Fundamentals And Applications Of Perturbation Methods In .
F UNDAMENTALSANDA PPLICATIONSOFP ERTURBATION M ETHODSINF LUID DYNAMICSFundamentals and Applications ofPerturbation Methods in Fluid DynamicsTheory and Exercises - JMBC Course - 2018Sjoerd RienstraSingularity is almost invariably a clue(Sherlock Holmes, The Boscombe Valley Mystery)107-03-2018
F UNDAMENTALSANDA PPLICATIONSOFP ERTURBATION M ETHODS2INF LUID DYNAMICS07-03-2018
Contents1 Mathematical Modelling and Perturbation Methods92 Modelling and Scaling2.113Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .132.1.1 What is a model? Some philosophical considerations. . . . . . . . . . . . . . . .2.1.2 Types of models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .13152.1.3 Perturbation methods: the continuation of modelling by other means . . . . . . .162.1.4 Nondimensionalisation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.1.5 Example: a systematic derivation of the Korteweg-de Vries Equation . . . . . . .18202.2Modelling, Nondimensionalisation and Scaling: Assignments. . . . . . . . . . . .232.2.1 Travel time in cities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2.2 Membrane resonance. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .23232.2.3 Ship drag: wave and viscosity effects. . . . . . . . . . . . . . . . . . . . . . .242.2.4 Sphere in viscous flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2.5 Cooling of a cup of tea. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .24252.2.6 The velocity of a rowing boat. . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2.7 A sessile drop with surface tension. . . . . . . . . . . . . . . . . . . . . . . . . .25252.2.8 The drag of a plate sliding along a thin layer of lubricant. . . . . . . . . . . . . .2.2.9 The suspended cable . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .26262.2.10 Electrically heated metal . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2.11 Traffic waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .27282.2.12 The Korteweg-de Vries equation . . . . . . . . . . . . . . . . . . . . . . . . . .2.2.13 Just an equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .28292.2.14 The pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2.15 Heat convection and diffusion . . . . . . . . . . . . . . . . . . . . . . . . . . . .29302.2.16 Heat conduction in a long bar . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2.17 A simple balloon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .30312.2.18 A pulsating sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .2.2.19 Similarity solutions for non-linear and linear diffusion . . . . . . . . . . . . . . .32332.2.20 Falling through the center of the earth. . . . . . . . . . . . . . . . . . . . . . .34. . . . . . . . . . . . . . . . . . . . . . . . . . . .342.2.21 Energy consumption of a car3
F UNDAMENTALSANDA PPLICATIONSOFP ERTURBATION M ETHODSINF LUID DYNAMICS3 Asymptotic Analysis353.1 Asymptotic approximations and expansions . . . . . . . . . . . . . . . . . . . . . .3.1.1 Asymptotic approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .35353.1.2 Asymptotic expansions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .363.2Basic definitions and theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . .383.3Asymptotic Expansions: Applications . . . . . . . . . . . . . . . . . . . . . . . . .433.3.1 General procedure for algebraic equations . . . . . . . . . . . . . . . . . . . . .3.3.2 Example: roots of a polynomial . . . . . . . . . . . . . . . . . . . . . . . . . . .43443.4 Asymptotic Expansions: Assignments . . . . . . . . . . . . . . . . . . . . . . . . .3.4.1 Asymptotic order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47473.4.2 Asymptotic expansions in ε . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3.4.3 Asymptotic sequences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .47483.4.4 Asymptotic expansions in x and ε . . . . . . . . . . . . . . . . . . . . . . . . .3.4.5 Solving algebraic equations asymptotically . . . . . . . . . . . . . . . . . . . . .48493.4.6 Solving differential equations asymptotically . . . . . . . . . . . . . . . . . . . .3.4.7 A water-bubbles mixture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .53543.4.8 A chemical reaction-diffusion problem (regular limit) . . . . . . . . . . . . . . .554 Method of Slow Variation574.1 Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.1.1 General procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .57574.1.2 Example: heat flow in a bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . .584.2Method of Slow Variation: Assignments . . . . . . . . . . . . . . . . . . . . . . . .614.2.1 Heat flux in a bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.2.2 Lubrication flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .61614.2.3 Quasi 1D gas dynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .4.2.4 Webster’s horn . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .62634.2.5 Shallow water waves along a varying bottom . . . . . . . . . . . . . . . . . . . .4.2.6 A laterally heated bar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .64655 Method of Lindstedt-Poincaré5.167Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .675.1.1 Secular behaviour with naive expansion . . . . . . . . . . . . . . . . . . . . . . .5.1.2 General Procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .67685.1.3 Example: the pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .685.2 Method of Lindstedt-Poincaré: Assignments . . . . . . . . . . . . . . . . . . . . .5.2.1 A quadratically perturbed harmonic oscillator . . . . . . . . . . . . . . . . . . .71715.2.2 A weakly nonlinear harmonic oscillator. . . . . . . . . . . . . . . . . . . . . .5.2.3 A weakly nonlinear, quadratically perturbed harmonic oscillator4. . . . . . . . .717107-03-2018
F UNDAMENTALSANDA PPLICATIONS5.2.4 A coupled nonlinear oscillatorOFP ERTURBATION M ETHODSINF LUID DYNAMICS. . . . . . . . . . . . . . . . . . . . . . . . . . .715.2.5 A weakly nonlinear 4th order oscillator . . . . . . . . . . . . . . . . . . . . . . .5.2.6 A weakly nonlinear oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . .72725.2.7 The Van der Pol oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .5.2.8 A variant of the Van der Pol oscillator . . . . . . . . . . . . . . . . . . . . . . .72725.2.9 Another weakly nonlinear oscillator72. . . . . . . . . . . . . . . . . . . . . . . .6 Matched Asymptotic Expansions6.173Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .736.1.1 Singular perturbation problems . . . . . . . . . . . . . . . . . . . . . . . . . . .6.1.2 Matched Asymptotic Expansions . . . . . . . . . . . . . . . . . . . . . . . . . .73736.2 Matched Asymptotic Expansions: Assignments . . . . . . . . . . . . . . . . . . . .6.2.1 Non-uniform approximations and boundary layers . . . . . . . . . . . . . . . . .85856.2.2 Boundary layers and integration . . . . . . . . . . . . . . . . . . . . . . . . . .6.2.3 Friedrichs’ model problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . .85856.2.4 Singularly perturbed ordinary differential equations . . . . . . . . . . . . . . . .6.2.5 A hidden boundary layer structure . . . . . . . . . . . . . . . . . . . . . . . . .85866.2.6 A singularly perturbed nonlinear problem . . . . . . . . . . . . . . . . . . . . .6.2.7 A singularly perturbed linear problem . . . . . . . . . . . . . . . . . . . . . . .86866.2.8 A boundary layer problem6.2.9 Sign and scaling problems. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .86876.2.10 The Michaelis-Menten model . . . . . . . . . . . . . . . . . . . . . . . . . . . .876.2.11 Groundwater flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.2.12 Stirring a cup of tea . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .88896.2.13 Fisher’s travelling wave problem . . . . . . . . . . . . . . . . . . . . . . . . . .6.2.14 Nonlinear diffusion in a semi-conductor . . . . . . . . . . . . . . . . . . . . . .89906.2.15 Heat conduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.2.16 Polymer extrusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .90916.2.17 Torsion of a thin-walled tube . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.2.18 A visco-elastic medium forced by a piston . . . . . . . . . . . . . . . . . . . . .91926.2.19 Heat conduction in fluid flow through a slit . . . . . . . . . . . . . . . . . . . . .6.2.20 The sag of a slender plate supported at the ends . . . . . . . . . . . . . . . . . .93936.2.21 Heat conduction along cylinder walls . . . . . . . . . . . . . . . . . . . . . . . .6.2.22 Cooling by radiation of a heat conducting plate . . . . . . . . . . . . . . . . . .94966.2.23 The stiffened catenary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .6.2.24 A boundary layer problem with x-dependent coefficients . . . . . . . . . . . . .97986.2.25 A catalytic reaction problem in 1D . . . . . . . . . . . . . . . . . . . . . . . . .6.2.26 A cooling problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .98996.2.27 Visco-elastic fibre spinning . . . . . . . . . . . . . . . . . . . . . . . . . . . . .996.2.28 The weather balloon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 102507-03-2018
F UNDAMENTALSANDA PPLICATIONSOFP ERTURBATION M ETHODS6.2.29 A chemical reaction-diffusion problem (singular limit)INF LUID DYNAMICS. . . . . . . . . . . . . . 1056.2.30 An internal boundary layer (Oxford, OCIAM, 2003) . . . . . . . . . . . . . . . . 1056.2.31 The Van der Pol equation with strong damping . . . . . . . . . . . . . . . . . . . 1056.2.32 A beam under tension resting on an elastic foundation . . . . . . . . . . . . . . . 1057 Multiple Scales, WKB and Resonance7.1107Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1077.1.1 Multiple Scales: general procedure . . . . . . . . . . . . . . . . . . . . . . . . . 1077.1.2 A practical example: a damped oscillator . . . . . . . . . . . . . . . . . . . . . . 1087.1.3 The air-damped resonator. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1107.1.4 The WKB Method: slowly varying fast time scale . . . . . . . . . . . . . . . . . 1127.1.5 Higher dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1147.1.6 Weakly nonlinear resonance problems . . . . . . . . . . . . . . . . . . . . . . . . 1157.2 Multiple Scales, WKB and Resonance: Assignments . . . . . . . . . . . . . . . . . 1187.2.1 Non-stationary Van der Pol oscillator . . . . . . . . . . . . . . . . . . . . . . . . 1187.2.2 The air-damped, unforced pendulum . . . . . . . . . . . . . . . . . . . . . . . . 1187.2.3 The air-damped pendulum, harmonically forced near resonance . . . . . . . . . . 1187.2.4 Relativistic correction for Mercury . . . . . . . . . . . . . . . . . . . . . . . . . 1197.2.5 Weakly nonlinear advection-diffusion . . . . . . . . . . . . . . . . . . . . . . . 1197.2.6 Golden Ten: an application of multiple scales . . . . . . . . . . . . . . . . . . . 1207.2.7 Modal sound propagation in slowly varying ducts . . . . . . . . . . . . . . . . . 1257.2.8 A nearly resonant weakly nonlinear forced harmonic oscillator . . . . . . . . . . 1267.2.9 A non-linear beam with small forcing . . . . . . . . . . . . . . . . . . . . . . . 1267.2.10 Acoustic rays in a medium with a varying sound speed. . . . . . . . . . . . . . 1277.2.11 Homogenisation as a Multiple Scales problem . . . . . . . . . . . . . . . . . . . 1277.2.12 The non-linear pendulum with slowly varying length . . . . . . . . . . . . . . . 1287.2.13 Asymptotic behaviour of solutions of Bessel’s equation . . . . . . . . . . . . . . 1287.2.14 Kapitza’s Pendulum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1297.2.15 Doppler effect of a moving sound source . . . . . . . . . . . . . . . . . . . . . . 1307.2.16 Vibration modes in a slowly varying elastic beam . . . . . . . . . . . . . . . . . 1317.2.17 An aging spring . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1318 Integral Asymptotics1338.1Integrals and Watson’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1338.2Laplace’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1368.3Method of Stationary Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1378.4Method of Steepest Descent or Saddle Point Method . . . . . . . . . . . . . . . . . 1398.5Applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1418.5.1 Group velocity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1418.5.2 Doppler effect of a moving sound source. . . . . . . . . . . . . . . . . . . . . . . 141607-03-2018
F UNDAMENTALS8.6ANDA PPLICATIONSOFIntegral Asymptotics: AssignmentsP ERTURBATION M ETHODSINF LUID DYNAMICS. . . . . . . . . . . . . . . . . . . . . . . . . . 1438.6.1 Integrals and Watson’s Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . 1438.6.2 Laplace’s Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1458.6.3 Method of Stationary Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1468.6.4 Method of Steepest Descent or Saddle Point Method . . . . . . . . . . . . . . . . 1479 Some Mathematical Auxiliaries1499.1Phase plane . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1499.2Newton’s equation9.3Normal vectors of level surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1539.4Trigonometric relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15210 Special Functions15510.1 Bessel Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15510.2 Gamma Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15610.3 Dilogarithm and Exponential Integral . . . . . . . . . . . . . . . . . . . . . . . . . 15710.4 Error Function. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15711 Units, Dimensions, Dimensionless Numbers159Quotes163Bibliography165707-03-2018
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Chapter 1Mathematical Modelling andPerturbation MethodsMathematical modelling is an art. It is the art of portraying a real, often physical, problem mathematically, by sorting out the whole spectrum of effects that play or may play a role, and then making ajudicious selection by including what is relevant and excluding what is too small. This selection iswhat we call a model or theory. Models and theories, applicable in a certain situation, are not isolated islands of knowledge provided with a logical flag, labelling it valid or invalid. A model is neverunique, because it depends on the type, quality and accuracy of answers we are aiming for, and ofcourse the means (time, money, numerical power, mathematical skills) that we have available.Normally, when the problem is rich enough, this spectrum of effects does not simply consist of twoclasses important and unimportant, but is a smoothly distributed hierarchy varying from essentialeffects via relevant and rather relevant to unimportant and absolutely irrelevant effects. As a result, inpractically any model there will be effects that are small but not small enough to be excluded. We canignore their smallness, and just assume that all effects that constitute our model are equally important.This is the usual approach when the problem is simple enough for analysis or a brute force numericalsimulation.Figure 1.1: Concept of hierarchy (turbofan engine)There are situations, however, where it could be wise to utilise the smallness of these small but important effects, but in such a way, that we simplify the problem without reducing the quality of themodel. Usually, an otherwise intractable problem becomes solvable and (most importantly) we gaingreat insight in the problem.9
F UNDAMENTALSANDA PPLICATIONSOFP ERTURBATION M ETHODSINF LUID DYNAMICSPerturbation methods do this in a systematic manner by using the sharp fillet knife of mathematics in general, and asymptotic analysis in particular.From this perspective, perturbation methods are ways of modelling withother means and are therefore much more important for the understanding and analysis of practical problems than they’re usually credited with.David Crighton [14] called “Asymptotics - an indispensable complement tothought, computation and experiment in applied mathematical modelling”.Examples are numerous: simplified geometries reducing the spatial dimension, small amplitudes allowing linearization, low velocities and long timescales allowing incompressible description, small relative viscosity allowing inviscid models, zero or infinite lengths rather than finite lengths, etc.The question is: how can we use this gradual transition between models of different level. Of course,when a certain aspect or effect, previously absent from our model, is included in our model, the changeis abrupt and big: usually the corresponding equations are more complex and more difficult to solve.This is, however, only true if we are merely interested in exact or numerically exact solutions. But anexact solution of an approximate model is not better than an approximate solution of an exact model.x 2 4 10 6 x 5x2 4Figure 1.2: Compare “exact” and approximate models.So there is absolutely no reason to demand the solution to be more exact than the corresponding model.If we accept approximate solutions, based on the inherent small or large modelling parameters, we dohave the possibilities to gradually increase the complexity of a model, and study small but significanteffects in the most efficient way.The methods utilizing systematically this approach are called perturbations methods. Usually, a distinction is made between regular and singular perturbations. A (loose definition of a) regular perturbation is one in which the solutions of perturbed and unperturbed problem are everywhere close to eachother.We will find many applications of this philosophy in continuous mechanics (fluid mechanics, elasticity), and indeed many methods arose as a natural tool to understand certain underlying physicalphenomena. We will consider here four methods relevant in continuous mechanics: (1) the method of1007-03-2018
F UNDAMENTALSANDA PPLICATIONSOFP ERTURBATION M ETHODSINF LUID DYNAMICSslow variation and (2) the method of Lindstedt-Poincaré as examples of regular perturbation methods;then (3) the method of matched asymptotic expansions and (4) the method of multiple scales (withas a special case the WKB method) as examples of singular perturbation methods. In (1) the typicallength scale in one direction is much greater than in the others, while in (2) the relevant time scale isunknown and part of the problem. In (3) several approximations, coupled but valid in spatially distinctregions, are solved in parallel. Method (4) relates to problems in which several length scales act in thesame direction, for example a wave propagating through a slowly varying environment.In order to quantify the used small effect in the model, we will always int
Fundamentals and Applications of Perturbation Methods in Fluid Dynamics Theory and Exercises - JMBC Course - 2018 Sjoerd Rienstra Singularity is almost invariably a clue (Sherlock Holmes, The Boscombe Valley Mystery) 1 07-03-2018. FUNDAMENTALS AND APPLICATIONS OF PERTURBATION METHODS IN FLUID DYNAMICS
Quantum Mechanical Perturbation Theory Quantum mechanical perturbation theory is a widely used method in solid-state physics. Without the details of derivation, we shall list a number of basic formulas of time-independent (stationary) and time-dependent perturbation theory below. For simp
Quantum mechanics 2 - Lecture 2 Igor Luka cevi c UJJS, Dept. of Physics, Osijek 17. listopada 2012. Igor Luka cevi c Perturbation theory. Contents Time-independent nondegenerate perturbation theory Time-independent degenerate perturbation theory Time-
4.1 Perturbation theory, Feynman diagrams As as been presented for QED, a natural scheme is to assume that g is small and perform aseriesexpansioninpowersofg. This amounts to consider that the interaction terms are small, and represents a small perturbation of the free theory. Thus we expand the interaction term in the functional integral exp 4 .
CMB anisotropy galaxy distribution and bulk flow Can predict LSS in terms of primordial curvature perturbation (x) existing at T 1MeV. So (x) is an observable . Other effects 10%, probably ' 0 tensor perturbation, cosmic strings, textures, isocurvature (matter or ) perturbation. KEK 2010 - p.3/21
4 Classical perturbation theory 4 . Celestial Mechanics greatly motivated the advances of perturbation . different mathematical settings by Arnold ([16]) and Moser ([37]). The overall theory is known with the acronym of KAM theory
Development of the perturbation stationary bicycle robotic system System description The Perturbation Stationary Bicycle Robotic system (PerStBiRo system) is a mechatronic device weighing 90 kg that provides lateral roll-angle (lateral tilt) balance perturbations during hands-on or hands-free bicycling in a safe environment (see Fig. 1).
similarities with perturbation theory of quantum mechanics (Rae 1992). They need useful approaches for formulating nonstationary or stochastic system dynamics. Of course, the system behaviors are represented differently. Perturbation theory assumes well-defined system behavior, e.g. the wave equation (
3 PRACTICE TEST 01 May 2004 Question 1-10 All mammals feed their young. Beluga whale mothers, for example, nurse their calves for some twenty months, until they are about to give birth again and their young are able to
