Perturbation Equations Research Articles

Axion as a fuzzy dark matter candidate: proofs in different gauges

Axion as a coherently oscillating massive scalar field is known to behave as a zero-pressure irrotational fluid with characteristic quantum stress on a small scale. In relativistic perturbation theory, the case was proved in the axion-comoving gauge up to fully nonlinear and exact order. Our basic assumption is that the field is oscillating with Compton frequency and the Compton wavelength is smaller than the horizon scale. Here, we revisit the relativistic proof to the linear order in the other gauge conditions. We show that the same equation for density perturbation known in the non-relativistic treatment can be derived in two additional gauge conditions: the zero-shear gauge and the uniform-curvature gauge. The uniform-expansion gauge fails to get the aimed equation, and the quantum stress term is missing in the synchronous gauge. For comparison, we present the relativistic density perturbation equations in the zero-pressure fluid in these gauge conditions. Except for the comoving and the synchronous gauge, the equations strikingly differ from the axion case. We clarify that the relativistic analysis based on time averaging is valid for scales larger than the Compton wavelength. Below the Compton wavelength, the field is not oscillating, and our oscillatory ansatz does not apply. We suggest an equation valid in all scales in the comoving gauge. For comparison, we review the non-relativistic quantum hydrodynamics and present the Schrödinger equation to first-order post-Newtonian expansion in the cosmological context.

Secular perturbations of the translational-rotational motion in a nonstationary three-body problem

A non-stationary three-body problem with axisymmetric dynamic structure, shape, and variable compression is considered. The Newtonian interaction force is characterized by an approximate expression of the force function accurate to the second harmonic. The masses of bodies change isotropically at different rates. The axes of inertia of the proper coordinate system of nonstationary axisymmetric three bodies coincide with the principal axes of inertia of the bodies, and it is assumed that their relative orientations remain unchanged during the evolution. Differential equations of translational and rotational motion of three non-stationary axisymmetric bodies with variable masses and dimensions in the relative coordinate system with the origin in the center of the more massive body are presented. The analytical expression for the Newtonian force function of the interaction of three bodies with variable masses and dimensions is given. The canonical equations of translational-rotational motion of three bodies in the Delaunaye-Andouye analogues have been obtained. The equations of secular perturbations of translational-rotational motion of unsteady axisymmetric three-bodies in the analogue of the Delaunaye-Andouye osculating elements have been obtained. The problem is investigated by methods of perturbation theory.

On the convective wave equation for the investigation of combustor stability using FEM-methods

Solving the Helmholtz equation with spatially resolved finite element method (FEM) approaches is a well-established and cost-efficient methodology to numerically predict thermoacoustic instabilities. With the implied zero Mach number assumption all interaction mechanisms between acoustics and the mean flow velocity including the advection of acoustic waves are neglected. Incorporating these mechanisms requires higher-order approaches that come at massively increased computational cost. A tradeoff might be the convective wave equation in frequency domain, which covers the advection of waves and comes at equivalent cost as the Helmholtz equation. However, with this equation only being valid for uniform mean flow velocities it is normally not applicable to combustion processes. The present paper strives for analyzing the introduced errors when applying the convective wave equation to thermoacoustic stability analyses. Therefore, an acoustically consistent, inhomogeneous convective wave equation is derived first. Similar to Lighthill’s analogy, terms describing the interaction between acoustics and non-uniform mean flows are considered as sources. For the use with FEM approaches, a complete framework of the equation in weak formulation is provided. This includes suitable impedance boundary conditions and a transfer matrix coupling procedure. In a modal stability analysis of an industrial gas turbine combustion chamber, the homogeneous wave equation in frequency domain is subsequently compared to the Helmholtz equation and the consistent Acoustic Perturbation Equations (APE). The impact of selected source terms on the solution is investigated. Finally, a methodology using the convective wave equation in frequency domain with vanishing source terms in arbitrary mean flow fields is presented.

Axial perturbations of hairy Gauss-Bonnet black holes with a massive self-interacting scalar field

We study the axial quasinormal modes of hairy black holes in Gauss-Bonnet gravity with a massive self-interacting scalar field. Two coupling functions of the scalar field to the Gauss-Bonnet invariant are adopted with one of them leading to black hole scalarization. The axial perturbations are studied via time evolution of the perturbation equation, and the effect of the scalar field mass and the self-interaction constant on the oscillation frequency and damping time is examined. We study as well the effect of nonzero scalar field potential on the critical point at which the perturbation equation loses hyperbolicity in the case of black hole scalarization. The results show that the nonzero scalar field potential extends the range of parameters where such a loss of hyperbolicity is observed thus shrinking the region of stable black hole existence. This will have an important effect on the nonlinear dynamical simulation studies in massive scalar Gauss-Bonnet gravity.

Nonlinear analysis of flame hydrodynamic instability at large gas expansion ratio

In the context of the large thermal-expansion approximation, we derive an equation describing flame front dynamics under conditions of Darrieus-Landau instability. We show that the second-order theory leads to system of two evolution equations for the flame front perturbations and for the potential of the unburned mixture flow. In the limiting case of long evolution, the system of equations can be reduced to one equation with respect to the additive variable that is the sum of the front perturbations and the flow potential. The equation with respect to the additive variable at large gas expansion coefficients has the form of the Sivashinsky equation obtained for the case of small gas expansion coefficients.

A hybrid computational aeroacoustic model with application to turbulent flows over foil and bluff bodies

In this study, a versatile and general three-dimensional hybrid aeroacoustic model is presented for low Mach number turbulent flows. A dual-domain approach is adopted, with the fluid and acoustic fields being solved simultaneously on separate, partially overlapping grids. By solving both parts at the same time, the need to store large data sets is removed, which improves the efficiency and usability of the model. Incompressible large eddy simulations are used to resolve the flow field and compute the source terms, which are interpolated onto the acoustic domain using radial basis functions. The acoustic perturbation equations are then solved using a finite-volume Riemann-based solver. The model is validated using a series of cases involving transitional and turbulent flows over foil and bluff bodies. Trailing edge instability noise and rod-foil interaction noise are modelled, with the results being compared to a range of experimental sources. Excellent agreement is seen, demonstrating the robustness and accuracy of the method. The trailing edge studies also provide additional insights into the origin of secondary tones in the acoustic spectra.

Normal modes for two-dimensional gravitating kinks

We study small perturbations around an arbitrary static kink solution of a two-dimensional (2D) gravity-scalar system, where the gravity part is described by a subclass of 2D dilaton gravity theory, and the scalar matter field has generalized dynamics. We expand the action around an arbitrary static solution and keep terms up to the second order of the perturbations. After variation the linear-order action leads to background field equations, as expected. The quadratic action of the normal modes are obtained after fixing the gauge and using the constraint equation. The linear perturbation equations obtained from the quadratic action are consistent with those obtained by linearizing the field equations under the dilaton gauge. All the calculations are assisted by a Mathematica code, which is also provided as a supplementary material.

Stability of generalized Einstein-Maxwell-scalar black holes

We study the stability of static black holes in generalized Einstein-Maxwell-scalar theories. We derive the master equations for the odd and even parity perturbations. The sufficient and necessary conditions for the stability of black holes under odd-parity perturbations are derived. We show that these conditions are usually not similar to energy conditions even in the simplest case of a minimally coupled scalar field. We obtain the necessary conditions for the stability of even-parity perturbations. We also derived the speed of propagation of the five degrees of freedom and obtained the class of theories for which all degrees of freedom propagate at the speed of light. Finally, we have applied our results to various black holes in nonlinear electrodynamics, scalar-tensor theories and Einstein-Maxwell-dilaton theory. For the latter, we have also calculated the quasinormal modes.

Evanescent acoustic-gravity waves in a rotating stratified atmosphere

In this work, we focused on the study of the influence of the rotation of the Earth’s atmosphere on the properties of evanescent acoustic-gravity waves, which we studied earlier in the absence of rotation. It is shown that evanescent acoustic-gravity waves (AGW) with a continuous spectrum can exist in an atmosphere rotating with an angular frequency Ω below the frequency 2 Ω (the Coriolis parameter). It is also shown that the rotation of the atmosphere also leads to a modification of the previously discovered continuous spectrum of evanescent AGWs with frequencies higher than the Coriolis parameter, which fills the entire “forbidden” region in the diagnostic diagram between freely propagating acoustic and internal gravity waves. It is concluded that the AGW spectrum in the diagnostic diagram consists of regions of acoustic and gravity waves, as well as two regions of evanescent waves, and is continuous. The found new spectrum expands the full spectrum of evanescent waves and indicates the need to search for evanescent waves at ultra-low frequencies. The result is obtained for high-latitude regions from a system of linear hydrodynamic equations for perturbations that take into account the rotation of the Earth’s atmosphere, by imposing an additional spatial relation on the components of the perturbed velocity vector of the elementary volume of the medium, which proposed by us for the first time. This made it possible to obtain an infinite number of solutions describing evanescent acoustic-gravity waves propagating in an isothermal atmosphere. The specified connection between the components of the perturbed velocity is characterized by the α parameter, which can only take real values. It has been established that the detected spectrum of evanescent acoustic-gravity waves can exist only at 0 < α < 1 , while the previously found spectrum of these waves, modified by taking into account the Earth’s rotation, is realized at arbitrary values of α . Analytical and numerical analysis of the obtained solutions is carried out. It is shown that these solutions, at certain values of the parameter α , pass into the previously studied evanescent modes.

Relativistic star perturbations in Horndeski theories with a gauge-ready formulation

We present a general framework for studying the relativistic star perturbations on a static and spherically symmetric background in full Horndeski theories. We take a perfect fluid into account as a form of the Schutz-Sorkin action. Our formulation is sufficiently versatile in that the second-order actions of perturbations in odd- and even-parity sectors are derived without choosing particular gauge conditions, so they can be used for any convenient gauges at hand. The odd-parity sector contains one dynamical gravitational degree of freedom coupled to a time-independent four velocity of the fluid. In the even-parity sector there are three dynamical perturbations associated with gravity, scalar field, and matter sectors, whose equations of motion are decoupled from other nondynamical perturbations. For high radial and angular momentum modes, we obtain the propagation speeds of all dynamical perturbations and show that the perfect fluid in the even-parity sector has a standard sound speed affected by neither gravity nor the scalar field. Our general stability conditions and perturbation equations of motion can be directly applied to the stabilities of neutron stars and black holes as well as the calculations of their quasinormal frequencies.

Gravitational and electromagnetic radiation from an electrically charged black hole in general nonlinear electrodynamics

We derive the equations for the odd and even parity perturbations of coupled electromagnetic and gravitational fields of a black hole with an electric charge within the context of general nonlinear electrodynamics. The Lagrangian density is a generic function of the Lorentz invariant scalar quantities of the electromagnetic fields. We include the Hodge dual of the electromagnetic field tensor and the cosmological constant in our calculations. For each type of parity, we reduce the system of Einstein field equations coupled to nonlinear electrodynamics to two coupled Schr\"odinger-type wave equations, one for the gravitational field and one for the electromagnetic field. The stability conditions in the presence of the Hodge dual of the electromagnetic field are derived.

Convergence Results for the Double-Diffusion Perturbation Equations

We study the structural stability for the double-diffusion perturbation equations. Using the a priori bounds, the convergence results on the reaction boundary coefficients k1, k2 and the Lewis coefficient Le could be obtained with the aid of some Poincare´ inequalities. The results showed that the structural stability is valid for the the double-diffusion perturbation equations with reaction boundary conditions. Our results can be seen as a version of symmetry in inequality for studying the structural stability.

Pin-Gap Correction of Coaxial Calibration Standards for TRL or LRL Calibration

In this paper, a novel method is proposed by which to correct the pin gaps of coaxial calibration standards for Thru-Reflect-Line (TRL) or Line-Reflect-Line (LRL) calibration. This method is a post-process in which the pin-gap correction is applied to the measured S-parameters of the device under test (DUT) after TRL or LRL calibration. It is based on the perturbation equations derived from the sensitivity-coefficient approach. These equations enable us to obtain the correction quantity of the DUT-S-parameter due to the pin gaps. We verify the proposed method by simulation for a 2.4 mm coaxial line, and the result shows that the pin-gap correction works successfully. Because the method is based on the perturbation equations, the smaller the pin gap, the better the correction results. Nevertheless, we can achieve sufficient correction results within the range of the pin-gap to at least 10 times the nominal value. In addition, the proposed pin-gap correction can play the role of impedance renormalization, even though the two LRL calibration lines have different characteristic impedance. Finally, we demonstrate the correction method for a 2.4 mm coaxial LRL calibration kit through measurement.

Параметрическая неустойчивость одиночной капли и ансамбля капель при круговых вибрациях

This article discusses forced oscillations and parametric instability of a cylindrical drop and an ensemble of drops under circular vibrations. The drop is surrounded by an incompressible liquid of a different density and is sandwiched between two parallel plates. In equilibrium, the drop has the shape of a circular cylinder bounded in the axial direction by these plates. The dynamic and average shape of the drop is constructed. A system of amplitude equations for small perturbations of the forced oscillations is obtained and the parametric instability of the single drop is studied. By analogy, a system of equations was written to study the parametric instability for an arbitrary drop in an ensemble of interacting drops. The regions of instability are constructed both for interacting modes and for modes of a higher order. It is shown that in the case of nonzero interaction, the lower modes are more dangerous in the presence of frequency detuning.

Гравитационно-скалярная неустойчивость космологической модели на основе двухкомпонентной системы вырожденных скалярно заряженных фермионов с асимметричным Хиггсовым взаимодействием. I. Уравнения для возмущений

A mathematical model is formulated for the evolution of plane perturbations in a cosmological two-component statistical system of completely degenerate scalarly charged fermions with an asymmetric scalar Higgs interaction. A complete closed system of differential equations describing the unperturbed state of a homogeneous and isotropic system and a system of self-consistent evolution equations of small perturbations are constructed.

The off-shell recursion for gravity and the classical double copy for currents

We construct the off-shell recursion for gravity and the graviton current for the perturbative double field theory (DFT). We first formulate the perturbative DFT, which is equivalent but simpler to perturbative general relativity, to all-orders in fluctuations of generalised metric. The perturbative action and equations of motion (EoM) are derived to arbitrary order for pure gravity case. We then derive the graviton off-shell recursion, the gravity counterpart of the Berends-Giele recursion in Yang-Mills theory, through the so-called perturbiner method using the EoM of the perturbative DFT. We solve the recursion iteratively and obtain the graviton off-shell currents explicitly. We then discuss the classical double copy for the off-shell currents. We present the current KLT relation for gravity by extending the result proposed by Mizera and Skrzypek for the non-gravitational effective field theories. The relation represents graviton currents by squaring gluon currents with the KLT kernel up to gauge transformation and regular terms that do not have any pole. Finally we discuss the off-shell conservation of currents for nonlinear gauge choices.

SPICE OF SOLUTIONS TO SINGULARLY PERTURBED EQUATIONS

SPICE OF SOLUTIONS TO SINGULARLY PERTURBED EQUATIONS

Zonal direct-hybrid aeroacoustic simulation of trailing edge noise using a high-order discontinuous Galerkin spectral element method

We present the extension of a discontinuous Galerkin framework to zonal direct-hybrid aeroacoustic simulations. This extension provides the ability to simultaneously perform a zonal large eddy simulation (LES), solving the compressible Navier–Stokes equations, and an acoustic propagation simulation, solving the acoustic perturbation equations. In doing so, the acoustic sources are exchanged without using the file system, and the bottleneck of I/O operations is avoided. This approach is well suited for large-scale simulations done in high-performance computing. The zonal LES uses the recently introduced recycling rescaling anisotropic linear forcing as a turbulent inflow method. We present a methodology to model the required Reynolds stresses based on the distribution of the turbulent kinetic energy obtained from solving the Reynolds-averaged Navier–Stokes equations. We show at the example of a turbulent flow over a flat plate and a NACA 64418 trailing edge simulation that the chosen model of the Reynolds stresses is valid. Direct-hybrid simulation results of a NACA 0012 airfoil, including tonal self-noise and a NACA 64418 trailing edge, demonstrate the presented approach’s applicability. This zonal direct-hybrid simulation approach shows great potential for efficient hybrid computational aeroacoustic simulations in high-performance computing.

Spatial decay bound and structural stability for the double-diffusion perturbation equations.

In this paper, we study the double-diffusion perturbation equations when the flow is through a porous medium. If the initial conditions satisfy some constraint conditions, the Saint-Venant type spatial decay of solutions for double-diffusion perturbation equations is obtained. Based on the spatial decay bound, the structural stability for the double-diffusion perturbation equations is also established.

Using the first-order, second-order, and third-order interpolation perturbation theory from quantum mechanics of modern physics to predict the CGM sensor device’s fasting plasma glucose value in the early morning, while analyzing their associated waveform shape similarities over a 14-month period based on GH-Method: math-physical medicine (No. 561)

The author utilizes the first-order, second-order, and third-order interpolation perturbation equations from quantum mechanics of modern physics to his medical research work, which he has previously written a few medical articles on this topic. This equation is the simplest application using one selected “perturbation factor” to generate perturbed results with high prediction accuracy and waveform shape similarity. During November 2021, he has applied the statistics regression analysis model to analyze the relationship among many biomarkers where he wrote ~20 medical articles. In this study, he predicts his fasting plasma glucose (FPG) value from his body temperature (BT) and body weight (BW) in the early morning. As a comparison, he chose two separate perturbation values of 0.4 for BT perturbation factor and 0.6 for BW perturbation factor to calculate and predict his FPG value. He then compares the predicted FPG dataset against his measured sensor FPG dataset. This comparison study also contains the following two final measurement yardsticks to confirm the usefulness of the perturbed method. The first yardstick is to verify the prediction accuracies of the perturbed FPG dataset against his measured FPG dataset. The second yardstick is to examine the waveform shape similarity via the calculated correlation coefficient between the predicted or perturbed, FPG curve and the measured FPG curve. In summary, the purpose of this study is to investigate the prediction accuracy and the waveform shape similarities between a perturbed or predicted FPG waveform and his measured FPG waveform over a 14-month period, which is a total of 420 days from 10/1/2020 to 11/24/2021. He utilizes the first-order, second-order, and third-order of interpolation perturbation equations with two different perturbation factor values of 0.4 for BT and 0.6 for BW. The author has selected these two slightly different slopes, i.e., perturbation values, to study the sensitivity results. The two conclusions drawn from this research work are listed as follows: First, the 3 perturbation equations for the BT and BW cases offered an identical match for waveform shape similarity with a 100% correlation coefficient. Second, the 3 perturbation equations also provided extremely high prediction accuracies as outlined: Prediction Accuracy for BT Case (Perturbation Value 0.4) First-order: Accuracy = 96% Second-order: Accuracy = 95% Third-order: Accuracy = 94%Prediction Accuracy for BW case (Perturbation Value 0.6) First-order: Accuracy = 98% Second-order: Accuracy = 96% Third-order: Accuracy = 95% All 6 prediction accuracies are higher than 94%. It seems that the higher the slope (perturbation value), the higher prediction accuracy will be achieved.