Theoretical study of particle and energy balance equations in locally bounded plasmas

In this study,new particle and energy balance equations have been developed to predict the electron temperature and density in locally bounded plasmas.Classical particle and energy balance equations assume that all plasma within a reactor is completely confined only by the reactor walls.However,in industrial plasma reactors for semiconductor manufacturing,the plasma is partially confined by internal reactor structures.We predict the effect of the open boundary area(A′_(L,eff))and ion escape velocity(u_(i))on electron temperature and density by developing new particle and energy balance equations.Theoretically,we found a low ion escape velocity(u_(i)/u_(B)≈0.2)and high open boundary area(A′_(L,eff)/A_(T,eff)≈0.6)to result in an approximately 38%increase in electron density and an 8%decrease in electron temperature compared to values in a fully bounded reactor.Additionally,we suggest that the velocity of ions passing through the open boundary should exceedω_(pi)λ_(De)under the condition E^(2)_(0)?(Φ/λ_(De))^(2).

Solution of general dynamic equation for nanoparticles in turbulent flow considering fluctuating coagulation

A new averaged general dynamic equation （GDE） for nanoparticles in the turbulent flow is derived by considering the combined effect of convection, Brownian diffusion, turbulent diffusion, turbulent coagulation, and fluctuating coagulation. The equation is solved with the Taylor-series expansion moment method in a turbulent pipe flow. The experiments are performed. The numerical results of particle size distribu- tion correlate well with the experimental data. The results show that, for a turbulent nanoparticulate flow, a fluctuating coagulation term should be included in the averaged particle GDE. The larger the Schmidt number is and the lower the Reynolds number is, the smaller the value of ratio of particle diameter at the outlet to that at the inlet is. At the outlet, the particle number concentration increases from the near-wall region to the near-center region. The larger the Schmidt number is and the higher the Reynolds num- ber is, the larger the difference in particle number concentration between the near-wall region and near-center region is. Particle polydispersity increases from the near-center region to the near-wall region. The particles with a smaller Schmidt number and the flow with a higher Reynolds number show a higher polydispersity. The degree of particle polydispersity is higher considering fluctuating coagulation than that without considering fluctuating coagulation.

ON EQUATION OF DISCRETE SOLID PARTICLES' MOTION IN ARBITRARY FLOW FIELD AND ITS PROPERTIES

The forces on rigid particles moving in relation to fluid having been studied and the equation of modifications of their expressions under different flow conditions discussed, a general form of equation for discrete particles' motion in arbitrary flow field is obtained. The mathematical features of the linear form of the equation are clarified and analytical solution of the linearized equation is gotten by means of Laplace transform. According to above theoretical results, the effects of particles' properties on its motion in several typical flow field are studied, with some meaningful conclusions being reached.

Fresnel Equations Derived Using a Non-Local Hidden-Variable Particle Theory

Problem: The Fresnel equations describe the proportions of reflected and transmitted light from a surface, and are conventionally derived from wave theory continuum mechanics. Particle-based derivations of the Fresnel equations appear not to exist. Approach: The objective of this work was to derive the basic optical laws from first principles from a particle basis. The particle model used was the Cordus theory, a type of non-local hidden-variable (NLHV) theory that predicts specific substructures to the photon and other particles. Findings: The theory explains the origin of the orthogonal electrostatic and magnetic fields, and re-derives the refraction and reflection laws including Snell’s law and critical angle, and the Fresnel equations for s and p-polarisation. These formulations are identical to those produced by electromagnetic wave theory. Contribution: The work provides a comprehensive derivation and physical explanation of the basic optical laws, which appears not to have previously been shown from a particle basis. Implications: The primary implications are for suggesting routes for the theoretical advancement of fundamental physics. The Cordus NLHV particle theory explains optical phenomena, yet it also explains other physical phenomena including some otherwise only accessible through quantum mechanics (such as the electron spin g-factor) and general relativity (including the Lorentz and relativistic Doppler). It also provides solutions for phenomena of unknown causation, such as asymmetrical baryogenesis, unification of the interactions, and reasons for nuclide stability/instability. Consequently, the implication is that NLHV theories have the potential to represent a deeper physics that may underpin and unify quantum mechanics, general relativity, and wave theory.

Determination of Photon and Elementary Particles Rest Masses Using Maxwell’s Equations and Generalized Potential Dependent Special Relativity

The nature and origin of the photon and elementary rest masses are some of the challeng-ing problems that physics face. The approaches used to solve these problems are complex and time-consuming. Specifically, the photon rest mass pays attention to theoretical physi-cists. Many experimental works show that the photon rest mass is non zero. This problem can be solved using generalized potential dependent special relativity, which has been de-rived using simple arguments, and Maxwell’s equations, besides the conventional Einstein energy-momentum relation. The results obtained show that the rest mass of photons and elementary particles are strongly dependent on the vacuum energy and a universal con-stant. This result conforms with the models that predict time decaying vacuum energy as-sociated with production of smaller rest mass particles followed by larger masses. The two potential dependent mass expressions conform with the cosmological models that suggest the photon is generated first by assuming the universe consisting of total constant vacuum with decaying cosmological part and mass generating part. Using Maxwell’s equations, beside plank and De Broglie hypothesis together with special relativity energy-momentum relation the photon rest mass is estimated. It was shown that the photon rest mass is ex-tremely small compared to the electron mass.

The Natures of Microscopic Particles Depicted by Nonlinear Schrodinger Equation in Quantum Systems

When the microscopic particles was depicted by linear Schrodinger equation, we find that the particles have only a wave feature, thus, a series of difficulties and intense disputations occur in quantum mechanics. These problems excite us to consider the nonlinear interactions among the particles or between the particle and background field, which is completely ignored in quantum mechanics. Thus we use the nonlinear Schrodinger equation to describe the natures of microscopic particles. In this case the natures and features of microscopic particles are considerably different from those in quantum mechanics, where the microscopic particles are localized and have truly a wave-particle duality. Meanwhile, they satisfy both the classical dynamics equation and Lagrangian and Hamilton equations and obey the conservation laws of mass, energy and momentum. These natures and features are due to the nonlinear interactions, which are generated in virtue of the interaction between the moved particles and background field through the mechanisms of self-trapping, self-focus and self-condensation. Finally, we verified experimentally the localization and wave-corpuscle features of microscopic particles described by the nonlinear Schrodinger equation using the properties of water soliton and optical-soliton depicted also by the nonlinear Schrodinger equation in water and optical fiber, respectively. Therefore, the new nonlinear quantum theory established on the basis of nonlinear Schrodinger equation is correct and credible. From this investigation we can not only solve difficulties and problems disputed for about a century by plenty of scientists in quantum mechanics but also promote the development of physics and enhance the knowledge and recognition levels to the essences of microscopic matter.

Experimental study on movement of particles in centrifugal impeller

The slurry pump is the key component of a dredger. Solid particles have strong influence on the performance of a slurry pump. The movement of solid particles in a centrifugal impeller was studied using particle image velocimetry(PIV) measurement. The experiments were conducted in a dredging pump model at Hohai University. Some transparent glass spheres with diameter of 0. 2-0. 4 mm were used as solid particles. The concentration and relative velocities of the particles were analyzed to investigate the particle trajectory. The results show that the concentration of the particles on the pressure surfaces of the blades is higher than on the suction surfaces,and the particles tend to move towards the suction surfaces. Moreover,the particles have faster relative velocities than the liquid phase through the flow channels of the impeller.

Multi-quadric Equations Interpolation and Its Applications to the Establishment of Crustal Movement Speed Field

In this paper,multi_quadric equations interpolation is used to establish a widely covered and valuable speed field model,with which the crustal movement image is obtained.

Relationship between self-propelled velocity and Brownian motion for spherical and ellipsoid particles

The Brownian motion of spherical and ellipsoidal self-propelled particles was simulated without considering the effect of inertia and using the Langevin equation and the diffusion coefficient of ellipsoidal particles derived by Perrin.The P´eclet number(Pe)was introduced to measure the relative strengths of self-propelled and Brownian motions.We found that the motion state of spherical and ellipsoid self-propelled particles changed significantly under the influence of Brownian motion.For spherical particles,there were three primary states of motion:1)when Pe<30,the particles were still significantly affected by Brownian motion;2)when Pe>30,the self-propelled velocities of the particles were increasing;and 3)when Pe>100,the particles were completely controlled by the self-propelled velocities and the Brownian motion was suppressed.In the simulation of the ellipsoidal self-propelled particles,we found that the larger the aspect ratio of the particles,the more susceptible they were to the influence of Brownian motion.In addition,the value interval of Pe depended on the aspect ratio.Finally,we found that the directional motion ability of the ellipsoidal self-propelled particles was much weaker than that of the spherical self-propelled particles.

MULTI-MONTE-CARLO METHOD FOR GENERAL DYNAMIC EQUATION CONSIDERING PARTICLE COAGULATION

Monte-Carlo （MC） method is widely adopted to take into account general dynamic equation （GDE） for particle coagulation, however popular MC method has high computation cost and statistical fatigue. A new Multi-Monte-Carlo （MMC） method, which has characteristics of time-driven MC method, constant number method and constant volume method, was promoted to solve GDE for coagulation. Firstly MMC method was described in details, including the introduction of weighted fictitious particle, the scheme of MMC method, the setting of time step, the judgment of the occurrence of coagulation event, the choice of coagulation partner and the consequential treatment of coagulation event. Secondly MMC method was validated by five special coagulation cases in which analytical solutions exist. The good agreement between the simulation results of MMC method and analytical solutions shows MMC method conserves high computation precision and has low computation cost. Lastly the different influence of different kinds of coagulation kernel on the process of coagulation was analyzed： constant coagulation kernel and Brownian coagulation kernel in continuum regime affect small particles much more than linear and quadratic coagulation kernel,whereas affect big particles much less than linear and quadratic coagulation kernel.

Implementation of the Integrated Green’s Function Method for 3D Poisson’s Equation in a Large Aspect Ratio Computational Domain

The solution of Poisson’s Equation plays an important role in many areas, including modeling high-intensity and high-brightness beams in particle accelerators. For the computational domain with a large aspect ratio, the integrated Green’s function method has been adopted to solve the 3D Poisson equation subject to open boundary conditions. In this paper, we report on the efficient implementation of this method, which can save more than a factor of 50 computing time compared with the direct brute force implementation and its improvement under certain extreme conditions.

The Advection Wave-in-Secondary Saturation Movement Equation and Its Application to Concentration Tension-Driven Saturation Kinetic Flow

The deterministic description of a wave of solution particle of efavirenz is given. Simulated pharmacokinetic data points from patients on efavirenz are used. The one dimensional wave equation is used to infer on transfer of vibrations due to tension between solution particles. The work investigates movement using wave analogy, but in a different variable space. Two important movement fluxes of a wave are derived an attracting one identified as tension conductivity and a dispersing one identified as tension diffusivity. The Wave Equation can be used to describe another spin-off movement flux formed induced by vibrations in solution particle.

Theoretical Computation of Nonlinear System Equations of Heavier Pellets Movements for Two Phase Flow

This paper presents nonlinear ordinary differential equations (ODES) of the heavier pellets movement for two phase flow, which actually represent a system of equations. The usual methods of solution such as Runge -Kutta method and it's datum results are discussed. This paper solves ODES of general form using variable mesh-length, linearizing the nonlinear terms by finite analysis method, fuilding an iteration sequence, and amending the nonlinear terms by iteration . The conditions of convergent operation of iteration solution is checked. The movement orbit and velocity of the pellets are calculated. Analysis of research results and it's application examples are illustrated.

Solution of the Poisson-Boltzmann Equation for a Cylindrical Particle with a Limited Length: Functional Theoretical Approach

With the help of the method of separation of variables and the Debye-Hüchel approximation, the Poisson-Boltzmann equation that describes the distribution of the potential in the electrical double layer of a cylindrical particle with a limited length has been firstly solved under a very low potential condition. Then with the help of the functional analysis theory this equation has been further analytically solved under general potential conditions and consequently, the corresponding surface charge densities have been obtained. Both the potential and the surface charge densities cointide with those results obtained from the Debye-Hüchel approximation when the very low potential of zeψ〈〈kT is introduced.

Schrdinger Equation of a Particle on a Rotating Curved Surface

We derive the Schr6dinger equation of a particle constrained to move on a rotating curved surface S. Using the thin-layer quantization scheme to confine the particle on S, and with a proper choice of gauge transformation for the wave function, we obtain the well-known geometric potentiM Vg and an additive Coriolis-induced geometric potential in the co-rotationM curvilinear coordinates. This novel effective potential, which is included in the surface Schr6dinger equation and is coupled with the mean curvature of S, contains an imaginary part in the general case which gives rise to a non-Hermitian surface Hamiltonian. We find that the non-Hermitian term vanishes when S is a minimal surface or a revolution surface which is axially symmetric around the rolling axis.

Particle Swarm Optimization for Solving Sine-Gordan Equation

The term‘optimization’refers to the process of maximizing the beneficial attributes of a mathematical function or system while minimizing the unfavorable ones.The majority of real-world situations can be modelled as an optimization problem.The complex nature of models restricts traditional optimization techniques to obtain a global optimal solution and paves the path for global optimization methods.Particle Swarm Optimization is a potential global optimization technique that has been widely used to address problems in a variety of fields.The idea of this research is to use exponential basis functions and the particle swarm optimization technique to find a numerical solution for the Sine-Gordan equation,whose numerical solutions show the soliton form and has diverse applications.The implemented optimization technique is employed to determine the involved parameter in the basis functions,which was previously approximated as a random number in the work reported till now in the literature.The obtained results are comparable with the results obtained in the literature.The work is presented in the form of figures and tables and is found encouraging.

Equation of Motion of a Spinning Test Particle in Gravitational Field

Based on the coupfing between the spin of a particle and gravitoelectromagnetic field, the equation of motion of a spinning test particle in gravitational field is deduced. From this equation of motion, it is found that the motion of a spinning particle deviates from the geodesic trajectory, and this deviation originates from the coupling between the spin of the particle and gravitoelectromagnetic field, which is also the origin of Lense-Thirring effects. In post-Newtonian approximations, this equation gives the same results as those of Mathisson-Papapetrou equation. Effect of the deviation of geodesic trajectory is detectable.

A Particle Method for the b-Equation

In this paper, we apply the particle method to solve the numerical solution of a family of non-li-near Evolutionary Partial Differential Equations. It is called b-equation because of its bi-Hamiltonian structure. We introduce the particle method as an approximation of these equations in Lagrangian representation for simulating collisions between wave fronts. Several numerical examples will be set to illustrate the feasibility of the particle method.

Modeling of Surface Tension and Viscosity for Non-electrolyte Systems by Means of the Equation of State for Square-well Chain Fluids with Variable Interaction Range

The equation of state(EOS)for square-well chain fluid with variable range(SWCF-VR) developed in our previous work based on statistical mechanical theory for chemical association is employed for the correlations of surface tension and viscosity of common fluids and ionic liquids(ILs).A model of surface tension for multi-component mixtures is presented by combining the SWCF-VR EOS and the scaled particle theory and used to produce the surface tension of binary and ternary mixtures.The predicted surface tensions are in excellent agreement with the experimental data with an overall average absolute relative deviation(AAD)of 0.36%.A method for the calculation of dynamic viscosity of common fluids and ILs at high pressure is presented by combining Eyring’s rate theory of viscosity and the SWCF-VR EOS.The calculated viscosities are in good agreement with the experimental data with the overall AAD of 1.44% for 14 fluids in 84 cases.The salient feature is that the molecular parameters used in these models are self-consistent and can be applied to calculate different thermodynamic properties such as pVT,vapor-liquid equilibrium,caloric properties,surface tension,and viscosity.

Exact Solutions of the Klein-Gordon Equation with a New Anharmonic Oscillator Potential

We solve the Klein-Cordon equation with a new anharmonic oscillator potential and present the exact solutions. It is shown that under the condition of equal scalar and vector potentials, the Klein-Cordon equation could be separated into an angular equation and a radial equation. The angular solutions are the associated-Legendre polynomial and the radial solutions are expressed in terms of the confluent hypergeometric functions. Finally, the energy equation is obtained from the boundary condition satisfied by the radial wavefunctions.