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The standard inertial Navier–Stokes equations consisting of the conservation equations for mass, momentum and energy are often used to investigate the motion of a compressible fluid around an object that is in arbitrary motion. The non-inertial form of the Navier–Stokes equations can be used to accurately capture the acceleration effects that arise from the unsteady motion. The acceleration source terms that arise in the conservation equation for momentum have been extensively documented. In this paper, an Eulerian approach for deriving the apparent forces is presented to transform the governing conservation equation for energy into a non-inertial reference frame that is in arbitrary motion. The Eulerian approach is based on successive Galilean transformations between an inertial frame, an orientation-preserving non-inertial frame and a rotating non-inertial frame. The paper demonstrates that for an object in arbitrary motion, the rate of work done due to fictitious forces affects the rate of change of the total energy. The fictitious work arises in the kinetic energy equation while the internal energy and enthalpy equations remain invariant in the non-inertial frame. The present derivation is a step towards quantifying the contribution of the fictitious work terms to the heat transfer of a body that is accelerating/decelerating.

U radu su analizirane inercijalne oscilacije, kako do njih dolazi te njihove glavne karakteristike. Uvodno su opisana obilježja neinercijalnog sustava te razmotrene inercijalne sile. Prikazano je kako oscilacije nisu strogo vezane za neinercijalni, već se mogu zapaziti i u inercijalnom sustavu. Primjenjen je model Zemlje kao neinercijalnog sustava te su postavljene glavne jednadžbe koje opisuju inercijalne oscilacije. Prikazana su analitička rješenja iz kojih je pokazano kako svojstva oscilacija ovise o geografskoj širini. Uz to, jednadžbe su riješene numeričkim putem preko eksplicitne te implicitne metode. Dobiveni numerički rezultati, iz kojih su vidljive osobine i analizirano ponašanje rješenja svake od metoda, uspoređeni su s analitičkim rješenjima., The thesis analyses inertial oscillations, how they occur and their main characteristics. Introductory the properties of the non-inertial reference frame are described and inertial forces are considered, as well. It is shown that the oscillations are not strictly related to the non-inertial, but can also be observed in the inertial frame of reference. The model of the Earth as a non-inertial frame was applied and the main equations describing inertial oscillations were derived. Analytical solutions are presented from which it is shown that the properties of oscillations depend on the latitude. In addition, the equations are solved numerically using explicit and implicit methods. The obtained numerical results, from which the properties and analysed behavior of the solution of each of the methods are visible, are compared with the analytical solutions.

In this paper, the 3D numerical modelling of the sloshing phenomenon in rectangular tanks was conducted under the oblique excitation. The effect of the aspect ratio on the maximum side wall, bed hydrodynamic forces and the maximum water run up over the side wall investigated under the intelligent controls. The non-inertial reference frame method was used to apply the dynamic excitation to the fluid body, and the volume of fluid (VOF) method was used to define the water surface variations. The experimental benchmark test case was used to verify the non-inertial reference frame method and effect of the excitation direction, frequency and amplitude were studied. Results show that the non-inertial reference method is more accurate than other method to predict the sloshing phenomenon. Using Flow-3D software, intelligent controls and big data algorithms the optimal numerical results are obtained. Then to study the effect of the geometrical parameters on the sloshing phenomenon, three rectangular tanks with different length/width ratio were considered and for various frequencies ratio were loaded under harmonic excitations. Based on these considerations, for all cases the maximum pressure force on the side wall occurs when the excitation is parallel with the side wall, and the maximum water surface occurs for the longitudinal excitation. By increasing the length/width ratio the maximum pressure force on the side wall decreases and on the other hand the maximum pressure force on the tank bottom increases.

Publisher Summary This chapter illustrates an elative motion analysis to gain familiarity with the problem of maneuvering one spacecraft relative to another, especially when they are in close proximity. In a rendezvous maneuver, two orbiting vehicles observe one another from each of their own free-falling, rotating, clearly non-inertial frames of reference. To base impulsive maneuvers on observations made from a moving platform requires transforming relative velocity and acceleration measurements into an inertial frame else the true thrusting forces cannot be sorted out from the fictitious inertia forces that appear in Newton's law. The chapter presents the relative motion in an orbit, linearization of the equations of relative motion in orbit, the Clohessy–Wiltshire equations (CW equations), two-impulse rendezvous maneuver, and the relative-motion in close proximity circular orbits.

We discuss several approaches to determine the Casimir force in inertial frames of reference in different dimensions. On an example of a simple model involving mirrors in Rindler spacetime we show that Casimir's and Lifschitz's methods are inequivalent and only latter can be generalized to other spacetime geometries. For conformally coupled fields we derive the Casimir force in conformally flat spacetimes utilizing an anomaly and provide explicit examples in the Friedmann-Lemaitre-Robertson-Walker $(k=0)$ models., (21 pages, 1 figure)

Purpose This paper aims to improve the computational efficiency and to achieve high-order accuracy for the computation of helicopter rotor unsteady flows in forward flight during the industrial preliminary design stage. Design/methodology/approach The integral arbitrary Lagrangian–Eulerian form of unsteady compressible Navier–Stokes equations with low Mach number preconditioned pseudo time terms based on non-inertial frame of reference undergoing rotating and translating was derived and discretized in the framework of multi-block structured finite volume grid using three types of spatial reconstruction schemes, i.e. the third-order accurate monotonic upwind scheme for conservation laws, the fifth-order accurate weighted essentially non-oscillatory and the fifth-order accurate weighted compact nonlinear schemes. Findings The results show that the present non-inertial computational method can obtain comparable results with other methods, such as the dynamic overset method, and make sure that the higher-order spatial schemes can significantly improve the tip vortex resolution. Originality/value The computational grid used by the present method remained static during the whole unsteady computation process, with only local deformations induced by blade cyclic pitch and other operating motions, which greatly reduced the complexity of grid motion and enhanced the efficiency and robustness.

The problem of transformation of electromagnetic field from inertial frame of reference to non-inertial rotating frame of reference and vice versa is discussed. By the use of the tetrad method (moving reper method) it is shown that in cylindrical coordinates the usual Lorentz formulas can be applied. It is also shown that in cylindrical coordinates the form of Maxwell equations in rotating frame of reference is the same as in inertial one.

Spherical reservoirs, as objects perfect with respect to their weight, are used in spacecrafts, where thin-walled elements are joined by frames into multifunction structures. The junctions are local, which results in origination of stress concentration regions and the corresponding rigidity problems. The thin-walled elements are reinforced by frame to decrease the stresses in them. To simplify the analysis of the mathematical model of common deformation of the shell (which is a mathematical idealization of the reservoir) and the frame, the assumption that the frame axial line is inextensible is used widely (in particular, in the manual literature). The unjustified use of this assumption significantly distorts the concept of the stress-strain state. In this paper, an example of a lens-shaped structure formed as two spherical shell segments connected by a frame of square profile is used to carry out a numerical comparative analysis of the solutions with and without the inextensible frame hypothesis taken into account. The scope of the hypothesis is shown depending on the structure geometric parameters and the load location degree. The obtained results can be used to determine the stress-strain state of the thin-walled structure with an a priori prescribed error, for example, in research and experimental design of aerospace systems.

In this paper we study the occurrence of accelerating universe versus decelerating universe between the F(R) gravity frame (Jordan frame) and non-minimally coupled scalar field theory frame, and the minimally coupled scalar field theory frame (Einstein frame) for various models. As we show, if acceleration is imposed in one frame, it will not necessarily correspond to an accelerating metric when transformed in another frame. As we will demonstrate, this issue is model and frame-dependent but it seems there is no general scheme which permits to classify such cases., To appear in Physics Letters B. Some corrections added

The article presents approaches to the formation of a general computational scheme for modeling (simulating) the particle motion on an axisymmetric rotating curved surface with a vertical axis of rotation. To describe the complex particle motion over a given surface, the fundamental equation of particle dynamics in a non-inertial reference frame was used, and by projecting it onto the axes of cylindrical coordinates, the Lagrange&rsquo, s differential equations of the first kind were obtained. According to the proposed algorithm in C#, an application was developed that enables graphical and numerical control of the calculation results. The program interface contains six screen forms with tabular baseline data (input) and a table of a step-by-step calculation of results (output), particle displacement, velocity, and acceleration diagrams constructed along the axes of the system of cylindrical coordinates &rho, and z, graphical presentation of the generate of the surface of revolution and the trajectory of the absolute motion of a particle over the axisymmetric rotating surface developed in polar coordinates. Examples of the calculation of the particle motion are presented. The obtained results can be used for the study and design of machines, for example, centrifugal rotary chopper machines.

Mixing applications operating in the laminar regime are used in numerous industrial processes in the pharmaceutical, chemical, and food industries. The aim of this paper is to introduce a numerical model adapted to solid-liquid mixing situations in stirred tanks. The method presented herein is based on a Euler-Lagrange approach using the CFD-DEM method. This method couples computational fluid dynamics (CFD) for the fluid with the discrete element method (DEM) for the solid particles. We introduce a rotating frame of reference approach, which is the first of its kind for CFD-DEM. In this paper we discuss the main issues related to the modeling of complex rotating impeller geometries, we explain the various issues involved in conducting a CFD-DEM simulation in a non-inertial frame, we compare our model with experimental results obtained with a pitched blade turbine and, lastly, we use our model to study solid-liquid mixing with a double helical ribbon.

It is commonly thought that the Michelson-Morley experiment from 1887 and Kennedy-Thorndike experiment from 1932 demonstrated that the universal frame of reference (ether) does not exist and that the velocity of light in vacuum is absolutely constant. The analysis of this experiment led to the creation of the Special Theory of Relativity (STR). The article explains why Michelson-Morley and Kennedy-Thorndike experiments could not detect the universal frame of reference. In this article, a different transformation of time and position than the Lorentz transformation is derived on the basis of the geometric analysis of the Michelson-Morley and Kennedy-Thorndike experiments. The transformation is derived based on the assumption that the universal frame of reference (UFR) exists. UFR is a frame of reference in which the velocity of light is constant in every direction. In inertial frames of reference moving in the UFR, the velocity of light may be different. The article has derived the formula for relative speed and patterns for the maximum and minimum speed of light that can be measured in the inertial system. Finally, the anisotropy of the microwave background radiation has been explained by using the presented theory. According to the body kinematics model presented in this article, anisotropy of cosmic microwave background is the Doppler effect for observer moving in the UFR.

The purpose of this research is to show that Foaucault pendulum as well as other Coriolis effects, which are normally studied in a rotating coordinate system, can also be analyzed in a fixed reference frame. To this end, Foucault pendulum and other Coriolis effects are studied in inertial reference frames. The approach is simple, yet rigorous, and the results are exactly the same as those obtained in non-inertial reference frames but without resorting to some of the assumptions that are needed in rotating coordinate systems.

The mechanical energy change of a system in an inertial frame of reference equals work done by the total nonconservative force in the same frame. This relation is covariant under the Galilean transformations from inertial frame S to S′, where S′ moves with constant velocity relative to S. In the presence of nonconservative forces, such as normal and tension forces, the mechanical energy of a system can be conserved in S and not be conserved in S′. In this paper we find useful relations between the mechanical energy changes in two inertial frames of reference.

In Newtonian mechanics, the non-inertial reference frames is a generalization of Newton’s laws to any reference frames. While this approach simplifies some problems, there is often little physical insight into the motion, in particular into the effects of the Coriolis force. The fictitious Coriolis force can be used by anyone in that frame of reference to explain why objects follow curved paths. In this paper, a mathematical solution based on differential equations in non-inertial reference is used to study different types of motion in rotating system. In addition, the experimental data measured on a turntable device, using a video camera in a mechanics laboratory was conducted to compare with mathematical solution in case of parabolically curved, solving non-linear least-squares problems, based on Levenberg-Marquardt’s and Gauss-Newton algorithms.

Most computational fluid dynamics simulations are, at present, performed in a body-fixed frame, for aeronautical purposes. With the advent of sharp manoeuvre, which may lead to transient effects originating in the acceleration of the centre of mass, there is a need to have a consistent formulation of the Navier–Stokes equations in an arbitrarily moving frame. These expressions should be in a form that allows terms to be transformed between non-inertial and inertial frames and includes gravity, viscous terms, and linear and angular acceleration. Since no effects of body acceleration appear in the inertial frame Navier–Stokes equations themselves, but only in their boundary conditions, it is useful to investigate acceleration source terms in the non-inertial frame. In this paper, a derivation of the energy equation is provided in addition to the continuity and momentum equations previously published. Relevant dimensionless constants are derived which can be used to obtain an indication of the relative significance of acceleration effects. The necessity for using computational fluid dynamics to capture nonlinear effects remains, and various implementation schemes for accelerating bodies are discussed. This theoretical treatment is intended to provide a foundation for interpretation of aerodynamic effects observed in manoeuvre, particularly for accelerating missiles. Funding agencies: DRDB [KT466921, KT528944, KT470887]

In this paper, we consider a small ball placed in a liquid-filled transparent cylinder sealed at both ends. When the tube is inclined and spun about the vertical axis like a centrifuge, a heavy ball (more dense than the liquid) will accelerate to either the top or bottom end of the tube depending on its release position while a light ball (less dense than the liquid) will instead tend towards an equilibrium position in the tube. Experimental measurements of the ball's motion show large disagreements with Stokes' theory of drag force on the ball. The formation of a rotating column of liquid induced by the Coriolis force, known as a Taylor column, significantly increases the drag on the ball. In this paper, we aim to investigate the equilibrium position as well as the motion path of the ball, with a numerical solution to the 3D time-dependent Navier–Stokes equation in the rotating frame of the tube, as a function of the inclination angle and angular velocity of rotation of the tube, which are experimentally verified for a ball in a water-filled tube. Increasing both the angular velocity and inclination angle will cause the equilibrium position to shift closer to the bottom of the tube, a faster motion of the ball, and larger deflections to the side of the tube. The formation of the Taylor column was also visualised in the rotating frame with dye injection.

We construct an alternative uniformly accelerated reference frame based on the 3+1 formalism in adapted coordinates. In this frame, time-dependent redshift drift exists between co-moving observers, which differs from that in Rindler coordinates. This phenomenon can be tested in laboratory and improve our understanding of non-inertial frames., Comment: v1: 28 pages, 4 figures; v2: 29 pages, 4 figures. Published in Chinese Physics C

In a flat spacetime with one spatial dimension compactified, inertial reference frames are not all equivalent, but there are the preferred ones. This paper investigates the nonequivalence of inertial frames and also that of uniformly accelerated frames in connection with the response of the Unruh-DeWitt detector coupled to a massless scalar field. The detector's transition rates of both excitation and de-excitation are studied in depth for three different cases: (i) the detector moving at an arbitrary constant velocity, (ii) moving with a constant acceleration in the compact direction, and (iii) moving with a constant acceleration in noncompact directions. The instantaneous transition rate in relation to the switching function is also taken into account., 33 pages, 2 figures; v3: intensively revised, version to appear in PRD

The interest of computer-aided investigations of structure motions in maritime applications has become more relevant during the last years. The present work deals with the 6DOF (six degrees of freedom) motion-simulation of a free floating vessel in a seaway. A comparison between the calculation of forces and moments in the body-fixed system using the Kirchhoff-Motion-Equations and the calculation within the inertial coordinate system is shown. For this purpose, the relevant equations are defined in the inertial reference frame. Special attention applies to the matrices of mass and moment of inertia. Due to hydrodynamic effects in maritime applications and the transformation from a moving body-fixed reference frame, these matrices are time varying in the inertial reference frame. Finally, a comparison of the registered ship motions in the body-fixed and the inertial reference frame is presented.

We study the Bell's inequality and multipartite entanglement generation for initially maximally entangled states of free Dirac field in a non inertial frame and asymptotically flat Robertson–Walker space-time. For two qubit case, we show that the Bell's inequality always is violated as measured by the accelerated observers which are in the causally connected regions. On the other hand, for those observers in the causally disconnected regions inequality is not violated for any values of acceleration. The generated three qubit state from two qubit state due to acceleration of one parties has a zero 3-tangle. For a three qubit state, the inequality violated for measurements done by both causally connected and disconnected observers. Initially GHZ state with non zero 3-tangle, in accelerated frame, transformed to a four qubit state with vanishing 4-tangle value. On the other hand, for a W-state with zero 3-tangle, in non inertial frame, transformed to a four qubit state with a non-zero 4-tangle acceleration dependent. In an expanding space-time with asymptotically flat regions, for an initially maximally entangled state, the maximum value of violation of Bell's inequality in the far past decreased in the far future due to cosmological particle creation. For some initially maximally entangled states, the generated four qubit state due to expansion of space-time, has non vanishing 4-tangle.

The analysis of relative motion of two spacecraft in Earth-bound orbits is usually carried out on the basis of simplifying assumptions. In particular, the reference spacecraft is assumed to follow a circular orbit, in which case the equations of relative motion are governed by the well-known Hill–Clohessy–Wiltshire equations. Circular motion is not, however, a solution when the Earth’s flattening is accounted for, except for equatorial orbits, where in any case the acceleration term is not Newtonian. Several attempts have been made to account for the $$J_2$$ effects, either by ingeniously taking advantage of their differential effects, or by cleverly introducing ad-hoc terms in the equations of motion on the basis of geometrical analysis of the $$J_2$$ perturbing effects. Analysis of relative motion about an unperturbed elliptical orbit is the next step in complexity. Relative motion about a $$J_2$$ -perturbed elliptic reference trajectory is clearly a challenging problem, which has received little attention. All these problems are based on either the Hill–Clohessy–Wiltshire equations for circular reference motion, or the de Vries/Tschauner–Hempel equations for elliptical reference motion, which are both approximate versions of the exact equations of relative motion. The main difference between the exact and approximate forms of these equations consists in the expression for the angular velocity and the angular acceleration of the rotating reference frame with respect to an inertial reference frame. The rotating reference frame is invariably taken as the local orbital frame, i.e., the RTN frame generated by the radial, the transverse, and the normal directions along the primary spacecraft orbit. Some authors have tried to account for the non-constant nature of the angular velocity vector, but have limited their correction to a mean motion value consistent with the $$J_2$$ perturbation terms. However, the angular velocity vector is also affected in direction, which causes precession of the node and the argument of perigee, i.e., of the entire orbital plane. Here we provide a derivation of the exact equations of relative motion by expressing the angular velocity of the RTN frame in terms of the state vector of the reference spacecraft. As such, these equations are completely general, in the sense that the orbit of the reference spacecraft need only be known through its ephemeris, and therefore subject to any force field whatever. It is also shown that these equations reduce to either the Hill–Clohessy–Wiltshire, or the Tschauner–Hempel equations, depending on the level of approximation. The explicit form of the equations of relative motion with respect to a $$J_2$$ -perturbed reference orbit is also introduced.

The basic concepts and laws of classical dynamics are analyzed from the standpoint of their adequacy to the physical nature of material bodies and to the dynamical processes. What Newton called absolute immovable space is the world ether that generates any forces, including the inertial forces imposing constraints on the magnitude of the acceleration. Inertial reference frames are frames moving translationally, rectilinearly, and uniformly relative to the ether. The physical nature of the principle of relativity is that the acceleration of a material point in all inertial reference frames coincides with its acceleration relative to the ether, and, therefore, Newton's dynamical equation retains the same form in any inertial reference frame. The principle of relativity is valid as long as Newton's dynamical equation is valid. In new formulations, the four laws of dynamics contain the concepts of ether and inertial forces. These laws are independent between themselves, and each of them expresses only one elementary truth.

Buçalışmada, bir serbest akım içerisinde yer alan ve salınım hareketi yapmayazorlanan bir silindirin etrafındaki akış, açık kaynaklı OpenFOAM ortamındageliştirilen bir çözücü ile sayısal olarak incelenmiştir. Akışkan hareketinindenklemleri, silindire yapışık, ataletsiz koordinat sistemi cinsinden yazılmışve sonlu hacimler yöntemi ile çözülmüştür. Sıkıştırılamaz, daimi olmayan, ikiboyutlu akış kabulleri altında çözüm yapılmıştır. Silindirin harmonik salınımı,tüm serbestlik dereceleri (dalıp çıkma, boyuna öteleme ve baş-kıç vurma) içinayrı ayrı ele alınmıştır. Çeşitli salınım frekansları taranarak, çevrikilitlenme davranışı yakalanmıştır. Hareketli silindir problemine dair bulunanliteratür ile uyumlu sonuçlar, yöntemin, rijit ve hareketli cisimlerin dış akışsimülasyonlarında kullanışlı bir araç olduğunu ortaya koymuştur., Inthis study, the flow past a cylinder which is forced to oscillate in a freestream is numerically investigated using a solver developed within theframework of the open-source toolbox OpenFOAM. Governing equations are writtenin the non-inertial reference frame fixed to the cylinder and solved using thefinite volume method. Flow is assumed to be incompressible, unsteady andtwo-dimensional. Harmonic oscillation of the cylinder is considered separatelyfor each degree-of-freedom (heaving, surging and pitching). By spanning severaloscillation frequencies, the lock-in behavior is captured. The agreementobtained in the moving cylinder problem demonstrates the convenience of theapproach in the general flow simulations of moving rigid bodies.

The paper presents a quaternion approach of giving a closed form solution of the motion in a central force field relative to a rotating reference frame. This new method involves two quaternion operators: the first one transforms the motion from a non-inertial reference frame to a inertial one with a very significant consequence of vanishing all the non-inertial terms (Coriolis and centripetal forces); the second quaternion operator provides the solution of the motion in the noninertial reference frame by applying it to the solution in the inertial reference frame. This process will govern the inverse transformation of the motion and is proved on two particular cases, the Foucault Pendulum and Keplerian motions problems relative to rotating reference frames.

The paper studies the motion of the Foucault Pendulum in a rotating non-inertial reference frame and provides a closed form vector solution determined by vector and matrix calculus. The solution is determined through vector and matrix calculus in both cases, for both forms of the law of motion (for the Foucault Pendulum Problem and its “Reduced Form”). A complex vector which transforms the motion equation in a first order differential equation with constant coefficients is used. Also, a novel kinematic interpretation of the Foucault Pendulum motion is given.

The paper studies the free vibrations of a robotic arm, located on a rotating platform. The robotic arm is modeled as a system with a finite number of degrees of freedom, consisting of elastically coupled rigid bodies. The differential equations of the free vibrations are obtained using the Lagrange equations formalism, in a generalized form, valid with respect to a non-inertial reference frame. The influence of the kinematic parameters of the platform motion upon the eigenfrequencies of the robotic arm is analyzed, considering two configurations of the system.