1. The document analyzes the effect of radiation on flame structure and extinction in a one-dimensional planar diffusion flame. A simplified configuration is adopted where quantities depend on the flame normal direction only.
2. The governing equations are transformed to a mass-weighted coordinate system using the Howarth transformation to simplify the treatment of density. Radiation is incorporated as a small perturbation using a dimensionless parameter defined as the product of the radiation absorption coefficient and a characteristic length scale.
3. Radiation lowers the flame temperature, affecting the reaction rate. The radiation absorption term in the energy equation is obtained from the radiation transport equation, appearing as a non-local integral that is treated using Green's functions.
N. Bilic - "Hamiltonian Method in the Braneworld" 3/3SEENET-MTP
This document discusses various models for dark energy and dark matter unification, including quintessence, k-essence, phantom quintessence, Chaplygin gas, and tachyon condensates. It provides field theory descriptions and equations of state for these models. It also discusses issues like the sound speed and structure formation problems that arise for some unified dark energy/dark matter models. Modifications to address these issues, such as generalized Chaplygin gas and variable Chaplygin gas models, are presented.
N. Bilic - "Hamiltonian Method in the Braneworld" 1/3SEENET-MTP
This document provides an overview of the Hamiltonian method in braneworld cosmology. It begins with introductory lectures on preliminaries like Legendre transformations, thermodynamics, fluid mechanics, and basic cosmology. It then covers braneworld universes, strings and branes, and the Randall-Sundrum model. The document concludes with applications of the Hamiltonian method to topics like quintessence, dark energy/matter unification, and tachyon condensates in braneworlds.
In this note, we derive (to third order in derivatives of the fluid velocity) a 2+1
dimensional theory of fluid dynamics that governs the evolution of generic long-
wavelength perturbations of a black brane or large black hole in four-dimensional
gravity with negative cosmological constant, applying a systematic procedure de-
veloped recently by Bhattacharyya, Hubeny, Minwalla, and Rangamani. In the
regime of validity of the fluid-dynamical description, the black-brane evolution
will generically correspond to a turbulent flow. Turbulence in 2+1 dimensions
has been well studied analytically, numerically, experimentally, and observation-
ally as it provides a first approximation to the large scale dynamics of planetary
atmospheres. These studies reveal dramatic differences between fluid flows in
2+1 and 3+1 dimensions, suggesting that the dynamics of perturbed four and
five dimensional large AdS black holes may be qualitatively different. However,
further investigation is required to understand whether these qualitative differ-
ences exist in the regime of fluid dynamics relevant to black hole dynamics.
I. Cotaescu - "Canonical quantization of the covariant fields: the Dirac fiel...SEENET-MTP
The document discusses the canonical quantization of covariant fields on curved spacetimes, specifically the de Sitter spacetime. It introduces covariant fields that transform under representations of the spin group SL(2,C) and have covariant derivatives ensuring gauge invariance. Isometries of the spacetime generate Killing vectors and induce representations of the external symmetry group, which is the universal covering group of isometries and combines isometries with gauge transformations. Generators of these representations provide conserved observables that allow canonical quantization analogous to special relativity. The paper focuses on applying this framework to the Dirac field on de Sitter spacetime.
N. Bilic - "Hamiltonian Method in the Braneworld" 2/3SEENET-MTP
This document discusses braneworld models and the Randall-Sundrum model. It begins by introducing the relativistic particle and string actions used to describe dynamics in higher dimensions. It then summarizes the two Randall-Sundrum models: RS I contains two branes separated in a fifth dimension to address the hierarchy problem, while RS II has the negative tension brane sent to infinity and observers on a single positive tension brane. Finally, it derives the RS II model solution, using Gaussian normal coordinates and imposing junction conditions at the brane.
This document discusses outgoing and ingoing Klein-Gordon waves near the event horizons of black holes. It first presents the Klein-Gordon equation of motion in the background of the Schwarzschild spacetime metric. Near the event horizon, the radial equation is approximated in the Regge-Wheeler coordinate, leading to oscillatory solutions. The time and radial solutions are then expressed in outgoing and ingoing coordinates, resulting in outgoing and ingoing waves with different analytic properties in the future and past event horizons.
Numerical Simulations on Flux Tube Tectonic Model for Solar Coronal HeatingRSIS International
This document discusses numerical simulations of the flux tube tectonic model for solar coronal heating. It begins with an introduction to the sun's corona and mechanisms for coronal heating. It then describes the 3D MHD code used to model the interaction and reconnection of magnetic flux tubes in the photosphere. The simulations show the formation of current sheets where flux tubes of opposite polarity come into contact. An analytical 1D model is also presented that can predict the behavior observed in the simulations, such as the outward migration of current sheets over time as photospheric shearing increases.
N. Bilic - "Hamiltonian Method in the Braneworld" 3/3SEENET-MTP
This document discusses various models for dark energy and dark matter unification, including quintessence, k-essence, phantom quintessence, Chaplygin gas, and tachyon condensates. It provides field theory descriptions and equations of state for these models. It also discusses issues like the sound speed and structure formation problems that arise for some unified dark energy/dark matter models. Modifications to address these issues, such as generalized Chaplygin gas and variable Chaplygin gas models, are presented.
N. Bilic - "Hamiltonian Method in the Braneworld" 1/3SEENET-MTP
This document provides an overview of the Hamiltonian method in braneworld cosmology. It begins with introductory lectures on preliminaries like Legendre transformations, thermodynamics, fluid mechanics, and basic cosmology. It then covers braneworld universes, strings and branes, and the Randall-Sundrum model. The document concludes with applications of the Hamiltonian method to topics like quintessence, dark energy/matter unification, and tachyon condensates in braneworlds.
In this note, we derive (to third order in derivatives of the fluid velocity) a 2+1
dimensional theory of fluid dynamics that governs the evolution of generic long-
wavelength perturbations of a black brane or large black hole in four-dimensional
gravity with negative cosmological constant, applying a systematic procedure de-
veloped recently by Bhattacharyya, Hubeny, Minwalla, and Rangamani. In the
regime of validity of the fluid-dynamical description, the black-brane evolution
will generically correspond to a turbulent flow. Turbulence in 2+1 dimensions
has been well studied analytically, numerically, experimentally, and observation-
ally as it provides a first approximation to the large scale dynamics of planetary
atmospheres. These studies reveal dramatic differences between fluid flows in
2+1 and 3+1 dimensions, suggesting that the dynamics of perturbed four and
five dimensional large AdS black holes may be qualitatively different. However,
further investigation is required to understand whether these qualitative differ-
ences exist in the regime of fluid dynamics relevant to black hole dynamics.
I. Cotaescu - "Canonical quantization of the covariant fields: the Dirac fiel...SEENET-MTP
The document discusses the canonical quantization of covariant fields on curved spacetimes, specifically the de Sitter spacetime. It introduces covariant fields that transform under representations of the spin group SL(2,C) and have covariant derivatives ensuring gauge invariance. Isometries of the spacetime generate Killing vectors and induce representations of the external symmetry group, which is the universal covering group of isometries and combines isometries with gauge transformations. Generators of these representations provide conserved observables that allow canonical quantization analogous to special relativity. The paper focuses on applying this framework to the Dirac field on de Sitter spacetime.
N. Bilic - "Hamiltonian Method in the Braneworld" 2/3SEENET-MTP
This document discusses braneworld models and the Randall-Sundrum model. It begins by introducing the relativistic particle and string actions used to describe dynamics in higher dimensions. It then summarizes the two Randall-Sundrum models: RS I contains two branes separated in a fifth dimension to address the hierarchy problem, while RS II has the negative tension brane sent to infinity and observers on a single positive tension brane. Finally, it derives the RS II model solution, using Gaussian normal coordinates and imposing junction conditions at the brane.
This document discusses outgoing and ingoing Klein-Gordon waves near the event horizons of black holes. It first presents the Klein-Gordon equation of motion in the background of the Schwarzschild spacetime metric. Near the event horizon, the radial equation is approximated in the Regge-Wheeler coordinate, leading to oscillatory solutions. The time and radial solutions are then expressed in outgoing and ingoing coordinates, resulting in outgoing and ingoing waves with different analytic properties in the future and past event horizons.
Numerical Simulations on Flux Tube Tectonic Model for Solar Coronal HeatingRSIS International
This document discusses numerical simulations of the flux tube tectonic model for solar coronal heating. It begins with an introduction to the sun's corona and mechanisms for coronal heating. It then describes the 3D MHD code used to model the interaction and reconnection of magnetic flux tubes in the photosphere. The simulations show the formation of current sheets where flux tubes of opposite polarity come into contact. An analytical 1D model is also presented that can predict the behavior observed in the simulations, such as the outward migration of current sheets over time as photospheric shearing increases.
Understanding the experimental and mathematical derivation of Heisenberg's Uncertainty Principle. Simple application for estimating single degree of freedom particle in a potential free environment is also discussed.
This document provides a basic refresher on Green's functions as applied to scalar field theories. It outlines the Lagrangian for a basic Higgs boson theory and considers perturbative solutions. The Green's function is expressed as a Fourier integral, and contour integration is used to derive forms for both massless and massive scalar fields. For a massless scalar, the Green's function is shown to be proportional to the difference of two delta functions representing forward and backward traveling waves.
Outgoing ingoingkleingordon 8th_jun19sqrdfoxtrot jp R
This document discusses solutions to the Klein-Gordon equation in Schwarzschild spacetime near a black hole's event horizon. The radial equation is approximated in the Regge-Wheeler coordinate, leading to oscillatory solutions. These solutions are then expressed as outgoing and ingoing waves, which have different properties in the future and past horizons. Near the horizon, the radial equation simplifies to an oscillatory form, allowing solutions describing outgoing and ingoing waves.
Higgsbosontoelectron positron decay_dsplyfoxtrot jp R
This document provides an overview of first order calculations to model the decay of a Higgs boson into an electron-positron pair. It begins by defining the initial and final states as one-particle and two-particle states raised from the vacuum. It then uses the time evolution operator and Dyson expansion to first order to relate the initial and final states. Next, it expresses the Higgs, electron, and positron fields as operators that are split into positive and negative energy modes. Finally, it applies the field operators to the initial state to begin calculating the matrix element for this decay process.
This document discusses solutions to the Emmons boundary layer problem with radiation heat loss. It presents the governing equations in Howarth transformed coordinates, which transforms the compressible flow equations into an incompressible form. It then discusses the Blasius solution approach using self-similarity to reduce the equations. With radiation included, the energy equation contains an additional term representing the radiation source. Extinction conditions are analyzed, showing that radiation lowers the flame temperature and weakens the flame as you move downstream. Self-similarity may break down for the energy equation when non-local radiation effects become significant.
This document outlines an approach to studying time correlations of conserved fields in anharmonic chains using nonlinear fluctuating hydrodynamics. It introduces the BS model, which has two conserved fields - displacement and potential energy. The dynamics of these fields can be approximated by a two-component stochastic Burgers equation. Classifying the universality classes of this equation's correlation functions allows insights into the original anharmonic chain model. Numerical results for specific potentials are also discussed.
- Thermodynamics describes the equilibrium states of systems and the spontaneous processes between those states.
- The first law of thermodynamics states that the total energy of an isolated system is conserved. It can be expressed as ΔU = Q + W, where ΔU is the change in internal energy, Q is heat, and W is work.
- For a closed system, the first law takes the form ΔU = Q - PΔV, where PΔV is the work done by expansion or compression. For a constant volume process where no work is done, ΔU = Q.
- The enthalpy H is a state function defined as H = U + PV. For a
1) The document provides an overview of the contents of Part II of a slideshow on modern physics, which covers topics such as charge and current densities, electromagnetic induction, Maxwell's equations, special relativity, tensors, blackbody radiation, photons, electrons, scattering problems, and waves.
2) It aims to provide a brief yet modern review of foundational concepts in electromagnetism and set the stage for introducing special relativity, quantum mechanics, and matter waves for undergraduate students.
3) The overview highlights that succeeding chapters will develop tensor formulations of electromagnetism and special relativity from first principles before discussing applications like blackbody radiation and early quantum models.
(1) The document discusses linear perturbations of the metric around a Schwarzschild black hole. It derives the Regge-Wheeler equation, which governs axial perturbations and takes the form of a wave equation with an effective potential.
(2) It shows that the Regge-Wheeler potential has a maximum just outside the event horizon. This allows it to be considered as a scattering potential barrier for wave packets.
(3) It concludes that Schwarzschild black holes are stable under smooth, compactly supported exterior perturbations, as these perturbations will remain bounded for all times according to properties of the Regge-Wheeler equation and solutions to the Schrodinger equation.
This document summarizes recent results from the STAR experiment regarding correlations and fluctuations in heavy ion collisions at RHIC. It discusses measurements of elliptic and directed flow that provide evidence for local equilibration and pressure gradients in the quark-gluon plasma. HBT interferometry measurements indicate a source elongated perpendicular to the reaction plane, consistent with initial collision geometry. Charge-dependent number correlations reveal modified hadronization in the quark-gluon plasma compared to pp collisions, suggesting local charge conservation effects during hadronization. Overall, the results provide insights into the equilibration and relevant degrees of freedom in the quark-gluon plasma.
This document discusses key thermodynamic relations that can be derived from limited experimental data. It begins by introducing eight thermodynamic properties - pressure, volume, temperature, internal energy, enthalpy, entropy, Helmholtz function and Gibbs function. Only pressure, volume and temperature are directly measurable. The document then derives equations relating these properties using partial differentiation and the first and second laws of thermodynamics. Maxwell relations are also discussed, which relate the partial derivatives of different properties with each other. Equations are derived for internal energy, enthalpy, and entropy to express them in terms of temperature, pressure and volume. Measurable quantities like heat capacities, coefficient of expansion and compressibility are also discussed.
This document provides an overview of quantum electrodynamics (QED). It begins by discussing cross sections and the scattering matrix, defining cross section as the effective size of target particles. It then derives an expression for cross section in terms of the transition rate and flux of incident particles. Next, it summarizes the derivation of the differential cross section and decay rate formulas in QED using relativistic quantum field theory and Feynman diagrams. It concludes by briefly reviewing the historical development of QED and the equivalence of the propagator approach and other formulations.
The document provides an overview of dimensional analysis and its applications in fluid mechanics. Dimensional analysis is a tool used to correlate analytical and experimental results and predict prototype behavior from model measurements. It involves identifying the fundamental dimensions of variables (e.g. mass, length, time) and establishing dimensionless relationships between variables using methods like Rayleigh's or Buckingham's π-theorem. Dimensional analysis is important for model analysis where dynamic similarity between a model and prototype must be achieved. The document discusses different model laws used to design models for similarity based on dominant forces like viscosity, gravity, etc. It also provides scale ratios required for geometric, kinematic and dynamic similarity.
1) The document discusses the mathematical analysis of a basic AC circuit consisting of a resistor and inductor connected in series and driven by an external sinusoidal voltage source.
2) Kirchhoff's voltage law is applied to derive the differential equation governing the circuit and the forced steady-state response is shown to be a sinusoidal current lagging the driving voltage by a phase angle.
3) Expressions are derived relating the phase lag to the circuit properties and defining the real and reactive power consumed based on the circuit response.
This document discusses energy-size reduction relationships in comminution processes. It outlines three main theories proposed to explain the relationship between energy input and size reduction: Rittinger's law, Kick's law, and Bond's law. The author proposes a new general form for the energy-size reduction relationship that is a linear combination of Kick-type and Rittinger-type laws. Empirical data from various studies is used to validate that this combined law provides a better fit than previous generalized laws, particularly in intermediate size ranges. The combined law also approximates to Kick-type and Rittinger-type laws at size extremes as previously observed. This combined law may have applications in computer simulations of comminution processes and mill
This document discusses the hydrodynamic equations that describe neutral gas and plasma, and how they are modified to become the magnetohydrodynamic (MHD) equations when a conducting fluid is in a magnetic field. It introduces the continuity, momentum, and entropy equations for neutral gas hydrodynamics. It then explains how these are updated to the MHD equations by adding magnetic forces and Ohm's law relating current and fields. The key MHD equations derived include equations for momentum, entropy, and the magnetic field evolving due to motion and diffusion.
By using the anharmonic correlated einstein model to define the expressions o...Premier Publishers
By using potential effective interaction in the anharmonic correlated Einstein model on the basis of quantum statistical theory with phonon interaction procedure, the expressions describing asymmetric component (cumulants) and thermodynamic parameters including the anharmonic effects contributions and by new structural parameters of cubic crystals have been formulated. These new parameters describe the distribution of atoms. The expansion of cumulants and thermodynamic parameters through new structural parameters has been performed. The results of this study show that, developing further the anharmonic correlated Einstein model it obtained a general theory for calculation cumulants and thermodynamic parameters in XAFS theory including anharmonic contributions. The expressions are described through new structural parameters that agree with structural contributions of cubic crystals like face center cubic (fcc), body center cubic (bcc).
This document summarizes key concepts in advanced thermodynamics including:
- Pressure-temperature and pressure-volume diagrams for pure fluids and the phase change curves and points they depict.
- Equations of state relating pressure, volume, and temperature for homogeneous fluids in equilibrium.
- Properties and examples of ideal gas behavior and the virial equation of state for real gases.
- Calculation of work, heat, internal energy, and enthalpy changes for various thermodynamic processes involving ideal gases including isothermal, adiabatic, constant pressure, and throttling processes.
N. Bilić: AdS Braneworld with Back-reactionSEENET-MTP
- A 3-brane moving in an AdS5 background of the Randall-Sundrum model behaves like a tachyon field with an inverse quartic potential.
- When including the back-reaction of the radion field, the tachyon Lagrangian is modified by its interaction with the radion. As a result, the effective equation of state obtained by averaging over large scales describes a warm dark matter.
- The dynamical brane causes two effects of back-reaction: 1) the geometric tachyon affects the bulk geometry, and 2) the back-reaction qualitatively changes the tachyon by forming a composite substance with the radion and a modified equation of state.
This document is a dissertation proposal by Rishideep Roy at the University of Chicago in November 2014. The proposal is to generalize results on extreme values and entropic repulsion for two-dimensional discrete Gaussian free fields to a more general class of Gaussian fields with logarithmic correlations. Specifically, the proposal plans to find the convergence in law of the maximum of these log-correlated Gaussian fields under minimal assumptions, as well as obtain finer estimates on entropic repulsion which relates to the behavior of these fields near hard boundaries. The proposal provides background on related works and outlines the key steps to be taken, including proving expectations and tightness of maxima, invariance of maximum distributions under perturbations, approximating the fields, and
This document provides an analysis of the structure and extinction of asymptotic flamelets based on the seminal paper by Norbert Peters in 1983. It begins with notation and assumptions, then derives the flamelet equation through non-dimensionalization and asymptotic expansions. This yields the flamelet structure equation, which is then simplified using the Schwab-Zeldovich coupling relation between temperature and species. The analysis results in an ordinary differential equation for the flamelet structure identical to that in Law's combustion text. Outer solutions and boundary conditions are then discussed to complete the derivation.
Convection involves determining the flow field, temperature field, and heat transfer coefficient (h) of a fluid. h can be determined from Newton's law of cooling, which relates heat flux to the temperature difference between a surface and fluid. A boundary layer exists near the surface where viscous effects dominate. Internal flows are confined by boundaries while external flows develop freely. Forced convection correlations relate the Nusselt, Reynolds, and Prandtl numbers to determine h. Average h over a surface can be determined by integrating local h values. The Reynolds analogy shows a relationship between frictional drag and convective heat transfer.
Understanding the experimental and mathematical derivation of Heisenberg's Uncertainty Principle. Simple application for estimating single degree of freedom particle in a potential free environment is also discussed.
This document provides a basic refresher on Green's functions as applied to scalar field theories. It outlines the Lagrangian for a basic Higgs boson theory and considers perturbative solutions. The Green's function is expressed as a Fourier integral, and contour integration is used to derive forms for both massless and massive scalar fields. For a massless scalar, the Green's function is shown to be proportional to the difference of two delta functions representing forward and backward traveling waves.
Outgoing ingoingkleingordon 8th_jun19sqrdfoxtrot jp R
This document discusses solutions to the Klein-Gordon equation in Schwarzschild spacetime near a black hole's event horizon. The radial equation is approximated in the Regge-Wheeler coordinate, leading to oscillatory solutions. These solutions are then expressed as outgoing and ingoing waves, which have different properties in the future and past horizons. Near the horizon, the radial equation simplifies to an oscillatory form, allowing solutions describing outgoing and ingoing waves.
Higgsbosontoelectron positron decay_dsplyfoxtrot jp R
This document provides an overview of first order calculations to model the decay of a Higgs boson into an electron-positron pair. It begins by defining the initial and final states as one-particle and two-particle states raised from the vacuum. It then uses the time evolution operator and Dyson expansion to first order to relate the initial and final states. Next, it expresses the Higgs, electron, and positron fields as operators that are split into positive and negative energy modes. Finally, it applies the field operators to the initial state to begin calculating the matrix element for this decay process.
This document discusses solutions to the Emmons boundary layer problem with radiation heat loss. It presents the governing equations in Howarth transformed coordinates, which transforms the compressible flow equations into an incompressible form. It then discusses the Blasius solution approach using self-similarity to reduce the equations. With radiation included, the energy equation contains an additional term representing the radiation source. Extinction conditions are analyzed, showing that radiation lowers the flame temperature and weakens the flame as you move downstream. Self-similarity may break down for the energy equation when non-local radiation effects become significant.
This document outlines an approach to studying time correlations of conserved fields in anharmonic chains using nonlinear fluctuating hydrodynamics. It introduces the BS model, which has two conserved fields - displacement and potential energy. The dynamics of these fields can be approximated by a two-component stochastic Burgers equation. Classifying the universality classes of this equation's correlation functions allows insights into the original anharmonic chain model. Numerical results for specific potentials are also discussed.
- Thermodynamics describes the equilibrium states of systems and the spontaneous processes between those states.
- The first law of thermodynamics states that the total energy of an isolated system is conserved. It can be expressed as ΔU = Q + W, where ΔU is the change in internal energy, Q is heat, and W is work.
- For a closed system, the first law takes the form ΔU = Q - PΔV, where PΔV is the work done by expansion or compression. For a constant volume process where no work is done, ΔU = Q.
- The enthalpy H is a state function defined as H = U + PV. For a
1) The document provides an overview of the contents of Part II of a slideshow on modern physics, which covers topics such as charge and current densities, electromagnetic induction, Maxwell's equations, special relativity, tensors, blackbody radiation, photons, electrons, scattering problems, and waves.
2) It aims to provide a brief yet modern review of foundational concepts in electromagnetism and set the stage for introducing special relativity, quantum mechanics, and matter waves for undergraduate students.
3) The overview highlights that succeeding chapters will develop tensor formulations of electromagnetism and special relativity from first principles before discussing applications like blackbody radiation and early quantum models.
(1) The document discusses linear perturbations of the metric around a Schwarzschild black hole. It derives the Regge-Wheeler equation, which governs axial perturbations and takes the form of a wave equation with an effective potential.
(2) It shows that the Regge-Wheeler potential has a maximum just outside the event horizon. This allows it to be considered as a scattering potential barrier for wave packets.
(3) It concludes that Schwarzschild black holes are stable under smooth, compactly supported exterior perturbations, as these perturbations will remain bounded for all times according to properties of the Regge-Wheeler equation and solutions to the Schrodinger equation.
This document summarizes recent results from the STAR experiment regarding correlations and fluctuations in heavy ion collisions at RHIC. It discusses measurements of elliptic and directed flow that provide evidence for local equilibration and pressure gradients in the quark-gluon plasma. HBT interferometry measurements indicate a source elongated perpendicular to the reaction plane, consistent with initial collision geometry. Charge-dependent number correlations reveal modified hadronization in the quark-gluon plasma compared to pp collisions, suggesting local charge conservation effects during hadronization. Overall, the results provide insights into the equilibration and relevant degrees of freedom in the quark-gluon plasma.
This document discusses key thermodynamic relations that can be derived from limited experimental data. It begins by introducing eight thermodynamic properties - pressure, volume, temperature, internal energy, enthalpy, entropy, Helmholtz function and Gibbs function. Only pressure, volume and temperature are directly measurable. The document then derives equations relating these properties using partial differentiation and the first and second laws of thermodynamics. Maxwell relations are also discussed, which relate the partial derivatives of different properties with each other. Equations are derived for internal energy, enthalpy, and entropy to express them in terms of temperature, pressure and volume. Measurable quantities like heat capacities, coefficient of expansion and compressibility are also discussed.
This document provides an overview of quantum electrodynamics (QED). It begins by discussing cross sections and the scattering matrix, defining cross section as the effective size of target particles. It then derives an expression for cross section in terms of the transition rate and flux of incident particles. Next, it summarizes the derivation of the differential cross section and decay rate formulas in QED using relativistic quantum field theory and Feynman diagrams. It concludes by briefly reviewing the historical development of QED and the equivalence of the propagator approach and other formulations.
The document provides an overview of dimensional analysis and its applications in fluid mechanics. Dimensional analysis is a tool used to correlate analytical and experimental results and predict prototype behavior from model measurements. It involves identifying the fundamental dimensions of variables (e.g. mass, length, time) and establishing dimensionless relationships between variables using methods like Rayleigh's or Buckingham's π-theorem. Dimensional analysis is important for model analysis where dynamic similarity between a model and prototype must be achieved. The document discusses different model laws used to design models for similarity based on dominant forces like viscosity, gravity, etc. It also provides scale ratios required for geometric, kinematic and dynamic similarity.
1) The document discusses the mathematical analysis of a basic AC circuit consisting of a resistor and inductor connected in series and driven by an external sinusoidal voltage source.
2) Kirchhoff's voltage law is applied to derive the differential equation governing the circuit and the forced steady-state response is shown to be a sinusoidal current lagging the driving voltage by a phase angle.
3) Expressions are derived relating the phase lag to the circuit properties and defining the real and reactive power consumed based on the circuit response.
This document discusses energy-size reduction relationships in comminution processes. It outlines three main theories proposed to explain the relationship between energy input and size reduction: Rittinger's law, Kick's law, and Bond's law. The author proposes a new general form for the energy-size reduction relationship that is a linear combination of Kick-type and Rittinger-type laws. Empirical data from various studies is used to validate that this combined law provides a better fit than previous generalized laws, particularly in intermediate size ranges. The combined law also approximates to Kick-type and Rittinger-type laws at size extremes as previously observed. This combined law may have applications in computer simulations of comminution processes and mill
This document discusses the hydrodynamic equations that describe neutral gas and plasma, and how they are modified to become the magnetohydrodynamic (MHD) equations when a conducting fluid is in a magnetic field. It introduces the continuity, momentum, and entropy equations for neutral gas hydrodynamics. It then explains how these are updated to the MHD equations by adding magnetic forces and Ohm's law relating current and fields. The key MHD equations derived include equations for momentum, entropy, and the magnetic field evolving due to motion and diffusion.
By using the anharmonic correlated einstein model to define the expressions o...Premier Publishers
By using potential effective interaction in the anharmonic correlated Einstein model on the basis of quantum statistical theory with phonon interaction procedure, the expressions describing asymmetric component (cumulants) and thermodynamic parameters including the anharmonic effects contributions and by new structural parameters of cubic crystals have been formulated. These new parameters describe the distribution of atoms. The expansion of cumulants and thermodynamic parameters through new structural parameters has been performed. The results of this study show that, developing further the anharmonic correlated Einstein model it obtained a general theory for calculation cumulants and thermodynamic parameters in XAFS theory including anharmonic contributions. The expressions are described through new structural parameters that agree with structural contributions of cubic crystals like face center cubic (fcc), body center cubic (bcc).
This document summarizes key concepts in advanced thermodynamics including:
- Pressure-temperature and pressure-volume diagrams for pure fluids and the phase change curves and points they depict.
- Equations of state relating pressure, volume, and temperature for homogeneous fluids in equilibrium.
- Properties and examples of ideal gas behavior and the virial equation of state for real gases.
- Calculation of work, heat, internal energy, and enthalpy changes for various thermodynamic processes involving ideal gases including isothermal, adiabatic, constant pressure, and throttling processes.
N. Bilić: AdS Braneworld with Back-reactionSEENET-MTP
- A 3-brane moving in an AdS5 background of the Randall-Sundrum model behaves like a tachyon field with an inverse quartic potential.
- When including the back-reaction of the radion field, the tachyon Lagrangian is modified by its interaction with the radion. As a result, the effective equation of state obtained by averaging over large scales describes a warm dark matter.
- The dynamical brane causes two effects of back-reaction: 1) the geometric tachyon affects the bulk geometry, and 2) the back-reaction qualitatively changes the tachyon by forming a composite substance with the radion and a modified equation of state.
This document is a dissertation proposal by Rishideep Roy at the University of Chicago in November 2014. The proposal is to generalize results on extreme values and entropic repulsion for two-dimensional discrete Gaussian free fields to a more general class of Gaussian fields with logarithmic correlations. Specifically, the proposal plans to find the convergence in law of the maximum of these log-correlated Gaussian fields under minimal assumptions, as well as obtain finer estimates on entropic repulsion which relates to the behavior of these fields near hard boundaries. The proposal provides background on related works and outlines the key steps to be taken, including proving expectations and tightness of maxima, invariance of maximum distributions under perturbations, approximating the fields, and
This document provides an analysis of the structure and extinction of asymptotic flamelets based on the seminal paper by Norbert Peters in 1983. It begins with notation and assumptions, then derives the flamelet equation through non-dimensionalization and asymptotic expansions. This yields the flamelet structure equation, which is then simplified using the Schwab-Zeldovich coupling relation between temperature and species. The analysis results in an ordinary differential equation for the flamelet structure identical to that in Law's combustion text. Outer solutions and boundary conditions are then discussed to complete the derivation.
Convection involves determining the flow field, temperature field, and heat transfer coefficient (h) of a fluid. h can be determined from Newton's law of cooling, which relates heat flux to the temperature difference between a surface and fluid. A boundary layer exists near the surface where viscous effects dominate. Internal flows are confined by boundaries while external flows develop freely. Forced convection correlations relate the Nusselt, Reynolds, and Prandtl numbers to determine h. Average h over a surface can be determined by integrating local h values. The Reynolds analogy shows a relationship between frictional drag and convective heat transfer.
Anomalous Diffusion Through Homopolar Membrane: One-Dimensional Model by Guilherme Garcia Gimenez and Adélcio C Oliveira* in Evolutions in Mechanical Engineering
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.
This document introduces the concept of gauge invariance and gauge field theories. It discusses both global and local gauge symmetries:
- Global gauge symmetries lead to conserved currents and charges via Noether's theorem. They result in massless scalar bosons if spontaneously broken.
- Local gauge symmetries require the introduction of gauge fields which transform in a way that cancels out non-invariant terms under local transformations. This avoids the massless bosons and allows gauge fields to acquire mass. Quantum electrodynamics possesses a local U(1) gauge symmetry.
- Non-Abelian gauge groups were introduced by Yang and Mills, allowing the construction of gauge field theories with non-commuting gauge groups. Covariant derivatives are
The document discusses the classical and quantum mechanical treatment of the simple harmonic oscillator.
Classically, the simple harmonic oscillator exhibits sinusoidal motion with a single resonant frequency, where the restoring force is proportional to displacement from equilibrium. Quantum mechanically, the energy levels of the 1D harmonic oscillator are quantized and equally spaced. The wave functions are solutions of the Schrodinger equation and can be written as products of Hermite polynomials and Gaussian functions. The energy eigenstates of the 1D harmonic oscillator are (n+1/2)ħω, where n is the vibrational quantum number.
185817220 7e chapter5sm-final-newfrank-white-fluid-mechanics-7th-ed-ch-5-solu...Abrar Hussain
This document summarizes the solutions to several problems involving dimensional analysis. Some key points:
- Problem 5.1 calculates the volume flow rate needed for transition to turbulence in a pipe based on given parameters.
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1. Asymptotic analysis of radiative extinction in
a one-dimensional planar di↵usion flame
Praveen Narayanan
January 28, 2009
1 Introduction
In this study, the role of radiation is analysed to understand its e↵ect on
flame structure and extinction. A simplified, one dimensional configuration
is adopted, in which pertinent quantities are all one-dimensional, with the ex-
ception of the velocity field. The two dimensional Howarth transformation is
used to transform the governing equations into mass weighted coordinates in
order to simplify the treatment of density. The chemical reaction source term
is handled in conventional fashion as a singularity, resolved using matched
asymptotic expansions. The manifestation of radiation is in lowering the
flame temperature which then a↵ects the reaction rate significantly. The
form of the inner expansions is not a↵ected by radiation, and is described by
a purely reactive-di↵usive structure. However, the strength of the reaction
term is dictated by reduction in temperature brought about by radiation.
The treatment of the radiatively active outer layer is achieved by observing
that radiative length scale, as dictated by the inverse of the absorption co-
e cient is large compared to the di↵usion length scales. Thus radiation can
be incorporated as a small perturbation, the functional form of which may
be su ciently described by the leading order (or unperturbed) temperatures
in the respective emission and absorption terms. The outer region can then
be solved using a Green function as a convolution. The radiation terms may
also be treated equivalently using a regular perturbation expansion using the
small parameter identified above.
The radiation absorption term in the energy equation is obtained from the
radiation transport equation, with use being made of there being azimuthal
symmetry about the polar axis formed by the flame normal in spherical co-
ordinates. As mentioned above, absorption is accommodated in the current
1
2. framework from the leading order temperatures, and requires no special treat-
ment. However, being a non-local quantity, it appears as a non-local integral
over space. It is mentioned in passing (perhaps naively) that the utility of
Green’s functions is especially apparent in this case with non-singular (and
non-local) radiation terms. From the author’s cursory perusal of Mathemat-
ical Physics literature, the use of Green’s functions is a common procedure
(notable examples are in electromagnetism, quantum mechanics, and astro-
physics) to solve di↵erential equations with source terms.
2 Governing equations
2.1 Howarth transformation
Density weighted coordinates are used, in conformity with Carrier, Fendell
and Marble [1] that greatly simplifies the analysis. The transformation is
described here in brief. For more extensive notes, Dr. Howard Baum’s [3]
document may be referred to.
Consider a two dimensional planar flame configuration in which the flame
normal is along the x direction; u and v being the velocities in the x and y
directions and ⇢ being the density. The equation of continuity is
@⇢
@t
+
@(⇢u)
@x
+
@(⇢v)
@y
= 0 (1)
A mass weighted coordinate is defined as
⇠ =
Z x
1
⇢
⇢1
dx (2)
where ⇢1 = ⇢(x = 1, y) = ⇢(x = 1) because of one-dimensionality in
this planar configuration. Transform the equations into this coordinate sys-
tem comprised of X = ⇠,Y = ⌘ and T = t with the following transformation
rules
x = x(⇠, Y, T) )
@
@x
=
@
@⇠
@⇠
@x
+
@
@Y
@Y
@x
+
@
@T
@T
@x
=
@
@⇠
@⇠
@x
=
⇢
⇢1
@
@⇠
(3)
y = y(⇠, Y, t) )
@
@y
=
@
@⇠
@⇠
@y
+
@
@Y
@Y
@y
+
@
@t
@y
@t
=
@
@⇠
@⇠
@y
+
@
@Y
(4)
2
3. t = t(⇠, Y, T) )
@
@t
=
@
@⇠
@⇠
@t
+
@
@Y
@Y
@t
+
@
@T
@T
@t
=
@
@⇠
@⇠
@t
+
@
@T
(5)
The continuity equation becomes, together with equation (2)
@
@x
(⇢u + ⇢1
@⇠
@t
) +
@(⇢v)
@y
= 0 (6)
Now one may define a stream function (x, y, t) as follows
⇢1
@
@x
= ⇢v; ⇢1
@
@x
= (⇢u + ⇢1
@⇠
@t
) (7)
Together with equation (2) the this gives
@
@⇠
= v (8)
One can now define a normal transformed velocity U(⇠, Y, t) as
@
@Y
= U (9)
A constant density version of the continuity equation may now be written as
@v
@Y
+
@U
@⌘
= 0 (10)
The velocity U is obtained from the equation (7) as
U =
1
⇢1
@⇠
@t
+
⇢
⇢1
u + v
@⇠
@y
(11)
The continuity equation may now into a form, after suitable manipulation,
to preseve the convective derivative so that one obtains
D
Dt
=
@
@T
+ U
@
@⇠
+ v
@
@Y
⌘
˜D
˜DT
(12)
2.2 Applying Howarth transform to governing equa-
tions
In order to obtain the governing equations, one-dimensionality is assumed, so
that all quantities (except the velocities) vary only in the x direction, which
3
4. is normal to the flame. The velocity in the direction normal to the flame in
this new coordinate system is obtained from the equation (10) as
U =
Z ⇠
0
@v
@Y
d⇠ (13)
If a constant strain @v/@Y = ", together with a reference velocity of zero at
⇠ = 0 is assumed (which are perfectly reasonable assumptions), one gets
U =
Z ⇠
0
@v
@Y
d⇠ = "⇠ (14)
In the undermentioned, T and Y are replaced later by t and y to retain
the usual notation. The transport operator is now transformed into mass
weighted coordinates.As mentioned in the foregoing, quantities of interest
are only assumed to depend on x and t. Furthermore, unity Lewis numbers
are assumed so that the di↵usion coe cient may be denoted by a quantity
D. The untransformed transport operator is
⇢L ⌘ ⇢
@
@t
+ ⇢u
@
@x
@
@x
✓
D⇢
@
@x
◆
@
@y
✓
D⇢
@
@y
◆
(15)
Upon introducing the transformation, and ignoring Y dependence, one gets
⇢
@
@t
+ ⇢u
@
@x
= ⇢
✓
@
@T
"⇠
@
@⇠
◆
(16)
The di↵usion terms are manipulated, using the transformation (2) as follows
@
@x
✓
⇢D
@
@x
◆
=
⇢
⇢1
@
@⇠
✓
D⇢
⇢
⇢1
@
@⇠
◆
=
⇢
⇢2
1
@
@⇠
✓
D⇢2 @
@⇠
◆ (17)
Now, if one makes the simplification that D⇢2
is nearly constant in flows of
interest, one may pull out this quantity as D⇢2
= D1⇢2
1. One may replace
T with t to obtain
L ⌘
@
@t
"⇠
@
@⇠
D1
@2
@⇠2
(18)
2.3 Non-dimensionalization
The governing equations for species mass and energy are
L(F) = ⇢ 1
˙!F
⇢L(Y ) = ⇢ 1
rs ˙!F
⇢L(h) = ⇢ 1
hc ˙!F + ⇢ 1
5 ·~qR
(19)
4
5. where F, Y are the fuel and oxidizer mass fractions respectively; h =
R T
T0
cpdT
is the enthalpy for material at temperature T, based on some reference en-
thalpy at temperature T0; hc is the heat of combustion per unit mass of
fuel (a positive quantity); ˙!F is the fuel reaction rate per unit volume; rs
is the stoichiometric fuel-air coe cient in the chemical reaction. The non-
dimensionalized quantities are superscripted by primes
⇢0
=
⇢
⇢1
; ⇠0
=
⇠
l
; t0
= "t
F0
=
F
F1
; Y 0
=
Y
rsF1
; h0
=
h
hc
(20)
In the foregoing equations, l is a reference length scale, and quantities sub-
scripted with 1 are reference quanties at the boundaries: F1 = F(⇠ = 1),
Y1 = Y (⇠ = 1), ⇢1 = ⇢(⇠ = 1). The di↵usion coe cient is defined based
on the aforementioned length and time-scales as
D1 = "l2
(21)
The reaction term is taken to be of the following form, with an activation
energy ✓ defined suitably.
˙! = B(⇢F)⌫F
(⇢Y )⌫O
exp
✓
✓
h
◆
(22)
In conjunction with the ideal gas law, with nearly constant pressures
P = ⇢RT = P1 (23)
one gets for the reaction term as
˙! = B
✓
P1cp
R hc
◆⌫F +⌫O
h (⌫F +⌫O)
exp
✓
✓
h
◆
(24)
On non-dimensionalizing, one gets
˙!
⇢"
=
B
⇢"
✓
P1cp
R hc
◆⌫F +⌫O
h (⌫F +⌫O)
exp
✓
✓
h
◆
= B
✓
P1
R
cp
hc
h 1
◆ 1 ✓
P1cp
R hc
◆⌫F +⌫O
h (⌫F +⌫O)
exp
✓
✓
h
◆
=
B
"
✓
P1cp
R hc
◆⌫F +⌫O 1
h (⌫F +⌫O 1)
exp
✓
✓
h
◆
(25)
5
6. From the above, one may identify the multiplying quantities on the right hand
side as the first Damk¨ohler number D1 (notation consistent with Carrier,
Fendell and Marble).
D1 =
B
"
✓
P1cp
R hc
◆⌫F +⌫O 1
(26)
The small parameters for the reaction zone may either be taken as the inverse
of the first Damk¨ohler number D1 for large Damk¨ohler number expansions
or as h2
⇤/✓ where h⇤ is the flame temperature in a large activation energy
expansion. The treatment is postponed until later. In either case, it will
be seen that the form of the inner expansion is not a↵ected by the presence
of radiation. The radiation terms are non-dimensionalized with a reference
absorption coe cient ¯ and a reference radiation intensity ¯I. This, in fact,
furnishes a small parameter for the radiation source term.
5 · ~qR ⇠ ¯¯Ib
⇤
⇠ ¯ T4
⇤ (27)
where the subscript ⇤ denotes quantities at the flame, located at ⇠⇤; Ib(⇠) =
⇡ T(⇠)4
is the black body radiation intensity. The emission term is inspected
to provide a suitable small quantity for radiation.
5 · qR ⇠ ¯ T4
⇤
Or
5 · qR
⇢ hc"
⇠
¯ T4
⇤
⇢⇤ hc"
= ¯l
T4
⇤
⇢⇤ hcl"
(28)
Plugging in typical values taken as T⇤ = 2000 K, ⇢⇤ = 0.2 kg/m3
, hc =
32 MJ/kg, one gets
5 · qR
⇢ hc"
⇠
¯l
l"
(5.67 ⇥ 10 8)(2000)4
0.2 ⇥ (32 ⇥ 106)
=
✓
0.14
"l
◆
l
(29)
In the current problem, l is a small number because the radiation length
scale (⇠ 1/) is expected to be much larger than the di↵usion length scale
(⇠ l). Thus one can now define a small parameter ✏ such that
✏ = l ⌧ 1 (30)
Furthermore, the quantity multiplying l in the equation (29) is assumed to
be O(1). This is a reasonable assumption for small flames, and large strain
rates. The intent is to demonstrate that the product
✓
0.14
"l
◆
¯l ⌧ 1 (31)
6
7. and not per se, that 0.14/("l) ⇠ O(1) in that one wants to be able to show
that the non-dimensionalized radiation source term is much smaller than
unity so that the unperturbed (or radiation free) temperatures may be used
as a first approximation for these terms. Radiation is then a known field and
may be used in calculating the flame temperature.
To get back to the main discussion it therefore transpires that in the
non-dimensional equations the radiation term is multiplied by a quantity of
O(✏). This will be used in the analysis later. In order to make the notation
consistent, the radiation term is now scaled in terms of enthalpy instead of
temperature. If one were to non-dimensionalize
0
=
¯
(32)
and drop the prime symbol in the equations, one gets for the emission term
qR,E
5 · qR,E
⇢" hc
= ¯l
✓
T4
hc"⇢
◆
= (¯l)
✓
hc"l
h4
c
c4
p
h4
⇢
◆
= (¯l)
✓
hc"l
◆ ✓
h4
ch4
c4
p
◆ ✓
RT
P1
◆
= (¯l)
✓
hc"l
◆ ✓
h4
ch4
c4
p
◆ ✓
RcpT
hc
hc
cpP1
◆
= (¯l)
✓
hc"l
◆ ✓
h4
ch4
c4
p
◆ ✓
h
hcR
cpP1
◆
= (¯l)
"✓
hc"l
◆ ✓
hc
cp
◆5 ✓
R
P1
◆
h5
#
⇠ O(¯l)
(33)
The absorption term is obtained in the document [2].
One may then write the radiation terms (which will now include both
emission and absorption) as
5 · ~qR
⇢" hc
= (¯l)f(h(⇠), (⇠)) = (¯l)g(⇠) ⇠ O(¯l) (34)
The energy equation becomes
L(h) = D1h (⌫F +⌫O 1)
F⌫F
Y ⌫O
exp
✓
✓
h
◆
+ (¯l)g(⇠) (35)
7
8. The equations for fuel and oxidizer mass fractions are obtained from Scwhab-
Zeldovich relationships.
L(h + F) = (¯l)g(⇠) (36)
L(h + Y ) = (¯l)g(⇠) (37)
3 Inner expansions
The inner expansions may be carried out either with a large Activation En-
ergy expansion, or a large Damk¨ohler number expansion. For purposes of
illustration, a large Damk¨ohler number expansion is chosen. The large acti-
vation energy expansion is explored later, and is handled analogously to that
of numerous previous investigations. Consider the steady energy equation
⇠
dh
d⇠
d2
h
d⇠2
= D1h (⌫F +⌫O 1)
F⌫F
Y⌫O
exp
✓
✓
h
◆
+ (¯l)g(⇠) (38)
As in Carrier, Fendell and Marble, one takes the the inverse of the first
Damk¨ohler number = D 1
1 as the small parameter. Propose a stretched
inner coordinate for the boundary layer as
⌘ =
⇠ ⇠⇤
b
(39)
where ⇠⇤ is the flame location and b is a number to be determined shortly.
One writes
F(⇠) = F(⌘)
Y (⇠) = Y (⌘)
h(⇠) = H (⌘)
(40)
All pertinent quantities are now expanded as a power series of b
to first
order. Note that fuel and oxidizer are vanishing in the inner layer to leading
order.
F = 0 + b
F1
Y = 0 + b
Y1
H = H⇤
b
H1
(41)
Upon inserting the expansion (41) into equation (38) and rearranging (with
higher order terms included only where deemed useful for illustration), one
8
9. gets
⇠⇤
b
d
d⌘
(H⇤
b
H1)
1
2b
d2
d⌘2
(H⇤
b
H1)
= 1 b(⌫F +⌫O)
h (⌫F +⌫O 1)
⇤ F⌫F
1 Y ⌫O
1 exp
✓
✓
h⇤
◆
+ (¯l)g(⇠⇤)
(42)
It emerges from the above on multiplying throughout by b
and rearranging,
that the radiation term will be O( b
(¯l) and is negligible compared to the
reaction term, which is O(1). Also, the convective term may be neglected
compared to the di↵usion term. It transpires that the inner equation does
not feature the radiation term to leading order. Matching O(1) terms gives
b =
1
⌫F + ⌫O + 1
(43)
and the inner equation becomes
d2
H1
d⌘2
= h (⌫F +⌫O 1)
⇤ exp
✓
✓
h⇤
◆
F⌫F
1 Y ⌫O
1 (44)
One may notice that although the form of the inner equation is unaltered,
compared to the radiation free case, the strength of the reaction is reduced
because of a drop in temperature in the term
h (⌫F +⌫O 1)
⇤ exp
✓
✓
h⇤
◆
(45)
Notably, a small decrease in the temperature will lead to a large drop in the
chemical reaction rate because of exponential behavior of the function. The
exercise therefore boils down to determining the flame temperature given by
h⇤ with radiation. This is obtained from the outer solutions. The flame
location ⇤ is also obtained from the outer solutions. To leading order, this is
unaltered from the case with no radiation.
4 Outer solutions
The outer solutions may be postulated as a perturbation from the solution
obtained without radiation. To that end, it the species mass concentrations
are una↵ected, but the presence of the radiation source term will see an al-
teration in the flame temperature. One may propose the following expansion
for the enthalpy
h(⇠) = h0
(⇠; a
) + (¯l)h1
(46)
9
10. where h0
(⇠; a
) is the solution obtained without radiation (and is a known
function), and a is a quantity that is to be determined accordingly for the
fuel and oxidizer sides of the flame (also known). It can be seen that insofar
as radiation is concerned, one only needs the leading order behavior of h in
the expansion proposed above. This may be inserted in the outer equations
to obtain the flame temperature in the outer region. The equations may then
be solved using a regular perturbation expansion with ¯l as the expansion
parameter, or, more elegantly, by the use of Green’s functions.
4.1 Flame location
The flame location is the same as without radiation, because one can match
leading order solutions from the local Schwab-Zeldovich expansions. The
procedure is identical to that described in Carrier, Fendell and Marble. For
the local Schwab-Zeldovich relations one uses
L(F0
+ h0
) = 0
L(Y 0
+ h0
) = 0
(47)
This gives an expression for the flame location as
erf
✓
⇠⇤
p
2
◆
=
F1 Y1
F1 + Y1
(48)
4.2 Obtaining the flame temperature
4.2.1 Regular Perturbation
The flame temperature, as mentioned in the foregoing, may be determined
from a regular perturbation expansion for the temperature as follows
h(⇠) = h0
(⇠; a
) + (¯l)h1
(⇠) (49)
In the foregoing, h1
is a function that is smooth upto its first derivative so
that
[h1
]⇠⇤ = [h1
]⇠⇤+
✓
dh
d⇠
◆
⇠⇤
=
✓
dh
d⇠
◆
⇠⇤+
(50)
The function g(⇠) is entirely known because it only contains leading order
terms. One can then obtain for the O(¯kl) terms
L(h1
) = g0 (51)
10
11. with boundary conditions
h1
⇠= 1 = 0
✓
dh1
d⇠
◆
⇠= 1
= 0
h1
⇠=1 = 0
✓
dh1
d⇠
◆
⇠=1
= 0
(52)
In the aforementioned equation, it is expected that the conditions on the
derivatives should hold trivially.
4.2.2 Green’s function
As an alternative to regular perturbation, one may also use a Green’s function
approach to solve for the relevant quantities in the outer layers. One of the
advantages with this formalism is that it is easy to set up, and involves
much less labor than a regular perturbation. It is noted that the approach
works because the governing equations are linear. The procedure is briefly
illustrated below. For a more elaborate discussion Carrier and Pearson [4] or
any other text on ordinary di↵erential equations may be referred to. Consider
a linear di↵erential equation with the operator L
L =
d
d⇠
✓
p(⇠)
d
d⇠
◆
+ q(⇠) (53)
acting on the variable y to give the inhomogeneous problem
L y = f(⇠) (54)
If one were to find a Green’s function G(⇠, s), which is continuous in ⇠ and
di↵erentiable at all points except ⇠ = s, satisfying the boundary conditions
of the inhomogeneous problem (54) so that
L G = (⇠ s) (55)
where (⇠ s) is the Dirac Delta function, then upon multiplying both sides
by f(⇠) (a known function) and integrating with respect to s one gets
Z 1
1
L Gf(⇠)ds =
Z 1
1
f(⇠) (s ⇠)ds
= f(⇠)
(56)
11
12. Since the operator L is linear one can take it out of the integral to obtain
L
Z 1
1
G(⇠, s)f(⇠)ds = f(⇠) (57)
Inspection of equation (57) reveals that the integral in the LHS is the sought
solution.
y =
Z 1
1
G(⇠, s)f(⇠)ds (58)
One only needs to find a Green’s function, which can be done without much
di culty, as is shown in the forthcoming.
4.2.3 Application of Green’s function to find outer solutions
In this problem, the function f in the foregoing is replaced by the known
function g(⇠). Thus, one can determine the outer solution from the convolu-
tion
h =
Z 1
1
G(⇠, s)(¯l)g(⇠)ds (59)
Since this has known boundary conditions, the problem can be solved. One
may therefore obtain h⇤ for use in the inner equations (which has the same
boundary conditions as the case without radiation) and solve for the inner
temperature H. Extinction analysis can then be carried out by increasing the
radiation losses, which translates to a parametric variation of the coe cient
l.
4.2.4 Obtaining Green’s function for given problem
The Green’s function for this problem is obtained by solving the equivalent
homogeneous problem, defined as
L(G(⇠, s)) = (⇠ s) (60)
satisfying the boundary conditions
G( 1, s) = h 1
G(1, s) = h1
(61)
It may be added, superfluously, that the Green’s function is constructed so
as to satisfy the boundary conditions of the original inhomogeneous prob-
lem. One is referred to the text by Carrier and Pearson [4] for details. The
12
13. Green’s function constructed above is continuous over the domain, but has
a discontinuous derivative at ⇠ = s. Rewrite the di↵erential equation as
d
d⇠
✓
p(⇠)
dG
d⇠
◆
= 0 (62)
where
p(⇠) = exp
✓
⇠2
2
◆
(63)
Integrate (62) to get
G(⇠, s) = h 1 + A(s)
Z ⇠
1
exp
✓
µ2
2
◆
dµ 1 < ⇠ < s
= h1 + B(s)
Z 1
⇠
exp
✓
µ2
2
◆
dµ s < ⇠ < 1
(64)
Call the linearly independent solutions obtained above, on either side of ⇠ = s
as 1 and 2.
A(s) 1(⇠) = G(⇠, s) h 1; 1 < ⇠ < s
B(s) 2(⇠) = G(⇠, s) h1; s < ⇠ < 1
(65)
The constants A(s) and B(s) (invariant with respect to ⇠) may be deter-
mined by imposing continuity of the Green’s function, and discontinuity of
its derivative at ⇠ = s. Imposing continuity of G at ⇠ = s one gets
h 1 + A(s) 1(s) = h1 + B(s) 2(s) (66)
The jump condition for the derivative at ⇠ = s is given by
G⇠(s+, s) G⇠(s , s) =
1
p(s)
= exp
✓
s2
2
◆
(67)
where the subscript ⇠ refers to di↵erentiation with respect to ⇠. Equations
(66) and (67) may be solved to give A(s) and B(s) as follows
A(s) =
h 1 h1 + 2(s+)
p
2⇡
B(s) =
h 1 h1 1(s )
p
2⇡
(68)
The outer solution to the energy equation may now be written in terms of
the Green’s function
h(⇠) =
Z 1
1
G(⇠, s)f(⇠)ds
=
Z ⇠
1
[h1 + B(s) 2(⇠)]f(⇠)ds +
Z 1
⇠
[h 1 + A(s) 1(⇠)]f(⇠)ds
(69)
13
14. 5 Conclusions
A tentative framework has been laid out for the asymptotic analysis of the
radiating flame. The problem extends upon the work by Carrier, Fendell
and Marble by adding radiation as a small perturbation. The resulting small
decrease in flame temperature is expected to lead to a more drastic decrease
in reaction rate, and eventually cause flame extinction. The formulation is
general insofar as to account for both emission and absorption. As further
work, the details will be worked out for the regular perturbations and the
Green’s functions, so as to subsequently perform an analysis of extinction.
Large activation energy asymptotic expansions will also be explored to
obtain an extinction criterion. Since only the inner equations are relevant for
the extinction analysis, one may see that it is readily amenable to the form
proposed by Li˜nan from which can be extracted a criterion as reported in
other investigations. However, it is critical to obtain the flame temperature
from the outer expansions in this case as well.
References
[1] Carrier, G., F., Fendell, F., E., Marble, F., E., “The e↵ect of
strain rate on di↵usion flames”, SIAM Journal of Applied Math-
ematics, 28, No. 2, March 1975.
[2] Narayanan, P., “Expression for radiant energy absorption for use
in energy equation”, Internal document.
[3] Baum, H. R., “Notes on the Howarth Transformation”, “Internal
document.
[4] Carrier, G. F., Pearson, C. E. “Ordinary Di↵erential Equations,
Theory and Practice”, SIAM Classics in Applied Mathematics,
1992.
14