Radial Equation Research Articles

The Influence of Complex Flow on Axial Force in Rotor-Stator Cavity of Centrifugal Compressor with Radial Ribs

Abstract Research shows radial ribs can reduce the axial force acting on the centrifugal compressor but do not involve its impact on the flow of the rotational-stationary disk cavity. By establishing a coupling model of the centrifugal compressor and the ribbed rotational-stationary disk cavity, the flow characteristics of the ribbed rotational-stationary disk cavity and the effect of the complex flow in the cavity on the axial force were studied. The ribbed rotational-stationary disk cavity is in line with the basic flow structure and characteristics of the Batchelor flow pattern, as demonstrated by the results. The radial balance equation derived based on this flow pattern is suitable for describing the radial motion of the fluid in the ribbed cavity; the rib can increase the tangential velocity in the rotating core center and elevate the proportion of core center in the cavity while weakening the effect of the boundary layer in the region close the rotational disk, which increases the swirl ratio in the cavity; based on the radial balance equation, the mechanism of radial ribs on reducing the axial force was proposed from the perspective of complex flow. The mechanism is as follows: By increasing the swirl ratio, the ribs cause the radial pressure gradient in the cavity to increase, and the average pressure acting on the rotational disk surface is reduced, which leads to a reduction in axial force.

Topological defects on relativistic quantum oscillator in wormhole space–time background

In this research work, the relativistic quantum dynamics of oscillator field in wormhole background with a topological defect produced by a cosmic string is investigated. We consider an example of Morris–Thorne-type wormhole including a osmic string and derived the wave equation of the relativistic quantum oscillator. Through the Heun equation, we solve analytically the radial equation and obtain the ground state energy level [Formula: see text] and the radial wave function [Formula: see text] as particular cases. In fact, it is shown that the eigenvalue solution of the oscillator field is influenced by the topological defect of cosmic string and shifted the result. Furthermore, the wormhole throat radius also modifies the energy levels and the wave function of the relativistic quantum oscillator.

Comment on: “Some aspects of the interaction of a magnetic quadrupole moment with an electric field in a rotating frame” [J. Math. Phys. 58, 102103 (2017)

We discuss results obtained recently for a quantum-mechanical model given by a neutral particle with a magnetic quadrupole moment in a radial electric field and a scalar potential proportional to the radial variable in cylindrical coordinates that also includes the noninertial effects of a rotating reference frame. We show that the conjectured allowed values of the cyclotron frequency are a mere artifact of the truncation of the power series used to solve the radial eigenvalue equation. Our analysis proves that the analytical expression for the eigenvalues is incorrect.

The greybody factor for the monopole and odd-parity modes of the Proca field in the Schwarzschild black hole spacetimes

The greybody factor is one of the famous results of the black hole perturbation theory, which describes the transmission probability of a particle radiated by a black hole into spatial infinity. In this work, we separated the angular parts of the equations of motion and derived the radial equations for the Proca field in the Schwarzschild black hole spacetime. The radial equations for the monopole and odd-parity modes are fully decoupled in the Schrödinger-like form. We study the greybody factor by determining the rigorous bound.

Rhizosheath formation depends on mucilage concentration and water content

Abstract Aims Root exudates contain polymers that form crosslinks and can create a jelly like substance known as mucilage, which adheres to soil and thus promotes the formation of rhizosheaths, i.e. soil that remains attached to the roots after gentle shaking. We hypothesized that rhizosheath formation is optimal at an intermediate chia seed mucilage concentration and water content, but that its formation is limited at both a high concentration of chia seed mucilage and under dry conditions as well as at a low concentration of chia seed mucilage and under wet conditions. We used an artificial root soil system in which soil moisture and mucilage concentrations could be varied independently from one another with respect to their effect on rhizosheath formation. Methods Jute cords were disposed in sandy loam soil and in quartz sand. In a subsequent study, they were also amended to different moisture contents with five different concentrations of mucilage (from 0 to 0.2 g dry mucilage g−1 water), before being isolated from chia and flaxseed mucilage after swelling of the respective seeds in distilled water for 15 min. Results We found that in dry soil, rhizosheath formation peaked at an intermediate chia seed mucilage concentration. This behavior was supported by our conceptual model of mucilage spreading and rhizosheath formation, which relies on a radial diffusion equation and assumes that at low mucilage concentration, molecule numbers are insufficient to support polymer-like networks that stick soil particles together. In a very concentrated gel, however, mucilage is too sticky to diffuse far into the soil. Increasing soil moisture promotes rhizosheath formation both in a low and a high mucilage concentration range, although only up to an intermediate volumetric water content of 0.15cm3 cm–3. Conclusions We conclude that both water and chia seed mucilage concentration are important drivers of rhizosheath formation. The effects are not additive but can combine to an optimum range, with a maximum formation of rhizosheaths observed in this study at 0.12 g mucilage g−1 rhizosphere water.

In horizon penetrating coordinates: Kerr black hole metric perturbation, construction and completion

We investigate the Teukolsky equation in horizon-penetrating coordinates to study the behavior of perturbation waves crossing the outer horizon. For this purpose, we use the null ingoing/outgoing Eddington–Finkelstein coordinates. The first derivative of the radial equation is a Fuchsian differential equation with an additional regular singularity to the ones the radial one has. The radial functions satisfy the physical boundary conditions without imposing any regularity conditions. We also observe that the Hertz-Weyl scalar equations preserve their angular and radial signatures in these coordinates. Using the angular equation, we construct the metric perturbation for a circularly orbiting perturber around a black hole in Kerr spacetime in a horizon-penetrating setting. Furthermore, we completed the missing metric pieces due to the mass M and angular momentum J perturbations. We also provide an explicit formula for the metric perturbation as a function of the radial part, its derivative, and the angular part of the solution to the Teukolsky equation. Finally, we discuss the importance of the extra singularity in the radial derivative for the convergence of the metric expansion.

Greybody factor for a non accelerated charged modified black hole in anti-de sitter regime

This paper investigates the greybody factor for a non accelerated black hole with modified Maxwell electrodynamics in an anti-de Sitter regime. For this purpose, we compute the radial equation for a massless scalar field with the help of Klein–Gordon equation. We then formulate effective potential by transforming this equation into Schrodinger wave equation. We analyze the graphical behavior of effective potential for different values of mass, parameter characterizing the modified Maxwell theory, anti-de Sitter radius and electromagnetic charge parameters. The exact solutions are computed at two different horizons, i.e., event and cosmological horizons through the radial equation. Furthermore, we match the obtained solutions in an intermediate regime to enhance feasibility of the greybody factor over the entire domain and check its behavior graphically. It is found that the greybody factor has a direct relation with the radius, electromagnetic charge as well as angular momentum of the black hole and an inverse relation with the anti-de Sitter radius and modification parameter. We conclude that the modified Maxwell solution reduces the emission rate of the black hole.

Continuation of global solution curves using global parameters

This paper provides both the theoretical results and numerical calculations of global solution curves, by continuation in global parameters. Each point on the solution curves is computed directly as the global parameter is varied, so that all of the turns that the solution curves make, as well as their different branches, appear automatically on the computer screen. For radial p-Laplace equations we present a simplified derivation of the regularizing transformation from P. Korman (2015), and use this transformation for more accurate numerical computations. While for p>2 the solutions are not of class C2, we show that they are of the form w(rp2(p−1)), where w(z) is of class C2. Bifurcation diagrams are also calculated for non-autonomous problems, and for the fourth order equations modeling elastic beams. We show that the first harmonic of the solution can also serve as a global parameter.

Acceptable solutions of the radial Schrödinger equation for a particle in a central potential

We revisit the discussion about the boundary condition at the origin in the Schrödinger radial equation for central potentials. We give a transparent and convincing reason for demanding the radial part R(r) of the wave function to be finite at r = 0, showing that if R(0) diverges the complete wave function ψ does not satisfy the full Schrödinger equation. If R(r) is singular, we show that the corresponding ψ follows an equation similar to Schrödinger's, but with an additional term involving the Dirac delta function or its derivatives at the origin. Although, in general, understanding some of our arguments requires certain knowledge of the theory of distributions, the important case of a behavior R ∝ 1/r near r = 0, which gives rise to a normalizable ψ, is especially simple: The origin of the Dirac delta term is clearly demonstrated by using a slight modification of the usual spherical coordinates. The argument can be easily followed by undergraduate physics students.

On blow up for a class of radial Hartree type equations

On blow up for a class of radial Hartree type equations

Regional outer core kinematics from the time dependence of intense geomagnetic flux patches

Observations of the geomagnetic field by surface observatories and dedicated satellite missions such as the Swarm constellation provide constraints on the dynamics in Earth's outer core. In particular, global core flow models estimated by inversion of the radial magnetic induction equation provide an image of the circulation of the electrically conductive fluid at the top of the core. However, in these models the poloidal flow is much less robust than the toroidal core flow. Here, we infer regional outer core kinematics from the temporal variability of high-latitude intense geomagnetic flux patches. We develop an algorithm to fit anisotropic 2D-Gaussians to the shape of those flux patches in order to infer their area, amplitude and level of anisotropy. The temporal variabilities of these properties are used to quantify contraction, expansion, amplification, weakening and horizontal shear. Comparisons with idealized kinematic scenarios based on synthetic field and flow models allow to infer regional outer core kinematics. We found that some geomagnetic flux patches exhibit expansion and weakening corresponding to fluid upwellings, whereas other patches exhibit contraction and intensification corresponding to downwellings. In both cases the patches' area and amplitude relations follow hyperbolic curves. Our results show that the geomagnetic flux patches are affected by upwelling more often than by downwelling during the historical period. Equatorially symmetric poloidal flow prior to ≈1910 is inferred for the western intense patches. Kinematic scenarios where the field and flow structures centers coincide failed to reproduce the geomagnetic flux patches behavior. We recover the flux concentration efficiency of intense geomagnetic flux patches with an upwelling that resides two times its radius size away from the center of the flux patch. We also found a significant level of anisotropy over long periods for the historical geomagnetic flux patches. Anisotropic magnetic flux patches that are elongated in the direction of the shear flow may explain the east-west oriented present-day field at high latitudes of the southern Hemisphere. Overall, stretching effects at the top of the core can be deduced from our analysis of regional SV and allow further inferences on the poloidal part of the core flow.

Improved Power Series Solution of Transversely Loaded Hollow Annular Membranes: Simultaneous Modification of Out-of-Plane Equilibrium Equation and Radial Geometric Equation

The ability to accurately predict the shape of a transversely loaded hollow annular membrane is essential to the design of bending-free hollow annular shells of revolution, which requires a further improvement in the hollow annular membrane solution to meet the needs of this accurate prediction. In this paper, the large deflection problem of a transversely loaded hollow annular membrane is reformulated by simultaneously modifying the out-of-plane equilibrium equation and radial geometric equation, and a newer and more refined power series solution is derived. The reason why the classical radial geometry equation induces errors is revealed. The convergence and asymptotic behavior of the power series solution obtained is analyzed numerically. The newly derived solution is compared with the two previously derived solutions graphically, showing that the newly derived solution performs basically as well as expected. In addition, the anticipated use of the hollow and not-hollow annular membrane solutions for the design application of bending-free annular shells of revolution is discussed.

Drift phase resolved diffusive radiation belt model: 2. implementation in a case of random electric potential fluctuations

In the first part of this work, we highlighted a drift-diffusion equation capable of resolving the magnetic local time dimension when describing the effects of trapped particle transport on radiation belt intensity. Here, we implement these general considerations in a special case. Specifically, we determine the various transport and diffusion coefficients required to solve the drift-diffusion equation for equatorial electrons drifting in a dipole magnetic field in the presence of a specific model of time-varying electric fields. Random electric potential fluctuations, described as white noise, drive fluctuations of trapped particle drift motion. We also run a numerical experiment that consists of tracking trapped particles’ drift motion. We use the results to illustrate the validity of the drift-diffusion equation by showing agreement in the solutions. Our findings depict how a structure initially localized in magnetic local time generates drift-periodic signatures that progressively dampen with time due to the combined effects of radial and azimuthal diffusions. In other words, we model the transition from a drift-dominated regime, to a diffusion-dominated regime. We also demonstrate that the drift-diffusion equation is equivalent to a standard radial diffusion equation once the distribution function is phase-mixed. The drift-diffusion equation will allow for radiation belt modeling with a better spatiotemporal resolution than radial diffusion models once realistic inputs, including localized transport and diffusion coefficients, are determined.

A Bayesian estimation of the Milky Way’s circular velocity curve using Gaia DR3

Aims. Our goal is to calculate the circular velocity curve of the Milky Way, along with corresponding uncertainties that quantify various sources of systematic uncertainty in a self-consistent manner. Methods. The observed rotational velocities are described as circular velocities minus the asymmetric drift. The latter is described by the radial axisymmetric Jeans equation. We thus reconstruct the circular velocity curve between Galactocentric distances from 5 kpc to 14 kpc using a Bayesian inference approach. The estimated error bars quantify uncertainties in the Sun’s Galactocentric distance and the spatial-kinematic morphology of the tracer stars. As tracers, we used a sample of roughly 0.6 million stars on the red giant branch stars with six-dimensional phase-space coordinates from Gaia Data Release 3 (DR3). More than 99% of the sample is confined to a quarter of the stellar disc with mean radial, rotational, and vertical velocity dispersions of (35 ± 18) km s−1, (25 ± 13) km s−1, and (19 ± 9) km s−1, respectively. Results. We find a circular velocity curve with a slope of 0.4 ± 0.6 km s−1 kpc−1, which is consistent with a flat curve within the uncertainties. We further estimate a circular velocity at the Sun’s position of vc(R0) = 233 ± 7 km s−1 and that a region in the Sun’s vicinity, characterised by a physical length scale of ∼1 kpc, moves with a bulk motion of VLSR = 7 ± 7 km s−1. Finally, we estimate that the dark matter (DM) mass within 14 kpc is log10 MDM(R < 14kpc)/ M⊙ =(11.2+2.0-2.3) and the local spherically averaged DM density is ρDM(RO)=(0.41+0.10-0.09) GeV cm-3 = (0.011+0.003-0.002) M⊙pc-3. In addition, the effect of biased distance estimates on our results is assessed.

Effects of [formula omitted]-dimensional circularly symmetric and static traversable wormhole with cosmic string on oscillator field

In this study, we examine the relativistic quantum dynamics of an oscillator field described by the Klein–Gordon equation in a circularly symmetric and static (1+2)-dimensional traversable wormhole space–time with cosmic strings. We derive the radial equation for the Klein–Gordon oscillator in this wormhole background and solved it using special functions. For specific cases, we present the ground state energy level E1,ℓ± and wave function ψ1,ℓ of the oscillator field, subject to a constraint on the oscillator frequency ω1,ℓ. In fact, it is shown that the eigenvalue solution of the oscillator field is influenced by both the cosmic string parameter α and the wormhole throat radius a, leading to shifts in the energy levels and wave functions.

Rational extensions of the Dunkl oscillator in the plane and exceptional orthogonal polynomials

It is shown that rational extensions of the isotropic Dunkl oscillator in the plane can be obtained by adding some terms either to the radial equation or to the angular one obtained in the polar coordinates approach. In the former case, the isotropic harmonic oscillator is replaced by an isotropic anharmonic one, whose wave functions are expressed in terms of [Formula: see text]-Laguerre exceptional orthogonal polynomials. In the latter, it becomes an anisotropic potential, whose explicit form has been found in the simplest case associated with [Formula: see text]-Jacobi exceptional orthogonal polynomials.

What Are the Balanced and Unbalanced Dynamics of a Tropical Cyclone?

Abstract Recently, Ji and Qiao took into account the unbalanced components and derived an extended Sawyer–Eliassen (SE) equation. This study developed a new derivation of this extended SE equation from the perspective of restoring forces, and gives a physical interpretation for the coefficients that appear in the SE equation. For an unbalanced vortex, we demonstrated that the thermodynamic fields are only determined by the distribution of gradient wind, and thus, the gradient wind and thermodynamic fields always remain in balance as the unbalanced vortex evolves. Consequently, we attributed the gradient wind imbalance to the agradient wind rather than to the thermodynamic fields. Subsequently, we explored the effect of the agradient wind on the secondary circulation, and showed that the agradient wind strengthens the secondary circulation in its vicinity, which can be explained as a consequence of the restoring forces and mass continuity. Furthermore, we speculated that the SE equation, together with the radial velocity equation, could reproduce the primary characteristic of the axisymmetric boundary layer dynamics by prescribing the parameterization of subgrid-scale turbulent mixing. Specifically, the noU_BL and noVa_BL experiments conducted by Fei et al. in an article published in 2021 were reinterpreted, and the oscillation wavelength of the agradient wind in the eyewall was approximated based on this framework. Additionally, a new numerical solution algorithm to overcome the hyperbolicity near the boundary layer was proposed. This study attempts to develop a complete dynamic theory for tropical cyclones in both qualitative and quantitative perspectives.

Topological effects on generalized Duffin-Kemmer-Petiau oscillator under Aharonov–Bohm flux field and Coulomb potential

In this paper, we study the relativistic quantum motions of the oscillator field of the wave equation under the influence of the Aharonov–Bohm (AB) flux field with a Coulomb vector potential in the background of the topological defects produced by a cosmic string and global monopole space-time. We derive the radial equation of the generalized Duffin-Kemmer-Petiau (DKP) oscillator in a static cosmic string space-time and solve it through the Heun function equation. Afterwards, we derive the radial equation of the same generalized DKP oscillator in a point-like global monopole background and obtain the eigenvalue solutions using the same procedure. The generalized oscillator field is studied by substituting the radial momentum operator ∂ r → (∂ r + i M ω η 0 f(r)), where f(r) is an arbitrary function other than linear and introduces a vector potential of Coulomb-types through a minimal substitution via ∂ μ → (∂ μ − i q A μ ) in the relativistic wave equation. It is shown that the eigenvalue solutions of the oscillator field are influenced by the topological defects of the cosmic string and point-like global monopole space-times and get them modified. Furthermore, we see that the eigenvalue solutions depend on the geometric quantum phase, and hence, shifted them more in addition to the topological defects that show the gravitational analogue to the Aharonov–Bohm effect for the bound-states.

Greybody factors emitted by a regular black hole in a non-minimally coupled Einstein–Yang–Mills theory

In this paper, we study the greybody factors (GFs) for fermions with different spins and bosons in the regular black hole (BH) predicted by a non-minimal Einstein–Yang–Mills (EYM) theory. We investigate the effect of magnetic charge on effective potentials and GFs. For this purpose, we consider the Dirac and Rarita–Schwinger, as well as Klein–Gordon equations. First, we study the Dirac equation in curved spacetime for massive and massless spin-1/2 fermions. We then separate the Dirac equation into sets of radial and angular equations. Using the analytical solution of the angular equation, the Schrödinger-like wave equations with potentials are derived by decoupling the radial wave equations via the tortoise coordinate. We also consider the Rarita–Schwinger equation for massless spin-3/2 fermions and derive the one-dimensional Schrödinger wave equation with gauge-invariant effective potential. For bosons, we study the Klein–Gordon equation in the regular non-minimal EYM BH. Afterward, semi-analytic methods were used to calculate the fermionic and bosonic GFs. Finally, we discuss the graphical behavior of the obtained effective potentials and bounds on the GFs. According to graphs, the GF is highly influenced by the potential’s shape, which is determined by the parameterization of the model. This is in line with quantum theory.

Research on Electromagnetic Vibration Characteristics of a Permanent Magnet Synchronous Motor Based on Multi-Physical Field Coupling

Background and Purpose: The stator vibration characteristics are comprehensively mastered by considering the influence of winding and the housing structure on the stator modes. This effect is neglected in the research field of electromagnetic vibration of permanent magnet synchronous motors (PMSMs). Methods: The radial air-gap flux density equations and PMSM’s electromagnetic force density are derived, and then the harmonic characteristics of electromagnetic force density are studied. An equivalent finite element model of the stator is proposed that investigates the impacts of the winding and stator housing rules on the stator modal frequency. Finally, the harmonic response and acoustic analyses of electromagnetic vibration are carried out based on multi-physics field coupling. Results: The equivalent radiated power distribution laws and acoustic field of motor electromagnetic vibration under transient operating conditions are obtained. The theoretical analysis results are consistent with the experimental results. Conclusion: The obtained results show that the spatial order of the radial electromagnetic force is not equal to the order of the radial mode of the motor stator. The reason for this is that structural resonance is induced when the frequency components of the spatial radial electromagnetic force are coupled with the intrinsic frequency of the stator.