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Locomotion over a washboard Sui, Yi 2015

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Locomotion over a WashboardbyYi SuiB.A., University of California, Berkeley, 2012A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Mathematics)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)January 2015© Yi Sui 2015AbstractThe purpose of this thesis is to study the problem when a microorganismswims very close to a shaped boundary. In this problem, we model the swim-mer to be a two-dimensional, infinite periodic waving sheet. For simplicity,we only consider the case where the fluid between the swimmer and thewashboard is Newtonian and incompressible. We assume that the swimmerpropagates waves along its body and propels itself in the opposite direction.We consider two cases in our swimming sheet problem and the lubricationapproximation is applied for both cases. In the first case, the swimmer has aknown fixed shape. Various values of wavenumber, amplitude of the restor-ing force and amplitude of the topography were considered. We found theinstantaneous swimming speed behaved quite differently as the wavenumberwas varied. The direction of the swimmer was also found to depend on theamplitude of the restoring force. We also found some impact of the topo-graphic amplitude on the relationship between average swimming speed andthe wavenumber. We extended the cosine wave shaped washboard to be amore general shape and observed how it affected the swimming behaviour.In the second case, the swimmer is assumed to be elastic. We were inter-ested to see how different values of wavenumber, stiffness and amplitude ofthe restoring force could change the swimming behaviour. With normalizedstiffness and wavenumber, we found the swimmer remained in a periodicstate with small forcing amplitude. While the swimmer reached a steadystate with unit swimming speed for high forcing amplitude. However, forother values of stiffness and wavenumber, we found the swimmer’s swimmingbehaviour was very different.iiPrefaceThis thesis is original, unpublished, independent work by the author, Y. Suiunder the supervision of Dr. N. J. Balmforth.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . xDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Literature Review . . . . . . . . . . . . . . . . . . . . . . . . 31.2 Our Main Problem . . . . . . . . . . . . . . . . . . . . . . . 81.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . 112 Fixed Shape Swimmer with Lubrication Theory Analysis 122.1 Lubrication Theory Model . . . . . . . . . . . . . . . . . . . 122.1.1 Periodicity in Pressure Condition . . . . . . . . . . . 182.1.2 Zero Net Force Condition . . . . . . . . . . . . . . . . 192.2 Swimming with a Fixed Shape . . . . . . . . . . . . . . . . . 202.2.1 Swimming Speed for Different Wavenumbers . . . . . 222.2.2 Swimming with High Amplitude Force . . . . . . . . 252.2.3 Average Speed of the Swimmer . . . . . . . . . . . . 272.2.4 General Shape of the Topography . . . . . . . . . . . 29ivTable of Contents3 Elastic Swimmer with Lubrication Theory Analysis . . . . 343.1 Mathematical Formulation . . . . . . . . . . . . . . . . . . . 343.2 Swimming with Finite Stiffness . . . . . . . . . . . . . . . . . 373.2.1 Steady Swimming . . . . . . . . . . . . . . . . . . . . 403.2.2 Larger Stiffness . . . . . . . . . . . . . . . . . . . . . 433.2.3 Various Values of Wavenumber . . . . . . . . . . . . . 474 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 554.1 Summary of Results . . . . . . . . . . . . . . . . . . . . . . . 554.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . 58Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60AppendixA Details of Integral Evaluation . . . . . . . . . . . . . . . . . . 64vList of Figures1.1 A sketch of the geometry of the problem. . . . . . . . . . . . 102.1 A plot of exact instantaneous swimming speed, U , againstthe phase translation with respect to the washboard, φ(t),for a = 0.5, b = 0.4 and κ = 1. . . . . . . . . . . . . . . . . . . 232.2 Plot of instantaneous swimming speed, U , against the phasetranslation with respect to the washboard, φ(t), for a = 0.5,b = 0.4 and κ = 1 (blue), 2 (red), 1/2 (black), 4/5 (blackwith crosses) and 4/9 (green). . . . . . . . . . . . . . . . . . . 242.3 Plot of instantaneous swimming speed, U , against the phasetranslation with respect to the washboard, φ(t), for b = 0.4,κ = 1 with various values of a indicated. The arrows indicatethe moving direction of the swimmer and the two stars on theline a = 1 indicate the stable and unstable equilibria. . . . . . 262.4 Bifurcation diagram of two equilibria phases against the var-ious value of a. The stable solutions are indicated in redcrosses, while the unusable solution in black circles. . . . . . . 272.5 A plot of average speed, U¯ , against various values of wavenum-ber, κ, with a = 0.5 and b = 0.4. Some values of thewavenumber are labelled in the figure. . . . . . . . . . . . . . 292.6 A plot of average speed, U¯ , against various values of topo-graphic amplitude, b, with a = 0.5 and different wavenumbersindicated. Note κ = 4/9 is in red and κ = 4/5 is in green. . . 302.7 A zoomed-in plot of average speed U¯ for κ = 1/n, for n =2, 3, . . . , 6 (shown from top to bottom) as b→ 12 . . . . . . . . 302.8 Geometry of the problem for α = 0.5. . . . . . . . . . . . . . 31viList of Figures2.9 Geometry of the problem for α = 2. . . . . . . . . . . . . . . 312.10 A plot for instantaneous swimming speed, U(t), against phasetranslation with respect to the solid wall, φ(t), for a = 0.5 ,b = 0.4 and α = 0 (solid), α = 0.5 (dashed) and α = 2(dash-dotted). . . . . . . . . . . . . . . . . . . . . . . . . . . 322.11 A plot for average swimming speed, U¯ , against various valuesof the amplitude for the topology, b, for a = 0.5, and α = 0(solid), α = 0.5 (dashed) and α = 2 (dash-dotted). . . . . . . 333.1 A plot of instantaneous swimming speed, U(t), for variousvalues of A with D = 1, κ = 1 and b = 0.4. The solid curveis for A = 2, dashed curve for A = 3 and dashed and dottedcurve for A = 3.8. . . . . . . . . . . . . . . . . . . . . . . . . . 373.2 A plot displays successive snapshots of the final periodic statefor various values of A with D = 1, κ = 1 and b = 0.4. Panel(a) is for A = 2, (b) for A = 3 and (c) for A = 3.8. . . . . . . 383.3 Mean speed for the initial value problem against A with D =1, κ = 1 and b = 0.4. The circles indicate the numericalsolutions with topography, and the dashed curve indicatesthe analytical solutions without topography. . . . . . . . . . . 393.4 Mean speed for the initial value problem against topographicamplitude b with D = 1, κ = 1, and three different values ofA are indicated. (A = 2 circles, A = 3 stars and A = 3.8 dots). 403.5 The position eigenvalue, X, of two steady state solution branchesagainst A for D = 1, κ = 1 and b = 0.4. The upper branch isthe stable solution, while the lower one is unstable. . . . . . . 433.6 Plots of final steady state snapshots of Y and h for D = 1,κ = 1 and b = 0.4 with two different values of A. The stablesteady state solution is shown by a solid line and the unstablesolution is shown by a dashed line. . . . . . . . . . . . . . . . 433.7 Plots of instantaneous swimming speed over time with D =100, κ = 1 and b = 0.4. Panel (a) is the result for a = 0.4,(b) for a = 0.8 and (c) for a = 1. . . . . . . . . . . . . . . . . 44viiList of Figures3.8 Plots show the translation of the swimmer Y over the wavywall in time for D = 100, κ = 1 and b = 0.4 with two differentvalues of a. Panel (a) is the result for a = 0.4 and (b) for a = 0.8. 453.9 A plot of steady state solutions for a = 1 when D = 100,κ = 1 and b = 0.4. The stable branch is illustrated in solid,and the unstable one is dashed. . . . . . . . . . . . . . . . . . 453.10 Mean swimming speed against various forcing amplitude afor large stiffness swimmer with κ = 1 and b = 0.4. Dots andcircles are the solutions with topography for D = 100 andD = 1 respectively. Dashed line are the solutions withouttopography (upper one for D = 100 and lower one for D =10). The solid line is the solution for infinite stiffness. . . . . 463.11 A plot of two steady state solutions against D when the forc-ing amplitude a kept to be 1, κ = 1 and b = 0.4. The upperbranch is a stable solution and the lower one is unstable. . . . 473.12 A plot show mean speed against b of a = 0.5 for four differentvalues of wavenumber (κ = 1/2 is represented by dots, κ = 1by circles, κ = 4/5 by stars and κ = 2 by crosses) withD = 1000. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 483.13 Solutions for κ = 2 and D = 1 versus varying A. Panel (a)shows mean speeds for the initial value problem. The starsare the result for κ = 2 and b = 0.4, the solid line showsthe result for κ = 2 without topography, and the dashed lineshows the result for κ = 1 and b = 0.4. Panel (b) displaysthe positional shift, X, versus varying A. The solid branchis the stable steady state solution and the dashed line is theunstable steady state solution. . . . . . . . . . . . . . . . . . 503.14 Profiles of the swimmer for two special choices of A. Panel(a) demonstrates the periodic solution for A = 5 and panel(b) displays the stable steady state solution (solid) and theunstable steady state solution (dashed) for A = 8. . . . . . . 50viiiList of Figures3.15 Solutions of the initial value problem for κ = 1/2 when D = 1.Panel (a) shows the mean speed for various A. Stars indicatethe stable solutions with b = 0.4, and plus are unstable solu-tions with b = 0.4. Solid line is the stable solution withoutwashboard and dashed line is the unusable solution withoutwashboard. The dash-dotted line is the solution for κ = 1.Panel (b) displays the swimmer’s profile for the periodic so-lution when A = 3. . . . . . . . . . . . . . . . . . . . . . . . . 513.16 Periodic solutions of κ = 1/2 with D = 1, A = 6 and b =0.4. Panel (a) shows the evolution of the swimmer’s surface,Y (ξ, t), on the (ξ, t) plane. Panel (b) displays the unstableperiodic profile for A = 6. Panel (c) shows the stable periodicprofile for A = 6. . . . . . . . . . . . . . . . . . . . . . . . . . 523.17 The solutions of the initial value problem for κ = 1/2 withD = 10 and D = 100. Panel (a) shows mean speed forD = 10 against a. Panel (b) shows mean speed for D = 100against a. In both panel (b) and (c), the stars mark the stablesolutions with topography. The solid and dashed line showthe stable and unstable periodic solutions without topographyrespectively. The dash-dotted line displays the result for κ =1. The inset of (b) shows the positional shift, X, of the fourlocked state solutions versus the forcing amplitude a for theswimmer swims with topography when D = 100. Panel (c)shows the four steady state solution profiles for A = 105 andD = 100. Only the solid curve is the stable solution and therest are three unstable solutions. . . . . . . . . . . . . . . . . 54ixAcknowledgementsI would like to thank my supervisor, Dr. Neil Balmforth, for all his generoussupport, help, and patience through the completion of this thesis work.Thanks to Neil for introducing me to the field of fluid dynamics. I wouldlike to thank Dr. Daniel Coombs for taking time reading through my thesis.I would like also thank other professors at UBC whom I have takencourses from. I also owe thanks to all my friends and staff members at theIAM and UBC math department. They have been very supportive for thepast two years. I will never forget these two wonderful years at UBC.Finally, special thanks to my families for their endless love, support andencouragement.xDedicationTo my parents for always being there for me.xiChapter 1IntroductionIf we look at a spermatozoon and an eel-like fish swimming in the water,we will see both of them swimming with wave-like movements, where theyboth produce a wave that travels backwards along the body, pushing thewater backwards and the entire body forwards [2]. One might ask, thoughthe phenomena of a spermatozoon and eel-like fish swimming look the same,are there actually different underlying physical mechanisms underlying whatwe have seen? The answer is yes and the difference can be easily explainedusing the quantity called the Reynolds number. Normally, the Reynoldsnumber for a swimming fish is on the order of 102, while for tiny organismlike spermatozoon it goes down to about 10−3 or less. When the eel-likefish swims in the water, its wave-like movements give rise to circulationsaround the body and then it driven forwards by the inertial forces set up inthe surrounding fluid. However, when a spermatozoon swims in the water,due to the small value of the Reynolds number, the force due to viscosity ismuch greater than the one due to inertia. Here comes the question: how cana body propel itself when inertial forces are small compared to the forcesdue to viscosity? [30, 33, 35, 42, 43].The study of the physics of locomotion at low Reynolds number has along history. Since the 1930s, the swimming of various microorganisms likeE. coli, paramecium, and spermatoza has been studied by lots of researchers[32]. Spermatozoon motility has attracted the most attention, mainly dueto the fact that spermatozoon swimming is simpler to model and due to thebiological importance of spermatozoa. Motile bacteria commonly have manyflagella distributed in various ways over the cell, and their different flagellamay rotate or corkscrew together as a flagellar bundle. This is essentiallythree dimensional and so it will be very hard to find a simple model to1Chapter 1. Introductionrepresent the swimming of these bacteria. However, spermatozoa have onlytwo parts: a head and a long flagellum. The flagellum beats, producingwaves that propel the body through the fluid. Due to this simple simplestructure, it is possible to model the swimming of spermatozoon using asimple two-dimensional model introduced by G. I. Taylor [1, 11, 35, 42]. Onthe other hand, the huge number of unsolved problems in the locomotion ofmammalian sperm also motivate people to continue to model the movementof spermatozoa. The goal of the research is ultimately to help physicians findthe cure for significant health problems such as infertility and unintendedpregnancies [26, 39].Significant contributions to the field of swimming microorganism havebeen made by many scientists, including G. I. Taylor, A. J. Reynolds, J.R. Blake, J. Gray, G. J. Hancock and D. F. Katz [4, 6, 14, 22, 25, 36,42, 43]. Even though some of this work is over 60 years old, their workstill motivates people to continue research on the field of locomotion at lowReynolds number.The purpose of this thesis is to extend work done by G. I. Taylor [42, 43]and D. F. Katz [25], by looking at the problem of a microorganism swimmingclose to a washboard in a Newtonian fluid. We will assume that the swimmerpropagates waves along its body and propels itself in the opposite direction.We consider the swimmer to be a two-dimensional and infinitely periodicthin sheet. Instead of a single plane wall presented close to the swimmerwhich has been looked at by Katz [25], we will examine a more complicatedgeometry where there is a washboard present. This geometry allows us tomodel the natural situation where a microorganism needs to swim over amore geometrically complex environment than a plane wall [34]. In thisthesis, we consider two cases of the swimming problem. In the first case, weassume the swimmer has a fixed shape given by a simple sine wave. In thesecond case, we assume the shape of the swimmer is unknown. However,we will suppose there is a force distributed along the body of the swimmer.This force will be obtained from a simple elastic beam theory. In both thesetwo cases, we assume the lubrication approximation. We examine differentvalues of wavenumber of the swimmer and amplitude of the force driving the21.1. Literature Reviewlocomotion of the swimmer to see how it affects the swimming behaviour.In particular, for the case of the fixed shape swimmer, we looked at differentshapes of the washboard.In this chapter, we will first do a literature review to outline some signif-icant previous work in the field of swimming microorganisms. We will thenset up the main problem we will consider in this thesis with an explanationof all the necessary assumptions we have used. Finally, we will outline thestructure of this thesis.1.1 Literature ReviewThe study of locomotion of animals has a very long history, and there are lotsof publications on this topic. Among them, J. Gray and his colleagues madeparticularly significant contributions to the topic of locomotion of fish in aseries of articles published on the Journal of Experimental Biology duringthe 1930s [2, 16–21, 33]. In this thesis, instead of looking at the swimmingfish, we would like to particularly study the swimming of microorganismsat low Reynolds number. Since 1950s, various researchers have paid lots ofattention to swimming microorganisms, and in particular they were inter-ested in analyzing how fast the organisms could swim. Here, we will outlinesome literature covering the topic of swimming microorganisms.In 1952, G. I. Taylor published a paper analyzing the swimming of mi-croorganisms with an asymptotic approach [42]. Taylor assumed that thespermatozoon swam in the viscous fluid as a two-dimensional, infinite, inex-tensible thin sheet. The amplitude of the swimming wave was taken to besmall. The waving surface was represented by a simple sine curvey0 = b sin(kx− σt). (1.1)The velocity of the wave is σ/k, the wave length is 2pi/k = λ, and t representstime. The amplitude b was assumed to be small compared to λ and thewaving sheet moved in the positive x direction. Solving the two dimensionalNavier-Stocks equations of Newtonian fluids (with inertia term neglected),31.1. Literature ReviewTaylor found that the wave in the sheet could not propel it through the fluidat the first order in b. However, when he carried on the calculation to thesecond order, he found a nonzero swimming speed. In dimensional units,the swimming speed V he found wasV kσ=12b2k2(1−1916b2k2). (1.2)This formula shows that the leading order of swimming speed is proportionalto the amplitude of the swimming wave squared. In the paper, he alsocalculated the mean value of the dissipation of energy W , findingW = 2b2σ2kµcos2(kx− σt) = b2σ2kµ. (1.3)In addition to these, Taylor also examined the case where there was a viscousfluid on both sides of the sheet. He found that the swimming speed V wasas the same rate as for the situation where there was fluid only on oneside. However, the rate of dissipation of energy increased to be 2W insteadof W . In addition to this paper, Taylor had a later paper in 1952 whichassumed the tail of spermatozoon was a flexible cylindrical shape instead ofa thin sheet. He first considered the case where the swimmer propagatedwaves backwards along the tail to achieve a motion. He found, in this case,the swimming velocity was proportional to the square of the ratio of theamplitude of the wave to the wave length. He then considered a secondcase where the tail propagated a wave of spiral form and found that in thissituation the body was propelled twice as fast as the case where the wavewas propagated along the tail [43].In 1965, A. J. Reynolds published a paper with an extension of the pre-vious work of Taylor [36]. In this paper, same as Taylor’s paper, he assumedthe swimming object was a two dimensional waving sheet with the profiley = b sin(kx + σt), the amplitude of the wave was small, and the methodof asymptotic analysis was applied. In this paper, he generalized Taylor’sstudy in three respects. First, instead of looking at low Reynolds number ob-ject where inertia got neglected, Reynolds assumed there is an finite inertia41.1. Literature Reviewof the perturbed fluid. Solving the corresponding Navier-Stocks equationswith appropriate boundary conditions applied, he found the second-orderpropulsive velocity to beVU=14α2[β − cosφcosφ− 1/β− 1], (1.4)where V is the propulsive speed, U is the wave velocity and α is bk. sis defined as U/νk, where ν is the viscosity, β = (1 + s2)1/4 and cosφ =[(1 + β2)/2β2]1/2. When there was a large viscosity, he sawVU→12α2, (1.5)which matched with what Taylor had found for low Reynolds number swim-mers. Second, he looked at the case where the waving sheet was extensiblewith the inertia of the fluids neglected. The velocity solution he foundmatched with what Taylor had found when the extensibility parameter δapproaches zero. Finally, he considered the case when the swimmer swamnear a solid wall. When the waving sheet swims in the centre of a channelof width 2h, the swimming velocity satisfiesVU=12α2[sinh2(kh) + k2h2sinh2(kh)− k2h2]. (1.6)While, when the waving sheet is in a channel at a distance h1 from one walland h2 from the other,VU=12α2[−(1 + d1 + d2) +(h1 − h2)(d1 − d2)(h1 + h2)], (1.7)where d = sinh2(kh)/[sinh2(kh)− k2h2].Another great extension of Taylor’s two dimensional waving sheet prob-lem was done by D. F. Katz. In [25], Katz solved the swimming sheetproblem using both biharmonic and lubrication analysis with valid assump-tions. He assumed the sheet swims inside of a channel, where one of thewalls was defined as y = h1 and another one was y = −h2. The sheet propa-51.1. Literature Reviewgated a wave along its body in the negative x direction with speed c, whichpropelled the sheet in the positive x direction with velocity V . He set thepoint on the sheet surface, (x0, y0), to bex0 = xm + a cos k[xm + (c− V )t] + d sin k[xm + (c− V )t], (1.8)y0 = b sin k[xm + (c− V )t], (1.9)where (xm, 0) is the mean position of (x0, y0). Notice this set-up is moregeneral than what G. I. Taylor did in [42]. By assuming that the amplitudeof the wave was small, Katz applied biharmonic analysis and found thatthe leading order of swimming speed is proportional to the amplitude ofthe swimming sheet squared, which matched with what had been found byTaylor. Thus, the presence of walls doesn’t alter the leading order swim-ming speed. Furthermore, he considered the case when a sheet swam in anarrow channel. Next, instead of biharmonic analysis, lubrication-theoryanalysis was applied. He found the leading order of swimming speed in thedimensional variables asV0c=3(h/b)2 + 2, (1.10)where h measures the distance between two walls. It indicated that to thelowest order, velocity of propulsion is bounded by the wave speed. Beyonddoing the asymptotic analysis of the two dimensional swimming sheet prob-lem, Katz also did lots of experiments to look at how spermatozoa swim inthe real world [27–29].In addition to the flagellate microorganisms, like spermatozoa, we havebeen looking at, there is another common type of microorganism calleda ciliate, which consists of a body that is covered by a great number ofhair-like organelles [30]. Many researchers have paid lots of attention toanalyzing the swimming of ciliates. However, due to the more complicatedstructure, the ciliates can’t be simply modelled by a two dimensional model.Instead, a three dimensional model will be needed. In early 1970s, J. R.Blake published a series of articles using a spherical envelope model. In the61.1. Literature Reviewthree dimensional spherical envelope model, he assumed the organism wasspherical. There was a surface covering the ends of the numerous undulatingcilia over the surface of the organism called the “envelope”. The no-slipcondition was applied at the surface of the waving envelope. Blake firstconsidered a finite spherical envelope approach and found the mean velocityU¯ was proportional to aσ, where a was the radius of the organism and σwas the angular frequency of movement of the cilium. By taking a finitevalued N in the Taylor series expansion with a reasonable a and σ valueplugged in, he found this model agreed with velocities experienced in nature[6]. Blake then considered an infinite spherical model and found that as theradius of the organism increased, the result of the infinite spherical modeltended to that for the waving sheet. And the velocity of the propulsionfor the infinite model was over twice that obtained for the finite sphericalmodel [4, 5]. Later in 1972, based on previous work, Blake wrote a coupleof articles considering an alternative cilia sublayer method to look at thevelocity of ciliary locomotion [7–9].The swimming problem of microorganisms is not only studied analyt-ically, but also studied numerically. A couple of papers written by L. J.Fauci and her team in the 1990s particularly looked at the swimming ofspermatozoon numerically [13, 15]. In both of the articles, Fauci studiedthe fluid dynamics of sperm motility using the immersed boundary method.In [15], she modelled when spermatozoon swam near a rigid or elastic walland also when both single and several organisms were present. After com-paring the numerical results with analytical results, she found her resultsmatched with what Katz had found [25]. In the Early 2000s, she and M.M. Hopkins did further numerical analysis of the swimming of geotactic,gyrotactic and chemotactic microorganisms [23].More recent publications of swimming microorganisms have examinedviscoelastic fluids instead of simple Newtonian fluids. It makes more physicalsense to look at complex fluids because in nature, the biological environmentmicroorganisms experience often features complex fluids due to the presenceof biopolymers [32, 45]. In [31], E. Lauga revisited Taylor’s swimming sheetproblem where the fluid was non-Newtonain. He considered the cases where71.2. Our Main Problemthe Deborah number of the fluid was moderate or high and where the prop-agating wave included normal and tangential motion. He found that in allcases the velocities of the swimming sheet were smaller than for motion ina Newtonian fluid. Further experimental observations done by X. N. Shenand P. E. Arratia [37] have supported the theoretical predictions describedin [31]. In addition to these, Balmforth et al. wrote a paper particularlylooking at the influence of shear thinning, shear thickening and yield stressfluids when microorganisms swim very close to a solid boundary [3].In a recent paper, Majmudar et al. pointed out that in nature swimmingmicroorganisms often need make their way through a fluid with obstacles.They did both experiments and numerical analysis to see how microorgan-isms successfully navigated around the obstacles and how obstacles changedthe swimming path and speed of the microorganism [34]. Inspired by theirwork and the work done by Balmforth et al. [3, 34], in this thesis, we con-sider the situation when the microorganism swims near a washboard. Wemodel the swimmer as a two dimensional, infinite periodic and inextensiblethin sheet. To be simple, we assume the fluid between the swimmer and thewavy wall is Newtonian and incompressible. In this thesis, we will considertwo cases: when the swimmer has a fixed shape and when the swimmer is anelastic swimmer. We are interested to see how different values of wavenum-ber, wave amplitude, and stiffness change the swimming behaviour.1.2 Our Main ProblemNow we know why we are interested in this problem and what others havedone to study locomotion at low Reynolds number, we are in a good positionto set up our main problem mathematically.In our problem, we consider the swimmer to be two-dimensional andinfinitely periodic, and assume that it swims close to a washboard. Forsimplicity, we would like to consider the fluid between the swimmer and thewashboard as Newtonian and incompressible. We assume that the swimmerpropagates waves along its body and propels itself in the opposite direction.We call the swimming speed in time t, U(t), and the wave speed C. It is81.2. Our Main Problemassumed in the problem that the swimmer propagates waves along its bodyin the negative x-direction, so propelling itself to the positive x-directionwith a velocity of U(t). Let the shape of the swimmer be y = Y (x, t) whichis centred at y = 0, and the washboard is fixed at the position y = h(x). Thedistance between the swimmer and the washboard will be called d(x, t) =Y (x, t)− h(x).The problem satisfies the two-dimensional Navier-Stokes equations andthe continuity equation:ρ(∂u∂t+ u · ∇u)= −∇p+∇ · τ + F, (1.11)∇ · u = 0. (1.12)Here ρ is the fluid density, u = (u, v) is the velocity field, p is the pressurewithin the fluid, τ is the deviatoric component of the stress tensor and Frepresents body forces acting on the fluid.In the fixed reference frame, the velocity field u = (u, v) along y = Y (x, t)satisfiesu(x, Y ) = U(t), (1.13)v(x, Y ) = Yt + UYx. (1.14)When y = h(x), the no-slip boundary condition givesu(x, h) = 0, (1.15)v(x, h) = 0. (1.16)Hence, full geometry of the problem looks like the following:91.2. Our Main Problemyy=h(x)d=Y−hWave C xu=Uv=dY/dt+UdY/dxu=0v=0y=Y(x,t) UFigure 1.1: A sketch of the geometry of the problem.Along with these equations, based on the physical situation we are look-ing at, we will make several assumptions to simplify the problem a little bitand help us to solve the system.Because we look at the behaviour of micro-organisms, low Reynolds num-ber will be considered due to great physical interests. That is,0 < Re 1.In this sense, the inertial terms on the left hand of the Navier-Stokes equa-tions can be neglected. It will be also assumed that there is no externalforce acting in the problem. i.e.F = 0.Since we will only deal with Newtonian fluid, the shear stress term, ∇ · τ ,can be simplified to µ∇2u, where µ is the viscosity of the fluid. It will alsobe assumed that the amplitude of the propagating wave is small comparedto its wavelength. It makes physical sense because it has been found thatfreshly ejaculated sperm tend to swim with low amplitude and symmetricflagellar beats [40]. Meanwhile, it is assumed that the problem is periodic.In particular, we will look at the periodicity of pressure p in x. Moreover,the equation of motion of the swimmer gives thatmass per length× acceleration = net force.101.3. Thesis OutlineBecause we are looking at the situation having no inertia, the left hand sideof the equation of the motion is equal to zero and therefore the net force onthe swimmer is zero.Note the set-up of the problem described above will apply to all chaptersin this thesis.1.3 Thesis OutlineIn Chapter Two, we will look at the problem of a fixed shape swimmingsheet swimming near a washboard. The lubrication theory approximationis applied. First, we consider the case when the washboard has a simplecosine shape. Various values of wavenumber, amplitude of the resortingforce and amplitude of the topography are considered. We will look atthe instantaneous swimming speed and the average swimming speed of theswimmer. Here, a simple case when the wavenumber is equal to one will bestudied analytically, while other cases will be done numerically. Later, wewill also look at the case when the washboard has a more general shape.In Chapter Three, we will look at the problem of an elastic swimmingsheet swimming near a sinusoidal washboard. The lubrication theory ap-proximation is applied. Various values of wave amplitude and wavenumberare considered. We also look at the effects of stiffness. Here, all the workwill be done numerically.In Chapter Four, a summary of what we have found will be includedwith a mention of the future work could be done.11Chapter 2Fixed Shape Swimmer withLubrication Theory AnalysisWe will apply lubrication theory analysis through out this chapter, similarto the work shown in [25] by Katz. In order to apply lubrication theoryhere, we will assume the mean distance between the swimmer and the wavywall, d¯, compared with the wavelength of the swimmer, k, is very small. InKatz’s paper, he assumed the swimmer had a fixed shape which is definedby a sine wave. Carrying on Katz’s analysis, we will focus on the situationwhere the swimmer has a fixed shape in this chapter.In the chapter, we will look at cases with a small amplitude or highamplitude force driving the locomotion of the swimmer. We will also lookat various values of wavenumber and various values of the amplitude of thetopography to see how they change the swimming behaviour.2.1 Lubrication Theory ModelRecall that in our problem, there is a swimmer swimming near the wash-board with Newtonian fluid in between. We begin by considering the prob-lem in the fixed reference frame.In order to apply the lubrication approximation, we have assumed themean distance between the swimmer and the wavy wall is very small com-pared to the wavelength of the swimmer, i.e.0 < d¯k =  1. (2.1)With the assumptions mentioned in Chapter One, our problem satisfies122.1. Lubrication Theory Modelthe conservation of momentum and mass equations asρ(∂u∂t+ u · ∇u)= −∇p+ µ∇2u, (2.2)∇ · u = 0. (2.3)The boundary conditions on the lower surface of the locomotor, y = Y (x, t),areu(x, Y ) = U(t), (2.4)v(x, Y ) = Yt + UYx. (2.5)The boundary conditions on the stationary washboard, y = h(x), areu(x, h) = 0, (2.6)v(x, h) = 0. (2.7)The normal force balance on the upper surface of the fluids dictates thatp = DYxxxx − f, (2.8)where D is a parameter measuring the stiffness of the swimmer, p is thepressure, Y is the shape of the swimmer, and f is the applied restoringforce, which drives the locomotion of the swimmer. We define the restoringforce, f , asf(x, t) = A cos k(x−∫ t0U(tˆ) dtˆ+ Ct), (2.9)where A denotes the force strength. We assume the restoring force f pro-duces a propagating wave along the swimmer to the negative x directionwith a speed of C. Note that equation (2.8) could be used to determine theshape of the swimmer Y .132.1. Lubrication Theory ModelMoreover, we assume the stationary washboard looks likeh(x) = b cosKx− d¯ (2.10)where b measures the amplitude of the washboard.Now we can non-dimensionalize the problem using the following non-dimensional variables:x˜ = kx, y˜ = y/d¯,t˜ = kCt, u˜ = u/C,v˜ = v/vc ⇒ vc = C,p˜ = p/pc, ⇒ pc =Cµkd¯2.Upon substituting these non-dimensional variables, the x-component of theequation of conservation of momentum becomes:ρ(C2k∂u˜∂t˜+ C2ku˜∂u˜∂x˜+C2d¯v˜∂u˜∂y˜)= −pck∂p˜∂x˜+ Cµk2∂2u˜∂x˜2+Cµd¯2∂2u˜∂y˜2⇒ Re 2(∂u˜∂t˜+ u˜∂u˜∂x˜+ v˜∂u˜∂y˜)= −∂p˜∂x˜+ 2∂2u˜∂x˜+∂2u˜∂y˜2,where Re is the Reynolds number defined asRe =ρCµk.As Re→ 0 and → 0, this givesp¯x˜ = u˜y˜y˜. (2.11)142.1. Lubrication Theory ModelThe y-component becomesρ(C2k∂v˜∂t˜+ C2ku˜∂v˜∂x˜+2C2d¯v˜∂v˜∂y˜)= −Pcd¯∂p˜∂y˜+ µ(k2C∂2v˜∂x˜2+Cd¯2∂2v˜∂y˜2)⇒ Re 4(∂v˜∂t˜+ u˜∂v˜∂x˜+ v˜∂v˜∂y˜)= −∂p˜∂y˜+ 4∂2v˜∂x˜2+ 2∂2v˜∂y˜2As Re→ 0 and → 0, this givesp˜y˜ = 0. (2.12)The continuity equation isCk∂u˜∂x˜+Cd¯∂v˜∂y˜= 0⇒ u˜x˜ + v˜y˜ = 0. (2.13)Moreover, the dimensionless equation of topography will beh(x) = b cosKx− d¯⇒ h˜(x˜) = b˜ cos(κx˜)− 1, (2.14)where κ = K/k is called the wavenumber and it is selected to be a rationalnumber in order to allow for a periodic geometry.The dimensionless form of the restoring force f isf(x, t) = A cos k(x−∫ t0U(tˆ) dtˆ+ Ct)⇒ f(x˜, t˜) = A˜ cos(x˜−∫ t˜0U˜(˜ˆt) d˜ˆt+ t˜). (2.15)Thus, we could get the dimensionless form of (2.8) by taking the non-dimensional A˜ and D˜ asD˜ =d¯3k5cµD, A˜ =Cµkd¯2A.Note: From now on, everything will be dimensionless. To make things152.1. Lubrication Theory Modelsimple, we will drop the tilde notation.From (2.12), we can see that the pressure p is independent of y, i.e.p(x, t). We can integrate (2.11) twice, it gives:u(x, y) =px2y2 + Ey + F.Since u(x, h) = 0 and u(x, Y ) = U ,⇒ E =UY − h−px2(Y + h),F = −pxh22+(px2(Y + h)−UY − h)h =pxY h2−UhY − h.It implies thatu(x, y) =pxy22+(UY − h−px2(Y + h))y +pxY h2−UhY − h=12px(y − Y )(y − h) +UY − h(y − h). (2.16)From (2.13), we know thatux + vy = 0⇒ vy = −ux.Integrating in y from h to Y , it gives that∫ Yhvy dy = −∫ Yhux dy, (2.17)Applying Leibniz integral rule givesddx(∫ Yhu dy)=∫ Yhux dy + u(Y )Yx − u(h)h′ =∫ Yhux dy + UYx,162.1. Lubrication Theory Modelsince u = 0 on y = h and u = U on y = Y . Thus, (2.17) can be rewritten as∫ Yhvy dy = UYx −ddx(∫ Yhu dy)⇒ v(Y ) = UYx −ddx(∫ Yhu dy),because v = 0 on y = h. Recall that the vertical velocity v on Y (x, t)satisfies v(Y ) = Yt + UYx. It follows thatYt = −ddx(∫ Yhu dy).Plugging the velocity equation of u found in (2.16) into above equation givesYt = −ddx[∫ Yh(12px(y − Y )(y − h) +UY − h(y − h))dy]=ddx(px12(Y − h)3 −U2(Y − h)), (2.18)in the fixed lab reference frame.We can change every thing into the moving reference frame, (xˇ, t), bydefiningx = xˇ+∫ t0U(tˆ) dtˆ ⇒dxdt=dxˇdt+ U.Now (2.18) could be rewritten using the moving reference frame asYt = UYxˇ +ddxˇ(pxˇ12(Y − h)3 −U2(Y − h))=ddxˇ(pxˇ12(Y − h)3 +U2(Y − h)). (2.19)172.1. Lubrication Theory ModelFurthermore, we can change (2.19) into the wave reference frame, (ξ, t), asYt = UYxˇ +ddx(px12(Y − h)3 −U2(Y − h))⇒ (1− U)Yξ =ddξ(pξ12(Y − h)3 −U2(Y − h)), (2.20)where the horizontal spatial coordinate in the moving reference frame isdefined asξ = xˇ+ t = x−∫ t0U(tˆ) dtˆ+ t.Integrating (2.20) over ξ leavesQ(t) + (1− U)Y =pξ12(Y − h)3 −U2(Y − h)⇒ Q(t) =pξ12(Y − h)3 −U2(Y + h)− Y, (2.21)where Q(t) is the flux in the wave frame.2.1.1 Periodicity in Pressure ConditionA key assumption mentioned in Chapter One is that the periodic conditionof pressure p in x.Rearranging (2.21) we can get a equation of pξ in terms of the flux, Q(t),aspξ =12(Y − h)3(Q(t) + Y −U2(Y + h)).Notice in the fixed reference frame, it ispx =12(Y − h)3(Q(t) + Y −U2(Y + h)). (2.22)182.1. Lubrication Theory ModelSince pressure p is periodic in x, we will have 〈px〉 = 0. Here,〈· · · 〉 =1L∫ L0(· · · ) dx,where L denotes the length of individual periodic section of the swimmerand is set to be 2pik. It indicates that the periodic sections of the swimmercontains k wavelengths of the forcing and K wavelength of the topographyin length L.2.1.2 Zero Net Force ConditionThe traction on the lower surface of the swimmer drives its locomotion,which gives an additional dynamical condition on the swimmer. Recall, inChapter One, we have stated that the equation of motion of the swimmer ismass per unit length× acceleration = net force.We know the force on the fluid surface due to the fluid stress at y = Y isequal to the force on swimmer due to the fluid, and the stress tensor of thefluids in the dimensional form looks likeσ ≡(−p+ 2µux µ (uy + vx)µ (uy + vx) −p+ 2µvy).In the lubrication limit, it can be simplified toσ ≡(−p µuyµuy −p).The outwards unit normal vector, n, along the swimmer Y isn =(−Yx, 1)√1 + Y 2x,⇒ σ · n =(−p µuyµuy −p)·(−Yx1)/√1 + Y 2x .192.2. Swimming with a Fixed ShapeAccording to the Newton’s Second Law, the acceleration of the swimmerand the net force on the swimmer in the x direction satisfyMU˙ = −〈p(x, t)Yx + µuy(x, Y, t)〉, (2.23)where M indicates the effective inertia of the swimmer and 〈· · · 〉 denotesthe x-average defined same as before. With periodic condition of p in x andapplying the Leibniz integral rule, we will haveMU˙ = −〈µuy(x, Y, t)− px(x, t)Y 〉. (2.24)Since we are looking at the problem for a micro-organism, the problem isconsidered to be inertialess, i.e. M → 0. Then, (2.23) is reduced to be anintegral constraint0 = 〈p(x, t)Yx + µuy(x, Y, t)〉.Non-dimensionalizing the above equation using the same non-dimensionalvariables defined in the previous section, we get a dimensionless form integralconstraint as0 = 〈p(x, t)Yx + uy(x, Y, t)〉. (2.25)2.2 Swimming with a Fixed ShapeIn this chapter, we focus on the case when the swimmer has a fixed shapeas discussed in paper [25] written by Katz, i.e. the swimmer has an infinitestiffness. In this situation, from (2.8), we get the following relation:p DYxxxx − f ⇒ 0 ≈ DYxxxx − f.202.2. Swimming with a Fixed ShapeSince f(x, t) = A cos(x−∫ t0 U(tˆ) dtˆ+ t), the above relation can be rewrit-ten asYxxxx =ADcos(x−∫ t0U(tˆ) dtˆ+ t). (2.26)Integrating (2.26) four times and applying the corresponding periodic bound-ary conditions, i.e. Y is periodic up to and including its third derivatives,givesY = a cos(x−∫ t0U(tˆ) dtˆ+ t)= a cos ξ, (2.27)where a = A/D and ξ is the horizontal spatial coordinate in the wavereference frame.Moreover, we can plug (2.22) into the periodic condition 〈px〉 = 0, andget〈12(Y − h)3(Q(t) + Y −U2(Y + h))〉= 0⇒ I03Q =(I13 −12I02)U − I13 , (2.28)whereImn ≡〈Y m(Y − h)n〉.We plug in (2.16) and (2.22) into (2.25) and get(12I02 − I13)Q+(I23 − I12 +13I01)U +(12I12 − I23)= 0. (2.29)Substituting (2.28) into (2.29), we arrive at an equation of instantaneousswimming speed U(t) in terms of integrals Imn asU(t) =I03(I23 −12I12)− I13(I13 −12I02)I03(I23 − I12 +13I01)−(I13 −12I02)2 . (2.30)212.2. Swimming with a Fixed Shape2.2.1 Swimming Speed for Different WavenumbersSince we have found an expression for instantaneous swimming speed attime t, we could use it to evaluate the swimmer’s swimming speed for dif-ferent values of wavenumber κ, and see when the swimmer gets its optimalswimming speed.We will assume the amplitude of the propelling wave a is not too big,so the swimmer doesn’t get pushed too close to the topography. In order tomake things simpler, we first try the case when the wavelength of the swim-mer, k, and the wavelength of the topography, K, are commensurate, i.e. thewavenumber κ = 1. Note that since both the shape of the swimmer, Y (x, t),and the shape of the topography, h(x), are fixed for the infinite stiffnesscase, we could solve all the Imn integrals exactly by using the fundamentaltechniques of integration.With Y = a cos ξ and h = b cosx− 1, the distance d in between isd = a cos(x+ φ)− b cosx+ 1 = C1 cos(x+ Φ) + 1, (2.31)where φ(t) measures the phase of translation of the swimmer with respectto the washboard. To sum these two cosine functions, we define tan Φ =sinφcosφ− baand C1 = a√(cosφ− ba)2 + sin2 φ < 1. We can evaluate all sixintegrals, Imn , one by one and getI01 =1√1− C12, (2.32)I03 =12·(2 + C12)(1− C12)52, (2.33)I02 =1(1− C12)32, (2.34)I13 =12pi(2pi(1− C12)32+ b cos Φ(−3C1pi(1− C12)52)−pi(2 + C12)(1− C12)52), (2.35)I12 = −C1(C1 + b cos Φ)(1− C12)32, (2.36)222.2. Swimming with a Fixed ShapeandI23 =12pi[−2pi(C12 + 2C1b cos Φ + 1)(1− C12)32+ b2 cos2 Φpi(1 + 2C12(1− C12)52)+ b2 sin2pi(1− C12)32− 2(−3C1pib cos Φ(1− C12)52)+pi(2 + C12)(1− C12)52], (2.37)where sin Φ and cos Φ are defined assin Φ =sinφ√sin2 φ+ (cosφ− ba)2and cos Φ =cosφ− ba√sin2 φ+ (cosφ− ba)2.Note the details of calculating the integrals of Imn are covered in AppendixA.Now we substitute all the integral values (2.32)-(2.37) into (2.30) andget an equation of swimming speed U in terms of φ(t). By taking a = 0.5and b = 0.4, we get a plot of instantaneous swimming speed U versus phasetranslation, φ(t), for κ = 1 shown in figure 2.1.0 1 2 3 4 5 60.20.40.6Phase φUFigure 2.1: A plot of exact instantaneous swimming speed, U , against thephase translation with respect to the washboard, φ(t), for a = 0.5, b = 0.4and κ = 1.We then check how the swimming behavior changes as wavenumberchanges. Again, we could calculate the exact solution of the swimmingspeed by evaluating each integral Inm as what we did for κ = 1; however,to save some time, we will use trapezoidal numerical integration on MAT-232.2. Swimming with a Fixed ShapeLAB instead. The instantaneous swimming speeds for a = 0.5 and b = 0.4with different values of wavenumber are represented in figure 2.2. It canbe seen that instantaneous swimming speeds behave quite differently forvarious values of wavenumber.0 1 2 3 4 5 60.20.40.6UPhase φ  Figure 2.2: Plot of instantaneous swimming speed, U , against the phasetranslation with respect to the washboard, φ(t), for a = 0.5, b = 0.4 andκ = 1 (blue), 2 (red), 1/2 (black), 4/5 (black with crosses) and 4/9 (green).There are some interesting limiting cases of instantaneous swimmingspeed, we will take a look:Case I (amplitude of the wavy wall approaches to zero, i.e. b→ 0):In this case, d→ Y and (2.28) is reduced to beQI03 + I02 −U2I02 = 0⇒ Q =I02(U2 − 1)I03. (2.38)Meanwhile, (2.29) is reduced to be(12I02 − I02)Q+(I01 − I01 +13I01)U +(12I01 − I01)= 0⇒ −12I02Q+13I01U −12I01 = 0. (2.39)Combing (2.38) and (2.39) and solving for the swimming speed, U , givesU →6I01I03 − 6(I02 )24I01I03 − 3(I02 )2. (2.40)242.2. Swimming with a Fixed ShapeRecall thatI01 =1√1− a2, I02 =1(1− a2)3/2, I03 =2 + a22(1− a2)5/2.Plugging in above equations into (2.40), it arrives thatU →6I01I03 − 6(I02 )24I01I03 − 3(I02 )2=3a2(1− a2)3/1 + 2a2(1− a2)3≡3a21 + 2a2. (2.41)It matches with the result of a flat lower surface found by Chan et al. [10].Case II (amplitude of the restoring force approaches to zero, i.e. a→ 0):From our swimming speed equation (2.30), we can see that, as a→ 0, U → 0.Physically a → 0 means that the force strength driving the locomotionapproaches to zero. As the force strength approaches to zero, it is clear thatthe swimming speed U will approach to zero.Case III (sum of the amplitude of the washboard and the amplitude ofrestoring force approaches to the mean distance between the swimmer andthe washboard, i.e. a+ b→ 1):As a + b → 1, the swimmer will touch the washboard at some momentsduring the swimming. It will limit the swimmer’s swimming speed. Thissituation is unphysical in the sense that a divergent force would be requiredto overcome the resulting lubrication pressures.2.2.2 Swimming with High Amplitude ForceNote that if a + b > 1, the swimmer will finally reach a locked swimmingstate. It means that the swimmer must stay with a steady swimming speedU = 1; otherwise, the swimmer will collide with the washboard. In thissituation, (2.30) will not be valid when the swimmer and the washboardcollide; however, for other values of phases, the equation (2.30) will still bevalid.Under high amplitude assumption, a plot of instantaneous swimmingspeed, U , over the acceptable values of phase φ for various value of a areshown in figure 2.3 (b = 0.4 and κ = 1 are taken in this example).252.2. Swimming with a Fixed Shape−1.5 −1 −0.5 0 0.5 1 1.50.911.1Phase φUa=0.8a=1.2a=1.1a=1a=0.9φu φsFigure 2.3: Plot of instantaneous swimming speed, U , against the phasetranslation with respect to the washboard, φ(t), for b = 0.4, κ = 1 withvarious values of a indicated. The arrows indicate the moving direction ofthe swimmer and the two stars on the line a = 1 indicate the stable andunstable equilibria.We can see from figure 2.3 that when a is slightly above the value of 1−b(like a = 0.8 and a = 0.9 in the plot), the swimming speed U(φ) remainsbelow one over the acceptable range of φ, which is below the wavespeed.It indicates that the swimmer always travels in the negative x-directionwhenever the swimmer is released in the acceptable range of phase. In thewave frame the swimmer travels in the opposite direction of the washboard.In this situation, the swimmer eventually touches with the washboard aftertravelling leftwards. Conversely, if a is fairly high (like a = 1.1 and a =1.2 in plot), the swimming speed U(φ) remains above one throughout theadmissible range of φ. So, the swimmer will always move to the right beforetouching with the washboard.Notice that there is a range of forcing amplitude between 0.9 and 1.1shown in figure 2.3 where instantaneous swimming speed reaches one at twodifferent phases. It suggests that there is a swimming direction change overthe acceptable phases. When U(φ) < 1, the swimmer travels to the leftfor the corresponding phases of φ; while it travels to the right wheneverU(φ) > 1. In figure 2.3, the two black stars on the curve a = 1 indicate thetwo phases which gives U(φ) = 1. Since when U > 1, the swimmer moves tothe right and to the left when U < 1, the left-hand equilibrium is unstable262.2. Swimming with a Fixed Shape(labeled as φu in the figure) and the star on the right hand side is stable(labeled as φs in the figure). When these two equilibria points exist, the fateof the swimmer will depend on the condition of initial phase. If the initialphase φ0 > φu, the swimmer will be converging to the stable fixed pointφs. Or, the swimmer will hit the washboard after moving to the left whenφ0 < φu.As shown in figure 2.4, the appearance of two equilibria indicates thatthere is a saddle-node bifurcation in phase φ within a certain range of forceamplitude a. However, it could be seen that the range where two equilibriaexists is very narrow. So, when a+ b > 1, it is most likely that the swimmerwill hit the washboard very soon, and result in a steady locomoting statewith U = 1 and contact.0.98 1 1.02 1.04 1.06 1.08−1−0.500.51aPhase φFigure 2.4: Bifurcation diagram of two equilibria phases against the variousvalue of a. The stable solutions are indicated in red crosses, while theunusable solution in black circles.2.2.3 Average Speed of the SwimmerWhen we look at the motion of an object, it is not only important to look atinstantaneous swimming speed, but also its overall swimming behavior. Forour swimming problem, we already had an expression of swimming speedU(φ) as shown in equation (2.30), and now we can use it to calculate theaverage swimming speed U¯ over the phase translation φ easily. However, itmakes more physical sense to look at the average speed over time instead272.2. Swimming with a Fixed Shapeof over the phase translation. Now we would like to do a simple change ofvariable and get a new expression of average speed over time.We have defined the phase translation over the washboard, φ, asφ = t−∫ t0U(t) dt⇒dφ1− U= dt,and we know∫ 2pi0 dφ = 2pi and∫ T0 dt = T .Changing variable, it gives an expression of time period in terms of φ asT =∫ 2pi0dφ1− U.Thus, we can convert average speed over φ to be over time t byU¯ ≡∫U dtperiod T=∫ 2pi0U(φ) dφ1− U(φ)/∫ 2pi0dφ1− U(φ). (2.42)So far, we only focused on the case when the wavenumber κ = 1. Here,we extend our work to look at more general settings of the wavenumber. Todo that, we assume the wavenumber to be a rational number, κ = K/k. Wethen compute the integrates in equation (2.42) over the fundamental period,L = 2pik, of both the forcing and the topography to get the average speedover time for various values of wavenumber. A plot of average Speeds, U¯ ,over various number of wavenumber, κ, is illustrated in figure 2.5 with thesame values of a and b as before. We can see that there is a complicated anddiscontinuous relationship between average speed, U¯ , and the wavenumber,κ. It doesn’t show a clear pattern that how the wavenumber will affect theswimmer’s swimming behavior. When κ < 1, with some choice of wavenum-ber, there is an increasing of average speed with higher wavenumber. Whileκ > 1, for some values of wavenumber, there is a decreasing of average speedwith higher wavenumber. We can see that when κ = 1/2, it gives a fairlyhigh speed. However, κ = 3, the swimming speed is comparably low. Notethat most of the rational wavenumbers lead to similar average swimmingspeed of about 0.47.282.2. Swimming with a Fixed Shape10−1 100 1010.40.450.50.55κAverage Speed1/4 1/3 1/2 13/43/2 2 3 4 65Figure 2.5: A plot of average speed, U¯ , against various values of wavenum-ber, κ, with a = 0.5 and b = 0.4. Some values of the wavenumber arelabelled in the figure.As shown in figure 2.6 and 2.7, there is some dependence of the topo-graphic amplitude (b-dependent) on the relationship between average speedand various values of wavenumber. For κ < 1, average speed is higher withbigger b value compared to the smaller number of b. In opposite, for κ > 1,average speed reaches a bigger number for smaller values of b. For κ = 1/n,where the integer n  1, geometrically, we could expect that the topogra-phy appears much like a flat underlying plane for a big fraction of the time.With a = 0.5, we can see from figure 2.7 that, as n gets bigger, there is amore rapid change of average swimming speed when b → 12 . Actually, forκ = 1/n, where n is big, we should expect that the average swimming speed,U¯ , increases to unity as b → 12 , which means that the swimmer reaches itslocked state.2.2.4 General Shape of the TopographySo far, we have assumed that the topography is represented by a simplecosine curve. However, in the real world, the swimming environment formicoorganisms is more complicated than this [34]. Here, we will extend theprevious work to a more general topography. Now we redefine the topogra-292.2. Swimming with a Fixed Shape0 0.1 0.2 0.3 0.4 0.50.40.50.6bAverage Speedκ=3/2κ=2κ=1/2κ=1Figure 2.6: A plot of average speed, U¯ , against various values of topographicamplitude, b, with a = 0.5 and different wavenumbers indicated. Noteκ = 4/9 is in red and κ = 4/5 is in green.0.48 0.485 0.49 0.495 0.50.550.60.650.7bAverage Speed κ=1/2κ=1/6Figure 2.7: A zoomed-in plot of average speed U¯ for κ = 1/n, for n =2, 3, . . . , 6 (shown from top to bottom) as b→ 12 .phy, h(x), ash(x) = b(1 + α) ·cosκx1 + α cosκx− 1, (2.43)where b measures the amplitude of the topography, κ is the wavenumber,and α is a parameter that controls the shape of the topography. Note thatwhen α = 0, the topography goes back to our original simple cosine shape.With this general topography, our old equation of instantaneous swimmingspeed U(t) in terms of integrals Imn becomes,U(t) =I03(I23 −12I12)− I13(I13 −12I02)I03(I23 − I12 +13I01)−(I13 −12I02)2 , (2.44)302.2. Swimming with a Fixed ShapewhereImn ≡〈Y m(Y − h)n〉,is still valid. We also have the equation of average speedU¯ ≡∫U dtperiod T=∫ 2pi0U(φ) dφ1− U(φ)/∫ 2pi0dφ1− U(φ)(2.45)as before.In the following, we will take a look how various values of α changes theswimmer’s swimming behaviour. Here, α = 0.5 and α = 2 are taken as anexample. When α = 0.5, the geometry of our problem looks likey = Y(x, t)yy = h(x)xFigure 2.8: Geometry of the problem for α = 0.5.When α = 2, the geometry of our problem looks likey y=Y(x,t)xy=h(x)Figure 2.9: Geometry of the problem for α = 2.Note: Here for α = 2, the topography is not a continuous wavy shape anymore. There is a gap between each periodic section of the topography.Now, we can use MATLAB to calculate the instantaneous swimmingspeed of the swimmer, U(t), for various values of α as we did for the simplecosine wave topography. A plot of the instantaneous swimming speed of312.2. Swimming with a Fixed Shapethe swimmer with forcing amplitude a = 0.5, topographic amplitude b =0.4, and wavenumber κ = 1 is shown in figure 2.10. We can see that theswimmer behaves differently when swimming against the topography withdifferent shapes. It is clear to see that, when α increases from 0 to 0.5, theinstantaneous swimming speed at zero phase decreases. When α increasesto two, the instantaneous swimming speed at zero phase is even lower. Theinstantaneous swimming speed is zero for the region when the topographyis discontinuous.0 2 4 600.20.40.6UφFigure 2.10: A plot for instantaneous swimming speed, U(t), against phasetranslation with respect to the solid wall, φ(t), for a = 0.5 , b = 0.4 andα = 0 (solid), α = 0.5 (dashed) and α = 2 (dash-dotted).The average swimming speed against different values of topographic am-plitude for various values of α is shown in figure 2.11. When α = 0, theaverage swimming speed increases as the amplitude of the topography in-creases. When α = 0.5, at first, the average swimming speed increases as theamplitude of the topography decreases. However, as the gap of the swim-mer and the topography gets very narrow, the average swimming speed getshigher. It is interesting to see that as the swimmer and the topography getvery close, for these two values of α, the average swimming speed convergesto the same value. For α = 2, the average swimming speed is always in-creasing with the increasing values of topographic amplitude. However, theaveraged swimming speed for this case is always lower than the cases α = 0and α = 0.5.322.2. Swimming with a Fixed Shape0 0.1 0.2 0.3 0.40.40.6bAverage SpeedFigure 2.11: A plot for average swimming speed, U¯ , against various valuesof the amplitude for the topology, b, for a = 0.5, and α = 0 (solid), α = 0.5(dashed) and α = 2 (dash-dotted).33Chapter 3Elastic Swimmer withLubrication Theory AnalysisIn this chapter, lubrication theory analysis is again applied. So, the as-sumption that the mean distance between the swimmer and the washboard,d¯, compared with the wavelength of the swimmer, k, is very small is againneeded. As in the previous chapter, there is a prescribed normal forcingdistributed along the body of the swimmer. However, different from before,we assume the swimmer responds to the applied force as an elastic beamwith a finite stiffness. Some work related to our elastic swimmer model hasbeen done previously. In [24], A. E. Hosoi and L. Mahadevan looked atthe dynamics of an elastic sheet over a thin layer of fluid under lubricationapproximation. In [12], G. J. Elfring and E. Lauga modelled the swimmingof spermatozoon using a flexible thin sheet.In this chapter, we will look at cases where there is a small amplitudeor a high amplitude restoring force applied. We will also look at varioussettings of wavenumbers and various values of stiffness to see how the swim-ming behaviour changes as things varies. We further compare the results inChapter Two with the results about this finite stiffness swimmer.3.1 Mathematical FormulationIn the previous chapter, we looked at the case when the shape of the swimmeris fixed. Here we move on to look at the case where there is an elasticswimmer with a finite stiffness.Since we still working on the same main problem mentioned in ChapterOne with the lubrication approximation applied, everything mentioned in343.1. Mathematical Formulationsection 2.1 will continue to be valid. Here, let’s recall what we have fromsection 2.1.The geometry of the problem in the fixed reference frame will still be thesame. Under the lubrication approximation, after non-dimensionalizing, wehave the conservation of momentum and the mass equations reduced to bepx = uyy, (3.1)py = 0, (3.2)ux + vy = 0 (3.3)The boundary conditions areu(x, Y ) = U(t), (x, Y ) = Yt + UYx, (3.4)u(x, h) = 0, v(x, h) = 0. (3.5)Again, if we integrate the conservation of momentum equations with theboundary conditions applied, we getu(x, y) =12px(y − Y )(y − h) +UY − h(y − h). (3.6)Plugging in (3.6) into (3.3) with boundary conditions applied, we getYt =ddx(px12(Y − h)3 −U2(Y − h)). (3.7)The topography is still defined as a cosine wave, likeh(x) = b cos(κx)− 1. (3.8)Moreover, we have the normal force balance condition on the upper surfaceof the swimmer asp = DYxxxx − f, (3.9)where D is a parameter measuring the stiffness of the swimmer, p is the353.1. Mathematical Formulationpressure, Y is the shape of the swimmer, and f is the applied restoringforce, which drives the locomotion of the swimmer. We define the restoringforce, f , asf(x, t) = A cos(x−∫ t0U(tˆ) dtˆ+ t), (3.10)where A denotes the force strength. We assume the restoring force f pro-duces a propagating wave along the swimmer to the negative x direction.Note that here we assume D has an finite value. Plugging in (3.9) into (3.7),we getYt =ddx((DYxxxx − f)x12(Y − h)3 −U2(Y − h)). (3.11)Since the periodicity in pressure condition and the equation of the motionof the swimmer are still valid, we will still haveMU˙ = −〈p(x, t)Yx + uy(x, Y, t)〉. (3.12)Plugging in (3.6) and (3.9) into (3.12), we get a second equationMU˙ = −〈UY − h−(DYxxxx − f)x2(Y − h) + (DYxxxx − f)Yx〉. (3.13)Putting (3.11) and (3.13) together, we get a new system of partial differentialequations describing the swimmer’s shape, Y , and swimming speed, U ,Yt =ddx((DYxxxx − f)x12(Y − h)3 −U2(Y − h)),MU˙ = −〈UY − h−(DYxxxx − f)x2(Y − h) + (DYxxxx − f)Yx〉.Because we consider the inertialess limit with M = 0, the left hand side of(3.13) will be zero. Furthermore, we assume initially the swimmer is a flatsheet sitting along the x axis with no velocity, i.e. Y (x, 0) = 0 and U(0) = 0.363.2. Swimming with Finite Stiffness3.2 Swimming with Finite StiffnessIn this section, we will solve the above system of partial differential equations(equation (3.11) and (3.13)) as an initial value problem in MATLAB. Recall,the initial condition will be Y (x, 0) = 0 and U(0) = 0. The derivativesinvolved in the equations are computed using the fast Fourier transformalgorithm. This system is firstly discretized on the uniform spatial grid andthe resulting coupled ordinary differential equations in time would be solvedusing a standard stiff integrator (MATLAB’s ode15s).We use D = 1, κ = 1 and b = 0.4 as an example for the initial valueproblem. The solution curves for various values of swimming forcing ampli-tude are shown in figure 3.1. It can be seen that after a very short transient,the solutions converge to a well-defined periodic state. From this figure,we can see that the higher value of the forcing amplitude is the higher theswimming speed and the longer the swimming period are.0 20 40 60−0.200.20.40.60.8tUFigure 3.1: A plot of instantaneous swimming speed, U(t), for various valuesof A with D = 1, κ = 1 and b = 0.4. The solid curve is for A = 2, dashedcurve for A = 3 and dashed and dotted curve for A = 3.8.Series snapshots of periodic states with various of A are shown in figure3.2 panels (a)-(c). We can see that, beginning with a flat sheet, the loco-motion makes the swimmer deform into a richer spatio-temporal pattern atthe final periodic state. Moreover, as we expected, the bigger the amplitudeof the restoring force is, the bigger the amplitude of the locomotive wave is.373.2. Swimming with Finite Stiffness−1−0.50Y and h−1−0.500.5Y and h0 2 4 6−101xY and h  (a) A=2(c) A=3.8(b) A=3Figure 3.2: A plot displays successive snapshots of the final periodic statefor various values of A with D = 1, κ = 1 and b = 0.4. Panel (a) is forA = 2, (b) for A = 3 and (c) for A = 3.8.Now we would like to look for the answer of a key question: whether thewashboard will help the swimmer’s to swim for this sample case? Recall,as Balmforth et al. have found out in a previous paper, in the limit of lowamplitude, the swimming speed could be solved analytically by a regularperturbation expansion [3], which givesU ≈3A2(1− e−t/M )144 +D2. (3.14)A plot of mean speed against various values of A is shown in figure 3.3.We can see that, for small values of forcing amplitude, the mean swimmingspeed increases quadratically as A increases. Moreover, for small valuesof forcing amplitude, the mean speed solution with topography matcheswith the solution without topography. It indicates the fact that, for smallamplitude locomotive wave over a washboard, the swimming speed getsaveraged out to the leading order, so the mean swimming speed shows nodifference with the case without topography in figure 3.3. For bigger forcing383.2. Swimming with Finite Stiffnessamplitude, mean swimming speed with topography increases much morerapidly than the case without topography. For our special choice of stiffness,wavenumber and topography amplitude, it can be seen that the mean speedreaches the wave speed, i.e. U = 1, when A is beyond 4, which correspondsto the locked swimming state. In this locked state, the swimmer will be veryclose to the wavy wall and the lower surface of the swimmer will move likea caterpillar tread over the washboard.0 1 2 3 4 500.20.40.60.81AMean SpeedFigure 3.3: Mean speed for the initial value problem against A with D =1, κ = 1 and b = 0.4. The circles indicate the numerical solutions withtopography, and the dashed curve indicates the analytical solutions withouttopography.A plot of mean speed against topographic amplitude for our specialchoice of D = 1 and κ = 1 is shown in figure 3.4. Much like the infi-nite stiffness and κ = 1 result shown in figure 2.6, mean speed increaseswith the topological amplitude for all these three values of A. When A = 2,as the topographic amplitude b increases, mean swimming speed increasesbut only by a very small amount. While, for A = 3, mean swimming speedincreases more rapidly than A = 2. It can be seen from the figure that whenb = 0.6, the swimmer has reached the locked state. When A = 3.8, as thetopographic amplitude increases, the swimmer also reaches its locked state,but sooner compared to the case A = 2. One can say there is some certainforcing amplitude dependence for the critical value of b when the swimmerfirst runs into the locked state. From this figure, we can conclude for thisspecial choice of D and κ, the swimmer propels itself more effectively over393.2. Swimming with Finite Stiffnessa washboard compared to the case when there is no topography presented.0 0.2 0.4 0.6 0.800.20.40.60.81bMean Speed  A=3.8A=3A=2Figure 3.4: Mean speed for the initial value problem against topographicamplitude b with D = 1, κ = 1, and three different values of A are indicated.(A = 2 circles, A = 3 stars and A = 3.8 dots).3.2.1 Steady SwimmingIt has been seen in section 2.2.2 that when the forcing amplitude gets beyonda critical value, the swimmer arrives a locked swimming state. After reachingthe locked state, the swimmer keeps a constant swimming speed same asthe wave speed, i.e. U = 1. So, the locked state is a steady solution inthe reference frame of the washboard or the wave. At the locked swimmingstate, (3.7), (3.9) and (3.12) are still satisfied by the steady swimmer.Since now the swimmer keep a steady swimming speed of U = 1, thetime derivative in (3.7) and (3.12) will be zero. That is0 =ddx(px12(Y − h)3 −U2(Y − h)), (3.15)0 = 〈uy + pYx〉. (3.16)Integrating (3.15) over x, we get a flux equation:Q =px12(Y − h)3 −U2(Y − h)⇒ Q =px12(Y − h)3 −Y − h2. (3.17)403.2. Swimming with Finite StiffnessFrom (3.9), we know thatpx = DYxxxxx +A sin(x−X), (3.18)where X measures the positional shift of the forcing pattern with respect tothe washboard.Substituting (3.18) into (3.17), we get a differential eigenvalue problemwhich the steady state swimmer satisfies asQ =112(Y − h)3[DYxxxxx +A sin(x−X)]−Y − h2, (3.19)where the eigenvalues we look for are Q and X.To solve this eigenvalue problem explicitly, we need two additional in-tegral constraints for the eigenvalues. The first one is the speed constraintwhich comes from (3.16), and the second one is from mass conservation.From (3.16) we have0 =〈UY − h+px2(Y − h)〉− 〈pxY 〉 ⇒ U =〈px(Y − Y−h2 )〉〈1Y−h〉 = 1.Thus, the integral speed constraint is〈(Y −Y − h2)[DYxxxxx +A sin(x−X)]〉=〈1Y − h〉. (3.20)The mass conservation gives that〈Yt〉 = 〈Qx〉 =〈ddx(px12(Y − h)3 −U2(Y − h))〉.Due to the periodic boundary condition in x direction, it gives 〈Yt〉 = 0. Itmeans that, over time, 〈Y 〉 is a constant. Thus, the integral mass conserva-tion constraint is〈Y 〉 = 0. (3.21)413.2. Swimming with Finite StiffnessThe locked state solutions can be computed directly from the eigen-value problem with these two integral constraints on MATLAB. To get thelocked state solutions, we will first solve the initial value problem by doing aspatial discretization with derivatives calculated using the fast Fourier trans-form. The corresponding time-dependent ordinary differential equation willbe solved by the stiff solver (MATLAB’s ode15s). Then, we will get thelocked solutions by converging the final solution of the initial value problemusing Newton’s Method.One sample of the locked state solutions for D = 1, κ = 1 and b = 0.4with various values of A is shown in figure 3.5. Figure 3.5 plots the eigen-value X along two steady state solutions against forcing amplitude A. Itcan be seen that the two branches of steady state solutions only exist whenthe forcing amplitude is beyond some critical value of A∗, which matcheswith what we have found in the previous subsection. It is clear that thetwo solutions merge to each other when they approach to the critical value.Finally, the steady state solution will disappear when the forcing amplitudefalls below the critical value. Note that, among the two locked state solu-tions, only the upper branch corresponds to the stable solution as a solutionof the initial value problem. One more thing we should notice from figure 3.5is the behavior of these two locked state solutions indicates a saddle-nodebifurcation. The critical threshold value, A∗, denotes the forcing amplitudeat which the periodic solutions first locked onto the washboard. To be morespecific, if we approaches the critical threshold A∗ from below, the periodicoscillatory solution will increase and reach the infinity. This type of saddle-node bifurcation is called saddle-node infinite-period bifurcation, and moreinformation about this type of bifurcation can be found in [38]. This type ofbifurcation is seen by Thiele and Knobloch in their study of the depinningprocess of driven drops on heterogenous substrates [44].The snapshots demonstrated the final profile of two steady state solutionsfor two different values of A are shown in figure 3.6. Comparing panel (a)with (b), it can be seen that as the forcing amplitude increases, the swimmergets pushed closer to the washboard in some sections. Shown in panel (b),the big forcing amplitude leaves a “bubble” of trapped fluid between the423.2. Swimming with Finite Stiffness0 10 20 30 4044.555.5AXFigure 3.5: The position eigenvalue, X, of two steady state solution branchesagainst A for D = 1, κ = 1 and b = 0.4. The upper branch is the stablesolution, while the lower one is unstable.swimmer and the washboard, which has a maximum volume for the stablesolution or a minimum volume for the unstable solution. The small capregion is called occluded region and the big “bubble” is called a blister.Analytic expressions of the occlusion and the blister could be calculatedusing a matched asymptotic expansions method (Takagi and Balmforth in[41]).0 2 4 6−101xY and h (a) A=40 2 4 6−1012xY and h (b) A=29Figure 3.6: Plots of final steady state snapshots of Y and h for D = 1, κ = 1and b = 0.4 with two different values of A. The stable steady state solutionis shown by a solid line and the unstable solution is shown by a dashed line.3.2.2 Larger StiffnessSo far, in the finite stiffness section, we only examined the case when D = 1.It will be interesting to take a look at larger stiffness and see how stiffness so-lutions relate to low stiffness solutions and whether larger stiffness solutionsshow some similarities to the infinite stiffness results. In this subsection, we433.2. Swimming with Finite Stiffnesslook at the case for the elastic swimmer with larger stiffness. However, forsimplicity, we will only consider the wavenumber to be one in the followingdiscussion.Figure 3.7 displays the instantaneous swimming speeds of the case D =100 with three different values of the forcing amplitude a. Shown in panel(a) and (b), when a = 0.4 and a = 0.8, the swimmer stays in a periodicswimming state after a very short transition period, which is similar withwhat we have seen for D = 1 (shown in figure 3.1). If the force is sufficientlybig, the swimmer gets locked onto the washboard and swims at the unitspeed again (displayed in panel (c)). It has been seen earlier that the biggerthe forcing, the bigger the swimming speed is.0 10 20 30 4000.51tU(t)(a) a=0.40 20 40 60 80 10000.51tU(t)(b) a=0.80 2 400.51tU(t) (c) a=1Figure 3.7: Plots of instantaneous swimming speed over time with D = 100,κ = 1 and b = 0.4. Panel (a) is the result for a = 0.4, (b) for a = 0.8 and(c) for a = 1.Figure 3.8 shows the swimmer’s profiles over time for two different valuesof a with D = 100, κ = 1 and b = 0.4. Panel (a) displays the case for a = 0.4.For this case, the swimmer translates like a simple sine wave. It also shows anoticeable decreasing of swimming speed when the swimmer travels over thecrest of the topography, which could be demonstrated in figure 3.7 panel (a).443.2. Swimming with Finite StiffnessRecall, we have found in the previous chapter that for the case of infinitestiffness with κ = 1 and b = 0.4, the swimmer collides with the washboardwhen a > 0.6. Presented in panel (b) of figure 3.8, the shape of the swimmerdeforms significantly when it approaches the crest of the topography, and ittranslates like a sine wave elsewhere. Two steady state solutions for a = 1are shown in figure 3.9. Notice that these two steady solutions are similar tothe results shown in figure 3.6 for D = 1. Moreover, these two steady statesolutions look nearly symmetric, which is similar to the case of the finitestiffness swimmer shown in figure 2.3.0 2 4 6−1−0.50Y and hx(a) a=0.4 0 2 4 6−1−0.500.5xY and h(b) a=0.8Figure 3.8: Plots show the translation of the swimmer Y over the wavy wallin time for D = 100, κ = 1 and b = 0.4 with two different values of a. Panel(a) is the result for a = 0.4 and (b) for a = 0.8.0 2 4 6−101xY and hFigure 3.9: A plot of steady state solutions for a = 1 when D = 100, κ = 1and b = 0.4. The stable branch is illustrated in solid, and the unstable oneis dashed.Figure 3.10 demonstrates the differences in mean speed for different val-ues of stiffness when the swimmer swims with or without topography. Whenthe swimmer swims over a washboard, we can see that the mean swimmingspeed is higher for D = 100 compared to D = 10. For low forcing ampli-453.2. Swimming with Finite Stiffnesstude, the swimming speed with topography (shown using markers) matcheswith the speed without the topography (shown in dashed line) for bothstiffness values. For high forcing amplitude, when D = 10, we observe a bigdifference between the swimming speed with topography and without to-pography. However, for D = 100, there is no big difference and it will reachunity swimming speed for swimming with topography and without topog-raphy. Note that for both D = 10 and D = 100, the swimmer reaches thelocked state for high forcing amplitude when swimming over the washboard.We also see that, for low forcing amplitude, mean swimming speed for finitestiffness swimmer matches with the speed for D = 100. However, for highforcing amplitude, the infinite stiffness swimmer will collide with the topog-raphy and be a steady state swimmer. For the finite stiffness swimmer, theswimmer will never touch the topography and the steady swimming stateconnects with the periodic swimming state through a saddle-node infinite-period bifurcation.0 0.5 100.20.40.60.81a=A/DMean Speed  D=10D=100Figure 3.10: Mean swimming speed against various forcing amplitude a forlarge stiffness swimmer with κ = 1 and b = 0.4. Dots and circles are thesolutions with topography for D = 100 and D = 1 respectively. Dashed lineare the solutions without topography (upper one for D = 100 and lower onefor D = 10). The solid line is the solution for infinite stiffness.To see how large stiffness steady state solutions connect with the steadystate solutions for finite stiffness swimmer, a figure of positional shift Xagainst stiffness D is shown in figure 3.11. In order to do a comparison463.2. Swimming with Finite Stiffnesswith steady state solutions for finite stiffness swimmer shown in figure 2.3,we kept a = A/D = 1. We can see that the stable steady state solutionconverges to a positional shift with respect to the washboard of about 0.5as D increases, and the unstable steady state solution converges to a shiftof about −0.5, which matches with what we found for the finite stiffnessswimmer (shown in figure 2.3).0 200 400 600−2−101DXa=A/D=1Figure 3.11: A plot of two steady state solutions against D when the forcingamplitude a kept to be 1, κ = 1 and b = 0.4. The upper branch is a stablesolution and the lower one is unstable.3.2.3 Various Values of WavenumberFor now, we have assumed wavenumber κ = 1. It will be meaningful toextend our discussion to a more general setting of the wavenumber andcheck whether the washboard helps increase the swimmer’s swimming speedin general.Figure 3.12 presents the mean swimming speeds for the high stiffnessswimmer with a fixed forcing amplitude a = 0.5 for various values of wavenum-ber against b. Here, we specially pick the stiffness D = 1000. Comparingwith the case of the fixed shape swimmer shown in figure 2.6, we can seea similar dependence of the topographic amplitude (b-dependent) on therelationship between average speed and various values of wavenumber isalso shown up here. When κ < 1, mean speed of the swimmer is increas-ing with the increasing values of the topographic amplitude. On the otherhand, when κ > 1, there is a higher mean speed of the swimmer for smaller473.2. Swimming with Finite Stiffnesstopographic amplitude.0 0.1 0.2 0.3 0.4 0.50.40.50.6bMean SpeedFigure 3.12: A plot show mean speed against b of a = 0.5 for four differentvalues of wavenumber (κ = 1/2 is represented by dots, κ = 1 by circles,κ = 4/5 by stars and κ = 2 by crosses) with D = 1000.In the following, we pick two special choices of wavenumber, i.e. κ = 2and κ = 1/2 to do further analysis. Here, we will do detailed analysis forboth these two values of wavenumber and compare their results with thecase when wavenumber κ = 1 and the fixed shape swimmer.Case I (κ = 2):Mean speed and the positional shift versus various values of A for κ = 2 withD = 1 and b = 0.4 are displayed in figure 3.13. Recall, in figure 2.6, it hasbeen shown that, with infinite stiffness, the swimming is less effective overthe washboard than without washboard when κ = 2. However, as shown infigure 3.13, the situation with finite stiffness for κ = 2 is more complicatedcomparing with the infinite stiffness case or κ = 1. Again, for low forcingamplitude, the solution is periodic. As an example, the swimmer’s profile forA = 5 is displayed in panel (a) of figure 3.14. In this regime, the swimmeractually swims faster without the washboard (indicated in figure 3.13 panel(a)). We could say the washboard retards the swimmer for this range offorcing amplitude. As we increase the forcing amplitude, the swimmingspeed increases abruptly to unity while the swimmer reaches the locked state(an example for A = 20 is shown in panel (c) of figure 3.13). An exampleof two steady state solutions when swimming with washboard for A = 8 isshown in figure 3.14 panel (b). In this high forcing amplitude region, mean483.2. Swimming with Finite Stiffnessspeed for swimming without topography decreases because the swimmer ispushed too close to the wall [3]. Notice that the swimmer reaches its lockedstate sooner for κ = 1 comparing with κ = 2.One interesting thing we should see from figure 3.13 in panel (a) is thatthe transition between periodic solution and locked state solution is discon-tinuous for κ = 2. Comparing the low amplitude periodic solution with thebifurcation diagram shown in figure 3.13 panel (b), it suggests that there isa region (A between 6 and 8) where the periodic and locked states co-exist.The periodic and locked state solutions for this region are shown in the meanspeed plot. It is also interesting to see that, for swimming with a topogra-phy, if the forcing amplitude goes beyond A = 22, we would see the lockedstate disappeared due to a second saddle-node infinite-period bifurcation.In this region, mean swimming speed decreases and the swimmer goes backto the periodic state (an example for A = 30 is shown in panel (d) of figure3.13).Case II (κ = 1/2):As shown in figure 2.6, the washboard helps with swimming of κ = 1/2 forthe infinite stiffness swimmer. As the topographic amplitude increases, theswimming speed increases. Someone may predict the same thing will happenfor the finite stiffness swimmer; however, the situation is more complicatedfor the finite stiffness swimmer than the infinite stiffness swimmer.The periodic solutions for κ = 1/2 and b = 0.4 when D = 1 are shownin figure 3.15. It is indicated in panel (a) of figure 3.15 that the elasticswimmer is not able to push itself neither more or less effectively over thewashboard than the flat wall. At low forcing amplitude, mean speed forκ = 1/2 is similar with the swimming speed for κ = 1. As forcing amplitudeincreases, mean speed for κ = 1 increases more rapidly than mean speedfor κ = 1/2. An example of the swimmer’s periodic swimming profile withA = 3 is shown in panel (b) of figure 3.15.Note that for high forcing amplitude, the subharmonic instability showsup for the periodic swimmer with κ = 1/2. So, there are two periodicsolution branches showing up in panel (a) of figure 3.15. An example ofsubharmonic instability solutions with A = 6 are displayed in figure 3.16.493.2. Swimming with Finite Stiffness0 10 20 3000.20.40.60.81AMean Speed(a)5 10 15 202.62.833.23.43.6AX(b)0 20 40−1012tU(c) A=200 20 40−0.500.51tU(d) A= 30Figure 3.13: Solutions for κ = 2 and D = 1 versus varying A. Panel (a)shows mean speeds for the initial value problem. The stars are the resultfor κ = 2 and b = 0.4, the solid line shows the result for κ = 2 withouttopography, and the dashed line shows the result for κ = 1 and b = 0.4.Panel (b) displays the positional shift, X, versus varying A. The solid branchis the stable steady state solution and the dashed line is the unstable steadystate solution.−2 0 2−1−0.500.5xY and h(a) A=5−2 0 2−1012xY and h (b) A=8Figure 3.14: Profiles of the swimmer for two special choices of A. Panel(a) demonstrates the periodic solution for A = 5 and panel (b) displays thestable steady state solution (solid) and the unstable steady state solution(dashed) for A = 8.503.2. Swimming with Finite Stiffness0 5 1000.51a=A/DMean Speed (a) D=1−6 −4 −2 0 2 4 6−1−0.500.5Y and hx(b) A=3Figure 3.15: Solutions of the initial value problem for κ = 1/2 when D = 1.Panel (a) shows the mean speed for various A. Stars indicate the stablesolutions with b = 0.4, and plus are unstable solutions with b = 0.4. Solidline is the stable solution without washboard and dashed line is the unusablesolution without washboard. The dash-dotted line is the solution for κ = 1.Panel (b) displays the swimmer’s profile for the periodic solution when A =3.From panel (a), we observed a clear change of the swimmer’s profile overtime. Initially, the swimmer begins with a two-humped wave. The swim-mer’s profile is shown in panel (b) for this unstable periodic solution. Then,the two humps merge to each other and combined to a single-humped waveat about time t = 550. The swimmer’s profile is shown in panel (c) for thestable periodic solution. This phenomenon is called a subharmonic instabil-ity. Note that this instability happens for both the cases with topographyand without topography, and mean speed solutions for both cases are pre-sented in figure 3.15 panel (a).When the stiffness gets higher, we found that the situation for the highstiffness swimmer is more complicated than the low stiffness swimmer. Themean speed for the swimmer with D = 10 is shown in figure 3.17 panel(a). Note that, when the swimmer swims over a washboard, only the stablesolutions are displayed in the plot. We see that, for this stiffness, it doesn’tseem the washboard helps with the swimming speed. For this stiffness, theswimmer never reaches the locked state. It is shown that a subharmonicinstability happens for swimming without washboard within the range a =1.1 and a = 1.4 (shown in the inset of panel (a)). If we pull the stiffness even513.2. Swimming with Finite Stiffnessξt  0 5 100200400600800−1012(a)−5 0 5−101xY and h(b)−5 0 5−10123xY and h(c)Figure 3.16: Periodic solutions of κ = 1/2 with D = 1, A = 6 and b = 0.4.Panel (a) shows the evolution of the swimmer’s surface, Y (ξ, t), on the (ξ, t)plane. Panel (b) displays the unstable periodic profile for A = 6. Panel (c)shows the stable periodic profile for A = 6.higher to be 100, the things get more interesting and complicated as shownin panel (c). For very low forcing amplitude, again mean speed is aboutthe same for swimming with washboard or without washboard. However,unlike what happens to the low stiffness case, for high forcing amplitude, thewashboard actually helps the swimmer to swimming more effectively. Theswimmer swimming with topography actually reaches the locked state forthis stiffness number which is not seen for low stiffness. In this situation, itis also seen that the κ = 1/2 swimmer with washboard swims faster than theκ = 1 swimmer for some values of forcing amplitude. Notice that, like before,the subharmonic instability is shown when a is between 0.7 and 1.1 for theswimmer without topography. The inset of (b) gives a bifurcation diagram ofthe positional shift, X, of the steady state solutions versus forcing amplitude523.2. Swimming with Finite Stiffnessa. Note the stable solution is shown in solid and the unstable solution isshown in dashed curve. It can be seen that the bifurcation structure ismuch more complicated for κ = 1/2 than what happens for κ = 2 andκ = 1. We take A = 105, i.e. a = 1.05, as an example and we can seefrom the bifurcation diagram that there are four steady state solutions forthis value (three unstable solutions and one stable solution). The swimmer’sprofiles of the steady state solutions at A = 105 are displayed in panel (c),where only the solid curve is the stable solution and the rest are unstablesolutions.533.2. Swimming with Finite Stiffness0 1 2 300.20.40.60.81a= A/DMean Speed1.2 1.40.80.850.9(a) D=100 0.5 100.20.40.60.81a=A/DMean Speed0.8 1 1.2810 X vs. a(b) D=100−5 0 5−1012xY and h (c)A=105Figure 3.17: The solutions of the initial value problem for κ = 1/2 withD = 10 and D = 100. Panel (a) shows mean speed for D = 10 against a.Panel (b) shows mean speed for D = 100 against a. In both panel (b) and (c),the stars mark the stable solutions with topography. The solid and dashedline show the stable and unstable periodic solutions without topographyrespectively. The dash-dotted line displays the result for κ = 1. The inset of(b) shows the positional shift, X, of the four locked state solutions versus theforcing amplitude a for the swimmer swims with topography when D = 100.Panel (c) shows the four steady state solution profiles for A = 105 andD = 100. Only the solid curve is the stable solution and the rest are threeunstable solutions.54Chapter 4ConclusionIn this thesis, we extended a problem analyzed by G. I. Taylor [42] andD. F. Katz [25] to the situation where there is a microorganism swimmingvery close to a washboard. In this problem, we model the swimmer to bea two-dimensional, infinite periodic waving sheet. For simplicity, we onlyconsider the case where the fluid between the swimmer and the washboardis Newtonian and incompressible. We assume that the swimmer propagateswaves along its body and propels itself in the opposite direction. In thischapter, we summarize all the results and described what work could bedone in the future.4.1 Summary of ResultsTo begin with our problem, we first assumed the swimmer had a knownfixed sine wave shape. We first looked for the instantaneous swimmingspeed of the swimmer with various values of wavenumber. Here the casewhen the wavenumber κ = 1 was calculated analytically, while other valuesof wavenumber was calculated numerically. With amplitude of the resortingforce a picked to be 0.5 and amplitude of the topography b picked to be0.4, we saw the instantaneous swimming speed behaved quite differentlyfor various values of the wavenumber. No clear pattern was shown here.When we looked at the limiting case where the amplitude of the wavy wallapproached to zero, i.e. b→ 0, we found the swimming speed U →3a21 + 2a2,which matched with the result of a flat lower surface found by Chan etal. [10]. It can easily be seen that as the amplitude of the restoring forceapproached to zero, i.e. a→ 0, the swimming speed U approached to zero.When the sum of the amplitude of the washboard and the amplitude of the554.1. Summary of Resultsrestoring force approached to the mean distance between the swimmer andthe washboard, i.e. a+ b→ 1, the swimmer will touch with the washboardat some moments which limited the swimmer’s swimming speed.Next, we particularly looked at the case when the swimmer swam witha high amplitude force which gave a + b > 1. In this situation, the in-stantaneous speed formula, i.e. equation (2.30), was still valid when therewas a open gap between the swimmer and the washboard. Taking b = 0.4and κ = 1 as an example, we found when a was slightly above the value of1 − b, the swimming speed remained below one over the acceptable rangeof φ, which meant the swimmer always travelled to the left. While a wasvery high, the swimming speed remained above one. So, the swimmer al-ways travelled to the right. It was found that there was a small region of abetween 0.9 and 1.1 where the instantaneous swimming speed reached oneat two different phases. It suggested that there was a swimming directionchange over time. So, in this situation, the swimming behaviour of theswimmer would depend on the condition of the initial phase.We also looked at the average swimming speed of the swimmer over dif-ferent values of wavenumber. When a = 0.5 and b = 0.4 was examined, itdidn’t shown a clear pattern that how the wavenumber affected the swim-mer’s swimming behaviour. However, it was interesting to see that whenκ = 3 average swimming speed was comparably low and κ = 1/2 gave afairly high swimming speed. It was also found there was some dependenceof the topographic amplitude b on the relationship between average speedand the various values of the wavenumber. When κ < 1, average swimmingspeed was higher with larger value of b. In opposite, for κ > 1, averageswimming speed was bigger for smaller values of b.Finally, we checked general shapes of the washboard, which was definedas h(x) = b(1 + α) cosκx1+α cosκx − 1. Here the α is the parameter controllingthe shape of the washboard. Note when α = 0, the washboard went back tobe the simple cosine shape washboard. We first looked at the instantaneousswimming speed of the swimmer with forcing amplitude a = 0.5, topographicamplitude b = 0.4, and wavenumber κ = 1. We saw the swimmer behaveddifferently when swimming against the topography with different shapes.564.1. Summary of ResultsWe found that as the values of α increased, the instantaneous swimmingspeed at zero phase decreased. We then checked the average swimmingspeed against different values of topographic amplitude for various values ofα. We found that, for α = 0 and 0.5, as the swimmer and the topographygot very close, the average swimming speed converged to the same value.The averaged swimming speed for α = 2 was always lower than the casesα = 0 and 0.5.In addition to the case when the swimmer had a fixed known shape, welooked at the case of an elastic swimmer in Chapter Three. We first pickedD = 1, κ = 1 and b = 0.4 as an example. When the amplitude of therestoring force was small, after a very short transient, the swimmer stayedin a well-defined periodic state. When the amplitude of the restoring forcewas small, the mean speed of the swimmer with topography matched withthe solution without topography. It increased quadratically as A increased.While for bigger forcing amplitude, the mean speed with topography in-creased much more rapidly than without topography. The mean speed ofthe swimmer with topography stayed with unity after some critical valueof forcing amplitude A∗. We specially looked at this high forcing ampli-tude swimmer and found there were two branches of steady state solutionsexisting after reaching this A∗. This phenomenon could be explained bythe saddle-node infinite-period bifurcation. We also tried various values ofamplitude of the topography and saw the mean speed increased with thetopographical amplitude for A = 2, 3 and 3.8.Next, we varied the values of the stiffness. When D = 100, same as D =1, the swimmer stayed as periodic when A is small. While reached the steadystate with a high forcing amplitude. Looking at the cases for D = 100 andD = 10, we saw, for small forcing amplitude, mean speed with topographymatched with the mean speed without topography for both stiffness. Whilefor high forcing amplitude, it showed a big difference between mean speedwith topography and without topography when D = 10. However, no bigdifference was shown for D = 100. As stiffness increased, the steady statesolutions converged to the steady state solutions of the fixed shape swimmer.Finally, we examined various values of wavenumber to see how it af-574.2. Future Workfected the swimming behaviour. When κ = 2, with D = 1, we saw, havinglow forcing amplitude, the swimmer swam periodically. In this regime, theswimmer actually swam faster without the washboard. As forcing ampli-tude increased, the swimmer reached the locked state when swimming withthe topography. After doing the bifurcation analysis, it was interesting tosee that when A was between 6 and 8, the periodic and locked swimmingsolutions co-existed. Due to the second saddle-node infinite-period bifurca-tion happening for this wavenumber, the swimmer went back to the peri-odic swimming state with a very high forcing amplitude. The situation forκ = 1/2 was way more complicated compared to κ = 2. We saw for thisvalue of wavenumber, the swimmer wasn’t able to push itself neither moreor less effectively over the washboard than the flat wall. At low forcing am-plitude, mean speed for κ = 1/2 was similar with the swimming speed forκ = 1. While for high forcing amplitude, subharmonic instability showedup. We tried pulling up the stiffness number for this wavenumber, and foundwhen D = 10, the washboard didn’t help with the swimming at all. WhenD = 100, for high forcing amplitude, unlike the small stiffness case, thewashboard actually helped the swimmer to swimming more effectively. Itwas also found that for this stiffness value the bifurcation structure was verycomplicated.4.2 Future WorkLots more work could be done beyond this thesis. It has been mentioned inthe literature that microorganisms frequently experience complex fluids[32,45]. However, this thesis only examined the situation of Newtonian fluids.From a biological perspective, it will be good if we can extend this workto complex fluids. Moreover, in this thesis, we assumed the swimmer is aninfinite length waving sheet. However, in the real world, swimmer alwayshas a finite length. It will be good if we could look at the situation of a finiteswimmer as shown in [3]. Another thing mentioned in the paper written byMajmudar et al. is that in nature there are many instances where swim-ming microorganisms must make their way through a fluid embedded with584.2. Future Workobstacles [34]. However, in this thesis, we mostly looked at the topographyas a cosine curve. A general shape of the topography was only looked atfor the case of the fixed shape swimmer. We should also extend our workto look at a general shape of the topography for the elastic swimmer. Tobe more realistic, we could even extend the work to feature complex fluidssetting and a general shape of the washboard.So far, our analysis has been limited to the case where there is a verythin gap between the swimmer and the topography. It will be interesting toexamine other cases. For example, we could attempt biharmonic analysis(done by Katz in [25]) for our problem. We could even further extend ourmodel to the case where the gap between the swimmer and the topographyis not small and we do not have a small amplitude swimmer. In the elasticswimmer chapter, we have found that for high values of forcing amplitude,the swimmer got pushed close to the washboard, while the big forcing am-plitude also left a bubble of trapped fluid between the swimmer and thewashboard. The small cap region is called an occluded region and the bigbubble is called a blister. We see the work of asymptotic analysis about this“occluded and blistered” region have been done for the case when swim-ming near a flat plane in [3]. Here we could follow the same analysis for ourproblem to get some inside understanding of this behaviour analytically, wecould further compare the analytic solution with our numerical solution.59Bibliography[1] B. Alberts, A. Johnson, J. Lewis, M. Raff, K. Roberts, and P. Walter.Molecular Biology of the Cell. Garland Science, New York, 2007.[2] R. M. Alexander. Locomotion of animals. Chapman and Hall, NewYork, 1982.[3] N. J. Balmforth, D. Coombs, and S. Pachman. Microelastohydrody-namics of swimming organisms near solid boundaries in complex fluids.Q. J. Mech. Appl. Maths, 63:267–294, 2010.[4] J. R. Blake. Infinite models for ciliary propulsion. J. Fluid. Mech.,49:209 – 222, 1971.[5] J. R. Blake. 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Soc. Lond. A, 211:225–239, 1952.[44] U. Thiele and E. Knobloch. Driven drops on heterogeneous substrates:onset of sliding motion. Phys. Rev. Lett., 97:204501.1–204501.4, 2006.[45] S. Yazdi, A. M. Ardekani, and A. Borhan. Locomotion of microorgan-isms near a no-slip boundary in a viscoelastic fluid. Phys. Rev. Lett.,90:043002.1 – 043002.11, 2014.63Appendix ADetails of Integral EvaluationWhen we calculate the exact solution of instantaneous swimming speed forthe fixed shape swimmer with κ = 1, there are six Imn integrals evaluationinvolved. In this appendix, we provide full calculation of these six integrals,which has been omitted in section 2.2.1.Recall in Chapter Two, the swimmer has a fixed shape Y = a cos ξ andthe washboard wall is defined as h = b cosx− 1. The distance d in betweenis given asd = a cos ξ − b cosx+ 1 = a cos(x+ φ)− b cosx+ 1= C1 cos(x+ Φ) + 1, (A.1)where φ(t) measures the phase of translation of the swimmer with respectto the washboard. To sum these two cosine functions, we define tan Φ =sinφcosφ− baand C1 = a√(cosφ− ba)2 + sin2 φ < 1.Now we will calculate six integrals one by one:I03 =12pi∫ 2pi01(C1 cos(x+ Φ) + 1)3dxNote: Φ is a phase shift in x. If we integrate over a whole period in x, thephase shift will not change the integration result, i.e.12pi∫ 2pi01(C1 cos(x+ Φ) + 1)3dx =12pi∫ 2pi01(C1 cos(x) + 1)3dx.64Appendix A. Details of Integral EvaluationThus,I03 =12pi∫ 2pi01(C1 cos(x) + 1)3dx =12pi·12d2dα2[1α∫ 2pi0dx1 + C1α cosx] ∣∣∣∣∣α=1.By symmetry, we have∫ 2pi0dx1 + C1α cosx= 2∫ pi0dx1 + C1α cosx.With a tangent substitution, i.e. u = tanx2, the above integral is changedto be∫ pi0dx1 + C1α cosx=∫ ∞02(1− C1α)(1 + u2) + 2C1αdu=2(1 + C1α)·√1− C1α1 + C1αarctan(√1− C1α1 + C1α· u)∣∣∣∣∣∞0=pi√1−(C1α)2.Here, we can getI01 =1√1− C12(A.2)and I03 =12pi·12d2dα2(2pi√α2 − C12)∣∣∣∣∣α=1=12·(2 + C12)(1− C12) 52. (A.3)Similarly, we haveI02 = −12pi·ddα[1α∫ 2pi0dx1 + C1α cosx] ∣∣∣∣∣α=1=1(1− C12) 32. (A.4)65Appendix A. Details of Integral EvaluationWe can separate I13 into two parts asI13 =12pi(∫ 2pi01(Y − h)2dx+∫ 2pi0h(Y − h)3dx),and calculate these two parts one by one. We have found out earlier that12pi∫ 2pi01(Y − h)2dx =1(1− C12) 32.Notice that12pi∫ 2pi0h(Y − h)3dx =12pi∫ 2pi0b cosx(C1 cos(x+ Φ) + 1)3dx− I03 ,and∫ 2pi0b cosx(C1 cos(x+ Φ) + 1)3dx= b cos Φ∫ 2pi0cos x¯(C1 cos x¯+ 1)3dx¯+ b sin Φ∫ 2pi0sin x¯(C1 cos x¯+ 1)3dx¯.Due to the 2pi-periodic condition on cos x¯, we haveb sin Φ∫ 2pi0sin x¯(C1 cos x¯+ 1)3dx¯ = 0.Thus, we only need to deal with the first integral.Notice that∫ 2pi0cos x˜(A cos x˜+ 1)3dx˜ =12d2dα2[1α∫ 2pi0cos x˜1 + Aα cos x˜dx˜] ∣∣∣∣∣α=1.Again, we can use a tangent substitution as what we did for I03 and getb cos Φ∫ 2pi0cos x¯(C1 cos x¯+ 1)3dx¯ = b cos Φ−3C1pi(1− C12) 52 .66Appendix A. Details of Integral EvaluationThus,I13 =12pi2pi(1− C12) 32+ b cos Φ−3C1pi(1− C12) 52−pi(2 + C12)(1− C12) 52 . (A.5)Similarly,I12 =12pi(∫ 2pi01Y − hdx+∫ 2pi0h(Y − h)2dx)= I01 +12pi∫ 2pi0h(Y − h)2dx.Again,12pi∫ 2pi0h(Y − h)2dx =12pi∫ 2pi0b cosx(C1 cos(x+ Φ) + 1)2dx− I02 ,and∫ 2pi0b cosx(C1 cos(x+ Φ) + 1)2dx=b cos Φ∫ 2pi0cos x˜(C1 cos x˜+ 1)2dx˜+ b sin Φ∫ 2pi0sin x˜(C1 cos x˜+ 1)2dx˜=b cos Φ[−ddα(1α∫ 2pi0cos x˜1 + C1α cos x˜dx˜)] ∣∣∣∣∣α=1= −2pi(C1b cos Φ + 1)(1− C12) 32.Thus, combing all the parts, we getI12 = −C1(C1 + b cos Φ)(1− C12) 32. (A.6)The very last integral we need to deal with is I23 , which again can beseparated into two parts asI23 =12pi(∫ 2pi0Y + h(Y − h)2dx+∫ 2pi0h2(Y − h)3dx)=12pi−2pi(C12 + 2C1b cos Φ + 1)(1− C12) 32+∫ 2pi0h2(Y − h)3dx .67Appendix A. Details of Integral EvaluationThe second integral could be separated into three more parts as∫ 2pi0h2(Y − h)3dx =∫ 2pi0b2 cos2(x¯− Φ)(C1 cos x¯+ 1)3dx¯+6C1pib cos Φ(1− C12) 52+pi(2 + C12)(1− C12) 52.The first integral on the right hand side of above equation could be evaluatedby tangent substitution again, and we get∫ 2pi0b2 cos2(x¯− Φ)(C1 cos x¯+ 1)3dx¯ = b2 cos2 Φpi1 + 2C12(1− C12) 52+ b2 sin2pi(1− C12) 32.Combing all the parts, it givesI23 =12pi[−2pi(C12 + 2C1b cos Φ + 1)(1− C12) 32+ b2 cos2 Φpi1 + 2C12(1− C12) 52+ b2 sin2pi(1− C12) 32− 2−3C1pib cos Φ(1− C12) 52+pi(2 + C12)(1− C12) 52], (A.7)wheresin Φ =sinφ√sin2 φ+(cosφ− ba)2and cos Φ =cosφ− ba√sin2 φ+(cosφ− ba)2.Hence, all six integrals we need for evaluating instantaneous swimming speedare solved and their solutions are listed as (A.2)-(A.7).68

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