DLVO pair-potential can describe the results quantitatively. Besides electrostatic interactions, entropic depletion effects that arise from (hard-core) exclusion play an important role in determining the behavior of multi-component colloidal suspensions. A standard theory for depletion forces is due to Asakura and Oosawa and is based on the ideal gas approximation. To go beyond this approximation, we have studied entropic forces in molecular dynamics simulations of systems of hard spheres (the effects of the solvent have been ignored). The effective depletion forces for these systems can be found either from equilibrium distribution functions or from direct momentum transfer calculations. Our results obtained by either method show qualitative differences from the Asakura-Oosawa forces, indicating a longer range, higher value at contact and most importantly a more complicated structure, comprising of several maxima and minima. Our calculations include the determination of effective forces between two spheres, a hard sphere and a wall, and the behavior of a hard sphere near a step-edge and a corner. We also demonstrate that such entropic forces do not necessarily satisfy pairwise additivity.

Tehver, Riina↗

Hugoniot equation of state of anorthite glass and lunar anorthosite

Twenty-one Hugoniot experiments were conducted on an amorphous material of anorthite composition, in the pressure range 8-120 GPa, using both routine and new methods. Two Hugoniot measurements at about 120 GPa were made on lunar gabbroic anorthosite (Apollo 15,418). Theoretical Hugoniots are constructed for both materials assuming they are disproportionate to their component oxides. These accurately predict the P-p behavior of the lunar anorthosite Hugoniot at 120 GPa and the anorthite glass Hugoniot above 50 GPa, but overestimate the shock temperatures of anorthite glass. The mixed oxide model fails to predict the release paths of either material. It is concluded that the mixed oxide model is a good description of the bulk properties of the high-pressure phases of anorthite, but does not represent the actual phases. A significant enrichment of calcic refractory material in the earth's lower mantle is not precluded by the bulk properties of the anorthite high-pressure phases.

Boslough, M. B.↗

Diffusional decay of striations

This paper investigates the amount of decay of growth striations in bulk crystals grown from melts and solutions, due to solid-state diffusion during the cooling of the crystal to room temperature. A model is developed based on the assumption of a sinusoidal variation in the solid composition at the interface of a semiinfinite ingot growing at constant velocity, and the damping behavior of this variation is followed as it moves away from the solid-liquid interface. Numerical results are presented showing that an increase in the diffusivity causes the striations to penetrate deeper into the solid and, consequently causes the distance for 99.9 percent decay to increase. Analytical results are also presented for the decay of striations in an isothermal ingot induced by a sinusoidal fluctuation in the rate of crystal growth.

Gray, R. T.↗

Fluid Film Bearing Code Development

The next generation of rocket engine turbopumps is being developed by industry through Government-directed contracts. These turbopumps will use fluid film bearings because they eliminate the life and shaft-speed limitations of rolling-element bearings, increase turbopump design flexibility, and reduce the need for turbopump overhauls and maintenance. The design of the fluid film bearings for these turbopumps, however, requires sophisticated analysis tools to model the complex physical behavior characteristic of fluid film bearings operating at high speeds with low viscosity fluids. State-of-the-art analysis and design tools are being developed at the Texas A&M University under a grant guided by the NASA Lewis Research Center. The latest version of the code, HYDROFLEXT, is a thermohydrodynamic bulk flow analysis with fluid compressibility, full inertia, and fully developed turbulence models. It can predict the static and dynamic force response of rigid and flexible pad hydrodynamic bearings and of rigid and tilting pad hydrostatic bearings. The Texas A&M code is a comprehensive analysis tool, incorporating key fluid phenomenon pertinent to bearings that operate at high speeds with low-viscosity fluids typical of those used in rocket engine turbopumps. Specifically, the energy equation was implemented into the code to enable fluid properties to vary with temperature and pressure. This is particularly important for cryogenic fluids because their properties are sensitive to temperature as well as pressure. As shown in the figure, predicted bearing mass flow rates vary significantly depending on the fluid model used. Because cryogens are semicompressible fluids and the bearing dynamic characteristics are highly sensitive to fluid compressibility, fluid compressibility effects are also modeled. The code contains fluid properties for liquid hydrogen, liquid oxygen, and liquid nitrogen as well as for water and air. Other fluids can be handled by the code provided that the user inputs information that relates the fluid transport properties to the temperature.

Source record↗

Effects of Heating Configuration and Operating Parameters on Heat Transfer and Interfacial Physics of Microgravity Flow Boiling With Subcooled Inlet Conditions –Experiments Onboard the International Space Station

This study is part of the Flow Boiling and Condensation Experiment (FBCE), a collaborative effort between the Purdue University Boiling and Two-Phase Flow Laboratory (PU-BTPFL) and the NASA Glenn Research Center. The FBCE fitted with the Flow Boiling Module (FBM) was launched to the International Space Station (ISS) in August 2021 and experiments were successfully performed from February to July 2022 to amass a large microgravity-flow-boiling database. This study is focused on heat transfer and flow visualization of microgravity flow boiling of n-Perfluorohexane in a rectangular channel of 5.0 mm height, 2.5 mm width (heated), and 114.6 mm length, with subcooled inlet conditions. High-speed-video photography is utilized to present flow patterns and temporal interfacial behavior. Heat transfer results are presented in the form of flow boiling curves and both parametric curves and streamwise profiles of wall temperature and heat transfer coefficient. Firstly, the parametric effects of mass velocity (199.4 – 3200.0 kg/m 2 s), inlet subcooling (0.2 – 46.0°C), and inlet pressure (124.2 – 176.7 kPa), on the aforementioned aspects are assessed for double-sided heating to establish them for a microgravity environment. Of these three parameters, mass velocity and inlet subcooling mostly determine the microgravity flow boiling behavior, while inlet pressure plays an insignificant role. Flow patterns for double-sided heating are more complex than those for single-sided heating due to interaction between the two vapor layers. Vapor interaction is minimized at high subcoolings and high mass velocities due to strong condensation offered by the subcooled bulk liquid layer separating them. Despite the different flow patterns, both single- and double-sided heating generally result in similar parametric trends and local heat transfer coefficients for similar operating conditions. Flow instabilities manifest as temporal flow anomalies and temperature oscillations, and their severity increases with increasing boiling number. Secondly, the effects of heating configuration are analyzed by comparing and contrasting several aspects of single- and double-sided heating data. The heat fluxes at which onset of nucleate boiling degradation (ONBD) and critical heat flux (CHF) occur are distinctly different for single- and double-sided heating. There exists a threshold inlet subcooling demarcating the dominance of flow acceleration and condensation effects in vapor removal from the near-wall region and replenishment of fresh liquid for boiling. Above the threshold, condensation from the near-wall region is dominant and single-sided heating yields higher heat fluxes, and below it, acceleration is dominant and double-sided yields higher heat fluxes. At mass velocity in the range of 200 – 2400 kg/m 2 s, the threshold inlet subcooling lies in the approximate range of 20 – 30°C (corresponding inlet quality of roughly -0.40 – -0.20).

Microgravity↗

Mechanism of lubrication by tricresylphosphate (TCP)

A pin-on-disk tribometer equipped with an induction heater was used to study the coefficient of friction as a function of temperature for tricresylphosphate (TCP) on continuous vacuum melted (CVM) M-50 tool steel when the TCP was present in a liquid reservoir (bulk lubrication), and when it was applied as a liquid layer directly to the disk (limited lubrication). Under limited lubrication conditions, experiments were performed in dry ( 100 ppm H2O) air, dry ( 20 ppm H2O) nitrogen, dry nitrogen with the disks heated to 700 C then cooled to room temperature before the TCP was applied and the measurements made (preheated disks), and moist nitrogen using preheated disks. When the coefficient of friction was plotted as a function of the disk temperature, the friction decreased at a characteristic temperature, T sub r whose observed values were 265 C for bulk lubrication conditions in dry air, 225 C for limited lubrication conditions in dry air, and 215 C for limited lubrication conditions in dry nitrogen. No decrease in friction was observed with preheated disks; instead a sharp failure temperature was observed at 218 C, which was taken as the temperature about which the behavior of TCP should be judged, X-ray photoelectron spectroscopy confirmed the presence of phosphate on the surface of the iron pins used in the tribometer under TCP lubrication. Depth profile studies support the idea that a chemical reaction occurs between the TCP and the metal surface at T sub r.

Faut, O. D.↗

Laboratory studies of HNO3/H2O mixtures at low temperatures

The possibility of stratospheric HN3 condensing out of the gas-phase at low temperatures has become important in the chemical explanations of the rapid loss of Antarctic ozone in spring. Consequently, knowledge about the behavior of the vapor pressures of water and HNO3 over HNO3/water mixtures at stratospheric temperatures is needed to determine if HNO3 could condense, and by how much the HNO3 vapor pressure could be depressed. Laboratory investigations of vapor pressures above HNO3/water mixtures are described. Vapor pressures were initially measured over liquid and frozen bulk mixtures contained in a glass still which was attached to a stainless steel vacuum chamber. The total pressure in the chamber was monitored with a precision pressure sensor, and the vapor pressures of HNO3, water and impurities were analyzed with a mass spectrometer beam system. In a second set of experiments, vapor deposits were made to produce ice and the mono- and trihydrate of HNO3. Further data and conclusions are presented.

Hanson, David↗

Implication of the Observable Spectral Cutoff Energy Evolution in XTE J1550-564

The physical mechanisms responsible for production of the non-thermal emission in accreting black holes should be imprinted in the observational appearances of the power law tails in the X-ray spectra from these objects. Variety of spectral states observed from galactic black hole binaries by it Rossi X-ray Timing Explorer (RXTE) allow examination of the photon upscattering under different accretion regimes. We revisit of RXTE data collected from the black hole X-ray binary XTE J1550-564 during two periods of X-ray activity in 1998 and 2000 focusing on the behavior of the high energy cutoff of the power law part of the spectrum. For the 1998 outburst the Iran- sition from the low-hard state to the intermediate state was accompanied by a gradual decrease in the cutoff energy which then showed a sharp reversal to a clear increasing trend during the further evolution towards the very high and high-soft states. However, the 2000 outburst showed only the decreasing part of this pattern. Notably, the photon indexes corresponding to the cutoff increase for the 1998 event are much higher than the index values reached during the 2000 rise transition. We attribute this difference in the cutoff' energy behav- for to the different partial contributions of the thermal and non-thermal (bulk motion) Comptonization in photon upscattering. Namely, during the 1998 event the higher accretion rate presumably provided more cooling to the Comptonizing media and thus reducing the effectiveness of the thermal upscattering process. Under these conditions the bulk motion takes a leading role in boosting the input soft photons. Monte Carlo simulations of the The physical mechanisms responsible for production of the non-thermal emission in accreting black holes should be imprinted in the observational apperances of the power law tails in the X-ray spectra from these objects. Variety of spectral states observed from galactic black hole binaries by it Rossi X-ray Timing Explorer (RXTE) allow examination of the photon upscattering under different accretion regimes. We revisit of RXTE data collected from the black hole X-ray binary XTE J1550-564 during two periods of X-ray activity in 1998 and 2000 focusing on the behavior of the high energy cutoff of the power law part of the spectrum. For the 1998 outburst the Iran- sition from the low-hard state to the intermediate state was accompanied by a gradual decrease in the cutoff energy which then showed a sharp reversal to a clear increasing trend during the further evolution towards the very high and high-soft states. However, the 2000 outburst showed only the decreasing part of this pattern. Notably, the photon indexes corresponding to the cutoff increase for the 1998 event are much higher than the index values reached during the 2000 rise transition. We attribute this difference in the cutoff' energy behav- for to the different partial contributions of the thermal and non-thermal (bulk motion) Comptonization in photon upscattering. Namely, during the 1998 event the higher accretion rate presumably provided more cooling to the Comptonizing media and thus reducing the effectiveness of the thermal upscattering process. Under these conditions the bulk motion takes a leading role in boosting the input soft photons. Monte Carlo simulations of the Comptonization in a bulk motion region near an accreting black hole by Laurent & Titarchuk (2010) strongly support this scenario. strongly support this scenario

Titarchuk, Lev↗

A Study of Time-dependent and Anisotropic Effects on the Deformation Response of Two Flywheel Designs

The influence of material time dependency and anisotropy in the context of two specific flywheel designs-preload and multi-directional composite (MDC)--is investigated. In particular, we focus on the following aspects: 1) geometric constraints, 2) material constraints, 3) loading type, and 4) the fundamental character of the time-dependent response, i.e., reversible or irreversible. The bulk of the results presented were obtained using a composite (PMC IM7/8552 at 135 C) material system. The material was characterized using a general multimechanism hereditary (viscoelastoplastic) model. As a general conclusion, the results have clearly shown that both the preload and the MDC rotor designs are significantly affected by time-dependent material behavior, which may impact the state of rotor balance and potentially reduce its operating life. In view of the results of the parametric studies and predictions made in the present study, the need for actual experimentation focusing on the time-dependent behavior of full-scale flywheel rotors is self-evident.

Saleeb, Atef F.↗

Surface chemistry, microstructure and friction properties of some ferrous-base metallic glasses at temperatures to 750 C

X-ray photoelectron spectroscopy analysis, transmission electron microscopy, diffraction studies, and sliding friction experiments were conducted with ferrous-base metallic glasses in sliding contact with aluminum oxide at temperatures from room to 750 C in a vacuum of 30 nPa. The results indicate that there is a significant temperature influence on the friction properties, surface chemistry, and microstructure of metallic glasses. The relative concentrations of the various constituents at the surface of the sputtered specimens were very different from the normal bulk compositions. Contaminants can come from the bulk of the material to the surface upon heating and impart boric oxide and silicon oxide at 350 C and boron nitride above 500 C. The coefficient of friction increased with increasing temperature to 350 C. Above 500 C the coefficient of friction decreased rapidly. The segregation of contaminants may be responsible for the friction behavior.

Miyoshi, K.↗

Measurement and calculation of levitation forces between magnets and granular superconductors

Recent developments indicate that exploitation of the phenomenon of magnetic levitation may become one of the most important near-term applications of high-T(sub c) superconductivity. Because of this, the interaction between a strong permanent magnet(PM) and bulk high-T(sub c) superconductor (HTSC) is currently a subject of much interest. We have studied central features of the mechanics of PM-HTSC systems of simple geometries. Here we report experimental results for the components of the levitation force, their associated stiffness and mechanical ac-loss. To analyze the observed behavior a theoretical framework based on critical-state considerations is developed. It will be shown that all the mechanical properties can be explained consistently at a quantitative level wing a minimum of model parameters.

Johansen, T. H.↗

Molecular Dynamics of ULTEM 9085 for 3D Manufacturing: Spectra, Thermodynamic Properties, and Shear Viscosity.

We present results of a molecular dynamic analysis of welding at the polymer-polymer interface. The analysis is performed for polyetherimide/ polycarbonate polymer blends. The work is motivated by the applications to 3D manufacturing in space. In the first part of the report, we discuss bulk and spectral characteristics of the amorphous polymer blends. The vibrational and infra-red spectra obtained using auto-correlation functions calculations in molecular dynamics are compared with the experimental spectra. The mechanical and thermal properties of the samples including heat capacity, bulk modulus, and thermal expansion coefficients are estimated and compared with experimental values. In the second part of the report, we discuss the result of molecular dynamical modeling of shear viscosity in a fully atomistic model of amorphous polymer blends with flat interface. The key result of the research is the demonstration of shear thinning behavior of the shear viscosity as a function of shear rate which is in good agreement with experimental data.

Luchinsky, Dmitry G.↗