Electron energy gain in the transverse electric mode of a coaxial waveguide filled with plasma by microwave radiation

The dynamics and acceleration of an electron in the field of a transverse electric wave propagating inside a cold plasma-filled waveguide by microwave radiation is analytically investigated by considering the existence of a coaxial metallic rod. The relativistic momentum and energy equations are solved for the electron when it is injected along the direction of mode propagation. The deflection angle and the acceleration gradient of the electron in the configuration are obtained. Besides, the results are graphically presented.     

High-order,finite-volume methods in mapped coordinates

We present an approach for constructing finite-volume methods for flux-divergence forms to any order of accuracy defined as the image of a smooth mapping from a rectangular discretization of an abstract coordinate space. Our approach is based on two ideas. The first is that of using higher-order quadrature rules to compute the flux averages over faces that generalize a method developed for Cartesian grids to the case of mapped grids. The second is a method for computing the averages of the metric terms on faces such that freestream preservation is automatically satisfied. We derive detailed formulas for the cases of fourth-order accurate discretizations of linear elliptic and hyperbolic partial differential equations. For the latter case, we combine the method so derived with Runge–Kutta time discretization and demonstrate how to incorporate a high-order accurate limiter with the goal of obtaining a method that is robust in the presence of discontinuities and underresolved gradients. For both elliptic and hyperbolic problems, we demonstrate that the resulting methods are fourth-order accurate for smooth solutions.     

Adaptive higher-order finite element methods for transient PDE problems based on embedded higher-order implicit Runge–Kutta methods

We present a new class of adaptivity algorithms for time-dependent partial differential equations (PDE) that combine adaptive higher-order finite elements (hp-FEM) in space with arbitrary (embedded, higher-order, implicit) Runge–Kutta methods in time. Weak formulation is only created for the stationary residual, and the Runge–Kutta methods are specified via their Butcher’s tables. Around 30 Butcher’s tables for various Runge–Kutta methods with numerically verified orders of local and global truncation errors are provided. A time-dependent benchmark problem with known exact solution that contains a sharp moving front is introduced, and it is used to compare the quality of seven embedded implicit higher-order Runge–Kutta methods. Numerical experiments also include a comparison of adaptive low-order FEM and hp-FEM with dynamically changing meshes. All numerical results presented in this paper were obtained using the open source library Hermes (http://www.hpfem.org/hermes) and they are reproducible in the Networked Computing Laboratory (NCLab) at http://www.nclab.com.     

Describing intrinsic noise in Chua's circuit

In this Letter we demonstrate that intrinsic inevitable noise effects, existing in realistic experiments with electronic circuits, are properly described theoretically using a Gaussian noise. For this we integrate numerically the equations of motion from the Chua circuit using a fourth-order stochastic Runge–Kutta integrator. Periodic structures in parameter space, related to periodic motion, start to be destroyed when the noise intensity is increased and vanish at a critical intensity value, for which only chaotic motion remains. We find the appropriate noise intensity interval which satisfactorily reproduces the parameter space from the corresponding experiment and it is in remarkable agreement with the estimated experimental noise. Present achievements should be applicable to describe noise effects in a wide number of electronic circuits.     

Instabilities Driven by an Intense Beam in a Plasma-Filled Dielectric-Lined Waveguide Immersed in a Finite Axial Magnetic Field

A linear analysis is described on stabilities driven by an intense relativistic electron beam in an infinitely long, plasma-filled, and dielectric-lined circular waveguide immersed in a finite strength axial magnetic field. A dispersion equation is derived from the cold fluid theory and solved numerically. Beam-plasma instabilities due to interaction between beam modes and the Trivelpiece-Gould modes appear as well as the Cherenkov and the cyclotron Cherenkov instabilities. Parametric researches are carried out varying magnetic field strength, plasma density, and dielectric constant. Effects of a finite magnetic field and plasma filling are discussed in connection with the possibilities of using this system as a microwave radiation source.     

Rapid communication: Transverse spin with coupled plasmons

The steady two-dimensional flow and heat transfer of a non-Newtonian power-law nanofluid over a stretching surface under convective boundary conditions and temperature-dependent fluid viscosity has been numerically investigated. The power-law rheology is adopted to describe non-Newtonian characteristics of the flow. Four different types of nanoparticles, namely copper (Cu), silver (Ag), alumina (Al ₂ O ₃) and titanium oxide (TiO ₂) are considered by using sodium alginate (SA) as the base non-Newtonian fluid. Lie symmetry group transformations are used to convert the boundary layer equations into non-linear ordinary differential equations. The transformed equations are solved numerically by using a shooting method with fourth-order Runge–Kutta integration scheme. The results show that the effect of viscosity on the heat transfer rate is remarkable only for relatively strong convective heating. Moreover, the skin friction coefficient and the rate of heat transfer increase with an increase in Biot number.     

左手介质矩形波导导模和表面模的场分布

从麦克斯韦方程组出发,结合电磁场的边界条件,推导出介质矩形波导导模的一般色散方程.对普通介质矩形波导和左手介质矩形波导的导模场分布分别进行了数值模拟.通过对比两种介质矩形波导的导模场分布的模拟结果,发现左手介质矩形波导Ex22模的场分布比普通介质波导Ex00模的场更集中在波导中部.同时,根据处理普通介质矩形波导的Marcatili方法.类比得到左手介质矩形波导表面模的色散方程,并数值模拟了低阶模的场分布,结果表明,能量主要集中在波导的四个角区以及波导的边缘.     

Efficiency enhancement of a two-beam free-electron laser using a nonlinearly tapered wiggler

A nonlinear and non-averaged model of a two-beam free-electron laser(FEL) wiggler that is tapered nonlinearly in the absence of slippage is presented.The two beams are assumed to have different energies,and the fundamental resonance of the higher energy beam is at the third harmonic of the lower energy beam.By using Maxwell’s equations and the full Lorentz force equation of motion for the electron beams,coupled differential equations are derived and solved numerically by the fourth-order Runge-Kutta method.The amplitude of the wiggler field is assumed to decrease nonlinearly when the saturation of the third harmonic occurs.By simulation,the optimum starting point of the tapering and the slopes for reducing the wiggler amplitude are found.This technique can be applied to substantially improve the efficiency of the two-beam FEL in the XUV and X-ray regions.The effect of tapering on the dynamical stability of the fast electron beam is also studied.     

Optimal Runge–Kutta schemes for discontinuous Galerkin space discretizations applied to wave propagation problems

We study the performance of methods of lines combining discontinuous Galerkin spatial discretizations and explicit Runge–Kutta time integrators, with the aim of deriving optimal Runge–Kutta schemes for wave propagation applications. We review relevant Runge–Kutta methods from literature, and consider schemes of order q from 3 to 4, and number of stages up to q + 4, for optimization. From a user point of view, the problem of the computational efficiency involves the choice of the best combination of mesh and numerical method; two scenarios are defined. In the first one, the element size is totally free, and a 8-stage, fourth-order Runge–Kutta scheme is found to minimize a cost measure depending on both accuracy and stability. In the second one, the elements are assumed to be constrained to such a small size by geometrical features of the computational domain, that accuracy is disregarded. We then derive one 7-stage, third-order scheme and one 8-stage, fourth-order scheme that maximize the stability limit. The performance of the three new schemes is thoroughly analyzed, and the benefits are illustrated with two examples. For each of these Runge–Kutta methods, we provide the coefficients for a 2N-storage implementation, along with the information needed by the user to employ them optimally.     

Efficiency enhancement of a two-beam free-electron laser using a nonlinearly tapered wiggler

A nonlinear and non-averaged model of a two-beam free-electron laser (FEL) wiggler that is tapered nonlinearly in the absence of slippage is presented. The two beams are assumed to have different energies, and the fundamental resonance of the higher energy beam is at the third harmonic of the lower energy beam. By using Maxwell's equations and the full Lorentz force equation of motion for the electron beams, coupled differential equations are derived and solved numerically by the fourth-order Runge-Kutta method. The amplitude of the wiggler field is assumed to decrease nonlinearly when the saturation of the third harmonic occurs. By simulation, the optimum starting point of the tapering and the slopes for reducing the wiggler amplitude are found. This technique can be applied to substantially improve the efficiency of the two-beam FEL in the XUV and X-ray regions. The effect of tapering on the dynamical stability of the fast electron beam is also studied.     

Magnetic field effect on the binding energy of a hydrogenic impurity in coaxial GaAs/ AlxGa1−xAs quantum well wires

We propose a coaxial cylindrical quantum well wire (QWW) system, in which two conducting cylindrical layers are separated by an insulating layer. The ground state binding energy of a hydrogenic impurity subjected to uniform magnetic field applied parallel to the wire axis is studied within a variational scheme as a function of the inner barrier thickness for two different impurity positions and various barrier potentials. The ground state energy and wave function in the presence of a magnetic field is directly calculated using the fourth-order Runge–Kutta method. It is found that the binding energy in critical barrier thickness shows a sharp increase or decrease depending on the impurity position and magnetic field strength. The main result is that a sharp variation in the binding energy, which may be important in device applications, depends strongly not only on the location of the impurity but also on the magnetic field and the geometry of the wire.     

Effect of electron–ion coupling on the electric microfield distribution in plasmas

The electronic and ionic effective potential of a fully ionized hydrogen plasma containing an impurity of electric charge (+Z _m e ) are calculated in a two‐component plasma model under semiclassical conditions using classical statistical mechanics with a regularized electron–ion interaction. These effective potentials are coupled in a system of nonlinear integral equations (or coupled differential equations), which is solved numerically with two methods, namely the fixed‐point method and the Runge–Kutta method. The Baranger–Moser electric microfield distributions are calculated and compared with those from molecular dynamics simulation. Agreement between theory and simulation is satisfactory, in general.     

Application of obliquely interfering TE10 modes for electron energy gain

Two fundamental TE₁₀ modes are considered to interfere at a small angle θ and then propagate along the z-axis in an evacuated rectangular waveguide. The electron trajectory in the resultant field and the expressions for energy gain and the acceleration gradient are obtained, when the electron is injected along the z-axis. A 50 keV electron gains 718 keV energy in a 4.0 cm × 2.5 cm waveguide, when the microwave with intensity of 1 × 10¹⁰ W/cm² and frequency 5.577 GHz is used and the modes superpose at an angle of 10°; here the maximum acceleration gradient is obtained as 251 MeV/m. The energy gain and acceleration gradient are decreased with increasing width of the waveguide and microwave frequency. Higher gradient and larger energy gain are obtained for the higher microwave intensity, smaller angle of superposition and also when the electron is injected with larger initial energy.     

Microbubble characteristic in a T-junction microchannel in microfluidic chip

Formation characteristic of microbubble in a T-junction microchannel in polydimethylsiloxane (PDMS) microfluidic chip is investigated. Microbubble formation experiments are carried out to investigate the effects of liquid flow rate, gas pressure and gas channel width on the detachment volume and formation time of microbubble. The growth process of the bubble is described by the form of force analysis, and the definition of the bubble detachment distance is introduced. The equivalent spherical theoretical model of bubble detachment diameter prediction is established, and the numerical solution of the model is obtained by the fourth-order Runge–Kutta method. The prediction results of theoretical model correlate well with the experimental data.     

Hydromagnetic flow of a variable viscosity nanofluid in a rotating permeable channel with hall effects

The flow, heat and mass transfer of water-based nanofluid are examined between two horizontal parallel plates in a rotating system. The effects of Brownian motion, thermophoresis, viscosity and Hall current parameters are considered. The governing partial differential equations are reduced to ordinary differential equations that are then solved numerically using the Runge–Kutta–Fehlberg method. Validation of numerical solution is achieved with an exact solution of primary velocity and found to be in good agreement. Results show that both surfaces experience opposite behavior regarding skin friction, Nusselt and Sherwood numbers in both primary and secondary flows. These physical quantities depend upon dimensionless parameters and numbers.     

Numerical study of magneto-convective heat and mass transfer from inclined surface with Soret diffusion and heat generation effects: A model for ocean magnetic energy generator fluid dynamics

A mathematical model is developed for steady state magnetohydrodynamic (MHD) heat and mass transfer flow along an inclined surface in an ocean MHD energy generator device with heat generation and thermo-diffusive (Soret) effects. The governing equations are transformed into nonlinear ordinary differential equations with appropriate similarity variables. The emerging two-point boundary value problem is shown to depend on six dimensionless thermophysical parameters - magnetic parameter, Grashof number, Prandtl number, modified Prandtl number, heat source parameter and Soret number in addition to plate inclination. Numerical solutions are obtained for the nonlinear coupled ordinary differential equations for momentum, energy and salinity (species) conservation, numerically, using the Nachtsheim–Swigert shooting iteration technique in conjunction with the Runge–Kutta sixth order iteration scheme. Validation is achieved with Nakamura's implicit finite difference method. Further verification is obtained via the semi-numerical Homotopy analysis method (HAM). With an increase in magnetic parameter, skin friction is depressed whereas it generally increases with heat source parameter. Salinity magnitudes are significantly reduced with increasing heat source parameter. Temperature gradient is decreased with Prandtl number and salinity gradient (mass transfer rate) is also reduced with modified Prandtl number. Furthermore, the flow is decelerated with increasing plate inclinations and temperature also depressed with increasing thermal Grashof number.     

Numerical solution and experiment of a self-Q-switched 946 nm Cr,Nd:YAG laser

A LD-pumped self-Q-switched 946 nm laser by using a co-doped Cr,Nd:YAG crystal as a gain medium as well as a saturable absorber is studied. The 946 nm self-Q-switched rate equations of co-doped crystal are solved numerically by Runge–Kutta method directly. The important parameters of lasers, such as average output power, threshold pump power, pulse width, pulses repetition rate, and optimal transmission of output coupler are obtained numerically. Experimentally, the maximum average output power up to 2.61 W, corresponding to a slope efficiency of 23.43%, was obtained in a simple and compact linear cavity. The optical-to-optical efficiency is 17.3% and the peak power is 7.57 kW with repetition rate of 23.78 kHz and pulse width of 14.5 ns. In the experiments, a high-quality fundamental transverse mode can be preserved in a large range of incident pump power. The numerical results of the 946 nm self-Q-switched Cr,Nd:YAG laser are in good agreement with the experimental results.     

Electron energy gain in the fundamental mode of an elliptical waveguide in the presence of static helical magnet by microwave radiation

The dynamics and energy gain of an electron in the field of a transverse electric wave propagating inside an elliptical waveguide is analytically investigated by considering the existence of a helical magnet in which the field is perpendicular to the axis of the waveguide and rotating as a function of position along the magnet. Besides, by solving the relativistic momentum and energy equations, the deflection angle and the acceleration gradient of the electron in the waveguide are obtained. It is shown that the electron is deflected due to the field components of the transverse electric mode of this microwave, and at the same time, it is accelerated by these fields. Furthermore, the expressions of the acceleration gradient and deflection angle for an electron in the transverse electric mode inside the plasma elliptical waveguide without a static helical magnet are presented, which was injected initially along the propagation direction of the microwave. The results are graphically presented.     

微波大气击穿阈值的理论研究

本文通过对使用有效场强(或均方根场强)得到的微波大气击穿阈值表达式进行讨论, 指出其推导中所做的假设及这些假设应用到微波大气击穿过程中存在的问题. 然后分别使用解析理论和数值模拟对微波大气击穿过程中的有效电子温度变化过程和击穿阈值进行研究, 并将其与直流电场进行比较. 分析发现在高气压下, 电子能量转移频率高, 有效电子温度随电场大幅振荡, 由于电离频率随有效电子温度的增长率大于电子能量损失随有效电子温度的增长率, 因此在高气压时, 微波大气击穿阈值低于使用有效场强的击穿阈值. 通过大量分析, 给出了理论推导和数值模拟得到的微波大气击穿阈值拟合表达式.     

Electron Motion in the Three-Dimensional Field of Undulator

In this paper we consider the motion of relativistic electrons in an ideal three-dimensional magnetic field of an undulator. The ideality of the magnetic field means that, on the undulator axis, the field is directed strictly vertically upward and has a strictly sinusoidal shape. In the overwhelming majority of cases, only this leading component of the field is taken into account in calculating the electron trajectory. In this paper, in the equations of motion of an electron in the magnetic field of an undulator, all three components of the field are taken into account, so that the undulator field under consideration satisfies the stationary Maxwell equations. In this case, the differential equations of motion of the electron are solved analytically with the help of perturbation theory, and not by the method of averaging over fast oscillations of the electron, as was done in a number of previous papers. These analytic expressions for trajectories describe the behavior of particles in the focusing magnetic field of an undulator much more completely. An analysis of these expressions shows that the behavior of electrons in such a three-dimensional field of the undulator is much more complicated than what follows from the equations obtained by the averaging method. In particular, there is a cross effect when changes in the initial vertical parameters of the electron trajectory cause changes in the horizontal component of its trajectory and vice versa. A comparison of the solutions obtained analytically with the results of numerical calculations of electron trajectories using the Runge–Kutta method demonstrates their high accuracy.