Radial Case Research Articles

Doppler dimming and brightening effects in solar prominences

ABSTRACT We explored the impact that Doppler dimming and brightening effects from bulk motions of solar prominences have on the formation of Ly $~\alpha$, H $~\alpha$, and Mg ii h line profiles. We compared two schemes in which these effects manifest; when the prominence is moving radially away from the solar surface (radial case), and when the prominence is moving parallel to the solar surface (horizontal case). To do this, we analysed 13 332 model profiles generated through the use of the 1D NLTE (i.e. departures from Local Thermodynamic equilibrium) radiative transfer (RT) code promweaver, built on the lightweaver NLTE RT framework to mimic the behaviour and output of the 1D NLTE RT code prom. We found that horizontal velocities are just as, or more important than radial velocities. This demonstrates that horizontal velocities need to be accounted for when attempting to do any sort of forward modelling.

Existence and symmetry breaking results for positive solutions of elliptic Hamiltonian systems

Abstract In this paper, we are interested in positive solutions of $$ \begin{align*}\left\{ \begin{array}{@{}ll} -\Delta u = a(x)v^{p-1}, \quad &\text{ in } \Omega,\\ -\Delta v = b(x)u^{q-1}, \quad &\text{ in } \Omega,\\ u,v>0, \quad &\text{ in } \Omega,\\ u=v=0, \quad &\text{ on } \partial\Omega, \end{array} \right. \end{align*} $$ where $\Omega $ is a bounded annular domain (not necessarily an annulus) in ${\mathbb {R}}^N (N \ge 3)$ and $ a(x), b(x)$ are positive continuous functions. We show the existence of a positive solution for a range of supercritical values of p and q when the problem enjoys certain mild symmetry and monotonicity conditions. We shall also address the symmetry breaking phenomena where the system is fully symmetric. Indeed, as a consequence of our results, we shall show that problem (1) has $\Bigl \lfloor \frac {N}{2} \Bigr \rfloor $ (the floor of $\frac {N}{2}$ ) positive non-radial solutions when $ a(x)=b(x)=1$ and $\Omega $ is an annulus with certain assumptions on the radii. In general, for the radial case where the domain is an annulus, we prove the existence of a non-radial solution provided $$ \begin{align*}(p-1)(q-1)> \Big(1+\frac{2N}{\lambda_H}\Big)^2\left(\frac{q}{p}\right),\end{align*} $$ where $\lambda _H$ is the best constant for the Hardy inequality on $\Omega .$ We remark that the best constant $\lambda _H$ for the Hardy inequality is just the characteristic of the domain, and is independent of the choices of p and $q.$ For this reason, the aforementioned inequality plays a major role to prove the existence and multiplicity of non-radial solutions when the problem is fully symmetric. Our proofs use a variational formulation on appropriate convex subsets for which the lack of compactness is recovered for the supercritical problem.

Scattering threshold for the focusing energy-critical generalized Hartree equation

Abstract This work investigates the asymptotic behavior of energy solutions to the focusing nonlinear Schrödinger equation of Choquard type i ∂ t u + Δ u + ∣ u ∣ p − 2 ( I α * ∣ u ∣ p ) u = 0 , p = 1 + 2 + α N − 2 , N ≥ 3 . i{\partial }_{t}u+\Delta u+{| u| }^{p-2}\left({I}_{\alpha }* {| u| }^{p})u=0,\hspace{1.0em}p=1+\frac{2+\alpha }{N-2},\hspace{1.0em}N\ge 3. Indeed, in the energy-critical spherically symmetric regime, one proves a global existence and scattering versus finite time blow-up dichotomy. Precisely, if the data have an energy less than the ground state one, two cases are possible. If the kinetic energy of the radial data is less than the ground state one, then the solution is global and scatters. Otherwise, if the data have a finite variance or is spherically symmetric and have a finite mass, then the solution is nonglobal. The main difficulty is to deal with the nonlocal source term. The argument is the concentration-compactness-rigidity method introduced by Kenig and Merle (Global well-posedness, scattering and blow-up for the energy-critical, focusing, non-linear Schrödinger equation in the radial case, Invent. Math. 166 (2006), no. 3, 645–675). This note naturally complements the work by Saanouni (Scattering theory for a class of defocusing energy-critical Choquard equations, J. Evol. Equ. 21 (2021), 1551–1571), where the scattering of the defocusing energy-critical generalized Hartree equation was obtained.

Scalar quasi-normal modes of accelerating Kerr-Newman-AdS black holes

We study linear scalar perturbations of slowly accelerating Kerr-Newman-anti-de Sitter black holes using the method of isomonodromic deformations. The conformally coupled Klein-Gordon equation separates into two second-order ordinary differential equations with five singularities. Nevertheless, the angular equation can be transformed into a Heun equation, for which we provide an asymptotic expansion for the angular eigenvalues in the small acceleration and rotation limit. In the radial case, we recast the boundary value problem in terms of a set of initial conditions for the isomonodromic tau function of Fuchsian systems with five regular singular points. For the sake of illustration, we compute the quasi-normal modes frequencies.

The Born approximation in the three-dimensional Calderón problem Ⅱ: Numerical reconstruction in the radial case

In this work we illustrate a number of properties of the Born approximation in the three-dimensional Calderón inverse conductivity problem by numerical experiments. The results are based on an explicit representation formula for the Born approximation recently introduced by the authors. We focus on the particular case of radial conductivities in the ball $ B_R \subset \mathbb{R}^3 $ of radius $ R $, in which the linearization of the Calderón problem is equivalent to a Hausdorff moment problem. We give numerical evidences that the Born approximation is well defined for $ L^{\infty} $ conductivities, and present a novel numerical algorithm to reconstruct a radial conductivity from the Born approximation under a suitable smallness assumption. We also show that the Born approximation has depth-dependent uniqueness and approximation capabilities depending on the distance (depth) to the boundary $ \partial B_R $. We then investigate how increasing the radius $ R $ affects the quality of the Born approximation, and the existence of a scattering limit as $ R\to \infty $. Similar properties are also illustrated in the inverse boundary problem for the Schrödinger operator $ -\Delta +q $, and strong recovery of singularity results are observed in this case.

Radial solutions and a local bifurcation result for a singular elliptic problem with Neumann condition

We study the problem − Δ u = λ u − u − 1 with a Neumann boundary condition; the peculiarity being the presence of the singular term − u − 1 . We point out that the minus sign in front of the negative power of u is particularly challenging, since no convexity argument can be invoked. Using bifurcation techniques we are able to prove the existence of solution ( u λ , λ ) with u λ approaching the trivial constant solution u = λ − 1 / 2 and λ close to an eigenvalue of a suitable linearized problem. To achieve this we also need to prove a generalization of a classical two-branch bifurcation result for potential operators. Next we study the radial case and show that in this case one of the bifurcation branches is global and we find the asymptotical behavior of such a branch. This results allows to derive the existence of multiple solutions u with λ fixed.

Optimization study of radial finned latent heat storage system employing novel circumferential perforations

Radial fins enhance the performance of Latent Heat Storage (LHS) systems. However, they limit the convection within the Phase Change Material (PCM). This limitation is addressed in this study by employing perforations to maximize the benefit of radial fins. This work proposes a novel ‘circumferential perforations’ on radial finned LHS system to enhance convection. A numerical approach is adopted to optimize the perforated fin geometry with ease. The numerical solution uses the enthalpy porosity technique to model the phase change and is validated experimentally. An optimization study is performed to determine the preferred location, thickness, overall geometry of perforation, HTF charging mode, and shell geometry. The circumferential perforations are found to clearly impact the melting by enhancing the convection heat transfer within PCM. The utilization of bottom charging has been observed to augment the usefulness of perforations significantly. Compared to the unperforated radial fins case, the optimized perforated fin heat exchanger has melted the 99 % of PCM in less than 50 minutes. This result is far higher compared with the existing perforated fin studies.

Boundary stabilization of the focusing NLKG equation near unstable equilibria: radial case

Boundary stabilization of the focusing NLKG equation near unstable equilibria: radial case

On classification of non-radiative solutions for various energy-critical wave equations

Non-radiative solutions of energy critical wave equations are such that their energy in an exterior region |x|>R+|t| vanishes asymptotically in both time directions. This notion, introduced by [7], has been key in solving the soliton resolution conjecture for these equations in the radial case. In the present paper, we first classify their asymptotic behaviour at infinity, showing that they correspond to a k-parameters family of solutions where k depends on the dimension. This generalises the previous results [8,6] in three and four dimensions. We then establish a unique maximal extension of these solutions.

Soliton Resolution for the Energy-Critical Nonlinear Wave Equation in the Radial Case

Soliton Resolution for the Energy-Critical Nonlinear Wave Equation in the Radial Case

Numerical analysis of thermal and hydrodynamic characteristics in aquaculture tanks with different tank structures

Improving hydrodynamic performance by changing the geometries of aquaculture tanks can facilitate salmon growth. Thus, this study proposed to redesign the aquaculture tank by varying the tank shape, numbers of inlets, and nozzle angles. The hydrodynamic and thermal characteristics of an aquaculture tank were numerically investigated via computational fluid dynamics simulations. The results indicated that the circular tanks mixed better than the octagonal tanks. Secondary vortices were more extensive in a two-inlet than a one-inlet tank. The vortex column in a two-inlet tank was smaller than that in the one-inlet tank. Therefore, the mixing performance of the two-inlet tank was better than that of the one-inlet tank. Further, the flow uniformity could be improved by optimizing the nozzle angle. Consequently, the mixing performance of the radial nozzle tank was better than that of the tangential nozzle tank because the secondary vortices were prominent and vortex column was smaller. A comparison of all the cases revealed the flow uniformity of the radial nozzle case was the highest. The volume percentages corresponding to the suitable velocity for salmon growth and velocity uniformity were 52.27% and 0.878, respectively. Furthermore, the vortex size was reduced and the velocity uniformity was increased by 2.23% by changing the nozzle angle to a radial angle. Therefore, changing the nozzle angle along the radial direction better improved the hydrodynamic characteristics.

Study on Multiphase Flow Characteristics During RH Refining Process Affected by Nonradial Arrangement of Gas‐Blowing Nozzle

Bath stirring, degassing, and decarburization in steel refining are strongly related to flow behaviors. The bubble plume produced in Ruhrstahl–Heraeus (RH) up‐snorkel plays an important role during refining, since it not only acts as a bubble pump, but also provides the reaction interface. Herein, it is aimed to form a new flow pattern in the up‐snorkel by using a nonradially arranged gas‐injection nozzle to enhance the 260 ton RH refining process. The results show that an upward spiral steel flow is produced, when nonradial gas‐injection nozzles are used in the up‐snorkel. Meanwhile, some bubbles moved toward the center region of the up‐snorkel, which may be caused by the centripetal effect in a rotational steel flow. This leads to a more uniform bubble distribution on the cross section of the snorkel, compared to that of the conventional case. Specifically, the circulation flow rate is increased by about 18.0%, and the mixing time are shortened by about 26.2% (criteria of ±5%), compared to that of the conventional case. In addition, the inclusion removal rate is increased by 0.5%, 4.8%, and 11.3% for the inclusion size of 20, 50, and 100 μm, respectively, compared to the conventional radial nozzle case.

Radially symmetric spiral flows of the compressible Euler-Poisson system for semiconductors

In this paper, we study the steady flows to the compressible Euler-Poisson system for semiconductors with the nonzero angular velocity in a radially symmetric way in an annulus. The main purpose here is to elucidate the effect of the angular velocity in the structure of the steady flows. We show the well-posedness of all kinds of types of radially symmetric spiral flows including radial subsonic/supersonic/transonic flows, and further give a specific classification of the flow patterns under the assumption of various boundary conditions at the inner and the outer circle. Additionally, different from the purely radial case, the uniqueness of radial subsonic flow can not be obtained due to the nonlocal effect caused by the angular velocity, consequently we prove the uniqueness of the radial subsonic solution in the case without the semiconductor effect or with a small current assumption. Moreover, some new patterns of spiral flows with or without shock are observed, such as a smooth transonic flow and a supersonic-supersonic shock flow for a large relaxation time parameter.

Regular solutions of chemotaxis-consumption systems involving tensor-valued sensitivities and Robin type boundary conditions

This paper deals with a parabolic–elliptic chemotaxis-consumption system with tensor-valued sensitivity [Formula: see text] under no-flux boundary conditions for [Formula: see text] and Robin-type boundary conditions for [Formula: see text]. The global existence of bounded classical solutions is established in dimension two under general assumptions on tensor-valued sensitivity [Formula: see text]. One of the main steps is to show that [Formula: see text] becomes tiny in [Formula: see text] for every [Formula: see text] and [Formula: see text] when [Formula: see text] is sufficiently small, which seems to be of independent interest. On the other hand, in the case of scalar-valued sensitivity [Formula: see text], there exists a bounded classical solution globally in time for two and higher dimensions provided the domain is a ball with radius [Formula: see text] and all given data are radial. The result of the radial case covers scalar-valued sensitivity [Formula: see text] that can be singular at [Formula: see text].

Prescribed blow-up sets for sequences of solutions to a non-local Q-curvature equation in ℝ3

Let [Formula: see text] be an open connected domain and [Formula: see text] stand for the Laplacian in [Formula: see text]. For a uniformly bounded sequence [Formula: see text] having a fixed sign and non-positive bi-harmonic functions [Formula: see text], we prove that there exists a sequence of solutions [Formula: see text] to [Formula: see text] with a total curvature [Formula: see text] such that [Formula: see text] for [Formula: see text] and [Formula: see text] for [Formula: see text], where [Formula: see text] is the zero set given by [Formula: see text] In particular, when [Formula: see text] is smoothly bounded and [Formula: see text], we obtain that [Formula: see text] blows up on [Formula: see text] for any [Formula: see text]. This extends the work of Hyder et al. who treated the analogue in even dimensions and complements a previous result in dimension three. We also consider a radial case with no blowing up phenomena and large total curvature.

Blow-Up Solutions of Two-Coupled Nonlinear Schrödinger Equations in the Radial Case

Blow-Up Solutions of Two-Coupled Nonlinear Schrödinger Equations in the Radial Case

Singularity formation and regularization at multiple times in the viscous Hamilton–Jacobi equation

The Cauchy–Dirichlet problem for the superquadratic viscous Hamilton–Jacobi equation (VHJ) from stochastic control theory, admits a unique, global viscosity solution. Solutions thus exist in the weak sense after appearance of singularity, which occurs through gradient blow-up (GBU) on the boundary. Whereas viscosity solution theory has been extensively applied to many PDEs, there seem to be less results on refined singular behavior of solutions. Although occurrence of two types of GBU, with or without loss of boundary condition (LBC), are known, detailed behavior after GBU has remained open except for a strongly restricted special class of one dimensional solutions. In this paper, in general dimensions, we construct solutions which undergo GBU with LBC arbitrarily many times and then recover regularity, as well as solutions without LBC at first GBU time. In one space dimension, we obtain the complete classification of viscosity solutions at each time, which extends to radial case in higher dimensions. Furthermore we show the existence of solutions exhibiting an arbitrarily given combination of GBU types with/without LBC at multiple times, in which a new type of behavior called “bouncing” is discovered. Global weak solutions of VHJ with multiple times singularity turn out to display larger variety of behaviors than in Fujita equation. We introduce a method based on an arbitrary number of critical parameters, whose continuity requires delicate arguments. Since we do not rely on any known special solution unlike in Fujita equation, our method is expected to apply to other equations. Singular behaviors at multiple times are completely new in the context of VHJ but also of stochastic control theory. Our results imply that for certain spatial distributions of rewards, if a controlled Brownian particle starts near the boundary, then the net gain attains profitable values on different time horizons but not on some intermediate times.

Nozzle Jet Deviation from Bucket Pitch Circle’s Effect on the Stability and Efficiency of Pelton Turbine

During the operation of a Pelton turbine, the centerline of the nozzle jet may deviate from the bucket pitch circle due to the low installation and maintenance accuracy, which will reduce the operating efficiency and the stability of the turbine and even cause severe vibrations and damages. Based on the VOF (Volume of Fluid) two-phase flow model and the SST k-ω turbulence model, the flow characteristics of a Pelton turbine were simulated with the nozzle jet deviating from the bucket pitch circle. The pressure pulsation inside the bucket and the force distribution of the runner were obtained, the turbine oscillation and efficiency were measured before and after the jet deviation, and the effects of the radial and axial deviations on the stability and efficiency of the Pelton turbine were analyzed. The results show that both the radial and axial deviations of the jet cause a significant increase in the axial force and the pressure pulsation amplitude of the turbine; the radial and tangential forces on the runner are slightly reduced; the maximum axial force on the runner is increased by 4 times and 2 times, respectively, after the axial and radial deviations within the maximum value allowed by the industry standard; and the efficiency of the turbine is reduced by 0.4% and 0.3%, respectively. The maximum relative amplitude of pressure pulsation in the radial offset case appears in the center of the bucket blade, while the axial offset case causes uneven pressure distribution on both sides of the diverter blade, uneven force on the bucket blade of the runner, and fatigue damage. By comparing the operation of the runner under the two offset cases, we can find that the axial offset of the jet has a greater impact on the stability of the runner than the radial offset, and the unit is more prone to vibration, increasing the risk of the unit lifting.

A Novel Technique to Convert Static Sling into a Dynamic Sling in Reconstruction of Total Lower Lip Defects Using Palmaris Longus Tendon with Free Radial Forearm Flap and Case Series.

Lower lip reconstruction has been a major challenge for the reconstructive surgeons since time immemorial. Various types of reconstruction had been described for the reconstruction of lower lip ranging from local flaps to free tissue transfer to free functioning muscle transfer. For complete lower lip defects, the free radial forearm flap with palmaris longus tendon has been the standard of reconstruction for many years. Literatures suggests various techniques for utilizing palmaris longus tendon sling as a static as well as dynamic structure. The limitation with static reconstruction is the loss of sling support with time leading to eversion and drooping of the reconstructed lower lip and drooling of saliva. In this article we describe a simpler and novel technique which converts static reconstruction into a dynamic one with a series of 5 patients. Aim of this prospectivestudy wasto evaluate the post operative functional and aesthetic outcome of our technique of total lower lip reconstruction using free radial forearm flap with palmaris longus sling reconstruction. The medical records included were demographics, including age, gender, and reason for the acquired defect, lip reconstructed, reconstructive method, flap survival, and outcomes. At 6months follow up, all patients had satisfactory outcome and were able to take oral diets and none of the patients complained of drooling, an inability to eat in a public setting, or microstomia. This technique which is a simpler modification can be helpful in converting the traditional static sling reconstruction of lower lip into a dynamic one resulting in good aesthetic and functional outcomes.

Nuclear design of a shielded cabinet for electronics: The ITER radial neutron camera case study

The Radial Neutron Camera (RNC) is a diagnostic system located in ITER Equatorial Port #1 providing several spatial and time-resolved parameters for the fusion power estimation, plasma control and physics studies. The RNC measures the uncollided 14 MeV and 2.5 MeV neutrons from deuterium-tritium (DT) and deuterium-deuterium (DD) fusion reactions through an array of neutron flux detectors located in collimated Lines of Sight. Signals from RNC detectors (fission chambers, single Crystal Diamonds and scintillators) need preamplification because of their low amplitude. These preamplifiers have to be as close as possible to the detectors in order to minimize signal degradation and must be protected against fast and thermal neutrons, gamma radiation and electromagnetic fields. The solution adopted is to host the preamplifiers in a shielded cabinet located in a dedicated area of the Port Cell, behind the Bioshield Plug. The overall design of the cabinet must ensure the necessary magnetic, thermal and nuclear shielding and, at the same, satisfy weight and allocated volume constraints and maintain its structural integrity. The present paper describes the nuclear design of the shielded cabinet, performed by means of 3D particle transport calculations (MCNP), taking into account the radiation streaming through the Bioshield penetrations and the cross-talk effect from the neighboring Lower and Upper Ports. We present the assessment of its nuclear shielding performances and analyze the compliancy with the alert thresholds for commercial electronics in terms of neutron flux and cumulated ionizing dose.