This document summarizes research into developing novel multi-velocity models for simulating gas-particle flows. It discusses limitations of traditional single-velocity Eulerian models and reviews recent work using kinetic theory and moment methods to derive multi-velocity formulations. The paper examines and compares different multi-velocity models, presents numerical implementations to demonstrate their behavior, and explores how they can overcome issues with single-velocity treatments.
Double General Point Interactions Symmetry and Tunneling TimesMolly Lee
This document presents original research on modeling double point barriers in quantum mechanics using generalized point interactions. It investigates the symmetry properties of double point barriers under parity transformations and derives the necessary conditions for the barriers to have well-defined parity. It also examines the limits of zero interbarrier distance for odd and even arrangements. Finally, it calculates the phase and tunneling times for barriers with defined parity and discusses whether the generalized Hartman effect occurs in the opaque limit.
This document summarizes a study that uses neural networks and direct numerical simulations to develop closure models for two-fluid equations describing multiphase bubbly flows. Direct numerical simulations were performed for a system of bubbles rising in liquid and these results were used to train a neural network model. The neural network model was able to reasonably predict the evolution of different initial bubble distributions and velocities compared to direct numerical simulations. This approach shows promise for using computational simulations to develop reduced-order models that can simulate multiphase flows without resolving all length and time scales.
Structure and transport coefficients of liquid Argon and neon using molecular...IOSR Journals
Molecular dynamics simulation was employed to deduce the dynamics property distribution function of Argon
and Neon liquid. With the use of a Lennnard-Jones pair potential model, an inter-atomic interaction function was observed
between pair of particles in a system of many particles, which indicates that the pair distribution function determines the
structures of liquid Argon. This distribution effect regarding the liquid structure of Lennard-Jones potential was strongly
affected such that its viscosity depends on density distribution of the model. The radial distribution function, g(r) agrees well
with the experimental data used. Our results regarding Argon and Neon show that their signatures are quite different at
each temperature, such that their corresponding viscosity is not consistent. Two sharps turning points are more
prominent in Argon, one at temperature of 83.88 Kelvin (K) with viscosity of -0.548 Pascal second (Pa-s) and the
other at temperature of 215.64 K with viscosity of -0.228 Pa-s.
In Argon and Neon liquid, temperature and density are inversely and directly proportional to diffusion
coefficient, in that order. This characteristic suggests that the observed non linearity could result from the non
uniform thermal expansion in liquid Argon and Neon, which are between the temperature range of 21.98 K and
239.52 K.
- The document investigates nonlinear properties of ion-acoustic waves in a relativistically degenerate quantum plasma using the quantum hydrodynamic model.
- It derives a nonlinear spherical Kadomtsev–Petviashvili equation using the reductive perturbation method to analyze how electron degeneracy affects the linear and nonlinear properties of ion-acoustic waves in this quantum plasma system.
- The key findings were that electron degeneracy significantly impacts the linear and nonlinear behavior of ion-acoustic waves in quantum plasmas.
This document presents new ideas in loop quantum gravity, including:
1. Deriving a relationship between time and vorticity using wavefunction continuity.
2. Introducing a new "Eikonal constraint" and showing how it removes acausality by gauging time to light cones.
3. Proposing a new form of the Hamiltonian constraint in the style of the Dirac equation, which reduces state fuzziness in line with Penrose's ideas.
This document summarizes key concepts from Chapter 3 of the source text. It begins by deriving a closed system of averaged hydrodynamic equations to describe turbulent flows in multicomponent media. These equations account for factors like compressibility, heat and mass transfer, and chemical reactions. The document then discusses choosing an appropriate averaging operator for the equations. It argues that using weighted-mean averaging simplified the form and analysis of the equations for compressible, variable density media like those found in astrophysics. This approach defines weighted means for some fluctuating quantities based on the local density. The document aims to establish a general phenomenological theory of turbulent transport in multicomponent mixtures at the first-order closure level.
This document summarizes a research paper that studied the quantum criticality of a two-channel pseudogap Anderson model. The paper investigates the quantum phase transition from a two-channel Kondo phase to a local moment phase using the non-crossing approximation and numerical renormalization group approaches. It finds novel power-law scalings in the linear and non-linear conductance at the quantum critical point. The scaling behaviors are distinct between equilibrium and non-equilibrium conditions, providing insights into non-equilibrium quantum criticality.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
Double General Point Interactions Symmetry and Tunneling TimesMolly Lee
This document presents original research on modeling double point barriers in quantum mechanics using generalized point interactions. It investigates the symmetry properties of double point barriers under parity transformations and derives the necessary conditions for the barriers to have well-defined parity. It also examines the limits of zero interbarrier distance for odd and even arrangements. Finally, it calculates the phase and tunneling times for barriers with defined parity and discusses whether the generalized Hartman effect occurs in the opaque limit.
This document summarizes a study that uses neural networks and direct numerical simulations to develop closure models for two-fluid equations describing multiphase bubbly flows. Direct numerical simulations were performed for a system of bubbles rising in liquid and these results were used to train a neural network model. The neural network model was able to reasonably predict the evolution of different initial bubble distributions and velocities compared to direct numerical simulations. This approach shows promise for using computational simulations to develop reduced-order models that can simulate multiphase flows without resolving all length and time scales.
Structure and transport coefficients of liquid Argon and neon using molecular...IOSR Journals
Molecular dynamics simulation was employed to deduce the dynamics property distribution function of Argon
and Neon liquid. With the use of a Lennnard-Jones pair potential model, an inter-atomic interaction function was observed
between pair of particles in a system of many particles, which indicates that the pair distribution function determines the
structures of liquid Argon. This distribution effect regarding the liquid structure of Lennard-Jones potential was strongly
affected such that its viscosity depends on density distribution of the model. The radial distribution function, g(r) agrees well
with the experimental data used. Our results regarding Argon and Neon show that their signatures are quite different at
each temperature, such that their corresponding viscosity is not consistent. Two sharps turning points are more
prominent in Argon, one at temperature of 83.88 Kelvin (K) with viscosity of -0.548 Pascal second (Pa-s) and the
other at temperature of 215.64 K with viscosity of -0.228 Pa-s.
In Argon and Neon liquid, temperature and density are inversely and directly proportional to diffusion
coefficient, in that order. This characteristic suggests that the observed non linearity could result from the non
uniform thermal expansion in liquid Argon and Neon, which are between the temperature range of 21.98 K and
239.52 K.
- The document investigates nonlinear properties of ion-acoustic waves in a relativistically degenerate quantum plasma using the quantum hydrodynamic model.
- It derives a nonlinear spherical Kadomtsev–Petviashvili equation using the reductive perturbation method to analyze how electron degeneracy affects the linear and nonlinear properties of ion-acoustic waves in this quantum plasma system.
- The key findings were that electron degeneracy significantly impacts the linear and nonlinear behavior of ion-acoustic waves in quantum plasmas.
This document presents new ideas in loop quantum gravity, including:
1. Deriving a relationship between time and vorticity using wavefunction continuity.
2. Introducing a new "Eikonal constraint" and showing how it removes acausality by gauging time to light cones.
3. Proposing a new form of the Hamiltonian constraint in the style of the Dirac equation, which reduces state fuzziness in line with Penrose's ideas.
This document summarizes key concepts from Chapter 3 of the source text. It begins by deriving a closed system of averaged hydrodynamic equations to describe turbulent flows in multicomponent media. These equations account for factors like compressibility, heat and mass transfer, and chemical reactions. The document then discusses choosing an appropriate averaging operator for the equations. It argues that using weighted-mean averaging simplified the form and analysis of the equations for compressible, variable density media like those found in astrophysics. This approach defines weighted means for some fluctuating quantities based on the local density. The document aims to establish a general phenomenological theory of turbulent transport in multicomponent mixtures at the first-order closure level.
This document summarizes a research paper that studied the quantum criticality of a two-channel pseudogap Anderson model. The paper investigates the quantum phase transition from a two-channel Kondo phase to a local moment phase using the non-crossing approximation and numerical renormalization group approaches. It finds novel power-law scalings in the linear and non-linear conductance at the quantum critical point. The scaling behaviors are distinct between equilibrium and non-equilibrium conditions, providing insights into non-equilibrium quantum criticality.
The International Journal of Engineering & Science is aimed at providing a platform for researchers, engineers, scientists, or educators to publish their original research results, to exchange new ideas, to disseminate information in innovative designs, engineering experiences and technological skills. It is also the Journal's objective to promote engineering and technology education. All papers submitted to the Journal will be blind peer-reviewed. Only original articles will be published.
Translation of four dimensional axes anywhere within the spatial and temporal boundaries of the universe would require quantitative values from convergence between parameters that reflect these limits. The presence of entanglement and volumetric velocities indicates that the initiating energy for displacement and transposition of axes would be within the upper limit of the rest mass of a single photon which is the same order of magnitude as a macroscopic Hamiltonian of the modified Schrödinger wave function. The representative metaphor is that any local 4-D geometry, rather than displaying restricted movement through Minkowskian space, would instead expand to the total universal space-time volume before re-converging into another location where it would be subject to cause-effect. Within this transient context the contributions from the anisotropic features of entropy and the laws of thermodynamics would be minimal. The central operation of a fundamental unit of 10-20 J, the hydrogen line frequency, and the Bohr orbital time for ground state electrons would be required for the relocalized manifestation. Similar quantified convergence occurs for the ~1012 parallel states within space per Planck’s time which solve for phase-shift increments where Casimir and magnetic forces intersect. Experimental support for these interpretations and potential applications is considered. The multiple, convergent solutions of basic universal quantities suggest that translations of spatial axes into adjacent spatial states and the transposition of four dimensional configurations any where and any time within the universe may be accessed but would require alternative perspectives and technologies.
1) The paper investigates whether quantum variations around geodesics could circumvent caustics that occur in certain space-times.
2) An action is developed that yields both the field equations and geodesic condition. Quantizing this action provides a way to determine the extent of the wave packet around the classical path.
3) It is shown that replacing plane wave solutions with wave packets in the path integral still yields acceptable results. Determining if the distribution matches expectation values and variances is key to establishing geodesic completeness with quantum variations.
The use of Cellular Automata is extended in various disciplines for the modeling of complex system procedures. Their inherent simplicity and their natural parallelism make them a very efficient tool for the simulation of large scale physical phenomena. We explore the framework of Cellular Automata to develop a physically based model for the spatial and temporal prediction of shallow landslides. Particular weight is given to the modeling of hydrological processes in order to investigate the hydrological triggering mechanisms and the importance of continuous modeling of water balance to detect timing and location of soil slips occurrences. Specifically, the 3D flow of water and the resulting water balance in the unsaturated and saturated zone is modeled taking into account important phenomena such as hydraulic hysteresis and evapotranspiration. In this poster the hydrological component of the model will be presented and tested against well established benchmark experiments [Vauclin et al, 1975; Vauclin et al, 1979]. Furthermore, we investigate the applicability of incorporating it in a hydrological catchment model for the prediction (temporal and spatial) of rainfall-triggered shallow landslides.
Lagrange's theorem describes fluid motion using a Lagrangian description that tracks individual fluid particles over time rather than describing the fluid properties at fixed spatial locations like the Eulerian description. The Lagrangian description follows Newton's laws of motion for individual particles, making it easier to apply concepts from solid mechanics. However, the Eulerian description is more commonly used in fluid mechanics problems because it is not practical to track every particle in complex flows. Lagrange's equations, derived using the calculus of variations, provide an alternative formulation of classical mechanics that has advantages over Newtonian mechanics such as applying to any coordinate system.
Collective modes in the CFL phase - New Journal of Physics 13 (2011) 055002Roberto Anglani
This document summarizes a study of collective modes in the color flavor-locked (CFL) phase of dense quark matter. The authors derive the effective Lagrangian for the Nambu-Goldstone (NG) boson associated with spontaneous breaking of quark number symmetry, and determine corrections to previous results. They also derive the kinetic Lagrangian for the Higgs mode and interaction terms between the Higgs and NG fields using a Nambu-Jona-Lasinio model. This provides an effective description of low-energy excitations in the CFL phase to understand properties of compact stars containing quark matter.
IJERD (www.ijerd.com) International Journal of Engineering Research and Devel...IJERD Editor
This document summarizes a research article that investigates relativistic effects on the linear and nonlinear propagation of electron plasma waves in dense quantum plasma with arbitrary temperature. The key points are:
1) A quantum hydrodynamic model is used to study how relativistic effects from streaming motion impact electron plasma waves in a finite-temperature quantum plasma.
2) Both compressive and rarefactive solitons can form in the plasma depending on factors like electron degeneracy and streaming velocity, when relativistic effects are considered.
3) Relativistic effects are found to significantly influence the linear and nonlinear properties of electron plasma waves in this dense, finite-temperature quantum plasma system.
Using resonant ultrasound spectroscopy (RUS), the author will determine the complete elastic constant matrices of two thermoelectric single crystal samples, Ce.75Fe3CoSb12 and CeFe4Sb12. RUS involves measuring the resonant frequencies of a sample's vibrations, which depend on the sample's elastic constants, shape, orientation, and density. The author aims to obtain the elastic moduli from a single RUS spectrum for each sample. Understanding the elastic properties may help identify better thermoelectric materials by correlating low elastic stiffness with low thermal conductivity and higher thermoelectric efficiency. The author will compute the resonant frequencies using the samples' properties and compare to measurements.
This document discusses the incompatibility between classical mechanics and electromagnetism. It shows that under a Galilean transformation, the wave equation governing electromagnetic waves takes on a different form in different reference frames, violating Galilean invariance. This means that the laws of electromagnetism depend on the choice of reference frame. As such, classical mechanics and electromagnetism cannot be unified without modifications to account for this issue.
Lattice boltzmann simulation of non newtonian fluid flow in a lid driven cavitIAEME Publication
This document summarizes a study that uses Lattice Boltzmann Method (LBM) to simulate non-Newtonian fluid flow in a lid driven cavity. The study explores the mechanism of non-Newtonian fluid flow using the power law model to represent shear-thinning and shear-thickening fluids. It investigates the influence of power law index and Reynolds number on velocity profiles and streamlines. The LBM code is validated against published results and shows agreement with established theory and fluid rheological behavior.
This document provides a detailed report on resolving the twin paradox in special relativity. It begins with an abstract summarizing the topic. Section 2 discusses relevant theory, including frames of reference, Lorentz transformations, and explanations for how acceleration resolves the paradox. Section 3 describes the experimental method, which involves two models for analyzing acceleration phases. Section 4 presents the results from both models graphically. Section 5 discusses the results and equations, considers alternative explanations, and discusses further areas of investigation. The conclusion reaffirms that accounting for acceleration through non-inertial frames resolves the paradox.
This document outlines the lattice Boltzmann method and its applications in multiphase flows. It discusses the history and development of the lattice Boltzmann method from lattice gas automata to the BGK kinetic model. A kinetic theory for multiphase flows is presented along with models for intermolecular interactions. A lattice Boltzmann multiphase model is developed based on the kinetic theory. The model is applied to simulate phase separation and Rayleigh-Taylor and Kelvin-Helmholtz instabilities. Challenges in further developing the lattice Boltzmann method are also noted.
A study-to-understand-differential-equations-applied-to-aerodynamics-using-cf...zoya rizvi
This document discusses computational fluid dynamics (CFD) and its application to aerodynamics. It begins by introducing CFD and the governing equations of fluid dynamics - the continuity, momentum, and energy equations. These partial differential equations can be used to model fluid flow. The document then examines the finite control volume approach and substantial derivative used to develop the Navier-Stokes equations from fundamental principles. An example application of CFD to aerodynamics is provided. The document aims to explain the methodology of CFD, including establishing the governing equations and interpreting results.
This document summarizes research on improving the flotation of ultrafine particles smaller than 0.08mm. It proposes using smaller air bubbles in turbulent flow to increase collision probability between particles and bubbles. A mathematical model is described showing that concentration of particles decreases exponentially with flow distance as particles coagulate with bubbles. Experimental results show a new flotation machine using this approach concentrates over 90% of lead in under 1m3/h, demonstrating effectiveness for extracting ultrafine particles.
The document discusses the basic laws of chemistry including:
1. Law of conservation of mass (Lavoisier's law) which states that in a closed system, the mass of substances before and after a reaction is the same.
2. Law of definite proportions (Proust's law) which states that the ratios of the masses of elements in a compound are always the same.
3. Law of multiple proportions (Dalton's law) which describes the ratios of the masses of one element that combine with a fixed mass of another element form whole number ratios.
4. Gay-Lussac's law of gaseous volumes which states that gases react in volumes that are whole number ratios of their coefficients in
Oxford graduate lectures on "Quantum Chromodynamics and LHC phenomenology" Pa...juanrojochacon
The document discusses the historical motivation for quantum chromodynamics (QCD). It describes how the discovery of many new strongly interacting particles in the mid-20th century led to the proposal of quarks as fundamental constituents. Quarks were proposed to have a new quantum number called color to explain experimental observations. Deep inelastic scattering experiments provided evidence that quarks are real particles and not just mathematical entities. The document outlines the basic properties of QCD, including its SU(3) symmetry and how this allows color-singlet hadrons to form from quarks.
This document presents a temporal stability analysis of a swirling gas jet discharging into an ambient gas. The analysis considers the inviscid limit of the compressible swirling jet flow. The dispersion relation parameters studied include swirl number, Mach number, coflow velocity, and molar weight ratio. The base flow is solved using a self-similar solution, and a pseudospectral method is used to discretize the governing equations. The stability analysis derives the eigenvalue problem from the linearized Navier-Stokes equations to determine how perturbations of the base flow will grow or decay over time.
The document outlines a marketing strategy for implementing a car loan scheme in South Kolkata by United Bank of India. It begins with demographic details of Kolkata's population. It then describes the eligibility criteria and terms of the car loan scheme. The strategy involves focusing marketing efforts on high income areas of South Kolkata through social media campaigns, direct mail, car melas, media advertising, and maintaining a prospect list. It emphasizes maintaining contact with automobile dealers and ensuring an easy application process with quick processing times to make the scheme successful.
Froilan Esplaguera Abella is a System Specialist currently working at NCS Pte. Ltd. supporting over 6,500 computers at Khoo Teck Puat Hospital. He has over 10 years of experience in customer service, technical support, and systems administration roles. His skills include Windows, networking, virtualization, patch management, and troubleshooting various hardware and software issues. He holds several certifications including ITIL, MCSA, and has a Bachelor's degree in Information Technology.
Technician A and B disagree on the proper procedure for retrofitting a vehicle's A/C system from R-12 to R-134a. Technician A states that the old mineral oil must be removed by flushing the system with R-11, while Technician B says 8 oz of PAG oil should be installed in all retrofitted systems. The document indicates that Technician B is correct since PAG oil is required for systems using R-134a refrigerant.
در مصاحبه های استخدامی چه سوالهایی را نباید پرسید.کسب و کار شما
وقتی شما به یک جلسه مصاحبه شغلی دعوت می شوید، یعنی شما شخصی هستید که در مراحل اولیه انتخاب شده اید. و اکنون زمان این است که برای جلسه نهایی و نتیجه گیری از تلاش خود، مصاحبه ای را با مدیریت خود داشته باشید.
محصول یا خدماتی که میتواند پاسخگوی این مشکل باشد، جایگاه ارزشی نامیده میشود. ارزش این محصول و یا خدمات از دیدگاه مشتری، در تعیین این جایگاه ارزشی ملاک قرار میگیرد.
Translation of four dimensional axes anywhere within the spatial and temporal boundaries of the universe would require quantitative values from convergence between parameters that reflect these limits. The presence of entanglement and volumetric velocities indicates that the initiating energy for displacement and transposition of axes would be within the upper limit of the rest mass of a single photon which is the same order of magnitude as a macroscopic Hamiltonian of the modified Schrödinger wave function. The representative metaphor is that any local 4-D geometry, rather than displaying restricted movement through Minkowskian space, would instead expand to the total universal space-time volume before re-converging into another location where it would be subject to cause-effect. Within this transient context the contributions from the anisotropic features of entropy and the laws of thermodynamics would be minimal. The central operation of a fundamental unit of 10-20 J, the hydrogen line frequency, and the Bohr orbital time for ground state electrons would be required for the relocalized manifestation. Similar quantified convergence occurs for the ~1012 parallel states within space per Planck’s time which solve for phase-shift increments where Casimir and magnetic forces intersect. Experimental support for these interpretations and potential applications is considered. The multiple, convergent solutions of basic universal quantities suggest that translations of spatial axes into adjacent spatial states and the transposition of four dimensional configurations any where and any time within the universe may be accessed but would require alternative perspectives and technologies.
1) The paper investigates whether quantum variations around geodesics could circumvent caustics that occur in certain space-times.
2) An action is developed that yields both the field equations and geodesic condition. Quantizing this action provides a way to determine the extent of the wave packet around the classical path.
3) It is shown that replacing plane wave solutions with wave packets in the path integral still yields acceptable results. Determining if the distribution matches expectation values and variances is key to establishing geodesic completeness with quantum variations.
The use of Cellular Automata is extended in various disciplines for the modeling of complex system procedures. Their inherent simplicity and their natural parallelism make them a very efficient tool for the simulation of large scale physical phenomena. We explore the framework of Cellular Automata to develop a physically based model for the spatial and temporal prediction of shallow landslides. Particular weight is given to the modeling of hydrological processes in order to investigate the hydrological triggering mechanisms and the importance of continuous modeling of water balance to detect timing and location of soil slips occurrences. Specifically, the 3D flow of water and the resulting water balance in the unsaturated and saturated zone is modeled taking into account important phenomena such as hydraulic hysteresis and evapotranspiration. In this poster the hydrological component of the model will be presented and tested against well established benchmark experiments [Vauclin et al, 1975; Vauclin et al, 1979]. Furthermore, we investigate the applicability of incorporating it in a hydrological catchment model for the prediction (temporal and spatial) of rainfall-triggered shallow landslides.
Lagrange's theorem describes fluid motion using a Lagrangian description that tracks individual fluid particles over time rather than describing the fluid properties at fixed spatial locations like the Eulerian description. The Lagrangian description follows Newton's laws of motion for individual particles, making it easier to apply concepts from solid mechanics. However, the Eulerian description is more commonly used in fluid mechanics problems because it is not practical to track every particle in complex flows. Lagrange's equations, derived using the calculus of variations, provide an alternative formulation of classical mechanics that has advantages over Newtonian mechanics such as applying to any coordinate system.
Collective modes in the CFL phase - New Journal of Physics 13 (2011) 055002Roberto Anglani
This document summarizes a study of collective modes in the color flavor-locked (CFL) phase of dense quark matter. The authors derive the effective Lagrangian for the Nambu-Goldstone (NG) boson associated with spontaneous breaking of quark number symmetry, and determine corrections to previous results. They also derive the kinetic Lagrangian for the Higgs mode and interaction terms between the Higgs and NG fields using a Nambu-Jona-Lasinio model. This provides an effective description of low-energy excitations in the CFL phase to understand properties of compact stars containing quark matter.
IJERD (www.ijerd.com) International Journal of Engineering Research and Devel...IJERD Editor
This document summarizes a research article that investigates relativistic effects on the linear and nonlinear propagation of electron plasma waves in dense quantum plasma with arbitrary temperature. The key points are:
1) A quantum hydrodynamic model is used to study how relativistic effects from streaming motion impact electron plasma waves in a finite-temperature quantum plasma.
2) Both compressive and rarefactive solitons can form in the plasma depending on factors like electron degeneracy and streaming velocity, when relativistic effects are considered.
3) Relativistic effects are found to significantly influence the linear and nonlinear properties of electron plasma waves in this dense, finite-temperature quantum plasma system.
Using resonant ultrasound spectroscopy (RUS), the author will determine the complete elastic constant matrices of two thermoelectric single crystal samples, Ce.75Fe3CoSb12 and CeFe4Sb12. RUS involves measuring the resonant frequencies of a sample's vibrations, which depend on the sample's elastic constants, shape, orientation, and density. The author aims to obtain the elastic moduli from a single RUS spectrum for each sample. Understanding the elastic properties may help identify better thermoelectric materials by correlating low elastic stiffness with low thermal conductivity and higher thermoelectric efficiency. The author will compute the resonant frequencies using the samples' properties and compare to measurements.
This document discusses the incompatibility between classical mechanics and electromagnetism. It shows that under a Galilean transformation, the wave equation governing electromagnetic waves takes on a different form in different reference frames, violating Galilean invariance. This means that the laws of electromagnetism depend on the choice of reference frame. As such, classical mechanics and electromagnetism cannot be unified without modifications to account for this issue.
Lattice boltzmann simulation of non newtonian fluid flow in a lid driven cavitIAEME Publication
This document summarizes a study that uses Lattice Boltzmann Method (LBM) to simulate non-Newtonian fluid flow in a lid driven cavity. The study explores the mechanism of non-Newtonian fluid flow using the power law model to represent shear-thinning and shear-thickening fluids. It investigates the influence of power law index and Reynolds number on velocity profiles and streamlines. The LBM code is validated against published results and shows agreement with established theory and fluid rheological behavior.
This document provides a detailed report on resolving the twin paradox in special relativity. It begins with an abstract summarizing the topic. Section 2 discusses relevant theory, including frames of reference, Lorentz transformations, and explanations for how acceleration resolves the paradox. Section 3 describes the experimental method, which involves two models for analyzing acceleration phases. Section 4 presents the results from both models graphically. Section 5 discusses the results and equations, considers alternative explanations, and discusses further areas of investigation. The conclusion reaffirms that accounting for acceleration through non-inertial frames resolves the paradox.
This document outlines the lattice Boltzmann method and its applications in multiphase flows. It discusses the history and development of the lattice Boltzmann method from lattice gas automata to the BGK kinetic model. A kinetic theory for multiphase flows is presented along with models for intermolecular interactions. A lattice Boltzmann multiphase model is developed based on the kinetic theory. The model is applied to simulate phase separation and Rayleigh-Taylor and Kelvin-Helmholtz instabilities. Challenges in further developing the lattice Boltzmann method are also noted.
A study-to-understand-differential-equations-applied-to-aerodynamics-using-cf...zoya rizvi
This document discusses computational fluid dynamics (CFD) and its application to aerodynamics. It begins by introducing CFD and the governing equations of fluid dynamics - the continuity, momentum, and energy equations. These partial differential equations can be used to model fluid flow. The document then examines the finite control volume approach and substantial derivative used to develop the Navier-Stokes equations from fundamental principles. An example application of CFD to aerodynamics is provided. The document aims to explain the methodology of CFD, including establishing the governing equations and interpreting results.
This document summarizes research on improving the flotation of ultrafine particles smaller than 0.08mm. It proposes using smaller air bubbles in turbulent flow to increase collision probability between particles and bubbles. A mathematical model is described showing that concentration of particles decreases exponentially with flow distance as particles coagulate with bubbles. Experimental results show a new flotation machine using this approach concentrates over 90% of lead in under 1m3/h, demonstrating effectiveness for extracting ultrafine particles.
The document discusses the basic laws of chemistry including:
1. Law of conservation of mass (Lavoisier's law) which states that in a closed system, the mass of substances before and after a reaction is the same.
2. Law of definite proportions (Proust's law) which states that the ratios of the masses of elements in a compound are always the same.
3. Law of multiple proportions (Dalton's law) which describes the ratios of the masses of one element that combine with a fixed mass of another element form whole number ratios.
4. Gay-Lussac's law of gaseous volumes which states that gases react in volumes that are whole number ratios of their coefficients in
Oxford graduate lectures on "Quantum Chromodynamics and LHC phenomenology" Pa...juanrojochacon
The document discusses the historical motivation for quantum chromodynamics (QCD). It describes how the discovery of many new strongly interacting particles in the mid-20th century led to the proposal of quarks as fundamental constituents. Quarks were proposed to have a new quantum number called color to explain experimental observations. Deep inelastic scattering experiments provided evidence that quarks are real particles and not just mathematical entities. The document outlines the basic properties of QCD, including its SU(3) symmetry and how this allows color-singlet hadrons to form from quarks.
This document presents a temporal stability analysis of a swirling gas jet discharging into an ambient gas. The analysis considers the inviscid limit of the compressible swirling jet flow. The dispersion relation parameters studied include swirl number, Mach number, coflow velocity, and molar weight ratio. The base flow is solved using a self-similar solution, and a pseudospectral method is used to discretize the governing equations. The stability analysis derives the eigenvalue problem from the linearized Navier-Stokes equations to determine how perturbations of the base flow will grow or decay over time.
The document outlines a marketing strategy for implementing a car loan scheme in South Kolkata by United Bank of India. It begins with demographic details of Kolkata's population. It then describes the eligibility criteria and terms of the car loan scheme. The strategy involves focusing marketing efforts on high income areas of South Kolkata through social media campaigns, direct mail, car melas, media advertising, and maintaining a prospect list. It emphasizes maintaining contact with automobile dealers and ensuring an easy application process with quick processing times to make the scheme successful.
Froilan Esplaguera Abella is a System Specialist currently working at NCS Pte. Ltd. supporting over 6,500 computers at Khoo Teck Puat Hospital. He has over 10 years of experience in customer service, technical support, and systems administration roles. His skills include Windows, networking, virtualization, patch management, and troubleshooting various hardware and software issues. He holds several certifications including ITIL, MCSA, and has a Bachelor's degree in Information Technology.
Technician A and B disagree on the proper procedure for retrofitting a vehicle's A/C system from R-12 to R-134a. Technician A states that the old mineral oil must be removed by flushing the system with R-11, while Technician B says 8 oz of PAG oil should be installed in all retrofitted systems. The document indicates that Technician B is correct since PAG oil is required for systems using R-134a refrigerant.
در مصاحبه های استخدامی چه سوالهایی را نباید پرسید.کسب و کار شما
وقتی شما به یک جلسه مصاحبه شغلی دعوت می شوید، یعنی شما شخصی هستید که در مراحل اولیه انتخاب شده اید. و اکنون زمان این است که برای جلسه نهایی و نتیجه گیری از تلاش خود، مصاحبه ای را با مدیریت خود داشته باشید.
محصول یا خدماتی که میتواند پاسخگوی این مشکل باشد، جایگاه ارزشی نامیده میشود. ارزش این محصول و یا خدمات از دیدگاه مشتری، در تعیین این جایگاه ارزشی ملاک قرار میگیرد.
)انگیزه سازی ، راز مدیریت موفق است.( قسمت اولکسب و کار شما
اقتصاد جهانی رو به سویی می رود که شما هر روز، با یک شرایط کاری و البته بحران هایی عجیب و غریب مالی باید دست و پنجه نرم کنید. شرایطی که شما را گاهی وادار می کند مثل یک جنگجو لباس رزم بپوشید و به جنگ با تمام مسائل و آشفتگی های مالی ، بدحسابی ها، بدهی ها، چک های به اجرا گذاشته بروید
سام از همان کودکی مهماننواز بود و بعدها نیز همین مهماننوازی در مقابل مشتریان او را موفق کرد. در در نیوپورت آرکانزاس را » بن فرانکلین « او پس از مرخص شدن از ارتش، امتیاز فروشگاه ۱۹۴۵ اوت شعبه دیگر ۱۵ والتون،۱۹۶۰ و اوایل دهه ۱۹۵۰ خریداری و یک ماه بعد آن را افتتاح کرد. در طول دهه در جلسه هیات مدیره پیشنهاد کرد که کالاها ۱۹۶۲ بر تعداد فروشگاههای خود افزود. وی در زمستان در فروشگاهها با تخفیف به فروش برسند؛ ولی هیاتمدیره نظر او را نپذیرفت.
مدیریت کسب و کار و حل مشکلات در کسب و کارکسب و کار شما
بسیاری از مدیران در کسب وکارهای خود زمانی که به مشکل در سازمان یا موسسه خود پی می برند، حال این مشکل در قسمت مالی باشد یا نیروی انسانی، به شیوه ی مدیریت خود شک می کنند. گمان می کنند در حرفه و کسب و کار خود ، که در مدیریت خود ضعیف عمل کرده اند و یا آن طور که باید نتوانسته اند مدیریت سازمان یافته داشته باشند.
Leticia de Leon has over 20 years of experience in financial analysis and operations management. She currently works as a Financial Analyst for Weston Solutions, Inc., where she assists with projects, cash flow analysis, budget monitoring, and reporting. Previously she was the Profit Center Controller for Weston Solutions, overseeing finance teams and locations across multiple states. She holds a Bachelor's degree in Business Administration in Accounting from the University of Texas at San Antonio.
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The caustic that occur in geodesics in space-times which are solutions to the gravitational field equations
with the energy-momentum tensor satisfying the dominant energy condition can be circumvented if
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and the geodesic condition, and its quantization provides a method for determining the extent of the wave
packet around the classical path.
The caustic that occur in geodesics in space-times which are solutions to the gravitational field equations
with the energy-momentum tensor satisfying the dominant energy condition can be circumvented if
quantum variations are allowed. An action is developed such that the variation yields the field equations
and the geodesic condition, and its quantization provides a method for determining the extent of the wave
packet around the classical path.
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1. An Investigation of Multi-Velocity Treatments for
Multi-Phase Flows Inspired by Gaskinetic Theory
Zakaria Ben Dhia1 and James McDonald2
1 Institute Polytechnique Privé, Université Libre de Tunis,
30 Avenue Kheireddine Pacha, 1002, Tunis, Tunisia
2 Department of Mechanical Engineering, University of Ottawa,
161 Louis Pasteur, Ottawa, Ontario, Canada, K1N 6N5
Email: Zakaria Ben Dhia, zbend027@uottawa.ca
ABSTRACT
This research entails the investigation of novel models
for an inert, dilute, disperse, particle flow to be cou-
pled to a gas. Such flows are important in many en-
gineering situations. Particle phases are often difficult
to model, as Lagrangian methods can be too costly and
many Eulerian methods suffer from model deficiencies
and mathematical artifacts. Often, Eulerian formula-
tions assume that all particles at a location and time
have the same velocity [12]. This assumption leads
to nonphysical results, including an inability to pre-
dict particle paths crossing and a limited number of
boundary conditions that can be applied [12, 13]. Re-
cently, multi-velocity treatments of multi-phase flows
have been proposed that are based on the field of gask-
inetic theory [3, 4, 5].
This work examines and compares multi-velocity for-
mulations for the prediction of gas-particle flows. De-
tails regarding derivation, the mathematical structure,
and physical behaviour of the resulting models are ex-
plored. Finally, a numerical implementation is pre-
sented and results for several flow problems that are
designed to demonstrate the fundamental behaviour of
the models are presented.
1 INTRODUCTION
The behaviours of multi-phase gas-particle flows are
important in many practical engineering situations. As
one example, in internal-combustion engines, fuel is
often injected as a spray of tiny droplets and, dur-
ing combustion, a cloud of tiny soot particles can be
formed. The accurate prediction of the evolution of the
particulate phases is essential in the analysis of these
situations. The design of clean and efficient combus-
tion technologies rests on an ability to make accurate
predictions and analyzes of these situations.
Mathematical models of particulate flows fall into two
categories: Lagrangian and Eulerian. In Lagrangian
treatments, the evolution of individual particles are
tracked. This is often prohibitively computationally
expensive, as, even at dilute concentrations, an im-
mensely large number of particles are often present. In
Eulerian formulations, the particle phase is treated as
a continuum and partial differential equations (PDEs)
are defined which govern the evolution of field vari-
ables with position and time as independent variables.
The most basic Eulerian formulation restricts all parti-
cle velocities at a given position and time to be equal
to that of the fluid. A slightly more flexible model can
be developed by allowing the particle velocity to devi-
ate from that of the gas, but restricting all particles at
a location and time to have the same velocity as each
other [12]. In many cases, this is an acceptable model,
especially in situations when drag forces between the
particles and the gas are dominant and particles tend to
all have velocities near that of the gas. In other situa-
tions, however, this is not true, and forcing all particles
to have the same velocity leads to artifacts in the model
and causes it to produce completely nonphysical solu-
tions [12, 13].
Multi-velocity formulations in which particles are
grouped into “families”, each of which are treated with
extensions of single-velocity treatments, have been
proposed. However, these treatments often group par-
ticles based on a predefined separation of velocity
space [11]. The result is that some of the deficien-
cies of a single-velocity formulation may be removed
for some situations, however all of the deficiencies
of a single-velocity formulation can still be observed
2. for any case that includes multiple particle velocities
within a region of velocity space that has been assigned
to a single family. Also, the resulting model equations
are not Galilean invariant as the segregation of velocity
space is done in a fixed reference frame.
In a quest for new multi-velocity treatments for
particle-gas flows that eliminate some or all of the ar-
tifacts of previous treatments, while maintaining desir-
able mathematical structures (such as Galilean invari-
ance), the field of gaskinetic theory [6] can be used as
a guide. The physical situation of a huge number of in-
dependent, practically indistinguishable solid particles
is very similar to the situation of a gas that is comprised
of an enormous number of indistinguishable atoms or
molecules. It is therefore reasonable to expect that so-
phisticated techniques from gaskinetic theory can be
adopted. Recently, models based on such an idea have
begun to be explored [3, 4, 5]. These models repre-
sent the distribution of particle velocities at a location
and time by a finite collection of delta or Gaussian
distributions. Moment equations are then formulated
which describe the evolution of statistical properties of
the particle-velocity distributions. The result is an ex-
panded system of first-order hyperbolic balance laws
that are amenable to solution using standard numerical
techniques. This is a special case of the technique of
moment closures which can be used to describe gen-
eral non-equilibrium gas flows [7, 9, 10]. Moment clo-
sures provide an extended set of hyperbolic partial dif-
ferential equations describing the transport of macro-
scopic fluid properties. In general, the solution of these
PDEs require considerably less effort than obtaining
solutions using a direct particle simulation method.
This paper explains the construction of multi-velocity
models for particle-laden gas flows based on kinetic
theory of gases. The physical behaviour and limita-
tions of the models are discussed and demonstrated.
Numerical solutions for several flow problems de-
signed to clearly demonstrate the limitations of tradi-
tional models are presented.
2 MOMENT METHODS
Moment closures arise from the field of gaskinetic
theory. This theoretical approach takes into account
the particle nature of gases by defining a probability
density function, F (xi,vi,t), in six-dimensional phase
space which specifies the probability of finding parti-
cles at a given location, xi, and time, t, having a par-
ticular velocity, vi. This treatment can obviously be
immediately applied to solid-particle flow by simply
replacing the gas particles with the particulate. Macro-
scopic “observable” properties of the particle are then
obtained by taking appropriate velocity moments of F .
This is done by integrating the product of the distri-
bution function and an appropriate velocity-dependent
weight, W(vi), over all velocity space,
W(vi)F =
∞
−∞
∞
−∞
∞
−∞
W(vi)F (xi,vi,t)d3
v. (1)
For example, σ = m F , σui = m viF , and Pij =
m cicjF . Here m is the mass of a particle, σ is the
particle density, ui is average (bulk) particle velocity,
ci = vi −ui is the particle random velocity, and Pij is an
anisotropic tensor that is related to the standard deriva-
tion of particles velocities at a location. The symbol P
is used for consistency with kinetic theory of gases,
where it is related to pressure. In this work, the fol-
lowing examples demonstrate the notation that will be
used to describe moments of arbitrary order:
Mx = m vxF , Mxxy = m vxvxvyF , (2)
where the number of times a coordinate appears in
the subscript of the symbol M denotes the number of
times that velocity component appears in the generat-
ing weight. Third- and fourth-order moments of the
random velocity are given the symbols
Qijk = m cicjckF and Rijkl = m cicjckclF . (3)
The Einstein summation convention is used for gen-
eral indices (i, j, k, etc.), but not for specific Cartesian
directions (x, y, z).
The evolution of the velocity distribution function is
described by the Boltzmann equation [2, 1, 6]. This is
a high-dimensional integro-differential equation for F
having the form
∂F
∂t
+vi
∂F
∂xi
+ai
∂F
∂vi
=
δF
δt
. (4)
Here ai is the acceleration due to external forces, drag
between the fluid and particulate is an example of such
a force. The term on the right hand side of the equa-
tion, δF
δt , is the collision integral and represents the
time rate of change of the distribution function pro-
duced by inter-particle collisions.
Transport equations governing the time evolution of
macroscopic quantities can be derived by evaluating
velocity moments of the Boltzmann equation given
above. This leads to Maxwell’s equation of change
which describes the evolution of the moment WF
by
∂
∂t
WF +
∂
∂xi
viWF = ∆ WF . (5)
Here the acceleration field is taken to be zero, as will
be the case throughout the present work. ∆ WF =
3. W δF
δt is the effect of collisions on the moment
quantity, and is also neglected herein. Neglecting
these two terms corresponds to the case of the flow
of non-interacting particles in a vacuum. W is the cho-
sen velocity-dependent weight that corresponds to the
macroscopic quantity of interest.
It is at this point that the problem of closure becomes
apparent. The time evolution of a given moment,
WF , is clearly dependent on the spatial divergence
of viWF , a moment of one higher order in terms
of the velocity, vi. This pattern is repeated, with the
time evolution of every moment being dependent on
a moment of one higher order in vi. In general, an
infinite number of moment equations is required to
fully describe the evolution of a macroscopic quantity,
and solving this infinite system is equivalent to solving
Eq. (4).
One technique used to obtain moment closure is to re-
strict the distribution function to an assumed form [7].
Restricting the form of the distribution function has the
effect of restricting the value of all higher-order mo-
ments to be functions of lower-order known moments,
thus furnishing a closing relationship in the moment
equations.
3 GOVERNING EQUATIONS
The classical, single-velocity model for particle flows
can be derived in a moment closure framework by re-
stricting the distribution function describing particle
velocities to be of the form
F1 = ω(xi,t) δ(vi − ˆvi(xi,t)), (6)
where ω(xi,t) and ˆvi(xi,t) are closure coefficients that
must be chosen to ensure consistency with Eq. (1), and
δ is the Dirac delta function. By substituting Eq. (6)
into Eq. (5), and choosing as velocity weights W1 = m,
and W2 = mvi, the following conservation model for
particle flow can be found:
∂U
∂t
+
∂Fk
∂xk
= 0 (7)
with
U =
σ
σui
and Fk =
σuk
σuiuk
. (8)
This system comprises one scalar and one vector equa-
tion for the conservation of mass and momentum of
particles. This model is very standard and has been ex-
tensively utilized [12, 13]. However, restricting all par-
ticles at one location to have the same velocity obvi-
ously renders this model inappropriate when the situa-
tion comprises multiple particle velocities at any point
in the flow. The practical deficiencies that this can
cause are demonstrated in Section 5.
3.1 Two-Velocity Models
Once particle flows are seen through the lens of gask-
inetic theory, an obvious extension to the model de-
scribed in Eq. (8) is to allow for multiple particle fam-
ilies with arbitrary velocities at a location. For exam-
ple, a two-velocity description is given by restricting
the distribution function to be
F2 = ω(1)
(xi,t) δ(vi − ˆv
(1)
i (xi,t))
+ω(2)
(xi,t) δ(vi − ˆv
(2)
i (xi,t)), (9)
where the superscript (1) and (2) refer to particle fam-
ilies one and two respectively. Once this distribution
function is chosen, all that remains in to chose an
appropriate set of velocity weights, W, and moment
equations can be constructed.
Such a two-velocity distribution function was consid-
ered previously by Desjardins et al. [3]. In this study,
they chose as generating weights (in two dimensions),
m, mvx, mvy, mv2
x, mv2
y, and a strange third-order
weight given by mv3
x + mv3
y. The result is a system of
six conservative moment equations of the form given
in Eq. (7) with solution vector and flux diad given as
U =
σ
Mx
My
Mxx
Myy
Mxxx +Myyy
and Fk =
Mk
Mxk
Myk
Mxxk
Myyk
Mxxxk +Myyyk
.
One peculiarity of this model is that it is not invari-
ant under rotation. This is because the model pro-
vides a treatment for two entries of the second-order
moment, Mxx and Myy, but not the “cross” term, Mxy.
These three second-order moments make up a tensor
and knowledge of all are required for rotation. Also,
the last entry in the solution vector, Mxxx + Myyy, (the
sum of two entries in a third-order tensor) is a some-
what strange choice. This is similar to choosing to
have the sum of the components of a vector as a so-
lution variable. This is obviously not a value that is
invariant under rotation.
Another issue with this chosen set of generating
weights is that, for some moment values, the locations
of the two delta functions in velocity space is ambigu-
ous (i.e. the closure coefficients in Eq. (9) cannot be
uniquely determined from the given state). For exam-
ple, for a state with a non-dimensionalized density of
4. σ = 1, bulk velocity of ui = 0, non-dimensionalized
Mxx = Myy = 1, and third-moment Mxxx +Myyy = 0, at
least two distribution functions can be found. For both
cases, ω(1) = ω(2) = 1
2 , however, the velocities of the
two particle families can be located either at
ˆv(1)
= ˆi+ ˆj and ˆv(2)
= −ˆi− ˆj,
or
ˆv(1)
= ˆi− ˆj and ˆv(2)
= −ˆi+ ˆj,
where ˆi and ˆj are the unit vectors in the x and y di-
rections respectively. In these ambiguous situations,
the closing fluxes cannot be uniquely determined as a
function of the solution vector and multiple consistent
values of Fk are possible in Eq. (7).
In the present study, the distribution function defined
in Eq. (9) is also adopted. However, the generating
weights are chosen such that the resulting system in
both Galilean invariant, and has a closure that is never
ambiguous. The generating weights chosen here are m,
mvi, mvivj, and mvivjvj. This results in the following
solution and flux vectors (for two dimensions):
U =
σ
Mx
My
Mxx
Mxy
Myy
Mxii
Myii
and Fk =
Mk
Mxk
Myk
Mxxk
Mxyk
Myyk
Mxkii
Mykii
. (10)
This is a system of eight first-order PDEs that includes
conservation of mass, momentum, all components of
the second-order tensorial moment, and the contracted
third-order moment. It may seem odd to use a system
of eight equations when there are only six free param-
eters, or closure coefficients, in the presumed distri-
bution function. However, it is found that the extra
information is necessary to maintain rotational invari-
ance and a flux definition that is unambiguous in all
possible cases.
The moments presented in Eq. (10) can be expressed
as functions of the “random” moments, defined in Sec-
tion 2, through the relations [6]:
Mx = σux ,
My = σuy ,
Mxx = Pxx +σu2
x ,
Mxy = Pxy +σuxuy ,
Myy = Pyy +σu2
y ,
Mxxx = Qxxx +3uxPxx +σu3
x ,
Mxxy = Qxxy +uyPxx +2uxPxy +σu2
xuy ,
Mxyy = Qxyy +uxPyy +2uyPxy +σuxu2
y ,
Myyy = Qyyy +3uyPyy +σu3
y ,
Mxxii = Rxxii +2uyQxxy +u2
yPxx
+ux(2Qxxx +2Qxii)+4uxuyPxy
+u2
x(6Pxx +Pyy)
+σu2
x(u2
x +u2
y),
Mxyii = Rxyii +uy(Qxii +2Qxyy)
+ux(Qyii +2Qxxy)
+3Pxy(u2
x +u2
y)+3uxuyPii
+σuxuy(u2
x +u2
y),
Myyii = Ryyii +2uxQxyy +u2
xPyy
+uy(2Qyyy +2Qyii)+4uxuyPxy
+u2
y(6Pyy +Pxx)
+σu2
y(u2
x +u2
y).
The random moments, σ, ux, uy, Pxx, Pxy, Pyy, Qxii, and
Qyii can all be determined from the solution vector. To
close the system, the moments Qxxx, Qxxy, Qxyy, Qyyy,
Rxxii, Rxyii, and Ryyii must be related to those in the
solution vector.
Once the distribution function given in Eq. (9) is
adopted and the moment coefficients are determined
from Eq. (1), the expressions for the needed random
moments can be integrated and are found to be
Qxxx = Qxii
Pxx
Pii
,
Qxxy = Qyii
Pxx
Pii
,
Qxyy = Qxii
Pyy
Pii
,
Qyyy = Qyii
Pyy
Pii
,
Rxxii =
Q2
xii
Pii
+
PxxPii
σ
,
Rxyii =
QxiiQyii
Pii
+
PxyPii
σ
,
Ryyii =
Q2
yii
Pii
+
PyyPii
σ
.
The system of equations defining the model is thus
closed.
In any implementation, special care must be taken to
avoid dividing by zero in these expressions. This can
simply be accomplished by adding a small tolerance
5. to all the denominators. One does not need to worry
about the sign, as the moments that appear in the de-
nominators are all non-negative.
3.2 Multi-Velocity Models
It should be noted that extensions to models with more
than two distinct velocities have been developed and
considered [4, 5]. In these cases, however, it is not
possible to write the fluxes as closed-form functions
of the solution vector. This is because it is not pos-
sible to analytically determine the weights and loca-
tions of the delta functions in the assumed distribu-
tion. Rather, a numerical inversion algorithm must be
used to determine the moment relation each time a flux
evaluation is needed. Though these models hold the
promise of improved accuracy in situations when par-
ticles at a location can have one of many velocities, the
added computational expense is significant. It is hoped
that the two-velocity model developed above brings a
good balance between physical accuracy and numeri-
cal cost.
4 NUMERICAL METHODS
In order to evaluate the predictive capabilities of the
developed model, a two-dimensional flow solver is
constructed. An upwind Godunov-type finite-volume
scheme with piece-wise limited-linear reconstruction
using the Venkatakrishnan slope limiter [14] is used.
Inter-cellular fluxes are determined through the ap-
proximate solution of a Riemann problem using a
flux function developed by Saurel et al. [12] for the
single-velocity formulation, or the HLL flux func-
tion [8] for the two-velocity model. Second-order
accurate predictor-corrector time marching is used.
Wave speeds of the system are approximated through
the calculation of the locations of the two delta func-
tions given in Eq. (9).
As stated above, the solution vector contains eight en-
tries while the distribution function has only six de-
grees of freedom. This means that, after a time step, it
is possible to have a solution vector that does not corre-
spond to any distribution of the form in Eq. (9). Thus,
after each time step, the solution vector is adjusted to
ensure consistency with the distribution function. To
be consistent with a two-velocity distribution, it can be
shown that PxxPyy = P2
xy. Thus, after each update, the
sign of Pxy is maintained, but its magnitude is changed
to be equal to PxxPyy. The angle that the line that con-
nects the two deltas in the distribution function makes
with the x axis is given by the relation
cos2
θ =
Pxx
Pii
,
and it can be shown that the third-order vector, Qij j
must point along this line. Therefore, after an update,
the magnitude of Qij j is maintained, but its direction
is corrected.
5 RESULTS
Two situations are chosen that demonstrate the advan-
tages of a two-velocity description. First, two cross-
ing beams of non-interacting particles is considered.
Single-velocity descriptions are incapable of describ-
ing such crossing of particle trajectories. Second, a
situation of one group of fast-moving particles which
overtakes and passes through a group of slower par-
ticles is studied. Solutions to both situations are ob-
tained for the single-velocity model, Eq. (8), and the
two-velocity model, Eq. (10).
In both of these cases, much of the domain contains
no particles. Numerical and round-off errors can result
in some areas developing slightly non-physical states
(σ slightly less than zero, for example). It is therefore
necessary to ensure the state in each cell is realizable
after each time step. This is simply done by setting
any negative values of σ to zero and to set Pxx = Pxy =
Pyy = 0 if the determinant of this tensor is found to be
negative (which it cannot physically be).
In addition to issues related to non-realizability, it is
also found that, when all moments in the solution vec-
tor are on the order of the truncation error and are es-
sentially random, the state can correspond to a distri-
bution function where the two velocities are also ran-
dom and can be arbitrarily high. Such high veloci-
ties are a serious concern, not only because they can
seriously restrict the time-step size, but because they
can actually lead to large errors in the computation of
the fourth-order moments contained in the flux vec-
tor. This is found to lead to large random oscillations
in the solution. In order to handle this issue, maxi-
mum bounds on the size of particle-family velocities
are employed.
For all cases considered, a mesh of 300 by 300 cells is
used. This yields meshes of 90,000 equally sized cells.
Non-dimensional variables are used in all cases.
5.1 Crossing Beams
The first case considered is that of two crossing jets
of particles. The domain is −0.35 < x < −0.35 and
6. −0.35 < y < −0.35. The boundary conditions are
transmissive everywhere except on the left boundary,
where two jets of particles enter. If −0.3 < y < −0.2
on this wall, a jet of particles, all with σ = 1, ux = 1,
and uy = 1, enters. If 0.2 < y < 0.3 another jet of parti-
cles with σ = 1, ux = 1, and uy = −1 enters. Solutions
were advanced in time until steady-state was reliably
achieved.
As the particles are assumed not to interact with each
other, the exact solution is that the two beams should
continue in straight lines and exit the domain. Com-
puted solution values for particle density σ and the
y component of momentum, My, are shown in Fig-
ure 1. As can be seen in Figures 1a and 1c, when a
single-velocity model is used, the beams cannot cross
and the particles simply “pile up” along the symme-
try line. The two-velocity model solutions, shown in
Figures 1b and 1d, correctly allows the beams to cross.
It is interesting to see that a region of zero y-direction
momentum correctly develops as the beams intersect.
Non-zero momenta re-emerge from this region as the
particle-flow continues.
5.2 Superimposed Families of Particles
The second case considered is that of one family of
non-interacting particles overtaking another. In this
case, at time zero, one family of particles with σ = 1
and ux = 1 spans the region −0.6 < x < −0.4 and
−0.1 < y < 0.1. A second family with σ = 1 and ux =
0.1 spans the region −0.25 < x < 0.15 and −0.2 < y <
0.2. Results at t = 0, t = 0.5, and t = 1 are shown in
Figure 2.
Again, the exact solution should show the faster fam-
ily of particles simply passing through the slower one,
completely unaffected. It can again be seen that the
single-velocity model is wholly unable to predict this
effect as particles again collect in a concentrated re-
gion. The results for the two-velocity formulation are
much better, though not perfect. It can be seen that
the faster particles do pass through the slower group,
however, there is a noticeable interaction. It is thought
at this time that this is largely due to numerical errors
incurred when the two families initially come in con-
tact with each other. Further study of this process is
needed.
6 CONCLUSIONS
A Galilean-invariant, unambiguous, two-velocity
model for particle transport has been shown. This
model has the promise of being more physically ac-
curate in situations when particles at a given location
and time do not all have the same (or nearly the same)
velocity. It is hoped that this model provides a good
balance of improved physical accuracy over the stan-
dard, single-velocity treatment and the more expensive
multi-velocity treatments where the distribution of par-
ticle velocities must be computed using a costly algo-
rithm for every flux evaluation.
It was shown that the two-velocity model does in-
deed produce superior results for canonical particle-
flow problems. The increase in computational cost is
modest when moving from a one-velocity model, but
the possible increase in physical accuracy is large.
Future work will involve the coupling of the particle
model to a fluid model for a background gas flow. The
field of gaskinetic theory provides a framework for this
coupling through the acceleration term in Eq. (4). This
extension should therefore be fairly straight forward to
develop and implement. Collision operators to model
the interaction of particles with each other is also the
subject of possible study.
ACKNOWLEDGEMENTS
This first author’s time in Ottawa to conduct this re-
search was supported by Mr. Achraf Ben Dhia. Both
authors are very grateful for this support.
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7. 0.00
2.00
1.00
(a) Single-velocity model, particle density
0.00
2.00
1.00
(b) Two-velocity model, particle density
-1
1
-0.8
-0.4
0
0.4
0.8
My
(c) Single-velocity model, y momentum
-1
1
-0.8
-0.4
0
0.4
0.8
My
(d) Two-velocity model, y momentum
Figure 1: Solutions for case of two crossing beams of non-interacting particles, computed using one- and two-
velocity formulations.
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8. 0.00
2.00
1.00
sigma
(a) Single-velocity model, t = 0.0
0.00
2.00
1.00
sigma
(b) Two-velocity model, t = 0.0
0.00
2.00
1.00
sigma
(c) Single-velocity model, t = 0.5
0.00
2.00
1.00
sigma
(d) Two-velocity model, t = 0.5
0.00
2.00
1.00
sigma
(e) Single-velocity model, t = 1.0
0.00
2.00
1.00
sigma
(f) Two-velocity model, t = 1.0
Figure 2: Solutions for case of one family of non-interacting particles overtaking another, computed using one-
and two-velocity formulations.