DOCSLIB.ORG

Rheology.Pdf

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Rheology.Pdf RHEOLOGY Edited by Juan De Vicente Rheology Edited by Juan De Vicente Published by InTech Janeza Trdine 9, 51000 Rijeka, Croatia Copyright © 2012 InTech All chapters are Open Access distributed under the Creative Commons Attribution 3.0 license, which allows users to download, copy and build upon published articles even for commercial purposes, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. After this work has been published by InTech, authors have the right to republish it, in whole or part, in any publication of which they are the author, and to make other personal use of the work. Any republication, referencing or personal use of the work must explicitly identify the original source. As for readers, this license allows users to download, copy and build upon published chapters even for commercial purposes, as long as the author and publisher are properly credited, which ensures maximum dissemination and a wider impact of our publications. Notice Statements and opinions expressed in the chapters are these of the individual contributors and not necessarily those of the editors or publisher. No responsibility is accepted for the accuracy of information contained in the published chapters. The publisher assumes no responsibility for any damage or injury to persons or property arising out of the use of any materials, instructions, methods or ideas contained in the book. Publishing Process Manager Daria Nahtigal Technical Editor Teodora Smiljanic Cover Designer InTech Design Team First published February, 2012 Printed in Croatia A free online edition of this book is available at www.intechopen.com Additional hard copies can be obtained from [email protected] Rheology, Edited by Juan De Vicente p. cm. ISBN 978-953-51-0187-1 Contents Preface IX Part 1 Polymers 1 Chapter 1 Polymer Rheology by Dielectric Spectroscopy 3 Clement Riedel, Angel Alegria, Juan Colmenero and Phillipe Tordjeman Chapter 2 Viscoelastic Properties for Sol-Gel Transition 29 Yutaka Tanaka Chapter 3 Polymer Gel Rheology and Adhesion 59 Anne M. Grillet, Nicholas B. Wyatt and Lindsey M. Gloe Chapter 4 Fracture Behaviour of Controlled-Rheology Polypropylenes 81 Alicia Salazar and Jesús Rodríguez Chapter 5 Poisson’s Ratio and Mechanical Nonlinearity Under Tensile Deformation in Crystalline Polymers 113 Koh-hei Nitta and Masahiro Yamana Chapter 6 Batch Foaming of Amorphous Poly (DL-Lactic Acid) and Poly (Lactic Acid-co-Glycolic Acid) with Supercritical Carbon Dioxide: CO2 Solubility, Intermolecular Interaction, Rheology and Morphology 133 Hongyun Tai Part 2 Pipe Flow and Porous Media 149 Chapter 7 Resistance Coefficients for Non-Newtonian Flows in Pipe Fittings 151 Veruscha Fester, Paul Slatter and Neil Alderman VI Contents Chapter 8 Measurement and Impact Factors of Polymer Rheology in Porous Media 187 Yongpeng Sun, Laila Saleh and Baojun Bai Part 3 Drilling Fluids 203 Chapter 9 Stability and Flow Behavior of Fiber-Containing Drilling Sweeps 205 Matthew George, Ramadan Ahmed and Fred Growcock Part 4 Food Rheology 239 Chapter 10 Solution Properties of κ-Carrageenan and Its Interaction with Other Polysaccharides in Aqueous Media 241 Alberto Tecante and María del Carmen Núñez Santiago Chapter 11 Influence of Acidification on Dough Rheological Properties 265 Daliborka Koceva Komlenić, Vedran Slačanac and Marko Jukić Part 5 Other Materials and Applications 293 Chapter 12 Influence of Electric Fields and Boundary Conditions on the Flow Properties of Nematic-Filled Cells and Capillaries 295 Carlos I. Mendoza, Adalberto Corella-Madueño and J. Adrián Reyes Chapter 13 Rheological Behaviors and Their Correlation with Printing Performance of Silver Paste for LTCC Tape 321 Rosidah Alias and Sabrina Mohd Shapee Preface Rheology involves the study of the deformation and flow of matter. The goal is to establish relationships between stress and deformation for (non-Newtonian) materials where neither Newton's law nor Hooke's law suffice to explain their mechanical behaviour. Many materials exhibit a non-Newtonian behaviour and the area is relevant in many fields of study from industrial to technological applications such as concrete technology, geology, polymers and composites, plastics processing, paint flow, hemorheology, cosmetics, adhesives, etc ... In this Intech E-book, 13 chapters on various rheology related aspects are compiled. A number of current research projects are outlined as the book is intended to give the readers a wide picture of current rheology research balancing between fundamentals and applied knowledge. For this purpose, the chapters are written by experts from the Industry and Academia. The first part of the book is dedicated to polymers and deals with various advanced researches in this field. The first chapter, by Riedel et al. is devoted to the use of broadband dielectric spectroscopy to test macroscopic models for polymer dynamics. Chapters 2 and 3, by Tanaka and Grillet et al., revisit the fundamentals of gel formation and discuss the methods of characterizing polymer gels using rheological techniques to probe their adhesion and mechanical response. Chapter 4, by Salazar and Rodríguez focuses on the fracture behavior of controlled-rheology polypropylenes. Chapter 5, by Nitta and Yamana attempts to clarify the relationship between tensile mechanical nonlinearity and geometrical nonlinearity resulting from the transient change in Poisson's ratio in crystalline polymers. This first section finishes with Chapter 6, by Tai, and analyzes the rheological properties of poly(lactic acid) and poly(lactic acid-co-glycolic acid)/CO2 mixtures. The second part of the book covers two important flow fields of outstanding interest in petroleum industry. On the one hand, chapter 7, by Fester et al. deals explicitly with pipe flows. In particular, they review the loss coefficient data for laminar flow on non- Newtonian fluids in pipe fittings. Chapter 8, by Sun et al. focuses on enhanced oil recovery methods and explores the flow behavior of polymer solutions through a porous media. X Preface The third part of the book is devoted to drilling fluids. Chapter 9, by George et al. concerns the rheology of fiber-containing fluids specific to the oil and gas industry. They focus on drilling sweeps that are used to improve borehole cleaning. The fourth part of the book covers some relevant aspects of food rheology. The gelation mechanism of -Carrageenan and its interaction with starch and non-starch polysaccharides is reviewed in chapter 10 by Tecante and Nuñez-Santiago. Later, in chapter 11, by Komlenic et al., it is demonstrated that rheological properties of dough strongly depend on acidification. The last part of the book deals with liquids crystals and ceramic microelectronic devices. Chapter 12, by Mendoza et al. reviews recent theoretical results on the rheology of systems consisting of a flow-aligning nematic contained in cells and capillaries under a variety of different flow conditions and under the action of applied electric fields. Finally, chapter 13, by Alias and Shapee, stresses the impact of silver paste rheology in the fabrication of ceramic microelectronic devices (low temperature co-fired ceramic LTCC devices). The format of an E-book is chosen to enable fast dissemination of new research, and to give easy access to readers. The chapters can be read individually. I would like to express my gratitude to all the contributing authors that have made this book become a reality. I wish to thank also Intech staff and their team members for the opportunity to publish this work, in particular, Ana Pantar, Tajana Jevtic and Daria Nahtigal for their support which has made my job as editor an easy and satisfying one. Finally, I gratefully acknowledge financial support by the Ministerio de Ciencia e Innovación (MICINN MAT 2010-15101 project, Spain), by the European Regional Development Fund (ERDF), and by the projects P10-RNM-6630 and P11-FQM-7074 from Junta de Andalucía (Spain). Juan de Vicente University of Granada Spain Part 1 Polymers 10 Polymer Rheology by Dielectric Spectroscopy Clement Riedel1, Angel Alegria1, Juan Colmenero1 and Phillipe Tordjeman2 1Universidad del Pais Vasco / Euskal Herriko Unibertsitatea 2Université de Toulouse, INPT 1Spain 2France 1. Introduction The aim of this chapter is to discuss the main models describing the polymer dynamics at macroscopic scale and present some of the experimental techniques that permit to test these models. We will notably show how Broadband Dielectric Spectroscopy (BDS) permits to obtain rheological information about polymers, i.e. permits to understand how the matter flows and moves. We will focus our study the whole chain motion of cis-polyisoprene 1,4 (PI). Due to dipolar components both parallel and perpendicular to the chain backbone, PI exhibits a whole chain dielectric relaxation (normal mode) in addition to that associated with segmental motion. The Rouse model (Rouse, 1953), developed in the fifties, is well known to describe rather correctly the whole chain dynamics of unentangled polymers. In the past, the validity of the Rouse model has ever been instigated by means of different experimental techniques (BDS, rheology, neutron scattering) and also by molecular dynamics simulations. However this study is still challenging because in unentangled polymer the segmental dynamics contributions overlap significantly with the whole chain dynamics. In this chapter, we will demonstrate how we have been able to decorelate the effect of the α-relaxation on the normal mode in order to test how the Rouse model can quantitatively describes the normal mode measured by BDS and rheology. The introduction of polydispersity is a key point of this study. The reptational tube theory, first introduced by de Gennes de Gennes (1979) and developed by Doi and Edwads Doi & Edwards (1988) describes the dynamics of entangled polymers where the Rouse model can not be applied. Many corrections (as Contour Length Fluctuation or Constraints Release) have been added to the pure reptation in order to reach a totally predictive theory McLeish (2002); Viovy et al. (2002).
Load more

Recommended publications
  • 10. Collisions • Use Conservation of Momentum and Energy and The
    10. Collisions • Use conservation of momentum and energy and the center of mass to understand collisions between two objects. • During a collision, two or more objects exert a force on one another for a short time: -F(t) F(t) Before During After • It is not necessary for the objects to touch during a collision, e.g. an asteroid flied by the earth is considered a collision because its path is changed due to the gravitational attraction of the earth. One can still use conservation of momentum and energy to analyze the collision. Impulse: During a collision, the objects exert a force on one another. This force may be complicated and change with time. However, from Newton's 3rd Law, the two objects must exert an equal and opposite force on one another. F(t) t ti tf Dt From Newton'sr 2nd Law: dp r = F (t) dt r r dp = F (t)dt r r r r tf p f - pi = Dp = ò F (t)dt ti The change in the momentum is defined as the impulse of the collision. • Impulse is a vector quantity. Impulse-Linear Momentum Theorem: In a collision, the impulse on an object is equal to the change in momentum: r r J = Dp Conservation of Linear Momentum: In a system of two or more particles that are colliding, the forces that these objects exert on one another are internal forces. These internal forces cannot change the momentum of the system. Only an external force can change the momentum. The linear momentum of a closed isolated system is conserved during a collision of objects within the system.
    [Show full text]
  • The Impact of Food Viscosity on Eating Rate, Subjective Appetite, Glycemic Response and Gastric Emptying Rate
    The Impact of Food Viscosity on Eating Rate, Subjective Appetite, Glycemic Response and Gastric Emptying Rate Yong Zhu1, Walter H. Hsu2, James H. Hollis1* 1 Department of Food Science and Human Nutrition, Iowa State University, Ames, Iowa, United States of America, 2 Department of Biomedical Sciences, Iowa State University, Ames, Iowa, United States of America Abstract Understanding the impact of rheological properties of food on postprandial appetite and glycemic response helps to design novel functional products. It has been shown that solid foods have a stronger satiating effect than their liquid equivalent. However, whether a subtle change in viscosity of a semi-solid food would have a similar effect on appetite is unknown. Fifteen healthy males participated in the randomized cross-over study. Each participant consumed a 1690 kJ portion of a standard viscosity (SV) and a high viscosity (HV) semi-solid meal with 1000 mg acetaminophen in two separate sessions. At regular intervals during the three hours following the meal, subjective appetite ratings were measured and blood samples collected. The plasma samples were assayed for insulin, glucose-dependent insulinotropic peptide (GIP), glucose and acetaminophen. After three hours, the participants were provided with an ad libitum pasta meal. Compared with the SV meal, HV was consumed at a slower eating rate (P = 0.020), with postprandial hunger and desire to eat being lower (P = 0.019 and P,0.001 respectively) while fullness was higher (P,0.001). In addition, consuming the HV resulted in lower plasma concentration of GIP (P,0.001), higher plasma concentration of glucose (P,0.001) and delayed gastric emptying as revealed by the acetaminophen absorption test (P,0.001).
    [Show full text]
  • On Nonlinear Strain Theory for a Viscoelastic Material Model and Its Implications for Calving of Ice Shelves
    Journal of Glaciology (2019), 65(250) 212–224 doi: 10.1017/jog.2018.107 © The Author(s) 2019. This is an Open Access article, distributed under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives licence (http://creativecommons.org/licenses/by-nc-nd/4.0/), which permits non-commercial re-use, distribution, and reproduction in any medium, provided the original work is unaltered and is properly cited. The written permission of Cambridge University Press must be obtained for commercial re- use or in order to create a derivative work. On nonlinear strain theory for a viscoelastic material model and its implications for calving of ice shelves JULIA CHRISTMANN,1,2 RALF MÜLLER,2 ANGELIKA HUMBERT1,3 1Division of Geosciences/Glaciology, Alfred Wegener Institute Helmholtz Centre for Polar and Marine Research, Bremerhaven, Germany 2Institute of Applied Mechanics, University of Kaiserslautern, Kaiserslautern, Germany 3Division of Geosciences, University of Bremen, Bremen, Germany Correspondence: Julia Christmann <[email protected]> ABSTRACT. In the current ice-sheet models calving of ice shelves is based on phenomenological approaches. To obtain physics-based calving criteria, a viscoelastic Maxwell model is required account- ing for short-term elastic and long-term viscous deformation. On timescales of months to years between calving events, as well as on long timescales with several subsequent iceberg break-offs, deformations are no longer small and linearized strain measures cannot be used. We present a finite deformation framework of viscoelasticity and extend this model by a nonlinear Glen-type viscosity. A finite element implementation is used to compute stress and strain states in the vicinity of the ice-shelf calving front.
    [Show full text]
  • 1 Fluid Flow Outline Fundamentals of Rheology
    Fluid Flow Outline • Fundamentals and applications of rheology • Shear stress and shear rate • Viscosity and types of viscometers • Rheological classification of fluids • Apparent viscosity • Effect of temperature on viscosity • Reynolds number and types of flow • Flow in a pipe • Volumetric and mass flow rate • Friction factor (in straight pipe), friction coefficient (for fittings, expansion, contraction), pressure drop, energy loss • Pumping requirements (overcoming friction, potential energy, kinetic energy, pressure energy differences) 2 Fundamentals of Rheology • Rheology is the science of deformation and flow – The forces involved could be tensile, compressive, shear or bulk (uniform external pressure) • Food rheology is the material science of food – This can involve fluid or semi-solid foods • A rheometer is used to determine rheological properties (how a material flows under different conditions) – Viscometers are a sub-set of rheometers 3 1 Applications of Rheology • Process engineering calculations – Pumping requirements, extrusion, mixing, heat transfer, homogenization, spray coating • Determination of ingredient functionality – Consistency, stickiness etc. • Quality control of ingredients or final product – By measurement of viscosity, compressive strength etc. • Determination of shelf life – By determining changes in texture • Correlations to sensory tests – Mouthfeel 4 Stress and Strain • Stress: Force per unit area (Units: N/m2 or Pa) • Strain: (Change in dimension)/(Original dimension) (Units: None) • Strain rate: Rate
    [Show full text]
  • Law of Conversation of Energy
    Law of Conservation of Mass: "In any kind of physical or chemical process, mass is neither created nor destroyed - the mass before the process equals the mass after the process." - the total mass of the system does not change, the total mass of the products of a chemical reaction is always the same as the total mass of the original materials. "Physics for scientists and engineers," 4th edition, Vol.1, Raymond A. Serway, Saunders College Publishing, 1996. Ex. 1) When wood burns, mass seems to disappear because some of the products of reaction are gases; if the mass of the original wood is added to the mass of the oxygen that combined with it and if the mass of the resulting ash is added to the mass o the gaseous products, the two sums will turn out exactly equal. 2) Iron increases in weight on rusting because it combines with gases from the air, and the increase in weight is exactly equal to the weight of gas consumed. Out of thousands of reactions that have been tested with accurate chemical balances, no deviation from the law has ever been found. Law of Conversation of Energy: The total energy of a closed system is constant. Matter is neither created nor destroyed – total mass of reactants equals total mass of products You can calculate the change of temp by simply understanding that energy and the mass is conserved - it means that we added the two heat quantities together we can calculate the change of temperature by using the law or measure change of temp and show the conservation of energy E1 + E2 = E3 -> E(universe) = E(System) + E(Surroundings) M1 + M2 = M3 Is T1 + T2 = unknown (No, no law of conservation of temperature, so we have to use the concept of conservation of energy) Total amount of thermal energy in beaker of water in absolute terms as opposed to differential terms (reference point is 0 degrees Kelvin) Knowns: M1, M2, T1, T2 (Kelvin) When add the two together, want to know what T3 and M3 are going to be.
    [Show full text]
  • Computational Rheology (4K430)
    Computational Rheology (4K430) dr.ir. M.A. Hulsen [email protected] Website: http://www.mate.tue.nl/~hulsen under link ‘Computational Rheology’. – Section Polymer Technology (PT) / Materials Technology (MaTe) – Introduction Computational Rheology important for: B Polymer processing B Rheology & Material science B Turbulent flow (drag reduction phenomena) B Food processing B Biological flows B ... Introduction (Polymer Processing) Analysis of viscoelastic phenomena essential for predicting B Flow induced crystallization kinetics B Flow instabilities during processing B Free surface flows (e.g.extrudate swell) B Secondary flows B Dimensional stability of injection moulded products B Prediction of mechanical and optical properties Introduction (Surface Defects on Injection Molded Parts) Alternating dull bands perpendicular to flow direction with high surface roughness (M. Bulters & A. Schepens, DSM-Research). Introduction (Flow Marks, Two Color Polypropylene) Flow Mark Side view Top view Bottom view M. Bulters & A. Schepens, DSM-Research Introduction (Simulation flow front) 1 0.5 Steady Perturbed H 2y 0 ___ −0.5 −1 0 0.5 1 ___2x H Introduction (Rheology & Material Science) Simulation essential for understanding and predicting material properties: B Polymer blends (morphology, viscosity, normal stresses) B Particle filled viscoelastic fluids (suspensions) B Polymer architecture macroscopic properties (Brownian dynamics (BD), molecular dynamics (MD),⇒ Monte Carlo, . ) Multi-scale. ⇒ Introduction (Solid particles in a viscoelastic fluid) B Microstructure (polymer, particles) B Bulk rheology B Flow induced crystallization Introduction (Multiple particles in a viscoelastic fluid) Introduction (Flow induced crystallization) Introduction (Multi-phase flows) Goal and contents of the course Goal: Introduction of the basic numerical techniques used in Computational Rheology using the Finite Element Method (FEM).
    [Show full text]
  • 20. Rheology & Linear Elasticity
    20. Rheology & Linear Elasticity I Main Topics A Rheology: Macroscopic deformation behavior B Linear elasticity for homogeneous isotropic materials 10/29/18 GG303 1 20. Rheology & Linear Elasticity Viscous (fluid) Behavior http://manoa.hawaii.edu/graduate/content/slide-lava 10/29/18 GG303 2 20. Rheology & Linear Elasticity Ductile (plastic) Behavior http://www.hilo.hawaii.edu/~csav/gallery/scientists/LavaHammerL.jpg http://hvo.wr.usgs.gov/kilauea/update/images.html 10/29/18 GG303 3 http://upload.wikimedia.org/wikipedia/commons/8/89/Ropy_pahoehoe.jpg 20. Rheology & Linear Elasticity Elastic Behavior https://thegeosphere.pbworks.com/w/page/24663884/Sumatra http://www.earth.ox.ac.uk/__Data/assets/image/0006/3021/seismic_hammer.jpg 10/29/18 GG303 4 20. Rheology & Linear Elasticity Brittle Behavior (fracture) 10/29/18 GG303 5 http://upload.wikimedia.org/wikipedia/commons/8/89/Ropy_pahoehoe.jpg 20. Rheology & Linear Elasticity II Rheology: Macroscopic deformation behavior A Elasticity 1 Deformation is reversible when load is removed 2 Stress (σ) is related to strain (ε) 3 Deformation is not time dependent if load is constant 4 Examples: Seismic (acoustic) waves, http://www.fordogtrainers.com rubber ball 10/29/18 GG303 6 20. Rheology & Linear Elasticity II Rheology: Macroscopic deformation behavior A Elasticity 1 Deformation is reversible when load is removed 2 Stress (σ) is related to strain (ε) 3 Deformation is not time dependent if load is constant 4 Examples: Seismic (acoustic) waves, rubber ball 10/29/18 GG303 7 20. Rheology & Linear Elasticity II Rheology: Macroscopic deformation behavior B Viscosity 1 Deformation is irreversible when load is removed 2 Stress (σ) is related to strain rate (ε ! ) 3 Deformation is time dependent if load is constant 4 Examples: Lava flows, corn syrup http://wholefoodrecipes.net 10/29/18 GG303 8 20.
    [Show full text]
  • Guide to Rheological Nomenclature: Measurements in Ceramic Particulate Systems
    NfST Nisr National institute of Standards and Technology Technology Administration, U.S. Department of Commerce NIST Special Publication 946 Guide to Rheological Nomenclature: Measurements in Ceramic Particulate Systems Vincent A. Hackley and Chiara F. Ferraris rhe National Institute of Standards and Technology was established in 1988 by Congress to "assist industry in the development of technology . needed to improve product quality, to modernize manufacturing processes, to ensure product reliability . and to facilitate rapid commercialization ... of products based on new scientific discoveries." NIST, originally founded as the National Bureau of Standards in 1901, works to strengthen U.S. industry's competitiveness; advance science and engineering; and improve public health, safety, and the environment. One of the agency's basic functions is to develop, maintain, and retain custody of the national standards of measurement, and provide the means and methods for comparing standards used in science, engineering, manufacturing, commerce, industry, and education with the standards adopted or recognized by the Federal Government. As an agency of the U.S. Commerce Department's Technology Administration, NIST conducts basic and applied research in the physical sciences and engineering, and develops measurement techniques, test methods, standards, and related services. The Institute does generic and precompetitive work on new and advanced technologies. NIST's research facilities are located at Gaithersburg, MD 20899, and at Boulder, CO 80303.
    [Show full text]
  • Rheology and Thermodynamics of Starch-Based Hydrogel –Mixtures
    RHEOLOGY AND THERMODYNAMICS OF STARCH-BASED HYDROGEL –MIXTURES DISSERTATION zur Erlangung des Grades eines „Doktor der Naturwissenschaften“ im Promotionsfach Chemie am Fachbereich Chemie, Pharmazie und Geowissenschaften der Johannes Gutenberg -Universität in Mainz Natalie Ruß geb. in Kirchheim -Bolanden Mainz, April 2016 1. Berichterstatter: Prof. Dr. Thomas A. Vilgis 2. Berichterstatter: PD Dr. Wolfgang Schärtl Tag der mündlichen Prüfung: 17. Mai 2016 Prüfungskommission: Prof. Dr. Thomas A. Vilgis PD Dr. Wolfgang Schärtl Prof. Dr. Angelika Kühnle Prof. Dr. Heiner Detert “A Thing is a Thing, not what is said of that Thing.” II Abstract Abstract The present work presents two different ways to achieve physically modified tapioca starch and the influence on mechanical properties of an aqueous starch paste is investigated. A completely cold soluble tapioca starch powder is produced by spray drying a previously gelatinized starch paste. Rehydrating the received white powder forms a viscous paste with significant loss in elasticity. Heating of a native tapioca starch suspension in contrast yields highly viscous paste with dominating elastic behavior. The combination of the spray dried tapioca starch with nongelling food hydrocolloids, such as xanthan gum, ι-carrageenan, and guar gum restores the mechanical properties and creates new starch-based thickening agents with stable structure. Rheological measurements of a gelatinized native tapioca starch paste compared to a rehydrated paste made from the spray dried starch show significant differences in viscosity and viscoelastic properties, which depend on temperature, amplitude, frequency, or shear rate. Further rheological, optical, and scattering investigations indicate weakening of the amylose network structure generated by the harsh shear and heat conditions during the spray drying process.
    [Show full text]
  • Heat and Energy Conservation
    1 Lecture notes in Fluid Dynamics (1.63J/2.01J) by Chiang C. Mei, MIT, Spring, 2007 CHAPTER 4. THERMAL EFFECTS IN FLUIDS 4-1-2energy.tex 4.1 Heat and energy conservation Recall the basic equations for a compressible fluid. Mass conservation requires that : ρt + ∇ · ρ~q = 0 (4.1.1) Momentum conservation requires that : = ρ (~qt + ~q∇ · ~q)= −∇p + ∇· τ +ρf~ (4.1.2) = where the viscous stress tensor τ has the components = ∂qi ∂qi ∂qk τ = τij = µ + + λ δij ij ∂xj ∂xi ! ∂xk There are 5 unknowns ρ, p, qi but only 4 equations. One more equation is needed. 4.1.1 Conservation of total energy Consider both mechanical ad thermal energy. Let e be the internal (thermal) energy per unit mass due to microscopic motion, and q2/2 be the kinetic energy per unit mass due to macroscopic motion. Conservation of energy requires D q2 ρ e + dV rate of incr. of energy in V (t) Dt ZZZV 2 ! = − Q~ · ~ndS rate of heat flux into V ZZS + ρf~ · ~qdV rate of work by body force ZZZV + Σ~ · ~qdS rate of work by surface force ZZX Use the kinematic transport theorm, the left hand side becomes D q2 ρ e + dV ZZZV Dt 2 ! 2 Using Gauss theorem the heat flux term becomes ∂Qi − QinidS = − dV ZZS ZZZV ∂xi The work done by surface stress becomes Σjqj dS = (σjini)qj dS ZZS ZZS ∂(σijqj) = (σijqj)ni dS = dV ZZS ZZZV ∂xi Now all terms are expressed as volume integrals over an arbitrary material volume, the following must be true at every point in space, 2 D q ∂Qi ∂(σijqi) ρ e + = − + ρfiqi + (4.1.3) Dt 2 ! ∂xi ∂xj As an alternative form, we differentiate the kinetic energy and get De
    [Show full text]
  • Lecture 5: Magnetic Mirroring
    !"#$%&'()%"*#%*+,-./-*+01.2(.*3+456789* !"#$%&"'()'*+,-".#'*/&&0&/-,' Dr. Peter T. Gallagher Astrophysics Research Group Trinity College Dublin :&2-;-)(*!"<-$2-"(=*%>*/-?"=)(*/%/="#* o Gyrating particle constitutes an electric current loop with a dipole moment: 1/2mv2 µ = " B o The dipole moment is conserved, i.e., is invariant. Called the first adiabatic invariant. ! o µ = constant even if B varies spatially or temporally. If B varies, then vperp varies to keep µ = constant => v|| also changes. o Gives rise to magnetic mirroring. Seen in planetary magnetospheres, magnetic bottles, coronal loops, etc. Bz o Right is geometry of mirror Br from Chen, Page 30. @-?"=)(*/2$$%$2"?* o Consider B-field pointed primarily in z-direction and whose magnitude varies in z- direction. If field is axisymmetric, B! = 0 and d/d! = 0. o This has cylindrical symmetry, so write B = Brrˆ + Bzzˆ o How does this configuration give rise to a force that can trap a charged particle? ! o Can obtain Br from " #B = 0 . In cylindrical polar coordinates: 1 " "Bz (rBr )+ = 0 r "r "z ! " "Bz => (rBr ) = #r "r "z o If " B z / " z is given at r = 0 and does not vary much with r, then r $Bz 1 2&$ Bz ) ! rBr = " #0 r dr % " r ( + $z 2 ' $z *r =0 ! 1 &$ Bz ) Br = " r( + (5.1) 2 ' $z *r =0 ! @-?"=)(*/2$$%$2"?* o Now have Br in terms of BZ, which we can use to find Lorentz force on particle. o The components of Lorentz force are: Fr = q(v" Bz # vzB" ) (1) F" = q(#vrBz + vzBr ) (2) (3) Fz = q(vrB" # v" Br ) (4) o As B! = 0, two terms vanish and terms (1) and (2) give rise to Larmor gyration.
    [Show full text]
  • Chapter 15 - Fluid Mechanics Thursday, March 24Th
    Chapter 15 - Fluid Mechanics Thursday, March 24th •Fluids – Static properties • Density and pressure • Hydrostatic equilibrium • Archimedes principle and buoyancy •Fluid Motion • The continuity equation • Bernoulli’s effect •Demonstration, iClicker and example problems Reading: pages 243 to 255 in text book (Chapter 15) Definitions: Density Pressure, ρ , is defined as force per unit area: Mass M ρ = = [Units – kg.m-3] Volume V Definition of mass – 1 kg is the mass of 1 liter (10-3 m3) of pure water. Therefore, density of water given by: Mass 1 kg 3 −3 ρH O = = 3 3 = 10 kg ⋅m 2 Volume 10− m Definitions: Pressure (p ) Pressure, p, is defined as force per unit area: Force F p = = [Units – N.m-2, or Pascal (Pa)] Area A Atmospheric pressure (1 atm.) is equal to 101325 N.m-2. 1 pound per square inch (1 psi) is equal to: 1 psi = 6944 Pa = 0.068 atm 1atm = 14.7 psi Definitions: Pressure (p ) Pressure, p, is defined as force per unit area: Force F p = = [Units – N.m-2, or Pascal (Pa)] Area A Pressure in Fluids Pressure, " p, is defined as force per unit area: # Force F p = = [Units – N.m-2, or Pascal (Pa)] " A8" rea A + $ In the presence of gravity, pressure in a static+ 8" fluid increases with depth. " – This allows an upward pressure force " to balance the downward gravitational force. + " $ – This condition is hydrostatic equilibrium. – Incompressible fluids like liquids have constant density; for them, pressure as a function of depth h is p p gh = 0+ρ p0 = pressure at surface " + Pressure in Fluids Pressure, p, is defined as force per unit area: Force F p = = [Units – N.m-2, or Pascal (Pa)] Area A In the presence of gravity, pressure in a static fluid increases with depth.
    [Show full text]