Newtonian Part Research Articles

Generalized flow-composed symplectic methodsfor post-Newtonian Hamiltonian systems

Due to the nonseparability of the post-Newtonian (PN) Hamiltonian systems of compact objects, the symplectic methods that admit the linear error growth and the near preservation of first integrals are always implicit as explicit symplectic methods have not been currently found for general nonseparable Hamiltonian systems. Since the PN Hamiltonian has a particular formulation that includes a dominant Newtonian part and a perturbation PN part, we present the generalized flow-composed Runge-Kutta (GFCRK) method with a free parameter λ to PN Hamiltonian systems. It is shown that the GFCRK method is symplectic once the underlying RK method is symplectic, and it is symmetric once the underlying RK method is symmetric under the setting λ = 1/2. Numerical experiments with the 2PN Hamiltonian of spinning compact binaries demonstrate the higher accuracy and efficiency of the symplectic GFCRK method than the underlying symplectic RK method in the case of weak PN effect. Meanwhile, the numerical results also support higher efficiency of the symplectic GFCRK method than the semi-explicit mixed symplectic method of the same order.

Apparent Parity Violation in the Observed Galaxy Trispectrum.

Recent measurements of the four-point correlation function in large-scale galaxy surveys have found apparent evidence of parity violation in the distribution of galaxies. This cannot happen via dynamical gravitational effects in general relativity. If such a violation arose from physics in the early Universe it could indicate important new physics beyond the standard model, and would be at odds with most models of inflation. It is therefore now timely to consider the galaxy trispectrum in more detail. While the intrinsic four-point correlation function, or equivalently the trispectrum, its Fourier counterpart, is parity invariant, the observed trispectrum must take redshift-space distortions into account. Although the standard Newtonian correction also respects parity invariance, we show that subleading relativistic corrections do not. We demonstrate that these can be significant at intermediate linear scales and are dominant over the Newtonian parity-invariant part around the equality scale and above. Therefore when observing the galaxy four-point correlation function, we should expect to detect parity violation on large scales.

Oscillatory Marangoni flow in a deep layer of a Carreau fluid

We study the spreading dynamics of an insoluble and non-diffusive surfactant on the free surface of a deep layer of a Carreau fluid. When a non-uniform distribution of surfactant is imposed on the free surface of an initially motionless fluid, variations in surface tension are induced, causing the fluid motion from regions of lower surface tension to those of higher tension. Such movement redistributes the surfactant concentration until a uniform condition is reached. The two-dimensional momentum and convection–diffusion equations are employed to determine the hydrodynamics in the deep fluid layer and the evolution of the surfactant on the fluid surface. The modified vorticity-stream function formulation is used to solve numerically the hydrodynamic field, where the non-Newtonian dependent variables of the problem are decomposed into a Newtonian part and a non-Newtonian contribution. One of the most critical variables in practical applications regarding this phenomenon concerns the time required for the surfactant to reach a uniform distribution over the interface. The results show that the dimensionless parameters that control the decay of the variations in surfactant concentration in time are the Reynolds number Re, the fluid behavior index n, and ε, reflecting the influence of the inertia of the fluid. The results show that the temporal decay of the surfactant concentration on the fluid surface increases significantly for high values of the Reynolds number, it is more attenuated in pseudoplastic fluids than in Newtonian fluids, and the surfactant concentration decay exhibits asymmetric oscillations when inertial effects increase.

Should we teach free-body diagrams before or after Newton’s Laws?

There are two interesting lesson sequences for teaching force and motion in high-school physics. These are teaching free-body diagrams before Newton’s laws (FbN) and teaching Newton’s laws before free-body diagrams (NbF). Both sequences were found in physics textbooks. Different authors adopted the sequence that they believe it would affect student understanding better. However, some physics experts did not agree with this. It is therefore interesting to know if we should teach with the FbN or NbF sequence. This motivates us to study the effect of such lesson sequences on student understanding of force and motion. The sample group was grade-10 students from two physics courses in 2020. One course was taught with the FbN sequence (29 students) and the other with the NbF sequence (34 students). Their understanding was evaluated by using an assessment test which consisted of three parts including (1) Newtonian concept, (2) problem solving, and (3) free-body diagrams. The result shows that for the Newtonian concept part, the average scores are 11% for the FbN and 13% for the NbF sequence. The average scores of the problem-solving part are 13% and 9% and those of the free-body diagram part are 41% and 48% for the FbN and NbF sequences, respectively. The scores of all parts between the two sequences were not significantly different. In addition, student difficulties found in all parts were similar. However, a larger number of students who could provide the equation of motion (F = ma) in the problem-solving part was found in the FbN sequence. We might conclude that teaching free-body diagrams before or after Newton’s laws did not affect student understanding in the topic of force and motion. Detail of student difficulties in both sequences will be further discussed.

A first integral to the partially averaged Newtonian potential of the three-body problem

We consider the partial average, i.e. the Lagrange average with respect to just one of the two mean anomalies, of the Newtonian part of the perturbing function in the three-body problem Hamiltonian. We prove that such a partial average exhibits a non-trivial first integral. We show that this integral is fully responsible for certain cancellations in the averaged Newtonian potential, including a property noticed by Harrington in the 1960s. We also highlight its joint role (together with certain symmetries) in the appearance of the so-called Herman resonance. Finally, we discuss an application and an open problem.

Angular momentum of phonons and its application to single-spin relaxation

We reexamine the relaxation process of a single spin embedded in an elastic medium, a problem studied recently by Garanin and Chudnovsky (GC) [Phys. Rev. B 92, 024421 (2015)] from the viewpoint of angular-momentum transfer. Using Noether's theorem, we identify two distinct angular momenta of the medium, one Newtonian discussed by GC and the other field-theoretical, both of which consist of an orbital part and a spin part. For both angular momenta, we found that the orbital part is as essential as the spin part in the relaxation process. In particular, the angular-momentum transfer from the (real) spin to the Newtonian orbital part may be considered as an incipient rotation that leads to the Einstein--de Haas effect.

ANALYSIS OF HYDRODYNAMIC LUBRICATION OF JOURNAL BEARINGS WITH OIL OF NON-NEWTONIAN PROPERTIES

In this paper, the presented issue concerns hydrodynamic lubrication of the journal bearings with the oil of non-Newtonian properties. For the analysis of the hydrodynamic lubrication, a constitutive model of the third order was assumed. The assumed model consist of a Newtonian part –pI+ηA1 and non-Newtonian part β3·tr(A1 2)A1. The main part of this paper concerns transformation, nondimensionalization, and an estimation of the order of magnitude of the equation, which describes the apparent viscosity. Apparent viscosity describes changes in the dynamic viscosity with shear rate. In this way, the prepared model of apparent viscosity is used in momentum equations. These equations are integrated in order to designate components of the velocity vector. By substitution of the proper boundary conditions, a modified Reynolds type equation is obtained. A further stage of the research will be proceeding of the numerical calculations of the hydrodynamic pressure distribution followed by the designation of the carrying capacity, friction force, and friction coefficient while taking changes of the viscosity from shear rate into account (apparent viscosity).

Dynamics of capsules enclosing viscoelastic fluid in simple shear flow

Previous studies on capsule dynamics in shear flow have dealt with Newtonian fluids, while the effect of fluid viscoelasticity remains an unresolved fundamental question. In this paper, we report a numerical investigation of the dynamics of capsules enclosing a viscoelastic fluid and which are freely suspended in a Newtonian fluid under simple shear. Systematic simulations are performed at small but non-zero Reynolds numbers (i.e. $Re=0.1$) using a three-dimensional front-tracking finite-difference model, in which the fluid viscoelasticity is introduced via the Oldroyd-B constitutive equation. We demonstrate that the internal fluid viscoelasticity presents significant effects on the deformation behaviour of initially spherical capsules, including transient evolution and equilibrium values of their deformation and orientation. Particularly, the capsule deformation decreases slightly with the Deborah number De increasing from 0 to $O(1)$. In contrast, with De increasing within high levels, i.e. $O(1{-}100)$, the capsule deformation increases continuously and eventually approaches the Newtonian limit having a viscosity the same as the Newtonian part of the viscoelastic capsule. By analysing the viscous stress, pressure and viscoelastic stress acting on the capsule membrane, we reveal that the mechanism underlying the effects of the internal fluid viscoelasticity on the capsule deformation is the alterations in the distribution of the viscoelastic stress at low De and its magnitude at high De, respectively. Furthermore, we find some new features in the dynamics of initially non-spherical capsules induced by the internal fluid viscoelasticity. Particularly, the transition from tumbling to swinging of oblate capsules can be triggered at very high viscosity ratios by increasing De alone. Besides, the critical viscosity ratio for the tumbling-to-swinging transition is remarkably enlarged with De increasing at relatively high levels, i.e. $O(1{-}100)$, while it shows little change at low De, i.e. below $O(1)$.

Investigation of blood flow rheology using second-grade viscoelastic model (Phan-Thien-Tanner) within carotid artery.

Hemodynamic factors, such as Wall Shear Stress (WSS), play a substantial role in arterial diseases. In the larger arteries, such as the carotid artery, interaction between the vessel wall and blood flow affects the distribution of hemodynamic factors. The fluid is considered to be non-Newtonian, whose flow is governed by the equation of a second-grade viscoelastic fluid and the effects of viscoelastic on blood flow in carotid artery is investigated. Pulsatile flow studies were carried out in a 3D model of carotid artery. The governing equations were solved using finite volume C++ based on open source code, OpenFOAM. To describe blood flow, conservation of mass and momentum, a constitutive relation of simplified Phan-Thien-Tanner (sPTT), and appropriate relations were used to explain shear thinning behavior. The first recirculation was observed at t = 0.2 s, in deceleration phase. In the acceleration phase from t = 0.3 s to t = 0.5 s, vortex and recirculation sizes in bulb regions in both ECA and ICA gradually increased. As is observed in the line graphs based on extracted data from ICA, at t = 0.2 s, τyy is the maximum amount of wall shear stress and τxy the minimum one. The maximum shear stress occurred in the inner side of the main branch (inner side of ICA and ECA) because the velocity of blood flow in the inner side of the bulb region was maximum due to the created recirculation zone in the opposite side in this area. The rheology of blood flow and shear stress in various important parts (the area that are in higher rates of WSS such as bifurcation region and the regions after bulb areas in both branches, Line1-4 in Fig. 7) were also analyzed. The investigation of velocity stream line, velocity profile and shear stress in various sections of carotid artery showed that the maximum shear stress occurred in acceleration phase and in the bifurcation region between ECA and ICA which is due to velocity gradients and changes in thinning behavior of blood and increasing strain rate in Newtonian stress part.

Interpreting the direction of the gravity gradient tensor eigenvectors: The main tidal force and its relation to the curvature parameters of the equipotential surface

Rotating the gravity gradient tensor about a vertical axis by an appropriate angle allows one to express its components as functions of the curvatures of the equipotential surface. The description permits the identification of the gravity gradient tensor as the Newtonian tidal tensor and part of the tidal potential. The identification improves the understanding and interpretation of gravity gradient data. With the use of the plunge of the eigenvector associated with the largest eigenvalue or plunge of the main tidal force, it is possible to estimate the location and depth of buried gravity sources; this is illustrated in model data and applied to FALCON airborne gravity gradiometer data from the Canning Basin, Australia.

Interpreting the direction of the gravity gradient tensor eigenvectors: Their relation to curvature parameters of the gravity field

Rotating the gravity gradient tensor about a vertical axis by an appropriate angle allows one to express its components as functions of the curvatures of the equipotential surface. The description permits the identification of the gravity gradient tensor as the Newtonian tidal tensor and part of the tidal potential. The identification improves the understanding and interpretation of gravity gradient data. With the use of the plunge of the eigenvector associated with the largest eigenvalue or plunge of the main tidal force, it is possible to estimate the location and depth of buried gravity sources; this is developed in model data and applied to FALCON airborne gravity gradiometer data from the Canning Basin, Australia.

Is the Hawking Quasilocal Energy “Newtonian”?

The Misner–Sharp–Hernandez mass defined in general relativity and in spherical symmetry has been recognized as having a Newtonian character in previous literature. In order to better understand this feature we relax spherical symmetry and we study the generalization of the Misner–Sharp–Hernandez mass to general spacetimes, i.e., the Hawking quasilocal mass. The latter is decomposed into a matter contribution and a contribution coming solely from the Weyl tensor. The Weyl tensor is then decomposed into an electric part (which has a Newtonian counterpart) and a magnetic one (which does not), which further splits the quasilocal mass into “Newtonian” and “non-Newtonian” parts. Only the electric (Newtonian) part contributes to the quasilocal mass.

Do Newtonian large-scale structure simulations fail to include relativistic effects?

The Newtonian simulations describing the formation of large-scale structures do not include relativistic effects. A new approach to this problem is proposed, which consists of splitting the Hawking-Hayward quasilocal energy of a closed spacelike 2-surface into a ``Newtonian'' part due to local perturbations and a ``relativistic'' part due to the cosmology. It is found that the Newtonian part dominates over the relativistic one as time evolves, lending support to the validity of Newtonian simulations.

Frame dragging and Eulerian frames in general relativity

The physical interpretation of cold dark matter perturbations is clarified by associating Bertschinger's Poisson gauge with a Eulerian/observer's frame of reference. We obtain such an association by using a Lagrangian approach to relativistic cosmological structure formation. Explicitly, we begin with the second-order solution of the Einstein equations in a synchronous/comoving coordinate system---which defines the Lagrangian frame, and transform it to a Poissonian coordinate system. The generating vector of this coordinate/gauge transformation is found to be the relativistic displacement field. The metric perturbations in the Poissonian coordinate system contain known results from standard/Eulerian Newtonian perturbation theory, but contain also purely relativistic corrections. On sub-horizon scales these relativistic corrections are dominated by the Newtonian bulk part. These corrections however set up non-linear constraints for the density and for the velocity which become important on scales close to the horizon. Furthermore, we report the occurence of a transverse component in the displacement field, and find that it induces a non-linear frame dragging as seen in the observer's frame, which is sub-dominant at late-times and sub-horizon scales. Finally, we find two other gauges which can be associated with a Eulerian frame. We argue that the Poisson gauge is to be preferred because it comes with the simplest physical interpretation.

Electrophoretic velocity of spherical particles in Quemada fluids

The biomicrofluidic devices utilizing electrophoresis for sample manipulation are usually encountered with non-Newtonian behavior of working fluids. Hence, developing theoretical models capable of predicting the electrophoretic velocity of colloidal particles in non-Newtonian fluids is of high importance for accurate design and active control of these devices. The present investigation is dealing with the electrophoresis of a spherical particle in a biofluid obeying the Quemada rheological model. The sphere radius is considered to be significantly larger than the Debye length. Moreover, it is assumed that the particle zeta potential is small so that the Debye–Hückel linearization is allowable. An approximate analytical solution is obtained for the particle electrophoretic velocity for the conditions that the deviations of the fluid rheological behavior from the predictions of the Newton's law of viscosity are small. A parametric study along with inspection of some practical examples assuming the blood as the working fluid reveals that the non-Newtonian contribution to the electrophoretic velocity is comparable with the Newtonian part and therefore cannot be neglected. It is observed that the non-Newtonian part of electrophoretic velocity is an increasing function of the hematocrit level and the Deborah number and is independent of the particle size.

Precise determination of ocean tide loading gravity effect for absolute gravity stations in coastal area of China: Effects of land–sea boundary and station coordinate

Gravity effect due to ocean tide loading (OTL) is an important signal and correction in various gravimetric applications. In this paper, we assessed the OTL gravity effects at four absolute gravity (AG) stations in coastal China from several perspectives. The integrated Green's function of the Newtonian part was derived analytically and that of the elastic part was computed based on the PREM earth model. Ocean tide (OT) records near the four AG stations were used to enhance the accuracy of the global ocean tide model of NAO99b and the regional model of NAO99jb for the innermost field (<0.02°). The high-resolution digital elevation model (DEM) from the Shuttle Radar Topography Mission (SRTM) was used to define the land–sea boundary in the optimized near field (<1.0°). Results show that the high-resolution land–sea boundary is indispensible for OTL gravity modeling when the shoreline near a coastal station is complex. The SRTM-based OTL model outperforms the GMT-based model shoreline in terms of the agreement between modeled and observed gravity residuals at the four stations. The final gravity residuals, corrected for OTL, at the four stations are significantly smaller than those without OTL corrections. We give examples of accuracy requirements in coordinates at Qingdao for different station heights. At a station height of 80m and to ensure a 0.1μgal accuracy in OTL modeling, the required accuracies in the horizontal and vertical coordinate are 2.5 and 1. 3m, respectively. For a new coastal station and an expected OTL accuracy, one should inspect the variation of OTL due to coordinate variation to find a best strategy to determine the required accuracy of coordinate.

Numerical simulation of generalized Oldroyd-B and generalized Newtonian fluid flows

This paper is dealing with numerical simulation of generalized Newtonian and generalized Oldroyd-B fluids with the aim of blood flow simulation. The Newtonian model of a fluid cannot capture all the phenomena in many fluids with complex microstructure, such as polymers, suspensions (also many biological fluids) and granular materials. The motion of polymeric fluids is described by the conservation of mass and momentum. One shall assume that the fluid is incompressible and temperature variations are negligible. When one considers viscoelastic behavior of polymeric fluids, the extra stress tensor depends not only on the current motion of the fluid, but also on the history of the motion. In this case the extra stress tensor is decomposed into its Newtonian part and its elastic part. Components of the elastic part of the extra stress tensor are computed using the Oldroyd-B constitutive equation. Time derivative of the pressure is added into the continuity equation (Artificial compressibility method) and the arising system of equations is discretized in space with Finite volume method and numerically solved in time with Runge-Kutta method. Numerical methods are tested in the geometry of constricted channel.

Influence of the cosmological constant on the motion of Magellanic Clouds in the gravitational field of Milky Way

Using the pseudo-Newtonian (PN) potential reflecting properties of the Schwarz-schild-de Sitter spacetime, we estimate the influence of the repulsive cosmological constant Λ ∼ 1.3 × 10−56cm−2 implied by recent cosmological tests onto the motion of both Small and Large Magellanic Clouds (SMC and LMC) in the gravitational field of the Milky Way. Considering detailed modelling of the gravitational field of the Galaxy disc, bulge and cold dark matter halo, the trajectories of SMC and LMC constructed for the PN potential with the cosmological constant are confronted to those given for Λ = 0. In the realistic model of the extended cold dark matter halo its edge and related total mass are taken at typical values reflecting recent diversity in the total Galaxy mass estimates. In all cases, strong influence of the cosmological constant, on 10% level or higher, has been found for motion of both SMC and LMC. Inside the halo, the Newtonian part of the PN potential is exact enough, while outside the halo the PN potential can give relevant relativistic corrections. The role of the cosmological constant is most conspicuous when binding mass is estimated for the satellite galaxies. We have found a strong influence of cosmic repulsion on the total binding mass for both galaxies. For SMC there is the binding mass MSMCΛ = 0 = 7.07 × 1011M⨀ and MSMCΛ > 0 = 8.61 × 1011M⨀, while even much higher increase is found for LMC, where MLMCΛ = 0 = 1.50 × 1012M⨀ and MLMCΛ > 0 = 2.21 × 1012M⨀, putting serious doubts on the possibility that the LMC is bounded by the Milky Way. However, the estimates of binding masses are strongly influenced by initial velocity of SMC and LMC; we took the values inferred for the IAU MW rotation velocity ∼ 220 km/s. Our results indicate very important role of the cosmic repulsion in the motion of interacting galaxies, clearly demonstrated in the case of the satellite SMC and LMC galaxies moving in the field of Milky Way. In some cases, the effect of the cosmic repulsion can be even comparable to the effects of the dynamical friction and the Andromeda Galaxy.

How accurate is the cancelation of the first even zonal harmonic of the geopotential in the present and future LAGEOS-based Lense-Thirring tests?

The strategy followed so far in the performed or proposed tests of the general relativistic Lense-Thirring effect in the gravitational field of the Earth with laser-ranged satellites of LAGEOS type relies upon the cancelation of the disturbing huge precessions induced by the first even zonal harmonic coefficient J_2 of the multipolar expansion of the Newtonian part of the terrestrial gravitational potential by means of suitably designed linear combinations of the nodes \Omega of more than one spacecraft. Actually, such a removal does depend on the accuracy with which the coefficients of the combinations adopted can be realistically known. Uncertainties of the order of 2 cm in the semimajor axes a and 0.5 milliarcseconds in the inclinations I of LAGEOS and LAGEOS II, entering the expression of the coefficient c_1 of the combination of their nodes used so far, yield an uncertainty \delta c_1 = 1.30 10^-8. It gives an imperfectly canceled J_2 signal of 10.8 milliarcseconds per year corresponding to 23% of the Lense-Thirring signature. Uncertainties of the order of 10-30 microarcseconds in the inclinations yield \delta c_1=7.9 10^-9 which corresponds to an uncanceled J_2 signature of 6.5 milliarcseconds per year, i.e. 14% of the Lense-Thirring signal. Concerning a future LAGEOS-LAGEOS II-LARES combination with coefficients k_1 and k_2, the same uncertainties in a and the less accurate uncertainties in I as before yield \delta k_1=1.1 10^-8, \delta k_2=2 10^-9; they imply a residual J_2 combined precession of 14.7 milliarcseconds per year corresponding to 29% of the Lense-Thirring trend. Uncertainties in the inclinations at about 10 microarcseconds level give \delta k_1=5 10^-9, \delta k_2 = 2 10^-9; the uncanceled J_2 effect is 7.9 milliarcseconds per year, i.e. 16% of the relativistic effect.

Global symplectic structure-preserving integrators for spinning compact binaries

This paper deals mainly with the application of the second-order symplectic implicit midpoint rule and its symmetric compositions to a post-Newtonian Hamiltonian formulation with canonical spin variables in relativistic compact binaries. The midpoint rule, as a basic algorithm, is directly used to integrate the completely canonical Hamiltonian system. On the other hand, there are symmetric composite methods based on a splitting of the Hamiltonian into two parts: the Newtonian part associated with a Kepler motion, and a perturbation part involving the orbital post-Newtonian and spin contributions, where the Kepler flow has an analytic solution and the perturbation can be calculated by the midpoint rule. An example is the second-order mixed leapfrog symplectic integrator with one stage integration of the perturbation flow and two semistage computations of the Kepler flow at every integration step. Also, higher-order composite methods such as the Forest-Ruth fourth-order symplectic integrator and its optimized algorithm are applicable. Various numerical tests including simulations of chaotic orbits show that the mixed leapfrog integrator is always superior to the midpoint rule in energy accuracy, while both of them are almost equivalent in computational efficiency. Particularly, the optimized fourth-order algorithm compared with the mixed leapfrog scheme provides good precision and needs no expensive additional computational time. As a result, it is worth performing a more detailed and careful examination of the dynamical structure of chaos and order in the parameter windows and phase space of the binary system.