Planar sheath and presheath
|
|
- Jessica Davidson
- 6 years ago
- Views:
Transcription
1 5/11/1 Flui-Poisson System Planar sheath an presheath 1 Planar sheath an presheath A plasma between plane parallel walls evelops a positive potential which equalizes the rate of loss of electrons an ions. The potential profile in the plasma can be foun by integrating Poisson's equation from the plasma miplane to the wall. Moels are neee for the electron an ion ensities. It is customary to moel the electron ensity using a Mawellian istribution for which the ensity is given by a Boltzmann factor. The ions are moele using flui equations with a source term. The ion current to the wall is J an the charge ensity of ions is foun by iviing J by the flui velocity u. 1. Equations i Potential profile ( ) J ( ) e( ) o noq ep u( ) Te nu R Poisson's equation Continuity equation with source R It is assume that the ions are create uniformly throughout the plasma volume with a rate R per unit volume per unit time. For our one imensional plasma in a steay state, the current at a istance from the miplane must be sufficient to take away the ions that are create between the miplane an, which implies J = R. This result is obtaine by integrating once the continuity equation. The momentum equation is: t nu u nu nu u q m E where E is the electric fiel. Using the continuity equation, we can simplify the momentum equation: u nu q ne Ru m Note that the new ions are a sink for momentum because they are create with zero velocity an must be given momentum Ru per unit time in orer to move at the flui velocity. These ifferential equations can be solve simultaneously using Runge-Kutta or some other metho. It is useful to change to imensionless variables: / D R RD / nc s q / Te E ee / D T e J J / n o qc u i u / c s s J R ni J / u e n e where D T e / nq
2 5/11/1 Flui-Poisson System Planar sheath an presheath The equations for the problem, in imensionless form are then: ( ( ep ( E ) J i ) ( ) ( ) ) ( ) u E i Through substitutions, the momentum equation can be rewritten: u E R u n i u E u u In the Mathca version of the equation, the tiles are roppe for convenience.. Bounary conitions We must begin the integration slightly away from = to avoi the ivision by zero that woul occur in the momentum equation. u increases approimately linearly with, thus the term u/ has a finite value near the origin. There must be starting values for, E, an u a short istance from the origin. Symmetry requires that shoul be a function of even powers of. A first approimation for is then ( ) Thus E ( ) where is a constant to be etermine. 1 n e is assume at the miplane. From Poisson's equation we fin that (1 ) n i at the miplane. There is a slight ecess of ions at the center because the ions are less mobile. Using that J R n i u we fin for points near the origin R u R n This last relation provies us with a starting value for u at a small istance from the origin. The following three relations give us the starting values for E,, an u at the starting istance : u R E i 3. A relation between R an A further look at the momentum equation, using values vali near the origin, shows that R an cannot be chosen inepenently. The momentum equation can be solve for E to obtain R E ( ) n i Thus it follows that, but previously we showe R (1 ) E ( ) When R is sufficiently small: R
3 5/11/1 Flui-Poisson System Planar sheath an presheath 3 4. A relation between the plasma size an R A result of the sheath moel is that the current at the wall is approimately.5 in imensionless units. This means R =.5 at the wall. If the wall is a istance L from the miplane, RL =.5 an R =.5 / L, approimately. Thus the value of R to use can be foun from the size of the plasma L. 5. Defining the problem First we choose a size for the plasma of 1 Debye lengths: L 1 Then.5 R R α R α L We must start the integration a short istance from the origin: We use a istance 1% of the epecte istance L. 1 We put the starting values of, E an u into a vector y. Recall that the starting values are E u R The starting values will go in a vector y with the components, E an u: ystart α α R is y E is y1 u is y ystart is y E, is y1 u is y We will en the integration a little further than the estimate istance L: en ceil( L 16) 6. The erivatives for the Runge-Kutta integrator The meaning of this is: DY( y ) y 1 R e y y y y 1 y /= -E E/ = n i - n e = J/u - ep() u/ = E/u - u/ npoints en 1 the number of gri points. Note that the spacing is one Debye length. i 1 npoints
4 5/11/1 Flui-Poisson System Planar sheath an presheath 4 7. Solution by aaptive Runge-Kutta: M Rkaapt ystart ( ennpointsdy) For convenience, we assign the values in the answer matri M to the variable names use above: i Φ i u i M i M i1 M i3 n ei e Φ i n ii R i u i M () E() u() Below is a plot of the potential profile in imensionless units. The vertical scale is q/t an the horizontal scale is istance in Debye lengths. Potential profile Φ i Φ.618 Our estimate istance L is approimately where = -.5. L Ion velocity u as a function of istance. u i The ions become supersonic, u > 1, at approimately the bounary between the presheath an the sheath. u.916 L The bounary between the presheath an the sheath is somewhat arbitrary.
5 5/11/1 Flui-Poisson System Planar sheath an presheath 5 Electron an ion ensities 1.8 n ii n ei Note that the electron ensity falls more rapily as the wall is approache. The plasma is approimately quasineutral, up to the istance for which = -.5. Ion current to the wall R i L This trivial looking curve remains the same as L is change. In imensionless units, J is approimately.5 at the wall, always. The constancy of the curve shows that the ion current to the wall is nearly an invariant. The value of J =.5 n q c s is often calle the ion saturation current ensity. Try it: Change L from 1 to 1. It may be necessary to alter the efintion of en by a few Debye lengths so that the graph of escens to about -4, a typical value. Notes: 1. It is ifficult to fin plots of this type in the literature.. There are problems with numerical stability if L is mae larger than about 3. Stability is improve by using the Bulirsch-Stoer integrator instea of the Runge-Kutta integrator. Simply substitute: M Bulstoer( ystartennpointsdy)
6 5/11/1 Flui-Poisson System Planar sheath an presheath 6 3. There are iscussions in the literature of solutions mae by patching together solutions in the quasineutral region (mae assuming n i = n e eactly, sometimes calle the "plasma approimation") an solutions mae using Poisson's equation at the bounary. Moern computers can solve the full set of equations an there is no longer any motivation for patche solutions. 4. Eperimentalists usually assign zero potential to the walls, thus the center of the plasma is at a positive potential. Theoretical work is often one with the center of the plasma taken as the zero of the potential scale. The zero point of the potential scale is arbitrary. 5. At what place shoul the integration be ene? The potential in the center of a plasma is typically 3 to 5 T e /q greater than the wall potential. In moels, a value for is chosen so that the rate of loss of electrons an ions is equal. The simplest approimation is to fin where the ion current J = Rs equal to the ranom current of electrons, reuce by the factor ep[q/t e ]. This simple assumption, however, is not likely to be vali because the tail of the electron istribution is not necessarily Mawellian. 6. Can we have the integration en at = -4, for eample? There is no convenient way to have the integrator stop when a variable reaches a particular value. The loop below eecutes Runge-Kutta one step at a time an tests after each step whether or not has reache the specifie value. After each step, the new value for the variables is ae to the en of the answer matri N using the stack comman an the values for an ystart are upate so they can be use in the net iteration. 1 N M Rkaapt( ystart 1 DY) N M ystart submatri( M ) T M 1 while M 4 11 M Rkaapt( ystart 1 DY) ystart submatri( M ) T M 1 N stack( Nsubmatri( M1 1 3 )) N y shoul be a vector of one column, thus we use the matri transpose operation to convert the last row of the answer matri M to a column vector. Reference: Z. Sternovsky, Plasma Sources Sci. Technol. 14, 3-35 (5).
Dusty Plasma Void Dynamics in Unmoving and Moving Flows
7 TH EUROPEAN CONFERENCE FOR AERONAUTICS AND SPACE SCIENCES (EUCASS) Dusty Plasma Voi Dynamics in Unmoving an Moving Flows O.V. Kravchenko*, O.A. Azarova**, an T.A. Lapushkina*** *Scientific an Technological
More informationThermal conductivity of graded composites: Numerical simulations and an effective medium approximation
JOURNAL OF MATERIALS SCIENCE 34 (999)5497 5503 Thermal conuctivity of grae composites: Numerical simulations an an effective meium approximation P. M. HUI Department of Physics, The Chinese University
More informationLectures - Week 10 Introduction to Ordinary Differential Equations (ODES) First Order Linear ODEs
Lectures - Week 10 Introuction to Orinary Differential Equations (ODES) First Orer Linear ODEs When stuying ODEs we are consiering functions of one inepenent variable, e.g., f(x), where x is the inepenent
More informationChapter 9 Method of Weighted Residuals
Chapter 9 Metho of Weighte Resiuals 9- Introuction Metho of Weighte Resiuals (MWR) is an approimate technique for solving bounary value problems. It utilizes a trial functions satisfying the prescribe
More informationTable of Common Derivatives By David Abraham
Prouct an Quotient Rules: Table of Common Derivatives By Davi Abraham [ f ( g( ] = [ f ( ] g( + f ( [ g( ] f ( = g( [ f ( ] g( g( f ( [ g( ] Trigonometric Functions: sin( = cos( cos( = sin( tan( = sec
More information'HVLJQ &RQVLGHUDWLRQ LQ 0DWHULDO 6HOHFWLRQ 'HVLJQ 6HQVLWLYLW\,1752'8&7,21
Large amping in a structural material may be either esirable or unesirable, epening on the engineering application at han. For example, amping is a esirable property to the esigner concerne with limiting
More informationMAE 210A FINAL EXAM SOLUTIONS
1 MAE 21A FINAL EXAM OLUTION PROBLEM 1: Dimensional analysis of the foling of paper (2 points) (a) We wish to simplify the relation between the fol length l f an the other variables: The imensional matrix
More informationUNDERSTANDING INTEGRATION
UNDERSTANDING INTEGRATION Dear Reaer The concept of Integration, mathematically speaking, is the "Inverse" of the concept of result, the integration of, woul give us back the function f(). This, in a way,
More informationMathematical Review Problems
Fall 6 Louis Scuiero Mathematical Review Problems I. Polynomial Equations an Graphs (Barrante--Chap. ). First egree equation an graph y f() x mx b where m is the slope of the line an b is the line's intercept
More informationCS9840 Learning and Computer Vision Prof. Olga Veksler. Lecture 2. Some Concepts from Computer Vision Curse of Dimensionality PCA
CS9840 Learning an Computer Vision Prof. Olga Veksler Lecture Some Concepts from Computer Vision Curse of Dimensionality PCA Some Slies are from Cornelia, Fermüller, Mubarak Shah, Gary Braski, Sebastian
More informationProblem 3.84 of Bergman. Consider one-dimensional conduction in a plane composite wall. The outer surfaces are exposed to a fluid at T
1/10 bergman3-84.xmc Problem 3.84 of Bergman. Consier one-imensional conuction in a plane composite wall. The outer surfaces are expose to a flui at T 5 C an a convection heat transfer coefficient of h1000
More informationSeparation of Variables
Physics 342 Lecture 1 Separation of Variables Lecture 1 Physics 342 Quantum Mechanics I Monay, January 25th, 2010 There are three basic mathematical tools we nee, an then we can begin working on the physical
More informationarxiv: v2 [cond-mat.stat-mech] 11 Nov 2016
Noname manuscript No. (will be inserte by the eitor) Scaling properties of the number of ranom sequential asorption iterations neee to generate saturate ranom packing arxiv:607.06668v2 [con-mat.stat-mech]
More informationIntegration Review. May 11, 2013
Integration Review May 11, 2013 Goals: Review the funamental theorem of calculus. Review u-substitution. Review integration by parts. Do lots of integration eamples. 1 Funamental Theorem of Calculus In
More informationQuantum Mechanics in Three Dimensions
Physics 342 Lecture 20 Quantum Mechanics in Three Dimensions Lecture 20 Physics 342 Quantum Mechanics I Monay, March 24th, 2008 We begin our spherical solutions with the simplest possible case zero potential.
More informationThe Exact Form and General Integrating Factors
7 The Exact Form an General Integrating Factors In the previous chapters, we ve seen how separable an linear ifferential equations can be solve using methos for converting them to forms that can be easily
More informationAPPROXIMATE SOLUTION FOR TRANSIENT HEAT TRANSFER IN STATIC TURBULENT HE II. B. Baudouy. CEA/Saclay, DSM/DAPNIA/STCM Gif-sur-Yvette Cedex, France
APPROXIMAE SOLUION FOR RANSIEN HEA RANSFER IN SAIC URBULEN HE II B. Bauouy CEA/Saclay, DSM/DAPNIA/SCM 91191 Gif-sur-Yvette Ceex, France ABSRAC Analytical solution in one imension of the heat iffusion equation
More informationTAYLOR S POLYNOMIAL APPROXIMATION FOR FUNCTIONS
MISN-0-4 TAYLOR S POLYNOMIAL APPROXIMATION FOR FUNCTIONS f(x ± ) = f(x) ± f ' (x) + f '' (x) 2 ±... 1! 2! = 1.000 ± 0.100 + 0.005 ±... TAYLOR S POLYNOMIAL APPROXIMATION FOR FUNCTIONS by Peter Signell 1.
More information5-4 Electrostatic Boundary Value Problems
11/8/4 Section 54 Electrostatic Bounary Value Problems blank 1/ 5-4 Electrostatic Bounary Value Problems Reaing Assignment: pp. 149-157 Q: A: We must solve ifferential equations, an apply bounary conitions
More informationChapter 2 Governing Equations
Chapter 2 Governing Equations In the present an the subsequent chapters, we shall, either irectly or inirectly, be concerne with the bounary-layer flow of an incompressible viscous flui without any involvement
More informationConstruction of the Electronic Radial Wave Functions and Probability Distributions of Hydrogen-like Systems
Construction of the Electronic Raial Wave Functions an Probability Distributions of Hyrogen-like Systems Thomas S. Kuntzleman, Department of Chemistry Spring Arbor University, Spring Arbor MI 498 tkuntzle@arbor.eu
More informationSpring 2016 Network Science
Spring 206 Network Science Sample Problems for Quiz I Problem [The Application of An one-imensional Poisson Process] Suppose that the number of typographical errors in a new text is Poisson istribute with
More informationLinear First-Order Equations
5 Linear First-Orer Equations Linear first-orer ifferential equations make up another important class of ifferential equations that commonly arise in applications an are relatively easy to solve (in theory)
More informationAnalytical accuracy of the one dimensional heat transfer in geometry with logarithmic various surfaces
Cent. Eur. J. Eng. 4(4) 014 341-351 DOI: 10.478/s13531-013-0176-8 Central European Journal of Engineering Analytical accuracy of the one imensional heat transfer in geometry with logarithmic various surfaces
More informationHomework 7 Due 18 November at 6:00 pm
Homework 7 Due 18 November at 6:00 pm 1. Maxwell s Equations Quasi-statics o a An air core, N turn, cylinrical solenoi of length an raius a, carries a current I Io cos t. a. Using Ampere s Law, etermine
More informationChapter 2: One-dimensional Steady State Conduction
1 Chapter : One-imensional Steay State Conution.1 Eamples of One-imensional Conution Eample.1: Plate with Energy Generation an Variable Conutivity Sine k is variable it must remain insie the ifferentiation
More informationSummary: Differentiation
Techniques of Differentiation. Inverse Trigonometric functions The basic formulas (available in MF5 are: Summary: Differentiation ( sin ( cos The basic formula can be generalize as follows: Note: ( sin
More informationME338A CONTINUUM MECHANICS
global vs local balance equations ME338A CONTINUUM MECHANICS lecture notes 11 tuesay, may 06, 2008 The balance equations of continuum mechanics serve as a basic set of equations require to solve an initial
More informationMath 1271 Solutions for Fall 2005 Final Exam
Math 7 Solutions for Fall 5 Final Eam ) Since the equation + y = e y cannot be rearrange algebraically in orer to write y as an eplicit function of, we must instea ifferentiate this relation implicitly
More informationIntroduction to the Vlasov-Poisson system
Introuction to the Vlasov-Poisson system Simone Calogero 1 The Vlasov equation Consier a particle with mass m > 0. Let x(t) R 3 enote the position of the particle at time t R an v(t) = ẋ(t) = x(t)/t its
More informationSchrödinger s equation.
Physics 342 Lecture 5 Schröinger s Equation Lecture 5 Physics 342 Quantum Mechanics I Wenesay, February 3r, 2010 Toay we iscuss Schröinger s equation an show that it supports the basic interpretation of
More information6. Friction and viscosity in gasses
IR2 6. Friction an viscosity in gasses 6.1 Introuction Similar to fluis, also for laminar flowing gases Newtons s friction law hols true (see experiment IR1). Using Newton s law the viscosity of air uner
More informationθ x = f ( x,t) could be written as
9. Higher orer PDEs as systems of first-orer PDEs. Hyperbolic systems. For PDEs, as for ODEs, we may reuce the orer by efining new epenent variables. For example, in the case of the wave equation, (1)
More information1 dx. where is a large constant, i.e., 1, (7.6) and Px is of the order of unity. Indeed, if px is given by (7.5), the inequality (7.
Lectures Nine an Ten The WKB Approximation The WKB metho is a powerful tool to obtain solutions for many physical problems It is generally applicable to problems of wave propagation in which the frequency
More informationMathcad Lecture #5 In-class Worksheet Plotting and Calculus
Mathca Lecture #5 In-class Worksheet Plotting an Calculus At the en of this lecture, you shoul be able to: graph expressions, functions, an matrices of ata format graphs with titles, legens, log scales,
More informationLecture XII. where Φ is called the potential function. Let us introduce spherical coordinates defined through the relations
Lecture XII Abstract We introuce the Laplace equation in spherical coorinates an apply the metho of separation of variables to solve it. This will generate three linear orinary secon orer ifferential equations:
More informationChapter 6: Energy-Momentum Tensors
49 Chapter 6: Energy-Momentum Tensors This chapter outlines the general theory of energy an momentum conservation in terms of energy-momentum tensors, then applies these ieas to the case of Bohm's moel.
More informationDiagonalization of Matrices Dr. E. Jacobs
Diagonalization of Matrices Dr. E. Jacobs One of the very interesting lessons in this course is how certain algebraic techniques can be use to solve ifferential equations. The purpose of these notes is
More informationThermal Modulation of Rayleigh-Benard Convection
Thermal Moulation of Rayleigh-Benar Convection B. S. Bhaauria Department of Mathematics an Statistics, Jai Narain Vyas University, Johpur, Inia-3400 Reprint requests to Dr. B. S.; E-mail: bsbhaauria@reiffmail.com
More informationG4003 Advanced Mechanics 1. We already saw that if q is a cyclic variable, the associated conjugate momentum is conserved, L = const.
G4003 Avance Mechanics 1 The Noether theorem We alreay saw that if q is a cyclic variable, the associate conjugate momentum is conserve, q = 0 p q = const. (1) This is the simplest incarnation of Noether
More informationAverage value of position for the anharmonic oscillator: Classical versus quantum results
verage value of position for the anharmonic oscillator: Classical versus quantum results R. W. Robinett Department of Physics, The Pennsylvania State University, University Park, Pennsylvania 682 Receive
More informationTMA 4195 Matematisk modellering Exam Tuesday December 16, :00 13:00 Problems and solution with additional comments
Problem F U L W D g m 3 2 s 2 0 0 0 0 2 kg 0 0 0 0 0 0 Table : Dimension matrix TMA 495 Matematisk moellering Exam Tuesay December 6, 2008 09:00 3:00 Problems an solution with aitional comments The necessary
More informationThe Three-dimensional Schödinger Equation
The Three-imensional Schöinger Equation R. L. Herman November 7, 016 Schröinger Equation in Spherical Coorinates We seek to solve the Schröinger equation with spherical symmetry using the metho of separation
More informationTHE ACCURATE ELEMENT METHOD: A NEW PARADIGM FOR NUMERICAL SOLUTION OF ORDINARY DIFFERENTIAL EQUATIONS
THE PUBISHING HOUSE PROCEEDINGS O THE ROMANIAN ACADEMY, Series A, O THE ROMANIAN ACADEMY Volume, Number /, pp. 6 THE ACCURATE EEMENT METHOD: A NEW PARADIGM OR NUMERICA SOUTION O ORDINARY DIERENTIA EQUATIONS
More informationApplication of the homotopy perturbation method to a magneto-elastico-viscous fluid along a semi-infinite plate
Freun Publishing House Lt., International Journal of Nonlinear Sciences & Numerical Simulation, (9), -, 9 Application of the homotopy perturbation metho to a magneto-elastico-viscous flui along a semi-infinite
More informationMath Notes on differentials, the Chain Rule, gradients, directional derivative, and normal vectors
Math 18.02 Notes on ifferentials, the Chain Rule, graients, irectional erivative, an normal vectors Tangent plane an linear approximation We efine the partial erivatives of f( xy, ) as follows: f f( x+
More informationAssignment #3: Mathematical Induction
Math 3AH, Fall 011 Section 10045 Assignment #3: Mathematical Inuction Directions: This assignment is ue no later than Monay, November 8, 011, at the beginning of class. Late assignments will not be grae.
More informationOptimization of Geometries by Energy Minimization
Optimization of Geometries by Energy Minimization by Tracy P. Hamilton Department of Chemistry University of Alabama at Birmingham Birmingham, AL 3594-140 hamilton@uab.eu Copyright Tracy P. Hamilton, 1997.
More informationA note on asymptotic formulae for one-dimensional network flow problems Carlos F. Daganzo and Karen R. Smilowitz
A note on asymptotic formulae for one-imensional network flow problems Carlos F. Daganzo an Karen R. Smilowitz (to appear in Annals of Operations Research) Abstract This note evelops asymptotic formulae
More informationAnalytic Scaling Formulas for Crossed Laser Acceleration in Vacuum
October 6, 4 ARDB Note Analytic Scaling Formulas for Crosse Laser Acceleration in Vacuum Robert J. Noble Stanfor Linear Accelerator Center, Stanfor University 575 San Hill Roa, Menlo Park, California 945
More informationd dx But have you ever seen a derivation of these results? We ll prove the first result below. cos h 1
Lecture 5 Some ifferentiation rules Trigonometric functions (Relevant section from Stewart, Seventh Eition: Section 3.3) You all know that sin = cos cos = sin. () But have you ever seen a erivation of
More informationSection 2.7 Derivatives of powers of functions
Section 2.7 Derivatives of powers of functions (3/19/08) Overview: In this section we iscuss the Chain Rule formula for the erivatives of composite functions that are forme by taking powers of other functions.
More informationTwo Dimensional Numerical Simulator for Modeling NDC Region in SNDC Devices
Journal of Physics: Conference Series PAPER OPEN ACCESS Two Dimensional Numerical Simulator for Moeling NDC Region in SNDC Devices To cite this article: Dheeraj Kumar Sinha et al 2016 J. Phys.: Conf. Ser.
More informationLINEAR DIFFERENTIAL EQUATIONS OF ORDER 1. where a(x) and b(x) are functions. Observe that this class of equations includes equations of the form
LINEAR DIFFERENTIAL EQUATIONS OF ORDER 1 We consier ifferential equations of the form y + a()y = b(), (1) y( 0 ) = y 0, where a() an b() are functions. Observe that this class of equations inclues equations
More informationII. First variation of functionals
II. First variation of functionals The erivative of a function being zero is a necessary conition for the etremum of that function in orinary calculus. Let us now tackle the question of the equivalent
More informationA Novel Decoupled Iterative Method for Deep-Submicron MOSFET RF Circuit Simulation
A Novel ecouple Iterative Metho for eep-submicron MOSFET RF Circuit Simulation CHUAN-SHENG WANG an YIMING LI epartment of Mathematics, National Tsing Hua University, National Nano evice Laboratories, an
More informationSection 7.1: Integration by Parts
Section 7.1: Integration by Parts 1. Introuction to Integration Techniques Unlike ifferentiation where there are a large number of rules which allow you (in principle) to ifferentiate any function, the
More informationLecture contents. Metal-semiconductor contact
1 Lecture contents Metal-semiconuctor contact Electrostatics: Full epletion approimation Electrostatics: Eact electrostatic solution Current Methos for barrier measurement Junctions: general approaches,
More informationON THE OPTIMALITY SYSTEM FOR A 1 D EULER FLOW PROBLEM
ON THE OPTIMALITY SYSTEM FOR A D EULER FLOW PROBLEM Eugene M. Cliff Matthias Heinkenschloss y Ajit R. Shenoy z Interisciplinary Center for Applie Mathematics Virginia Tech Blacksburg, Virginia 46 Abstract
More informationIntroduction to variational calculus: Lecture notes 1
October 10, 2006 Introuction to variational calculus: Lecture notes 1 Ewin Langmann Mathematical Physics, KTH Physics, AlbaNova, SE-106 91 Stockholm, Sween Abstract I give an informal summary of variational
More informationSparse Reconstruction of Systems of Ordinary Differential Equations
Sparse Reconstruction of Systems of Orinary Differential Equations Manuel Mai a, Mark D. Shattuck b,c, Corey S. O Hern c,a,,e, a Department of Physics, Yale University, New Haven, Connecticut 06520, USA
More informationA General Analytical Approximation to Impulse Response of 3-D Microfluidic Channels in Molecular Communication
A General Analytical Approximation to Impulse Response of 3- Microfluiic Channels in Molecular Communication Fatih inç, Stuent Member, IEEE, Bayram Cevet Akeniz, Stuent Member, IEEE, Ali Emre Pusane, Member,
More informationUsing Quasi-Newton Methods to Find Optimal Solutions to Problematic Kriging Systems
Usin Quasi-Newton Methos to Fin Optimal Solutions to Problematic Kriin Systems Steven Lyster Centre for Computational Geostatistics Department of Civil & Environmental Enineerin University of Alberta Solvin
More informationChapter 4. Electrostatics of Macroscopic Media
Chapter 4. Electrostatics of Macroscopic Meia 4.1 Multipole Expansion Approximate potentials at large istances 3 x' x' (x') x x' x x Fig 4.1 We consier the potential in the far-fiel region (see Fig. 4.1
More informationarxiv:hep-th/ v1 3 Feb 1993
NBI-HE-9-89 PAR LPTHE 9-49 FTUAM 9-44 November 99 Matrix moel calculations beyon the spherical limit arxiv:hep-th/93004v 3 Feb 993 J. Ambjørn The Niels Bohr Institute Blegamsvej 7, DK-00 Copenhagen Ø,
More informationThe Principle of Least Action
Chapter 7. The Principle of Least Action 7.1 Force Methos vs. Energy Methos We have so far stuie two istinct ways of analyzing physics problems: force methos, basically consisting of the application of
More informationSIMULATION OF POROUS MEDIUM COMBUSTION IN ENGINES
SIMULATION OF POROUS MEDIUM COMBUSTION IN ENGINES Jan Macek, Miloš Polášek Czech Technical University in Prague, Josef Božek Research Center Introuction Improvement of emissions from reciprocating internal
More informationA SIMPLE ENGINEERING MODEL FOR SPRINKLER SPRAY INTERACTION WITH FIRE PRODUCTS
International Journal on Engineering Performance-Base Fire Coes, Volume 4, Number 3, p.95-3, A SIMPLE ENGINEERING MOEL FOR SPRINKLER SPRAY INTERACTION WITH FIRE PROCTS V. Novozhilov School of Mechanical
More informationCalculus Class Notes for the Combined Calculus and Physics Course Semester I
Calculus Class Notes for the Combine Calculus an Physics Course Semester I Kelly Black December 14, 2001 Support provie by the National Science Founation - NSF-DUE-9752485 1 Section 0 2 Contents 1 Average
More information23 Implicit differentiation
23 Implicit ifferentiation 23.1 Statement The equation y = x 2 + 3x + 1 expresses a relationship between the quantities x an y. If a value of x is given, then a corresponing value of y is etermine. For
More informationSYNCHRONOUS SEQUENTIAL CIRCUITS
CHAPTER SYNCHRONOUS SEUENTIAL CIRCUITS Registers an counters, two very common synchronous sequential circuits, are introuce in this chapter. Register is a igital circuit for storing information. Contents
More informationSpectral Flow, the Magnus Force, and the. Josephson-Anderson Relation
Spectral Flow, the Magnus Force, an the arxiv:con-mat/9602094v1 16 Feb 1996 Josephson-Anerson Relation P. Ao Department of Theoretical Physics Umeå University, S-901 87, Umeå, SWEDEN October 18, 2018 Abstract
More information3.7 Implicit Differentiation -- A Brief Introduction -- Student Notes
Fin these erivatives of these functions: y.7 Implicit Differentiation -- A Brief Introuction -- Stuent Notes tan y sin tan = sin y e = e = Write the inverses of these functions: y tan y sin How woul we
More informationinflow outflow Part I. Regular tasks for MAE598/494 Task 1
MAE 494/598, Fall 2016 Project #1 (Regular tasks = 20 points) Har copy of report is ue at the start of class on the ue ate. The rules on collaboration will be release separately. Please always follow the
More informationwater adding dye partial mixing homogenization time
iffusion iffusion is a process of mass transport that involves the movement of one atomic species into another. It occurs by ranom atomic jumps from one position to another an takes place in the gaseous,
More informationHeat Transfer from Arbitrary Nonisothermal Surfaces in a Laminar Flow
3 Heat Transfer from Arbitrary Nonisothermal Surfaces in a Laminar Flo In essence, the conjugate heat transfer problem consiers the thermal interaction beteen a boy an a flui floing over or insie it. As
More informationarxiv:physics/ v2 [physics.ed-ph] 23 Sep 2003
Mass reistribution in variable mass systems Célia A. e Sousa an Vítor H. Rorigues Departamento e Física a Universiae e Coimbra, P-3004-516 Coimbra, Portugal arxiv:physics/0211075v2 [physics.e-ph] 23 Sep
More informationShape functions in 1D
MAE 44 & CIV 44 Introuction to Finite Elements Reaing assignment: ecture notes, ogan.,. Summary: Prof. Suvranu De Shape functions in D inear shape functions in D Quaratic an higher orer shape functions
More informationx = c of N if the limit of f (x) = L and the right-handed limit lim f ( x)
Limit We say the limit of f () as approaches c equals L an write, lim L. One-Sie Limits (Left an Right-Hane Limits) Suppose a function f is efine near but not necessarily at We say that f has a left-hane
More informationEfficient Macro-Micro Scale Coupled Modeling of Batteries
A00 Journal of The Electrochemical Society, 15 10 A00-A008 005 0013-651/005/1510/A00/7/$7.00 The Electrochemical Society, Inc. Efficient Macro-Micro Scale Couple Moeling of Batteries Venkat. Subramanian,*,z
More informationControl of a PEM Fuel Cell Based on a Distributed Model
21 American Control Conference Marriott Waterfront, Baltimore, MD, USA June 3-July 2, 21 FrC1.6 Control of a PEM Fuel Cell Base on a Distribute Moel Michael Mangol Abstract To perform loa changes in proton
More information19 Eigenvalues, Eigenvectors, Ordinary Differential Equations, and Control
19 Eigenvalues, Eigenvectors, Orinary Differential Equations, an Control This section introuces eigenvalues an eigenvectors of a matrix, an iscusses the role of the eigenvalues in etermining the behavior
More informationNonlinear Dielectric Response of Periodic Composite Materials
onlinear Dielectric Response of Perioic Composite aterials A.G. KOLPAKOV 3, Bl.95, 9 th ovember str., ovosibirsk, 639 Russia the corresponing author e-mail: agk@neic.nsk.su, algk@ngs.ru A. K.TAGATSEV Ceramics
More informationUnit #6 - Families of Functions, Taylor Polynomials, l Hopital s Rule
Unit # - Families of Functions, Taylor Polynomials, l Hopital s Rule Some problems an solutions selecte or aapte from Hughes-Hallett Calculus. Critical Points. Consier the function f) = 54 +. b) a) Fin
More informationA Sketch of Menshikov s Theorem
A Sketch of Menshikov s Theorem Thomas Bao March 14, 2010 Abstract Let Λ be an infinite, locally finite oriente multi-graph with C Λ finite an strongly connecte, an let p
More informationMathematics 116 HWK 25a Solutions 8.6 p610
Mathematics 6 HWK 5a Solutions 8.6 p6 Problem, 8.6, p6 Fin a power series representation for the function f() = etermine the interval of convergence. an Solution. Begin with the geometric series = + +
More information12.11 Laplace s Equation in Cylindrical and
SEC. 2. Laplace s Equation in Cylinrical an Spherical Coorinates. Potential 593 2. Laplace s Equation in Cylinrical an Spherical Coorinates. Potential One of the most important PDEs in physics an engineering
More informationOne Dimensional Convection: Interpolation Models for CFD
One Dimensional Convection: Interpolation Moels for CFD ME 448/548 Notes Geral Recktenwal Portlan State University Department of Mechanical Engineering gerry@p.eu ME 448/548: D Convection-Di usion Equation
More informationBLOW-UP FORMULAS FOR ( 2)-SPHERES
BLOW-UP FORMULAS FOR 2)-SPHERES ROGIER BRUSSEE In this note we give a universal formula for the evaluation of the Donalson polynomials on 2)-spheres, i.e. smooth spheres of selfintersection 2. Note that
More informationA hyperbolic equation for turbulent diffusion
Nonlinearity 13 (2) 1855 1866. Printe in the UK PII: S951-7715()1271-X A hyperbolic equation for turbulent iffusion Sanip Ghosal an Joseph B Keller Center for Turbulence Research, Stanfor University, Stanfor,
More informationFluid Mechanics EBS 189a. Winter quarter, 4 units, CRN Lecture TWRF 12:10-1:00, Chemistry 166; Office hours TH 2-3, WF 4-5; 221 Veihmeyer Hall.
Flui Mechanics EBS 189a. Winter quarter, 4 units, CRN 52984. Lecture TWRF 12:10-1:00, Chemistry 166; Office hours TH 2-3, WF 4-5; 221 eihmeyer Hall. Course Description: xioms of flui mechanics, flui statics,
More informationMATH2231-Differentiation (2)
-Differentiation () The Beginnings of Calculus The prime occasion from which arose my iscovery of the metho of the Characteristic Triangle, an other things of the same sort, happene at a time when I ha
More informationCalculus of Variations
Calculus of Variations Lagrangian formalism is the main tool of theoretical classical mechanics. Calculus of Variations is a part of Mathematics which Lagrangian formalism is base on. In this section,
More informationSECTION 3.2 THE PRODUCT AND QUOTIENT RULES 1 8 3
SECTION 3.2 THE PRODUCT AND QUOTIENT RULES 8 3 L P f Q L segments L an L 2 to be tangent to the parabola at the transition points P an Q. (See the figure.) To simplify the equations you ecie to place the
More informationAn inductance lookup table application for analysis of reluctance stepper motor model
ARCHIVES OF ELECTRICAL ENGINEERING VOL. 60(), pp. 5- (0) DOI 0.478/ v07-0-000-y An inuctance lookup table application for analysis of reluctance stepper motor moel JAKUB BERNAT, JAKUB KOŁOTA, SŁAWOMIR
More informationensembles When working with density operators, we can use this connection to define a generalized Bloch vector: v x Tr x, v y Tr y
Ph195a lecture notes, 1/3/01 Density operators for spin- 1 ensembles So far in our iscussion of spin- 1 systems, we have restricte our attention to the case of pure states an Hamiltonian evolution. Toay
More informationLagrangian and Hamiltonian Mechanics
Lagrangian an Hamiltonian Mechanics.G. Simpson, Ph.. epartment of Physical Sciences an Engineering Prince George s Community College ecember 5, 007 Introuction In this course we have been stuying classical
More informationProblem 1 (20 points)
ME 309 Fall 01 Exam 1 Name: C Problem 1 0 points Short answer questions. Each question is worth 5 points. Don t spen too long writing lengthy answers to these questions. Don t use more space than is given.
More informationTHE VAN KAMPEN EXPANSION FOR LINKED DUFFING LINEAR OSCILLATORS EXCITED BY COLORED NOISE
Journal of Soun an Vibration (1996) 191(3), 397 414 THE VAN KAMPEN EXPANSION FOR LINKED DUFFING LINEAR OSCILLATORS EXCITED BY COLORED NOISE E. M. WEINSTEIN Galaxy Scientific Corporation, 2500 English Creek
More informationMath 1B, lecture 8: Integration by parts
Math B, lecture 8: Integration by parts Nathan Pflueger 23 September 2 Introuction Integration by parts, similarly to integration by substitution, reverses a well-known technique of ifferentiation an explores
More information