Course intro 18.354 18.354J Nonlinear Dynamics II: Continuum Systems Spring 2015 – Course Info Lectures: Instructor: Contact: Office Hours: Course website: Teaching assistant: TR 1-2:30 in E17-136 Jörn Dunkel dunkel@mit.edu 253-7826 (office phone) R 2:30-3:30 (E17-412) math.mit.edu/⇠dunkel/Teach/18.354/ TBD, Office: TBD Email: tbd This course introduces the basic ideas for understanding the dynamics of continuum systems, by studying specific examples from a range of di↵erent fields. Our goal will be to explain the general principles, and also to illustrate them via important physical e↵ects. A parallel goal of this course is to give you an introduction to mathematical modeling. The first part of the course will study di↵usion, to demonstrate how continuum descriptions arise from averaging microscopic degrees of freedom. The equations of motion that we derive for continuum systems are typically nonlinear partial di↵erential equations, for which it is very difficult to obtain analytical solutions. We shall therefore briefly foray into dimensional analysis to see how it is possible to obtain qualitative information about a system without having to solve the full equations of motion. We shall then study the ‘calculus of variations’, which is a minimization approach to finding solutions of continuous systems. We are familiar with these techniques for discrete The first part of the course will study di↵usion, to demonstrate how continuum descriptions arise from averaging microscopic degrees of freedom. The equations of motion that we derive for continuum systems are typically nonlinear partial di↵erential equations, for which it isNonlinear very difficult to Dynamics obtain analytical II: solutions. We shall therefore briefly foray 18.354J Continuum Systems into dimensional analysis to see how it is possible to obtain qualitative information about a system without having to solve the full equations of motion. Spring 2015 – Course Info We shall then study the ‘calculus of variations’, which is a minimization approach to Lectures: TR 1-2:30 We are in familiar E17-136 finding solutions of continuous systems. with these techniques for discrete Instructor: JörntoDunkel systems and now adapt the ideas help us with continuous systems. First, we examine the Contact: dunkel@mit.edu 253-7826 (office phone) classical brachiostrome problem posed by Bernoulli, and then consider an array of problems in many di↵erent physical systems (e.g., shapes of soap films, bending of elastic beams, etc.) Office Hours: R 2:30-3:30 (E17-412) InCourse the second half of the course we will examine hydrodynamic problems. We will discuss website: math.mit.edu/⇠dunkel/Teach/18.354/ derivations of Navier-Stokes-type equations study particular Teaching assistant: TBD, Office: and TBD Email: tbd solutions. In this context, we will introduce singular perturbation methods and survey hydrodynamic instabilities. In the last part, we will give an outlook on the mathematical description of solitons, active biological systems and topological defects. This course introduces the basic ideas for understanding the dynamics of continuum systems, by studying specific examples from a range of di↵erent fields. Our goal will be to GRADING explain the general principles, and also to illustrate them via important physical e↵ects. A Problem sets parallel• 35%: goal of this course is to give you an introduction to mathematical modeling. • 30%: Mid-term exam The first part of the course will study di↵usion, to demonstrate how continuum de• 5%: Project proposal scriptions arise from averaging microscopic • 30%: Final project presentation + reportdegrees of freedom. The equations of motion that we derive for continuum systems are typically nonlinear partial di↵erential equations, for which it is very difficult to obtain analytical solutions. We shall therefore briefly foray TEXTBOOKS into dimensional analysis to see how it is possible to obtain qualitative information about Although there are no textbooks which follow the precise spirit of this course, there are a system without having to solve the full equations of motion. literally hundreds of textbooks where the topics we will cover are discussed. For most Welectures, shall then study of variations’, is a minimization typed notesthe can‘calculus be downloaded from the which course webpage. Additionalapproach material to findingwill solutions of continuous Wethat are will familiar withfrequently these techniques for discrete be handed out in class.systems. One book be useful is: D. J. Acheson, systems and now adapt the ideas to help us with continuous systems. First, we examine the classical brachiostrome problem posed by Bernoulli, and then consider an array of problems in many di↵erent physical systems (e.g., shapes of soap films, bending of elastic beams, etc.) 18.354J II: Continuum In the second Nonlinear half of the courseDynamics we will examine hydrodynamic problems.Systems We will discuss derivations of Navier-Stokes-type equations and study particular solutions. In this context, Spring 2015 – Course Info we will introduce singular perturbation methods and survey hydrodynamic instabilities. In the last part, we will give an outlook on the mathematical description of solitons, active Lectures: TR 1-2:30 in E17-136 biologicalInstructor: systems and topological defects. Jörn Dunkel Contact: GRADING Office Hours: Course sets website: • 35%: Problem Teachingexam assistant: • 30%: Mid-term dunkel@mit.edu 253-7826 (office phone) R 2:30-3:30 (E17-412) math.mit.edu/⇠dunkel/Teach/18.354/ TBD, Office: TBD Email: tbd • 5%: Project proposal • 30%: Final project presentation + report This course introduces the basic ideas for understanding the dynamics of continuum TEXTBOOKS systems, by studying specific examples from a range of di↵erent fields. Our goal will be to explain the general and which also tofollow illustrate them via important physical e↵ects. Although there areprinciples, no textbooks the precise spirit of this course, there are A parallel goal of this course is to give youtheantopics introduction to mathematical modeling. literally hundreds of textbooks where we will cover are discussed. For most The first partnotes of the will studyfrom di↵usion, to demonstrate how continuum lectures, typed cancourse be downloaded the course webpage. Additional material dewill be handed out averaging in class. One book thatdegrees will be ofuseful frequently D. J. Acheson, scriptions arise from microscopic freedom. The is: equations of motion Elementary Dynamics, Oxford Pressnonlinear (1990). partial di↵erential equations, that we deriveFluid for continuum systemsUniversity are typically for which it is very difficult to obtain analytical solutions. We shall therefore briefly foray into dimensional analysis to see how it is possible to obtain qualitative information about a system without having to solve the full equations of motion. We shall then study the ‘calculus of variations’, which is a minimization approach to finding solutions of continuous systems. We are familiar with these techniques for discrete 18.354J Nonlinear Dynamics II: Continuum Systems Spring 2015 – Course Info Lectures: Instructor: Contact: Office Hours: Course website: Teaching assistant: TR 1-2:30 in E17-136 Jörn Dunkel dunkel@mit.edu 253-7826 (office phone) R 2:30-3:30 (E17-412) math.mit.edu/⇠dunkel/Teach/18.354/ TBD, Office: TBD Email: tbd HOMEWORK - PROBLEM This course introduces the basic SETS ideas for understanding the dynamics of continuum systems, by studying specific roughly examples from a rangeweeks. of di↵erent fields. Our will be to Homework will be assigned every two-three Each homework setgoal will contain analytical and computational problems, even the oddvia experiment. must A explain the general principles, and also to and illustrate them importantAssignments physical e↵ects. be handed by 1pm (start of give class)you on the due date. First late homework score parallel goal ofinthis course is to an introduction tounexcused mathematical modeling. will be multiplied No subsequent homework is accepted. You are deThe first part of by the3/4. course will study unexcused di↵usion, late to demonstrate how continuum welcome to discuss solution strategies and even solutions, but please write up the solution scriptions arise from averaging microscopic degrees of freedom. The equations of motion own. Be sure to support answer bynonlinear listing any relevant Theoremsequations, or by that on weyour derive for continuum systems your are typically partial di↵erential explaining important steps. Be as clear and concise as possible. I strongly encourage the for which it is very difficult to obtain analytical solutions. We shall therefore briefly foray computational problems to be written in MATLAB. into dimensional analysis to see how it is possible to obtain qualitative information about a system without having to solve the full equations of motion. MID-TERM EXAM We shall then study the ‘calculus of variations’, which is a minimization approach to There will be of a mid-term exam, which will place about of techniques the way through the finding solutions continuous systems. We take are familiar with3/4’s these for discrete course. The exam will be a take home, and will be in place of a homework set. There will 18.354J Nonlinear Dynamics II: Continuum Systems Spring 2015 – Course Info HOMEWORK - PROBLEM SETS Lectures: TR 1-2:30 in weeks.E17-136 Homework will be assigned roughly every two-three Each homework set will contain Instructor: Dunkel analytical and computationalJörn problems, and even the odd experiment. Assignments must beContact: handed in by 1pm (start of class) on the due date. First 253-7826 unexcused late homework score dunkel@mit.edu (office phone) will be multiplied by 3/4. No subsequent unexcused late homework is accepted. You are Office Hours: R 2:30-3:30 (E17-412) welcome to discuss solution strategies and even solutions, but please write up the solution website: math.mit.edu/⇠dunkel/Teach/18.354/ onCourse your own. Be sure to support your answer by listing any relevant Theorems or by explaining important steps. TBD, Be as clear and concise possible. I strongly encourage the Teaching assistant: Office: TBD as Email: tbd computational problems to be written in MATLAB. MID-TERM EXAM This course introduces the basic ideas for understanding the dynamics of continuum There will be a mid-term exam, which will take place about 3/4’s of the way through the systems, by studying specific examples from a range of di↵erent fields. Our goal will be to course. The exam will be a take home, and will be in place of a homework set. There will explain the principles, and also to illustrate them via important physical e↵ects. A be general no final exam. parallel goal of this course is to give you an introduction to mathematical modeling. FINAL The first partPROJECT of the course will study di↵usion, to demonstrate how continuum descriptionsThe arise from averaging microscopic degrees of fields, freedom. The equations ideas we will be discussing have applications to many many of which we will notof motion cover. for To give you a chance to explore area of interest to you,partial the course will requireequations, that we derive continuum systems arean typically nonlinear di↵erential project, in which you explore in depth something of interest to you and within the for which aitfinal is very difficult to obtain analytical solutions. We shall therefore briefly foray course’s scope. Final projects will be presented in class during the final two classes. into dimensional analysis to see how it is possible to obtain qualitative information about a system without havingDATES to solve the full equations of motion. IMPORTANT We shall then study the ‘calculus of variations’, which is a minimization approach to • Tues Feb 24 - Problem Set 1: DUE finding solutions of 12 continuous systems. with(1these techniques for discrete • Thur Mar - Problem Set 2: DUE •We Thurare Marfamiliar 12 - Proposal page) for final project. will be multiplied by 3/4. No subsequent unexcused late homework is accepted. You are welcome to discuss solution strategies and even solutions, but please write up the solution on your own. Be sure to support your answer by listing any relevant Theorems or by explaining important steps. Be as clear and concise as possible. I strongly encourage the computational problems to be written in MATLAB. 18.354J Nonlinear Dynamics II: Continuum Systems MID-TERM EXAM Spring 2015 – Course Info There will be a mid-term exam, which will take place about 3/4’s of the way through the Lectures: TR 1-2:30 in E17-136 course. The exam will be a take home, and will be in place of a homework set. There will Instructor: Jörn Dunkel be no final exam. Contact: dunkel@mit.edu 253-7826 (office phone) FINAL PROJECT Office Hours: R 2:30-3:30 (E17-412) The ideas we will be discussing have applications to many fields, many of which we will not Course website: math.mit.edu/⇠dunkel/Teach/18.354/ cover. To give you a chance to explore an area of interest to you, the course will require Teaching TBD, Office: TBD Email:to tbd a final project, inassistant: which you explore in depth something of interest you and within the course’s scope. Final projects will be presented in class during the final two classes. IMPORTANT DATES This course introduces the basic ideas for understanding the dynamics of continuum • Tues Feb 24 - Problem Set 1: DUE systems,• by specific from of (1di↵erent fields. Our goal will be to Thurstudying Mar 12 - Problem Set examples 2: DUE • Thur Mar a 12 range - Proposal page) for final project. DUEgeneral principles, and also to illustrate them via important physical e↵ects. A explain the • Thur Apr 2 - Problem Set 3: DUE parallel goal of this course is to give you an introduction to mathematical modeling. • Tues Apr 14 - Problem Set 4: DUE The •first the course will study di↵usion, to demonstrate how continuum deThur part Apr 16 of - Take-home Midterm exam. POSTED • Thur Aprfrom 23 - Take-home Midterm exam. DUE degrees of freedom. The equations of motion scriptions arise averaging microscopic • May 5/7 - Final project presentations. that we •derive continuum are typically nonlinear partial di↵erential equations, May 7 -for Final project report.systems DUE for which it is very difficult to obtain analytical solutions. We shall therefore briefly foray Note: The exact due dates for the P-sets may be subject to change into dimensional analysis to see how it is possible to obtain qualitative information about a system without having to solve the full equations of motion. We shall then study the ‘calculus of variations’, which is a minimization approach to finding solutions of continuous systems. We are familiar with these techniques for discrete Email: Phone: Office: Office hours: Course website: dunkel@mit.edu 253-7826 E17-412 Thursday 2:30-3:30 (E17-412) http://math.mit.edu/⇠dunkel/Teach/18.354/ 1. 2. 3. 4. — 5. 6. 7. T R T R T R T R Feb Feb Feb Feb Feb Feb Feb Feb 3 5 10 12 17 19 24 26 Introduction, overview & mathematical basics Dimensional analysis & scalings Hamiltonian dynamics & Kepler’s Laws Random walkers MIT MONDAY (PRESIDENTS DAY) Di↵usion equation: Fourier method Di↵usion equation: Green’s function method Linear stability analysis & pattern formation 8. 9. 10. 11. 12. 13. — T R T R T R TR Mar Mar Mar Mar Mar Mar Mar 3 5 10 12 17 19 23-27 Calculus of variations Surface tension Elasticity Deformation of a thin beam Towards hydrodynamics Navier-Stokes equations I SPRING VACATION 14. 15. 16. 17. 18. — 19. 20. 21. R T R T R T R T R Apr Apr Apr Apr Apr Apr Apr Apr Apr 2 7 9 14 16 20,21 23 28 30 Stokes limit & Oseen tensor Navier-Stokes equations II Singular perturbations Rotating flows & Taylor columns 2D hydrodynamics & conformal maps MIT HOLIDAY (PATRIOTS DAY) Hydrodynamic instabilities (overview) Solitons Active Matter PS1 due PS2 & proposal due PS3 due PS4 due Mid-term posted Mid-term due 9. 10. 11. 12. 13. — R T R T R TR Mar Mar Mar Mar Mar Mar 5 10 12 17 19 23-27 Surface tension Elasticity Deformation of a thin beam Towards hydrodynamics Navier-Stokes equations I SPRING VACATION 14. 15. 16. 17. 18. — 19. 20. 21. R T R T R T R T R Apr Apr Apr Apr Apr Apr Apr Apr Apr 2 7 9 14 16 20,21 23 28 30 Stokes limit & Oseen tensor Navier-Stokes equations II Singular perturbations Rotating flows & Taylor columns 2D hydrodynamics & conformal maps MIT HOLIDAY (PATRIOTS DAY) Hydrodynamic instabilities (overview) Solitons Active Matter 22. 23. 24. 25. T R T R May May May May 5 7 12 14 Final projects: student presentations Final projects: student presentations Bouncing droplets Topological defects & summary PS2 & proposal due PS3 due PS4 due Mid-term posted Mid-term due Project report due Q: What is Physical Applied Maths? A: PAM is like cooking... Often the ingredients (physical principles) are already known but not the way (mathematics/equations/couplings) to turn them into a nice dinner With some creativity, many new dishes (novel phenomena) can be created (discovered/understood) Wednesday, February 5, 14 Why study Applied Maths? • intellectual challenge • obtain general understanding of physical phenomena and the world around us • be able to make prediction about physical processes • development of general tools to be applied to other fields Wednesday, February 5, 14 Historical backdrop Some famous thoughts on Applied Maths 'Eureka, Eureka' (Archimedes) c. 287 BC - c. 212 BC Wednesday, February 5, 14 Some famous thoughts on Applied Maths 'Mathematic is written for mathematicians' (Nicolaus Copernicus) 19 February 1473 – 24 May 1543 1543 Some famous thoughts on Applied Maths? 'It would be better for the true physics if there were no mathematicians on earth' (Daniel Bernoulli) 8 February 1700 – 17 March 1782 Wednesday, February 5, 14 Some famous thoughts on Applied Maths 'Now I will have less distraction' (Leonhard Euler, upon losing the use of his right eye) 15 April 1707 – 18 September 1783 Some famous thoughts on Applied Maths 'I do not know' (Joseph-Louis Lagrange, summarizing his life's work) 25 January 1736 - 10 April 1813 Wednesday, February 5, 14 Some famous thoughts on Applied Maths 'Nature laughs at the difficulties of integration' (Pierre-Simon Laplace) 23 March 1749 – 5 March 1827 Wednesday, February 5, 14 Some famous thoughts on Applied Maths? 'Mathematicians are born, not made' (Henri Poincaré) 29 April 1854 – 17 July 1912 Wednesday, February 5, 14 Some famous thoughts on Applied Maths? 'Prediction is very difficult, especially about the future' (Niels Bohr) 7 October 1885 – 18 November 1962 Wednesday, February 5, 14 Some famous thoughts on Applied Maths? 'It is more important to have beauty in one's equations than to have them fit experiment' and 'This result is too beautiful to be false' (Paul Dirac) 8 August 1902 – 20 October 1984 Wednesday, February 5, 14 Some famous thoughts on Applied Maths? 'To those who do not know mathematics it is difficult to get across a real feeling as to the beauty, the deepest beauty, of nature' (Richard Feynman) May 11, 1918 – February 15, 1988 BSc 1939 Wednesday, February 5, 14 Course overview 18.354 • ODEs • linear/nonlinear PDEs for scalar & vector fields • perturbation theory • calculus of variations • Fourier transformations • complex numbers, conformal maps Dimensional analysis G.I. Taylor 1886-1975 Trinity nuclear test, July 1945 Life Magazine, August 20, 1945 The formation of a blast wave by a very intense explosion. II. The atomic explosion of 1945 BY SIR GEOFFREYTAYLOR,F.R.S. (Received 10 November 1949) [Plates 7 to 9] Photographs by J. E. Mack of the first atomic explosion in New Mexico were measured, and the radius, R, of the luminous globe or 'ball of fire' which spread out from the centre was determined for a large range of values of t, the time measured from the start of the explosion. The relationship predicted in part I, namely, that RI would be proportional to t, is surprisingly accurately verified over a range from R = 20 to 185 m. The value of R~t-1 so found was used in conjunction with the formulae of part I to estimate the energy E which was generated in the explosion. The amount of this estimate depends on what value is assumed for y, the ratio of the specific heats of air. Two estimates are given in terms of the number of tons of the chemical explosive T.N.T. which would release the same energy. The first is probably the more accurate and is 16,800 tons. The second, which is 23,700 tons, probably overestimates the energy, but is included to show the amount of error which might be expected if the effect of radiation were neglected and that of high temperature on the specific heat of air were taken into account. Reasons are, given for believing that these two effects neutralize one another. After the explosion a hemispherical volume of very hot gas is left behind and Mack's photographs were used to measure the velocity of rise of the glowing centre of the heated volume. This velocity was found to be 35 m./sec. Until the hot air suffers turbulent mixing with the surrounding cold air it may be expected to rise like a large bubble in water. The radius of the 'equivalent bubble' is calculated and found to be 293 m. The vertical velocity of a bubble of this radius is I /(g 29300) or 35-7 m./sec. The agreement with the measured value, 35 m./sec., is better than the nature of the measurements permits one to expect. COMPARISON WITH PHOTOGRAPHIC RECORDS OF THE FIRST ATOMIC EXPLOSION Two years ago some motion picture records by Mack (I947) of the first atomic explosion in New Mexico were declassified. These pictures show not only the shape of the luminous globe which rapidly spread out from the detonation centre, but also gave the time, t, of each exposure after the instant of initiation. On each series of photographs a scale is also marked so that the rate of expansion of the globe, or 'ball of fire', can be found. Two series of declassified photographs are shown in figure 6, plate 7. These photographs show that the ball of fire assumes at first the form of a rough sphere, but that its surface rapidly becomes smooth. The atomic explosive was fired at a height of 100 ft. above the ground and the bottom of the ball of fire reached the ground in less than 1 msec. The impact on the ground does not appear to have disturbed the conditions in the upper half of the globe which continued to expand as 1/5 a nearly perfect luminous hemisphere bounded by a sharp edge which must be taken as a shock wave. This stage of the expansion is shown in figure 7, plate 8 which corresponds with t = 15 msec. When the radius R of the ball of fire reached about 130 m., the intensity of the light was less at the outer surface than in the interior. At ⇤ ln R = ln c Vol 201. A. E [ 175 ] ⇥ ⌅ 2 + ln t 5 12 the two objects that is proportional to the product of t y proportional to the square of the separation of the tw Hamiltonian dynamics & e statement, itKepler’s is possible toproblem derive Kepler’s laws. s nd 2 law aw is the simplest to derive, and is a statement that the ving under a central force, such as gravity, is constant. um L of a particle with mass m and velocity u is dr L=r⇥m , dt ector position of the particle. The rate of change of ang Brownian motion By dividing both sides by A ⌧ the number of particles per un is seen to increase at the rate1 Random walks & diffusion n(x, t + ⌧ ) ⌧ In the limit ⌧ ! 0 and n(x, t) = @Jx @2n = D 2, @x @x For strictly one-dimensional systems, the boundary area is just a poin 7 Mark Haw David Walker Jx ! 0, this becomes @n = @t 1 [Jx (x + /2, t) Pattern formation dunkel@math.mit.edu e to the static angle of repose and subseque Zebra vs. granular media dunkel@math.mit.edu arxiv: 1208.4464 2d Swift-Hohenberg model tropic fourth-order model for a non-conserved scalar or ! reflection-symmetry ⇤(t, x), given by b = 2 ⇤ ⇤, (1) 2 2 0 ⇤t ⇥ =2 U (⇥) + 0 ⇥2 ⇥ (⇥ ) ⇥ 2 he time derivative, and ⇤ = ⌅2 is the d-dimensional ved from a Landau-potental U (⇤) a 2 b 3 c 4 U (⇤) = ⇤ + ⇤ + ⇤ , 2 3 4 (2) e rhs. of Equation (1) can also be obtained by variational ed energy functional. In the context of active suspensions, x) =fluctuations, ⇥ v tify local (t, energy local alignment, phase will assume throughout that the system is confined to d of volume (3) 0 trajectory of a swimming cell can arxiv:exhibit 1208.4464a pre 2d Swift-Hohenberg example,model the bacteria Escherichia coli [40] an broken tropic fourth-order model for a non-conserved scalar or ⌃ ij denotes the Cartesian components of the Levi-Civ reflection-symmetry ⇤(t, x), given by a summation convention for equal indices throughout. 2 b(1)= ⇤ ⇤, 2 2 2 0 2 ⇤t ⇥ = U (⇥) + 0 ⇥ ⇥ 2 (⇥ ) ⇥ he time derivative, and ⇤ = ⌅2 is the d-dimensional ved from a Landau-potental U (⇤) a 2 b 3 c 4 U (⇤) = ⇤ + ⇤ + ⇤ , 2 3 4 (2) e rhs. of Equation (1) can also be obtained by variational ed energy functional. In the context of active suspensions, x) =fluctuations, ⇥ v tify local (t, energy local alignment, phase will assume throughout that the system is confined to d of volume (3) 0 Calculus of variations What is the di↵erential equation satisfied by Y (x)? To answer this question, we compute the functional derivative What is the di↵erential equation satisfied by Y (x)? To answer this question, we compute he functional derivativeI[Y ] = lim 1 {I[f (x) + ✏ (x y)] I[f (x)]} ✏!0 ✏ Y Z x12 I[Y ] @f (x) + ✏ (x @fy)]0 I[f (x)]} == lim {I[f (x y) + (x y) dx ✏!0 ✏ Y 0 @Y Z xx21 @Y Z x2 @f @f 0 @f (x d y) @f+ = (x y) dx 0 = x1 @Y (x y)dx. (17) @Y 0 dx @Y Z xx21 @Y @f d @f = the Euler-Lagrange equations (x y)dx. (17) Equating this to zero, yields 0 @Y dx @Y x1 @f d @f 0 = Equating this to zero, yields the Euler-Lagrange equations @Y dx @Y 0 (18) @fY =d0 @f It should be noted that the condition I/ alone is not a sufficient condition for 0 = (18) 0 dx @Y a maximum. It is often possible, a minimum. In fact, the relation might @Y even indicate however, convince that no maximum for is thenot integral (e.g., the distancefor t should betonoted thatoneself the condition I/ Y = exists 0 alone a sufficient condition along a smooth path can be made as long as we like), and that our solution is a minimum. minimum. In fact, the relation might even indicate a maximum. It is often possible, To be rigorous, however, one should also consider the possibility that the minimum is merely owever, to convince oneself that no maximum exists for the integral (e.g., the distance a local minimum, or perhaps the relation I/ Y = 0 indicates a point of inflexion. long a smooth path can be made as long as we like), and that our solution is a minimum. Elasticity Elasticity photo: Andrej Kosmrlj http://newsoffice.mit.edu/2015/predicting-wrinkles-fingerprints-curved-surfaces-0202 collaboration with Reis lab (MechE) Surface tension Goldstein lab, Cambridge Large drop in microgravity e V is Z Z d/dt inside the integral sign we must take account pndS = rpdV. (7) of this. The Reyno ⇢u · ndS, ⇢dV.so, and it can be shown that for a deforming, (1)incompressible flu does V Z S Z S(t) are being deformed by the motion of the fluid, so if we want to take the d Du eaving this volume through the bounding surface S is where u(x, t) is the velocity of the fluid and=n is the norm ⇢udV ⇢ outward dV ntegral sign we must take account of this. The Reynolds transport theorem dt V (t) V (t) Dt Z Z Z can be shown that for a deforming, incompressibleZfluid element @⇢ ⇢u · ndS, Here dV = ⇢u · ndS(2) = r · (⇢u)d Z Z S V @t S V d Du Hydrodynamics D @ (8) ⇢udV = ⇢ dV = have + (u · r) Dt (t) V (t) the velocity ofdttheV fluid and is thehold outward normal. Hence we Thisn must for any arbitrary fluid element dV , thus Dt @t Z Z Z is called the convective derivative, and @⇢we shall discuss it’s significance @⇢ + r · (⇢u) dV = ⇢u · assuming ndS = thatr (3) = 0. F ·is(⇢u)dV. solely given by gravity, @t D S@ V @t V = + (u · r) Z Z(9) Dt @tThis is called the continuity equation. Du ⇢ dV = ( rp + ⇢g)dV for any arbitrary fluid element dV , thus V (t) Dt (t) For fluids like water, the density does not Vchange very much and vective derivative, and we shall discuss it’s significance in a moment. Hence, @⇢ to neglect the density variations. If we make this approximation F is solely given by gravity, + r Since · (⇢u)this = 0.must hold for any arbitrary fluid element (4) we arrive at to the incompressibility condition @t reduces Z Z Du Du (10) rp ⇢ dV = ( rp + ⇢g)dV = + g. e continuity Vequation. Dt Dt r · u⇢ = 0. (t) V (t) e water, the density does not change very much and we will often be tempted This, combined with equation (4), constitutes Eu hold forvariations. any arbitrary fluid element we arrive at the the ensity If we make this approximation thecontinuity continuity equation Like all approximations, this one is sometimes very good andthesom can be tidied up a little if we realise that the gravitational force, being ncompressibility condition will have to figure out where it fails. Du rp = written + g.as the gradient of a scalar potential r (11). It is therefore usual t Dt p⇢+ ! p. This implies that gravity simply modifies the pressure di r6.1.2 ·u = 0.Momentum (5) equations and does nothing to change the velocity. However, we cannot do this with the the continuity equation (4),have constitutes the Euler equations. Things if we a free surface (as we shall see later with water waves). mations, this one is sometimes very good and sometimes not so good. We So far we have more unknowns than equations (three velocity co p a little if we realise that the gravitational force, being conservative, can be Assuming the density is constant means we now have four equatio er Summer School 2011: Introduction to Low Reynolds Number Locomotion from Peko Hosoi’s Lecture) Typical Reynolds numbers Reynolds Numbers in Biology ynolds number is dimensionless group that characterizes the ratio of inertial to viscous t is defined as Macro⇥U L UL Re = = µ world is the density of the medium the organism is moving through; µ is the dynamic viscosity medium; is the kinematic viscosity; U is a characteristic velocity of the organism; and L racteristic length scale. When we discuss swimming biological organisms, we are usually to creatures that are moving through water (or through a fluid with material properties se to those of water). This means that the material properties µ and ⇥ are fixed1 and the s number is roughly determined by the size of the organism. al, the characteristic size of the organism and the characteristic swimming velocity are As a rule-of-thumb, the characteristic locomotion velocity, U , in biological organisms is o L by U L/second e.g. for people L 1 m and we move at U 1 m/s; bugs are about mm, and they move at about U 1 mm/s; for microorganisms L 100 µm and U 100 Obviously this is a very very very very rough estimate and one does not have to think very come up with exceptions (as is always the case in biology!). However, it serves as a good point to estimate the Reynolds numbers for various biological organisms as illustrated in ch in Figure ??. Note that even for organisms as small as ants, the Reynolds number is the order of 1 (which is not very low). In this lecture we will focus on Re ⇥dunkel@math.mit.edu 1 which is meters lds Numbers in Biology What happens at low Reynolds numbers ? number is dimensionless group that characterizes the ratio o fined as ⇥U L UL Re = = µ density of the medium the organism is moving through; µ is t ; is the kinematic viscosity; U is a characteristic velocity of stic length scale. When we discuss swimming biological organ eatures that are moving through water (or through a fluid with hose of water). This means that the material properties µ and ber is roughly determined by the size of the organism. e characteristic size of the organism and the characteristic sw rule-of-thumb, the characteristic locomotion velocity, U , in bi y U L/second e.g. for people L 1 m and we move at U 1 d they move at about U 1 mm/s; for microorganisms L 1 dunkel@math.mit.edu Low-Re (laminar) flow ertial (dynamic) pressure ⇤U 2 and viscous shearing str µU/L can be characterized by the Reynolds number4 Swimming at low Reynolds number R ⌅ U L⇤/µ = U L/⇥. Example: Swimming in water with ⇥ = 10 ( 6 m2 /s • fish/human: L ⌅ 1 m, U ⌅ 1 m/s ⇧ R ⌅ 106 . R • bacteria: L ⌅ 1 µm, U ⌅ 10 µm/s ⇧ R ⌅ 10 U L⇥/ ⇥ 1 Geoffrey Ingram Taylor 5 James Lighthill If the Reynolds number is very small, R ⇥ 1, t NSE (8) can be approximated by the Stokes equations 0 = µ ⌥2 u 0 = ⌥ · u. ⌥p + f , (10 (10 These equations+must still be endowed time-dependent BCs with appropria initial and boundary conditions, such as ,e.g.,6 Edward Purcell u(t, x) = 0, as Shapere & Wilczek p(t, x) = p⇥1987 , PRL |x| ⇤⌃ . (1 Superposition of singularities 2x stokeslet = symmetric dipole stokeslet rotlet -F F r̂ · F p(r) = + p0 2 4⇥r (8⇥µ) 1 vi (r) = [ ij + r̂i r̂j ]Fj r flow ~ r 1 F r 2 ‘pusher’ r 2 E.coli (non-tumbling HCB 437) Drescher, Dunkel, Ganguly, Cisneros, Goldstein (2011) PNAS Results w of itswe “puller” image. d. field Atwalls, distances rfocused <6µ mon theadipole model overestimates the bacterial flow field. (E) Experimentally measured flow to plane 50 µm from the top and bottom tancesurfaces 2 µm parallel to sample the wall. chamber, (F) Best fitand force-dipole model, (G) residual field. Notewhere the Bacterial cell body, of the recorded ∼ 2 and terabytes of flow 4 flowmovie fieldResults ofdata. an E. In coli “pusher” decays much faster, when swims close thecule surface, fortothe length of t theflow mea this data we identified ∼ 10 events when (non-tumbling HCB 437)rarea bacterium achieved by fittin theByTo measured andminisbest-fit force cellsBacterial swam in the for > surfaces. 1.5 s. tracking decays labeled, n flowfocal fieldplane far from resolve the the at variable locatio fluidcule tracers in each of the rare events, relating their position of the c decays of the flow speed u with flow field created by individual bacteria, we tracked gfp- sion of flu m surfaces. To resolve the minisfield (r >field 8 µm). and labeled, velocity to the position and orientation of the bacterium, dis non-tumbling E. coli as they swam through a suspenof the cell body (Fig. 1D) illustr walls, we dividual bacteria, tracked gfp- overforce the measured andaverage best-fit dipole we field The the 1C). specific fitting and performing anwe ensemble all tracers, re- (Fig. Howeve sionswam ofdecays fluorescent tracer particles. For measurements farcharacteristic from field displays the 1 dipole length ℓ =o of the flow speed u with distance r from thesurfaces center i as they through a suspensolved the time-averaged flow field in the E. coli swimming he minismeasure walls, we focused on a plane 50 µm from the top and bottom value of F is cons movie dat However, the force dipole flow sig oftothe cell (Fig. 1D) illustrate that the flow down 0.1% of body the mean swimming speed V0 = 22 ± 5 measured ticles. For measurements far from ckedplane gfpcell force bod surfaces of the sample chamber, and recorded ∼ 2 terabytes of 2 and resistive µm/s. As E. colidisplays rotate about their swimming direction, their cells field the characteristic 1/r4 decay oftoa the forceside dipole. measured flow ofswam thel a 50 suspenµmmovie from the top and bottom for the data. In field this in data we dimensions identified ∼is10cylindrically rare events when note that in the b time-averaged flow three fluid trace However, the force dipole flow significantly overestimates the cell1.5body, where the flow magnit far from achieved ber, and recorded ∼ 2 terabytes of cells swam in the focal plane for > s. By tracking the µm behind the ce symmetric.measured Our measurements capture all components of this flow to when the side of the cell body,ofand behind the 4 and veloci didentified bottom for the length the flagellar bun at varia fluid drag on the ∼ 10 rare events fluid tracers in each of the rare events, relating their position cylindrically symmetric flow, except the azimuthal flow due to cell body, where the flow magnitude u(r) is nearly constant and perfo abytes of field (r f achieved by fitting two opposite and velocity to the position and orientation of the bacterium, the rotation of the cell about its the body axis. The topology of ne for > 1.5fors. By tracking the length of the flagellar bundle. The force dipole fit was solved the nts when the spec the measured flow field (Fig. 1A) is the same as that of a and performing an ensemble average over all tracers, we reat variable locations along the sw are events, relating their position Bacterial flowdow fiel achieved by1B), fitting two opposite force monopoles (Stokeslets) dipole le plane kingforce the dipole flowtime-averaged (Fig. defined by solved the flow field in the E. coli swimming field (r > 8 µm). From the best and orientation of the bacterium, dipole flow descri at variable locations along the swimming direction to the far value of position µm/s. As plane down to 0.1% of the mean swimming speed V = 22 ± 5 0 the specific fitting routines fi with good and accura h i e average over all tracers, we refield (r > 8 µm). From the best fit, which is insensitive to and resi A E. coli r direction, their time-avera ℓF swimming acterium, µm/s. As 2rotate about their ˆ thisµm approximatio u(r)in= the 3(r̂.coli d) − 1 r̂, routines A= , r̂fitting = length , regions, [ℓ 1 ]= we dipole 1.9 and dip ow field E. swimming 2 the specific fitting and obtain the note tha2 |r| 8πηdimensions |r| is cylindrically s, we retime-averaged flow field in three symmetric a wall. Focusing value F is Fconsistent with opt dipole length 1.9±µm and dipole force = of 0.42 pN. This µm beh mean swimming speed V0 ℓ==22 5 capture symmetric. Our measurements allof components this wimming and applying the cylindrica and resistive force theory calculat fluid dra of Fforce, is consistent with optical trap measurements [45] where F isvalue the dipole ℓ the distance separating the force ut direction, their symmetric flow, except the azimuthal flow due to the = their 22 ±cylindrically 5swimming resulted in a sligh rotati note thatThe in the best fit, the cell and resistive force theory calculations [46]. It is interesting to η the viscosity the fluid, dˆ the orientation vector theof flow field struc the rotation ofisof the cell about itsunit body axis. topology on, pair, their three dimensions cylindrically the measu (swimming direction) the best bacterium, and rbehind the distance surfaces, the note thatflow inofthe fit, 1A) theµm cell drag Stokeslet isfrom 0.1 the measured field (Fig. is the same as that oflocated aof the center the cell ndrically nts capture all components of this force dipo Bacteria vector relative to the center of the dipole. Yet there are some ity of a no-slip sur µm behind the center of the cell body, possibly reflecting the force dipole flow (Fig. 1B), defined by fluid drag on the flagellar bundle tsexcept of this flow due Dunkel, Ganguly, Cisneros, Goldstein (2011) to PNAS Fig. differences 1.Drescher, Averagethe flow fieldazimuthal created by a single freely-swimming bacterium. (A) Experimentally measured flow field far from a surface. Stream lines indicate local direction offl dipole close to the cell body as shown by the residual of outward streamlin fluid drag on the flagellar bundle. flow. (B) Best fit force-dipole model, and (C) residual flow field, obtained by subtracting the best-fit dipole from the experimentally measured field. The presence of the flagella E.coli w due to weak ‘pusher’ dipole Volvox carteri somatic cell cilia 200 ㎛ daughter colony Drescher et al (2010) PRL dunkel@math.mit.edu Singular perturbations y di↵erential equation (Acheson, pp ntial equation 2 d u du ✏ 2+ = 1. dx dx t argue that we can neglect this term, the s u = x + C. e,the using the Cauchy-Riemann we see that Z-plane uniform flow pastequations a more wing-like shape? (Note that we have d some technical details here, such2as the requirement that dF/dz 6= 0 at any 2 @ @ @ @ would cause+a blow-up of the + velocity).= 0. = (6) 2 2 @x @y @x@y @y@x Conformal mappings le conformal maps e used for . We can therefore consider any analytic function (e.g., he real map is and imaginary parts and both of them satisfy Laplace’s equaZ = F (z) = z + b, (12) onents u and v are directly related to dw/dz, which is conveniently onds to a translation. Then there is dw @ @ = +Zi = F= u= ze iv.i↵ , (7) (13) (z) dz @x @x consider at anangle angle↵.↵ to The corresponding onds to auniform rotationflow through In the this x-axis. case, the complex potential for i↵ . Using the above relation, wast = au0cylinder ze i↵ . In this case dw/dz = u e 0 making angle ↵ with the stream is u0 cos ↵ and v = u0 sin✓↵. ◆ 2 R past i mine the complex potential for since we know that i↵ flow i↵ a cylinder W (Z) = u0 Ze + e ln Z. (14) Z 2⇡ ✓ ◆ R2 = u r + cos ✓, (8) i↵ 0 s expression could also include the term ln e = i↵ which I have neglected. r constant however and doesn’t change the velocity. al the complex potential re part is theofnon-trivial Joukowski transformation, ✓ ◆ 2 2 completely controls the dynamics. In the process of deriving this result we rn about a rather remarkable phenomenon in rotating fluid dynamics. Rotating flows The Taylor-Proudman theorem r a fluid rotating with angular velocity ⌦. The equation of motion in the fram e rotating with the fluid is @u 1 + u · ru + ⌦ ⇥ (⌦ ⇥ r) = rp⌦ + ⌫r2 u @t ⇢ r · u = 0. 2⌦ ⇥ u, re two additional terms: the first ⌦ ⇥ (⌦ ⇥ r) is the centrifugal acceleration, w e discussed before. This can be thought of as an augmentation to the pre tion, using the identity Taylor columns, 1 etc ⌦ ⇥ (⌦ ⇥ r) = 2 r(⌦ ⇥ r)2 . rth, we will simply absorb this into the pressure by writing p = p⌦ ⇢ r(⌦ ⇥ r)2 . http://ocw.mit.edu/ Taylor - column Solitons KdV equation Solitons credit: Christophe Finot the droplet surfaces to produce highly active two-dimensional nematic liquid crystals whose streaming flows are controlled by internally generated fractures and self-healing, as well as unbinding and annihilation of oppositely charged disclination defects. The resulting active emulsions exhibit unexpected properties, such as autonomous motility, which are not observed in their passive analogues. Taken together, these observations exemplify how assemblages of animate microscopic objects exhibit collective biomimetic Active matter a + + b PEG Depletion force Microtubules Kinesin clusters + Motor Time force c Dogic lab (Brandeis) Nature 2012 d to these d bundled m ters simul thus enha Motorrelative p microtub between m Active matter Dogic lab (Brandeis) Nature 2012 Active nematics REVIEW LETTERS week ending 31 MAY 2013 FIG. 2 (color online). Defect pair production in an active suspension of -1/2 microtubules and kinesin +1/2 (top) and the same phenomenon observed in our numerical simulation of an extensile nematic fluid with " ¼ 100 and ! ¼ "0:5. The experimental picture is reprinted with permission from T. Sanchez et al., Giomi et al PRL 2012 Nature (London) 491, 431 (2012). Copyright 2012, Macmillan. • optical effects • work hardening, etc Π2 (S 2 ) = Z. (7) Here, the 2 subscript says that we’re studying the second Homotopy group. It represents the fact that we are surrounding the defect with a 2-D spherical surface, rather number: FIG. 12: (a) Hedgehog defect. Magnets have no line defects (you can’t lasso a basketball), but do have point defects. ⃗ (x) = M0 x̂. You can’t Here is shown the hedgehog defect, M surround a point defect in three dimensions with a loop, but you can enclose it in a sphere. The order parameter space, remember, is also a sphere. The order parameter field takes the enclosing sphere and maps it onto the order parameter space, wrapping it exactly once. The point defects in magnets are categorized by this wrapping number: the second Homotopy group of the sphere is Z, the integers. (b) Defect line in a nematic liquid crystal. You can’t lasso the sphere, but you can lasso a hemisphere! Here is the defect corresponding to the path shown in figure 5. As you pass clockwise around the defect line, the order parameter rotates counterclockwise by 180◦ . This path on figure 5 would actually have wrapped around the right–hand side of the hemisphere. Wrapping around the left–hand side would have produced a defect which rotated clockwise by 180◦ . (Imagine that!) The path in figure 5 is halfway in between, and illustrates that these two defects are really not different topologically. ( Finally, why are these defect categories a group? group is a set with a multiplication law, not necess FIG. 13: Multiplying two loops. The product of two loo is given by starting from their intersection, traversing the fi loop, and then traversing the second. The inverse of a lo is clearly the same loop travelled backward: compose the t and one can shrink them continuously back to nothing. T definition makes the homotopy classes into a group. This multiplication law has a physical interpretation. If t defect lines coalesce, their homotopy class must of course given by the loop enclosing both. This large loop can deformed into two little loops, so the homotopy class of t coalesced line defect is the product of the homotopy clas of the individual defects. Two parallel defects can coalesce and heal, even thou each one individually is stable: each goes halfway arou the sphere, and the whole loop can be shrunk to zero. Π1 (RP 2 ) = Z2 . than the 1-D curve we used in the crystal.[13] You might get the impression that a strength 7 def is really just seven strength 1 defects, stuffed togeth You’d be quite right: occasionally, they do bunch but usually big ones decompose into small ones. T doesn’t mean, though, that adding two defects alwa gives a bigger one. In nematic liquid crystals, two li defects are as good as none! Magnets didn’t have a line defects: a loop in real space never surrounds so thing it can’t smooth out. Formally, the first homoto group of the sphere is zero: you can’t loop a basketb For a nematic liquid crystal, though, the order para ter space was a hemisphere (figure 5). There is a lo on the hemisphere in figure 5 that you can’t get rid by twisting and stretching. It doesn’t look like a lo but you have to remember that the two opposing poi on the equater really represent the same nematic orie tation. The corresponding defect has a director field which rotates 180◦ as the defect is orbited: figure 1 shows one typical configuration (called an s = −1/2 fect). Now, if you put two of these defects together, th cancel. (I can’t draw the pictures, but consider it a ch lenging exercise in geometric visualization.) Nematic li defects add modulo 2, like clock arithmetic in elementa school: Topological defects are discontinuities in order-parameter fields "umbilic defects" in a nematic liquid crystal “Quantum” HD Couder lab (Paris) Bush group