Newtonian Predictions Research Articles

Giant deviation of a relaxation time from generalized Newtonian theory in discontinuous shear thickening suspensions

We investigated the transient relaxation of a Discontinuous Shear Thickening suspension of cornstarch in water. We performed 2 types of relaxation experiments starting from a steady shear in a parallel plate rheometer, followed by either stopping the top plate rotation and measuring the transient torque relaxation, or removing the torque on the plate and measuring the transient tool rotation. We found that at low weight fraction $\phi_{eff}<58.8\pm0.4\%$, the suspensions exhibited a relaxation behavior consistent with a generalized Newtonian fluid. However, for larger weight fraction $58.8\% < \phi_{eff} < 61.0\%$, near the liquid-solid transition $\phi_c=61.0\pm0.7\%$, we found relaxation behaviors different from the generalized Newtonian model. The relaxation time in this range scales with the inverse of the critical shear rate at the onset of shear thickening. In this range the relaxation time was the same in both stress and rate controlled experiments, rather than the viscosity calculated from the relaxation time which is expected to be intrinsic material parameter in the generalized Newtonian model. The discrepancy between the measured relaxation times and the generalized Newtonian prediction was found to be up to $10^4$, and extrapolations diverge in the limit of $\phi_c$ as the generalized Newtonian prediction approaches 0. At the highest weight fractions, the relaxation time scales were measured to be on the order of $\sim 1$ s. The fact that this timescale is resolvable by the naked eye may be important to understanding some of the dynamic phenomenon commonly observed in these systems. We also showed that using the critical shear rate $\dot\gamma_c$ at the onset of shear thickening to characterize the effective weight fraction can more precisely characterize material properties near the critical point $\phi_c$, allowing us to resolve this transition so close to $\phi_c$.

Imprints of local lightcone \\ projection effects on the galaxy bispectrum. Part II

General relativistic imprints on the galaxy bispectrum arise from observational (or projection) effects. The lightcone projection effects include local contributions from Doppler and gravitational potential terms, as well as lensing and other integrated contributions. We recently presented for the first time, the correction to the galaxy bispectrum from all local lightcone projection effects up to second order in perturbations. Here we provide the details underlying this correction, together with further results and illustrations. For moderately squeezed shapes, the correction to the Newtonian prediction is ∼ 30% on equality scales at z∼ 1. We generalise our recent results to include the contribution, up to second order, of magnification bias (which affects some of the local terms) and evolution bias.

A general relativistic signature in the galaxy bispectrum: the local effects of observing on the lightcone

Next-generation galaxy surveys will increasingly rely on the galaxy bispectrum to improve cosmological constraints, especially on primordial non-Gaussianity. A key theoretical requirement that remains to be developed is the analysis of general relativistic effects on the bispectrum, which arise from observing galaxies on the past lightcone, as well as from relativistic corrections to the dynamics. As an initial step towards a fully relativistic analysis of the galaxy bispectrum, we compute for the first time the local relativistic lightcone effects on the bispectrum, which come from Doppler and gravitational potential contributions. For the galaxy bispectrum, the problem is much more complex than for the power spectrum, since we need the lightcone corrections at second order. Mode-coupling contributions at second order mean that relativistic corrections can be non-negligible at smaller scales than in the case of the power spectrum. In a primordial Gaussian universe, we show that the local lightcone projection effects for squeezed shapes at z ∼ 1 mean that the bispectrum can differ from the Newtonian prediction by ≳ 10% when the short modes are k ≲ (50 Mpc)−1. These relativistic projection effects, if ignored in the analysis of observations, could be mistaken for primordial non-Gaussianity. For upcoming surveys which probe equality scales and beyond, all relativistic lightcone effects and relativistic dynamical corrections should be included for an accurate measurement of primordial non-Gaussianity.

A proposed test of special-relativistic mechanics at low speed

We show that the difference between the Newtonian and special-relativistic predictions for the angular position increases linearly with time for a charged particle moving at low speed in a circular path in a constant uniform magnetic field. Numerical results suggest that it is possible to test the two different predictions experimentally.

Second-order cosmological perturbation theory and initial conditions forN-body simulations

We use gauge-invariant cosmological perturbation theory to calculate the displacement field that sets the initial conditions for $N$-body simulations. Using first and second-order fully relativistic perturbation theory in the synchronous-comoving gauge, allows us to go beyond the Newtonian predictions and to calculate relativistic corrections to it. We use an Einstein--de Sitter model, including both growing and decaying modes in our solutions. The impact of our results should be assessed through the implementation of the featured displacement in cosmological $N$-body simulations.

On the accuracy of N-body simulations at very large scales

We examine the deviation of Cold Dark Matter particle trajectories from the Newtonian result as the size of the region under study becomes comparable to or exceeds the particle horizon. To first order in the gravitational potential, the general relativistic result coincides with the Zel'dovich approximation and hence the Newtonian prediction on all scales. At second order, General Relativity predicts corrections which overtake the corresponding second order Newtonian terms above a certain scale of the order of the Hubble radius. However, since second order corrections are very much suppressed on such scales, we conclude that simulations which exceed the particle horizon but use Newtonian equations to evolve the particles, reproduce the correct trajectories very well. The dominant relativistic corrections to the power spectrum on scales close to the horizon are at most of the order of $\sim 10^{-5}$ at $z=49$ and $\sim 10^{-3}$ at $z=0$. The differences in the positions of real space features are affected at a level below $10^{-6}$ at both redshifts. Our analysis also clarifies the relation of N-body results to relativistic considerations.

Taming the post-Newtonian expansion: Simplifying the modes of the gravitational wave energy flux at infinity for a point particle in a circular orbit around a Schwarzschild black hole

(Abridged) High-order terms in the post-Newtonian (PN) expansions of various quantities for compact binaries exhibit a combinatorial increase in complexity, including ever-increasing numbers of transcendentals. Here we consider the gravitational wave energy flux at infinity from a point particle in a circular orbit around a Schwarzschild black hole, which is known to 22PN beyond the lowest-order Newtonian prediction, at which point each order has over 1000 terms. We introduce a factorization that considerably simplifies the spherical harmonic modes of the energy flux (and thus also the amplitudes of the spherical harmonic modes of the gravitational waves); it is likely that much of the complexity this factorization removes is due to curved-space wave propagation (e.g., tail effects). For the modes with azimuthal number l of 7 or greater, this factorization reduces the expressions for the modes that enter the 22PN total energy flux to pure integer PN series with rational coefficients, which amounts to a reduction of up to a factor of ~150 in the total number of terms in a given mode. The reduction in complexity becomes less dramatic for smaller l, due to the structure of the expansion, though the factorization is still able to remove all the half-integer PN terms. For the 22PN l = 2 modes, this factorization still reduces the total number of terms (and size) by a factor of ~10 and gives purely rational coefficients through 8PN. This factorization also improves the convergence of the series, though we find the exponential resummation introduced for the full energy flux by Isoyama et al. to be even more effective at improving the convergence of the individual modes, producing improvements of over four orders of magnitude over the original series for some modes. However, the exponential resummation is not as effective at simplifying the series, particularly for the higher-order modes.

Newtonian versus Special-Relativistic Statistical Predictions for Low-Speed Scattering

The statistical predictions of Newtonian and special-relativistic mechanics, which are calculated from an initially Gaussian ensemble of trajectories, are compared for a low-speed scattering system. The comparisons are focused on the mean dwell time, transmission and reflection coefficients, and the position and momentum means and standard deviations. We find that the statistical predictions of the two theories do not always agree as conventionally expected. The predictions are close if the scattering is non-chaotic but they are radically different if the scattering is chaotic and the initial ensemble is well localized in phase space. Our result indicates that for low-speed chaotic scattering, special-relativistic mechanics must be used, instead of the standard practice of using Newtonian mechanics, to obtain empirically-correct statistical predictions from an initially well-localized Gaussian ensemble.

Statistical Predictions for the Dynamics of a Low-Speed System: Newtonian versus Special-Relativistic Mechanics

The Newtonian and special-relativistic statistical predictions for the mean, standard deviation and probability density function of the position and momentum are compared for the periodically-delta-kicked particle at low speed. Contrary to expectation, we find that the statistical predictions, which are calculated from the same parameters and initial Gaussian ensemble of trajectories, do not always agree if the initial ensemble is sufficiently well-localized in phase space. Moreover, the breakdown of agreement is very fast if the trajectories in the ensemble are chaotic, but very slow if the trajectories in the ensemble are non-chaotic. The breakdown of agreement implies that special-relativistic mechanics must be used, instead of the standard practice of using Newtonian mechanics, to correctly calculate the statistical predictions for the dynamics of a low-speed system.

COMPARISON OF GENERAL RELATIVISTIC AND PSEUDO-NEWTONIAN DESCRIPTION OF MAGELLANIC-CLOUDS MOTION IN THE FIELD OF MILKY WAY

We test precision of the Cosmological Paczynski–Wiita (CPW) potential reflecting properties of the Schwarzschild–de Sitter (SdS) spacetimes in modeling dynamical phenomena related to galaxy motion. We consider a simplified model of Magellanic Clouds moving in the field of Milky Way as test particles. Time evolution of their position along trajectories obtained in the CPW framework using the notion of Newtonian time is compared to the one obtained in the fully general relativistic (GR) approach when the time evolution is expressed in terms of time related to the location of Earth in the Galaxy field. The differences in the position-evolution of the Magellanic Clouds obtained in the CPW and GR approaches are given for appropriately chosen values of the Milky Way mass. It is shown that the integrated relativistic corrections represent ~10-5 part of the Newtonian CPW predictions for the orbital characteristics of the motion and slightly grow with Galaxy mass growing, being at least by one order higher than the local scaling GR corrections. The integrated orbital GR corrections thus could be important only in very precise modeling of the motion of Magellanic Clouds. The CPW framework is used to show that, quite surprisingly, the influence of the cosmological constant on the Magellanic Clouds motion can be strong and significantly alters the trajectories of Magellanic Clouds and time evolution along them. The relative contribution of the cosmological constant is ~10-1 or higher. It is most profoundly demonstrated by the increase of the binding mass that represents 22% for Small Magellanic Cloud and even 47% for Large Magellanic Cloud, putting serious doubts on gravitational binding to the Milky Way in the later case.

Influence of Shear Thinning on Viscoelastic Fluid–Structure Interaction in a Two-Dimensional Collapsible Channel

We compute numerically the steady flow of a viscoelastic FENE-P fluid in a two-dimensional collapsible channel, in which a zero-thickness membrane under constant tension has replaced part of the top wall. This geometry has been studied extensively as a benchmark to examine fluid-structure interactions between a flowing Newtonian fluid and a deformable wall. The rheological behavior of the viscoelastic FENE-P fluid is described in terms of a conformation tensor model. The mesh equation and transport equations are discretized using the discrete elastic viscous stress split-traceless gradient (DEVSS-TG/SUPG) mixed finite-element method. The influence of viscoelasticity and shear thinning on flow patterns and stress profiles is examined, and comparisons with Newtonian predictions are reported. The existence of a limiting Weissenberg number beyond which computations fail is demonstrated. The extent of shear thinning is shown to be a key factor in determining the nature of the fluid-structure interaction.

Newtonian and special-relativistic predictions for the trajectories of a low-speed scattering system

Newtonian and special-relativistic predictions, based on the same model parameters and initial conditions for the trajectory of a low-speed scattering system are compared. When the scattering is chaotic, the two predictions for the trajectory can rapidly diverge completely, not only quantitatively but also qualitatively, due to an exponentially growing separation taking place in the scattering region. In contrast, when the scattering is nonchaotic, the breakdown of agreement between predictions takes a very long time, since the difference between the predictions grows only linearly. More importantly, in the case of low-speed chaotic scattering, the rapid loss of agreement between the Newtonian and special-relativistic trajectory predictions implies that special-relativistic mechanics must be used, instead of the standard practice of using Newtonian mechanics, to correctly describe the scattering dynamics.

Newtonian versus general-relativistic prediction for the trajectory of a bouncing ball system

The dynamics of a bouncing ball undergoing repeated inelastic impact with a sinusoidally-oscillating table is studied numerically. For the ball bouncing at low speed in a weak gravitational field, we show, contrary to expectation, that the trajectories predicted by Newtonian mechanics and general-relativistic mechanics from the same parameters and initial conditions will rapidly disagree completely if the trajectories are chaotic. When the two theories predict completely different chaotic trajectories for a low-speed weak-gravity system, we expect the general-relativistic prediction is empirically correct since general relativity continues to be verified in recent high-precision experiments.

The effect of viscoelasticity on a rising gas bubble

This paper investigates the role of viscoelasticity on the dynamics of rising gas bubbles. The dynamics of bubbles rising in a viscoelastic liquid are characterised by three phenomena: the trailing edge cusp, negative wake, and the rise velocity jump discontinuity. There is much debate in the literature over the cause of the jump discontinuity, which is observed once the bubble exceeds a certain critical volume. In this paper, the employment of some choice modelling assumptions allows insights into the mechanisms of the jump discontinuity which cannot be ascertained experimentally. The ambient fluid is assumed incompressible and the flow irrotational, with viscoelastic effects included through the stress balance on the bubble surface. The governing equations are solved using the boundary element method. Some Newtonian predictions are discussed before investigating the role of viscoelasticity. The model predicts the trademark cusp at the trailing end of a rising bubble to a high resolution. However, the irrotational assumption precludes the prediction of the negative wake. The corresponding absence of the jump discontinuity supports the hypothesis that the negative wake is primarily responsible for the jump discontinuity, as mooted in previous studies.

A brief history of disk color mixture

AbstractDisk color mixture, known since antiquity, represents a special kind of additive color mixture where the stimuli do not represent direct lights from light sources but light reflected from objects. Its complete elucidation took centuries, requiring means to measure reflectance properties and spectral powers of lights. Explanations of the cause of the mixture, the fact that perceptions are not instantaneous but take a certain amount of time were offered in the late 17th and early 18th centuries. The speed of disk rotation required to obtain additive fusion of reflected lights was determined in 1765. For a time in the second half of the 19th century disk mixture was the only means of establishing color matching equations and quantitatively defining color stimuli. The fact that its results neither agree with Newtonian predictions of spectral light mixture nor with the results of pigment mixture produced endless confusion in scientific and artistic circles in the Enlightenment. Today, there is no practical application of disk color mixture left and it has become a curiosity. © 2010 Wiley Periodicals, Inc. Col Res Appl, 2010

Newtonian and special-relativistic predictions for the trajectory of a slow-moving dissipative dynamical system

We show that the trajectories predicted by Newtonian mechanics and special relativistic mechanics from the same parameters and initial conditions for a slow-moving dissipative dynamical system will rapidly disagree completely if the trajectories are chaotic or transiently chaotic. There is no breakdown of agreement if the trajectories are non-chaotic, in contrast to the slow breakdown of agreement between non-chaotic Newtonian and relativistic trajectories for a slow-moving non-dissipative dynamical system studied previously. We argue that, once the two trajectory predictions are completely different for a slow-moving dissipative dynamical system, special relativistic mechanics must be used, instead of the standard practice of using Newtonian mechanics, to correctly study its trajectory.

Astronomical tests of relativity: beyond parameterized post-Newtonian formalism (PPN), to testing fundamental principles

AbstractBy the early 1970s, the improved accuracy of astrometric and time measurements enabled researchers not only to experimentally compare relativistic gravity with the Newtonian predictions, but also to compare different relativistic gravitational theories (e.g., the Brans-Dicke Scalar-Tensor Theory of Gravitation). For this comparison, Kip Thorne and others developed the Parameterized Post-Newtonian Formalism (PPN), and derived the dependence of different astronomically observable effects on the values of the corresponding parameters.Since then, all the observations have confirmed General Relativity. In other words, the question of which relativistic gravitation theory is in the best accordance with the experiments has been largely settled. This does not mean that General Relativity is the final theory of gravitation: it needs to be reconciled with quantum physics (into quantum gravity), it may also need to be reconciled with numerous surprising cosmological observations, etc. It is, therefore, reasonable to prepare an extended version of the PPN formalism, that will enable us to test possible quantum-related modifications of General Relativity.In particular, we need to include the possibility of violating fundamental principles that underlie the PPN formalism but that may be violated in quantum physics, such as scale-invariance, T-invariance, P-invariance, energy conservation, spatial isotropy violations, etc. In this paper, we present the first attempt to design the corresponding extended PPN formalism, with the (partial) analysis of the relation between the corresponding fundamental physical principles.

The Shear-Thinning Elastohydrodynamic Film Thickness of a Two-Component Mixture

Lubricant base oils are often blends of different molecular weight cuts to arrive at a specified ambient pressure viscosity and, to improve the temperature-viscosity behavior or to simply increase the viscosity, viscosity-modifying polymer additives are often added to the base oil. This paper investigates the effect of mixture rheology on elastohydrodynamic lubrication (EHL) film thickness using EHL contact measurements and a full numerical analysis for three synthetic lubricants including two single-component lubricants PAO650 and PAO100 and a mixture of these. The pressure and shear dependences of the viscosity of these lubricants were measured with high-pressure viscometers; viscosities were not adjusted to fit experiment. The point contact film thicknesses for these lubricants in pure rolling were measured using a thin-film colorimetric interferometry apparatus. Numerical simulations based on the measured rheology show very good agreement with the measurements of film thickness while the Newtonian prediction is up to twice the measurement. These results validate the use of realistic shear-thinning and pressure-viscosity models, which originate from viscosity measurements. It is conceivable that simulation may provide a means to “engineer” lubricants with the optimum balance of film thickness and friction through intelligent mixing of components.

Measuring Glassy and Viscoelastic Polymer Flow in Molecular-Scale Gaps Using a Flat Punch Mechanical Probe

This paper investigates molecular-scale polymer mechanical deformation during large-strain squeeze flow of polystyrene (PS) films, where the squeeze flow gap is close to the polymer radius of gyration (R(g)). Stress-strain and creep relations were measured during flat punch indentation from an initial film thickness of 170 nm to a residual film thickness of 10 nm in the PS films, varying molecular weight (M(w)) and deformation stress rate by over 2 orders of magnitude while temperatures ranged from 20 to 125 degrees C. In stress-strain curves exhibiting an elastic-to-plastic yield-like knee, the response was independent of M(w), as expected from bulk theory for glassy polymers. At high temperatures and long times sufficient to extinguish the yield-knee, the mechanical response M(w) degeneracy was broken, but no molecular confinement effects were observed during thinning. Creep measurements in films of 44K M(w) were well-approximated by bulk Newtonian no-slip flow predictions. For extrusions down to a film thickness of 10 nm, the mechanical relaxation in these polymer films scaled with temperature similar to Williams-Landel-Ferry scaling in bulk polymer. Films of 9000K M(w), extruded from an initial film thickness of 2R(g) to a residual film thickness of 0.5R(g), while showing stress-strain viscoelastic response similar to that of films of 900K M(w), suggestive of shear-thinning behavior, could not be matched to a constitutive flow model. In general, loading rate and magnitude influenced subsequent creep extrusion depth of high-M(w) films, with deeper final extrusions for high loading rates than for low loading rates. The measurements suggest that, for high-resolution nanoimprint lithography, mold flash or final residual film thickness can be reduced for high strain and strain rate loading of high-M(w) thin films.

Foundations of multiple-black-hole evolutions

We present techniques for long-term, stable, and accurate evolutions of multiple-black-hole spacetimes using the 'moving puncture' approach with fourth- and eighth-order finite-difference stencils. We use these techniques to explore configurations of three black holes in a hierarchical system consisting of a third black hole approaching a quasicircular black-hole binary, and find that, depending on the size of the binary, the resulting encounter may lead to a prompt merger of all three black holes, production of a highly elliptical binary (with the third black hole remaining unbound), or disruption of the binary (leading to three free black holes). We also analyze the classical Burrau three-body problem using full numerical evolutions. In both cases, we find behaviors distinctly different from Newtonian predictions, which has important implications for N-body black-hole simulations. For our simulations we use approximate analytic initial data. We find that the eighth-order stencils significantly reduce the numerical errors for our choice of grid sizes, and that the approximate initial data produces the expected waveforms for black-hole binaries with modest initial separations.