Drag and thrust effects of Viscoelastic fluidsbyGaurav GoyalB.Tech (Honours), Indian Institute of Technology, Madras, 2014A THESIS SUBMITTED IN PARTIAL FULFILLMENT OFTHE REQUIREMENTS FOR THE DEGREE OFMASTER OF APPLIED SCIENCEinThe Faculty of Graduate and Postdoctoral Studies(Mechanical Engineering)THE UNIVERSITY OF BRITISH COLUMBIA(Vancouver)December 2016c© Gaurav Goyal 2016AbstractViscoelastic fluids are non-Newtonian fluids exhibiting both viscous and elastic properties.Many fluids of practical importance (polymers, surfactants, mucus, shampoos etc.) displayviscoelastic effects to different degrees under a wide range of flow conditions and thus, thesefluids present a variety of problems. In this work, we study two problems at very differentflow conditions in viscoelastic fluids: a) the effect of swimming gait on bio-locomotion andb) characterizing the drag reducing fluids used for gravel-packing operations in the petroleumindustry.For the first problem, we give formulas for the swimming of simplified two-dimensionalbodies at low Reynolds numbers in complex fluids using the reciprocal theorem. By way ofthese formulas, we calculate the swimming velocity due to small-amplitude deformations on thesimplest of these bodies, a two-dimensional sheet, to explore general conditions on the swimminggait under which the sheet may move faster, or slower, in a viscoelastic fluid compared to aNewtonian fluid. We show that in general, for small amplitude deformations, a speed increasecan only be realized by multiple deformation modes in contrast to slip flows. Additionally, weshow that a change in swimming speed is directly due to a change in thrust generated by theswimmer.Later, we work with viscoelastic additives (xanthan and a zwitterionic viscoelastic surfac-tant, VES), widely used as drag reducers for gravel-packing applications. While the behaviorof xanthan is well characterized in the literature, much less is known about the VES charac-teristics, despite widespread use. We performed a number of rheological tests and flow-loopexperiments on VES solutions to understand the structural characteristics to make better pro-cess predictions. Unlike xanthan, which displays typical viscoelastic liquid characteristics, VESdisplays elastic gel-like behaviour. The gel-like behaviour suggests long and relatively unbreak-able chain lengths of the wormlike micelles in the VES at room temperature leading to gelationby entanglement. Also, a shear-thickening behaviour of VES samples at higher shear rates isobserved, possibly as a result of shear-induced structures. Finally, we present a novel repre-sentation scheme to depict the flow-loop results relating the rheological characterization whileobserving drag reduction.iiPrefaceThe research presented in this thesis is original work of the thesis author under the super-vision of Dr. Gwynn J. Elfring and Dr. Ian. A. Frigaard.A version of Chapter 2 is published in Journal of Non Newtonian Fluid Mechanics as ”G.J.Elfring and G. Goyal, The effect of gait on swimming in viscoelastic fluids, vol. 234, pp. 8-14,2016.”Chapter 3 is original and unpublished work of the thesis author.iiiTable of ContentsAbstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiPreface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iiiTable of Contents . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ivList of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viList of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiAcknowledgements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . viiiDedication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ix1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.1 Viscoelastic fluids . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11.2 Surfactant solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.3 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 Effect of gait on swimming in Viscoelastic fluids . . . . . . . . . . . . . . . . . 62.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62.2 Swimmer motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 72.3 The complex reciprocal theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 82.3.1 Small amplitude deformations . . . . . . . . . . . . . . . . . . . . . . . . 102.4 Viscoelastic fluid relations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112.5 Model Swimmers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5.1 Spherical swimmers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5.2 Cylindrical swimmers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122.5.3 Planar swimmers . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132.6 Generalized Sheet in a viscoelastic fluid . . . . . . . . . . . . . . . . . . . . . . . 142.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 Rheology and flow studies of drag reducing gravel packing fluids . . . . . . 203.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203.2 Experimental work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21ivTable of Contents3.2.1 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2.2 Sample preparation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213.2.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3.1 Rheometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.3.2 Flow loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 273.4 Discussion and Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31AppendicesA Oldroyd-B model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42A.1 The model equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42A.2 Small amplitude motion : Perturbations and Fourier series . . . . . . . . . . . . 43A.3 First order . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44A.4 General Relationship & Second order . . . . . . . . . . . . . . . . . . . . . . . . 44B The complex reciprocal theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 48C Rheology tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50D Flow loop data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52vList of Tables1.1 Commonly used constitutive models for VE fluids . . . . . . . . . . . . . . . . . . 3C.1 Methods and characteristics tested in Rheology . . . . . . . . . . . . . . . . . . . 50C.2 Apparent viscosity comparison between smooth and serrated parallel plate ge-ometry for VES (3.5%, 10.7 ppg CaCl2) . . . . . . . . . . . . . . . . . . . . . . . 50C.3 Rheological results for different gap values of serrated PP geometry with VES(3.5%, 10.7 ppg CaCl2) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51D.1 Flow loop measured and computed data for Xanthan (0.06 ppg) solution . . . . 54D.2 Flow loop measured and computed data for VES (3.5%, 10.7 ppg CaCl2) . . . . 57D.3 Flow loop measured and computed data for VES (4.5%, 9.2 ppg NaCl) . . . . . 59viList of Figures1.1 Pictorial denotations of two material responses: a) a dashpot (viscous fluid) andb) a spring (an elastic solid) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21.2 A series combination representing 1D linear Maxwell model . . . . . . . . . . . . 21.3 Schematic phase diagram for a surfactant solution [11] . . . . . . . . . . . . . . . 42.1 Schematic representation of a general swimmer. A swimmer is defined as bodywith whose surface deforms in time thereby effecting an instantaneous rigid-bodytranslation, U, and rotation, Ω. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73.1 A schematic of the flow loop . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 233.2 Viscosity vs shear rate for xanthan (0.04 ppg) at room temp. . . . . . . . . . . . 243.3 A SAOS frequency sweep (strain = 5 %) for xanthan (0.06 ppg) at room temp. . 243.4 A SAOS sequence (1 Hz, CS: 1 Pa) to monitor structure build up after 100 s−1,750 s−1 and 1100 s−1 for xanthan (0.06 ppg) at room temp. Here t = 0 indicatesthe time when the pre-shear is stopped. . . . . . . . . . . . . . . . . . . . . . . . 253.5 Photographs at room temperature of 20 ml VES samples for a) NaCl, 4.5%, b)NaCl, 5.5%, c) CaCl2, 3.5% d) CaCl2, 5.5% . . . . . . . . . . . . . . . . . . . . 253.6 A SAOS frequency sweep (CS = 1 Pa) for VES (5.5%, 11 ppg CaCl2) at roomtemp. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 263.7 Shear rheology profile of VES (3.5%, 10.7 ppg CaCl2) at room temp. . . . . . . . 263.8 A SAOS sequence (1 Hz, CS: 1 Pa) to monitor structure build up after 100 s−1,750 s−1 and 1000 s−1 for VES (3.5% 10.7 ppg CaCl2) at room temp. Here t = 0indicates the time when the pre-shear is stopped. . . . . . . . . . . . . . . . . . . 273.9 Friction factor vs Reynolds number for a xanthan (0.06ppg) and VES (3.5%, 10.7ppg CaCl2 and 4.5%, 9.2 ppg NaCl) solutions . . . . . . . . . . . . . . . . . . . . 29viiAcknowledgementsFirst and foremost, I would like to thank my thesis advisors, Prof. Gwynn Elfring and Prof.Ian A. Frigaard for being motivational, supportive and providing me an opportunity to gaina holistic research experience through working on theoretical and experimental approaches toproblems. I would also like to thank Prof. Savvas Hatzikiriakos and Prof. Mark Martinez forthe guidance and permitting the use of their experimental facilities.The members of the Complex fluids lab have contributed a lot to my academic and personalgrowth. I am grateful to the group: Zhiwei, Charu, Babak, Jaewoo, Parisa, Ali, Marjan for allthe wonderful time discussing topics ranging from the complex reciprocal theorem, viscoelasticfluids, rheometry to sports, food and world politics. I would like to acknowledge Dr. Mahmoud,Dr. Abbas and Dr. Maziyar for all the technical assistance with experiments and helpful dis-cussions on rheology and flow loop studies. The technical discussions with Dr. Mehmet Parlarand Dr. Bala Gadyar (Schlumberger Ltd.) were quite helpful to gain a practical understandingof the problem. Also, I would like to thank George Soong (PPC safety officer) for helping meget through the viciously viscous tanks of VES fluids. Finally, I am grateful for the fundingfrom MITACS Inc., Natural Science and Engineering Research Council of Canada (NSERC)and Schlumberger Ltd. to support my graduate studies.I consider myself fortunate to have wonderful family-like landlords (Uncle Hari & AuntyDJ), roommates (Fuhar, Manish, Raghav), badminton group (Sujay, Ankur, Nishant), travelgroup (Varun, Lakshana, Shubham, Namrata), residence advisors (Michael, Nick, Rob, Gavan,Harry) and a lot of friends who made the last two years enjoyable at UBC with many memorableexperiences. Finally, I would like to thank my family without whose support I could not havemade this experential academic journey from Barnala to Vancouver!viiiDedicationTo my parents, for their love and support.ixChapter 1Introduction1.1 Viscoelastic fluidsViscoelastic (VE) fluids are a common class of non-Newtonian fluids comprising variousbiological fluids (mucus, synovial fluid etc.), industrial fluids (polymers, surfactant solutionsetc.) and everyday examples like toothpaste, soft-serve ice cream, shampoos and gelatin. Theyexhibit both the viscous and elastic behaviour under a deformation: the fluid relaxes back likean elastic solid and dissipates deformation energy like a viscous liquid.To understand VE fluids, we first consider two limiting idealized cases i.e. a linear elasticsolid and a Newtonian viscous fluid to understand the distinction between the two extremelydifferent responses. Under a simple shear deformation caused by shear stress (τs), the strain(γs) developed in a linear elastic solid can be modelled using Hooke’s law and written asγs =τsG, (1.1)where G is the spring constant or the shear modulus of an elastic material. It implies that anelastic solid reacts instantly to the deformation stress and develops strain, which goes to zero assoon as the shear stress is removed. On the contrary, the strain (γd) developed in a Newtonianviscous fluid is related to the stress (τd) using Newtonian’s law asτd = ηγ˙d,⇒ γd = τdηt+ γd,0, (1.2)where η is viscosity of the fluid, γd,0 is the zero (initial) strain and the dot denotes the timederivative. It should be noted that the strain increases linearly with time and takes time tobuild up. Additionally, the built up strain is permanent and does not go to zero as the shearstress is removed. These linear elastic solids and Newtonian viscous fluids are often denotedwith springs and dashpots respectively as shown in fig. 1.1 because of their similar exhibitionof properties. Thus, a linear VE fluid can be thought of a combination of springs and dashpots.Maxwell model is one of the simplest combination of spring and dashpot as shown in fig.1.2. We consider the total deformation and stress to be γ and τp respectively. Analogous toelectrical circuits, γ and τp can be considered as the voltage and current respectively. Usingthe series constraint (γ = γs + γd and τp = τs = τd) and equations 1.1 & 1.2, the constitutive11.1. Viscoelastic fluidsa) b)Figure 1.1: Pictorial denotations of two material responses: a) a dashpot (viscous fluid) andb) a spring (an elastic solid)Figure 1.2: A series combination representing 1D linear Maxwell modelequation for the Maxwell model can be derived asτp + λτ˙p = ηpγ˙, (1.3)where λ = ηp/G is known as the relaxation time. In particular, this equation is not frameinvariant and can not be used for arbitrary flows. Thus, convected derivates are used to extracta frame invariant upper convected Maxwell model [1] which is written asτp + λ∇τp= ηpγ˙, (1.4)where the upper convected derivative for a tensor A is defined as∇A=∂A∂t+ u.∇A− (∇uT .A+ A.∇u). (1.5)A simple extension of upper convected Maxwell model is including the Newtonian solventcontribution (τs = ηγ˙s) which leads to an Oldroyd-B model for a viscoelastic fluid solutionswith a single relaxation time (see appendix A) and can be written asτ + λ1∇τ = η[γ˙ + λ2∇γ˙], (1.6)where τ = τs + τp, η = ηs + ηp, γ˙ = ∇u + ∇uT , λ1 = λ and λ2 = ηsλ/η < λ1, also knownas the retardation time. The Oldroyd-B model can also be derived exactly from the kinetictheory where the fluid molecules are modelled by elastic dumbbells consisting of an elastic21.2. Surfactant solutionsspring connecting two beads [2].We could have a variety of different models describing VE fluids equivalent to the variouspossible combinations of springs and dashpots. Therefore, there is no single model which coulddescribe all VE fluids unlike Navier-Stokes equations for Newtonian flows. For example, micellarsolutions with wormlike micelles also involve the breaking and recombination of chains whichmakes an Oldroyd-B model insufficient to explain the flow behaviour. Thus, a constitutivemodel is selected based on the the fluid microstructure as well as the flow conditions. Some ofthe common VE models are tabulated in table 1.1Model name CommentsOldroyd-B model [3] Extension of upper convected Maxwell modelFENE-P model [4] Elastic dumbbell springs with Finite Extensibility and Non-linear Elasticity-Peterlin approximationGiesekus model [5] Similar to Oldroyd B with a quadratic nonlinear term basedon configuration dependent molecular mobilityWhite-Metzner model [6] Modified upper convected Maxwell model to include shearthinning fluidsPhan-Thien-Tanner model[7]Similar to Giesekus with a different non linear term allow-ing an effective slip at network junctionsJohnson-Segalman-Oldroydmodel [8]Similar to Oldroyd-B model with (more general) Gordon-Scwalter convected derivative capturing nonaffine motionsVasquez-Cook-Mckinleymodel [9]Derived form kinetic theory based on a simplified versionof Cates model [10] for wormlike micellar solutionsTable 1.1: Commonly used constitutive models for VE fluids1.2 Surfactant solutionsSurfactant solutions, also referred as amphiphilic compounds generally exhibit viscoelasticcharacteristics and are distinguished with their structures containing a hydrophobic tail as wellas a hydrophilic head. Generally, the hydrophobic tail is a long chain of alkyl bases whereas ahydrophilic head is ionizable, polar or polarizable. Surfactants are often categorized accordingto charged groups on their hydrophilic head as non-ionic (no charge), anionic (negative), cationic(positive) or zwitterionic/amphoteric (positive and negative) surfactants.In an aqueous environment, the surfactant molecules orient in such a way that the con-tact between water and hydrophobic tail is avoided. There are only two ways to achieve this.Firstly, the molecules can try to leave the water phase and thus, approach the surface, aligningthe tails to the non-polar phase such as gas phase, a non-polar solid or a hydrophobic liquidphase. Secondly, the tails can realign in such a manner to form micelles where the tails areconcentrated in the centre with the heads at the surface in contact with the water phase. Asthe surfactants are continuously added to an aqueous environment, the first response of themolecules is to take the former approach of aligning at the surface (reducing surface tension)31.2. Surfactant solutionsuntil a critical concentration when the surface is largely covered and the latter approach ofmicellization becomes favourable. This critical concentration is known as Critical Micelle Con-centration (CMC). Above the CMC, the formed micellar structures and monomer molecules arein a state of thermodynamic equilibrium through breaking and recombination process. This isrepresented schematically as a phase diagram in fig. 1.3.CMCKrafft pointSolubility curveCMCIIRod-like micelles(Wormlike micelles)Spherical micellesMonomersTemperatureConcentrationFigure 1.3: Schematic phase diagram for a surfactant solution [11]The Krafft point is the minimum temperature required for micellization. As summarizedby Zakin et al [11], the surfactant is partially in crystal or gel form in the solution belowthe Kraft temperature. With further addition of micelles, there exists an another criticalconcentration (CMCII) above which the spherical micelles combine to form rod-like micelles(wormlike micelles) resembling long chains of polymers. With increasing concentration, thesewormlike micelles become large enough which can overlap and get entangled to form a densenetwork of wormlike micelles.Cates [10] proposed a primal linear viscoelastic model for wormlike micelles which con-sidered the reptation model for unbreakable chains [12, 13] modified to include the break-ing/recombinationof networks for stress relaxation. The two characteristic times involved inCates model are reptation time (τrep) and breaking time (τb). The τrep represents the timeneeded for a chain to reptate out of it’s original tube of average length L¯ whereas τb is definedas the mean time required for a chain of length L¯ to break into two or, equivalently, the lifetimeof micellar end before it recombines. These times scale with length asτrep ∝ L¯3, (1.7)τb ∝ L¯−1, (1.8)which physically can be understood as the longer chains are expected take longer time to reptate41.3. Thesis Outlineand shorter time to break. The relaxation times are estimated for the two limiting cases relatedto the relative magnitude of the characteristic times:1. τb << τrep - The chain breaking/recombination occurs multiple times favourably beforethe chain reptates out of it’s tube. The relaxation (stress decay) can be described witha single exponential and thus, a single terminal relaxation time which is the geometricalmean of characteristic times,τr = (τbτrep)1/2 ∼ L¯1 (1.9)Consider a tube segment newly formed by breakage/recombination of the molecules. Theτr relates to the time for the chain to reptate through this new tube segment before itloses an end by breakage/combination.2. τb >> τrep - The chain breaking/recombination is an unfavourable process and the life-time of wormlike micelles is long enough that it completes it’s reptation before any break-age/recombination occurs. The relaxation (stress decay) is dictated by reptation anddescribed as a nearly pure exponential with the terminal (longest) relaxation time similarto the reptation timescale,τr ∼ τrep ∼ L¯3. (1.10)1.3 Thesis OutlineThe thesis discusses two problems observed at very different flow conditions in viscoelasticfluids.Chapter 2 presents the mathematical modelling of bio-locomotion in complex fluids. Weexamine the conditions on the deformation of a two dimensional sheet to swim faster, or slowerin a viscoelastic fluid as compared to a Newtonian fluid. We also prescribe a slip velocity toobserve the pure elastic response and show speed change as a direct result of change in thrustby swimmer. Furthermore, the Oldroyd-B model equation and complex reciprocal theorem areelaborated mathematically in appendices A and B respectively.Chapter 3 examines the rheological characteristics and drag reduction effects of gravel pack-ing viscoelastic fluids, namely Xanthan and a zwitterionic surfactant solution. In appendixC,we mention the rheological methods performed along with wall slip & gap dependency resultsand appendix D tabulate the flow loop results for reference.5Chapter 2The effect of gait on swimming inviscoelastic fluids12.1 IntroductionThe locomotion of microscopic cells through viscous fluids is common in many areas ofbiology, from microbes searching for food [14, 15] or causing diseases [16], to sperm cells inmammalian reproduction [17]. The fluid forces generated by a deforming body at low Reynoldsnumbers are arguably well understood for Newtonian fluids [18, 19], but insight is far morelimited when considering bio-locomotion through fluids that display non-Newtonian rheology[20].Many biological fluids like blood, mucus, saliva and synovial fluid, demonstrate viscoelas-ticity and shear-thinning viscosity [21–23]. A viscoelastic fluid retains a memory of its flowhistory, while a shear-thinning fluid experiences a decrease in apparent viscosity with appliedstrain-rate and it is in such fluid environments that many microorganisms swim; Helicobac-ter pyroli in gastric mucus [24], and spermatozoa wading through cervical mammalian mucus[25] are common examples. Swimming in complex fluids can be substantially different than inNewtonian fluids, for example, locomotion in complex fluids is not constrained by the scalloptheorem [26] meaning reciprocal swimming strokes, which produce no net motion whatsoeverin Newtonian fluids, can propel a swimmer in complex fluids [27–29].Several recent articles have investigated changes in swimming kinematics due to nonlinearviscoelasticity theoretically [27, 30–43], numerically [44–49] and experimentally [50–55], whilecomprehensive reviews have summarized key findings in the theory [56], and experiments [57],of biolocomotion in complex fluids. The picture that emerges from recent studies on the effectsof viscoelastic fluids, is that whether a swimmer goes faster or slower depends on the type ofgait [44, 47] or the amplitude of the gait [46, 50]. Several studies have used the simplifiedmodels such as an infinite sheet [38, 39, 56] or an infinite helix [40] undergoing general small-amplitude deformations to shed light on how a particular gait (or which) may lead to faster(or slower) swimming in the presence of a nonlinearly viscoelastic fluid. It is in this vein thatwe proceed here, extending the recent results of Riley and Lauga [39] to a broader class ofboundary conditions on flat body. We discuss in detail why all individual deformation modes1A version of Chapter 2 has been published in Journal of Non Newtonian Fluid Mechanics. G.J. Elfring andG. Goyal, The effect of gait on swimming in viscoelastic fluids, vol. 234, pp. 8-14, 2016.62.2. Swimmer motionr0SS0rS U+⌦⇥ rS + uSFigure 2.1: Schematic representation of a general swimmer. A swimmer is defined as bodywith whose surface deforms in time thereby effecting an instantaneous rigid-body translation,U, and rotation, Ω.lead to slower swimming and how one may obtain faster swimming by determining the natureof the nonlinear response of the fluid. We also show that a prescribed slip velocity can lead toan entirely different viscoelastic response compared to a prescribed deformation. To do this,we first derive integral theorems for the locomotion of two-dimensional bodies in complex fluids[36], extending to non-Newtonian fluids recent results on the use of the reciprocal theorem forthe swimming of two-dimensional bodies in Newtonian fluids [58]. Through the use of theseintegral theorems one can show that slower swimming is directly the result of a reduction ofthrust in a viscoelastic fluid for any single (small amplitude) deformation mode.2.2 Swimmer motionThe motion of the surface, S(t), of a shape changing swimmer may be decomposed intotranslation, rotation and deformation as followsu(xS) = U + Ω× rS + uS , (2.1)where xS ∈ S(t). The first two terms represent rigid-body motion, U is the instantaneoustranslation, Ω is the angular velocity, while the third term uS is deformation due to a swimminggait [56, 59]. One often defines rS from the center of mass (extensive discussion is providedelsewhere [58–61]) and periodic deformations are written as deviations from a (simple) referencesurface S0 in a body-fixed frame asrS = r0 +∆r(r0, t), (2.2)and then uS = ∂∆r/∂t (see figure 2.1).We describe here the locomotion of microorganisms small enough that the Reynolds numberof the flows generated may be taken to be zero∇ · σ = −∇p+∇ · τ = 0, (2.3)where σ = −pI + τ is the stress tensor of a fluid with velocity, pressure and deviatoric stress72.3. The complex reciprocal theoremfields u, p, τ , respectively. Additionally the bodies themselves are considered instantaneouslyforce and torque free,F =∫Sn · σ dS = 0, (2.4)L =∫SrS × (n · σ) dS = 0, (2.5)where the surface S is a function of time and the normal to the surface n points into thefluid. For compactness we introduce six dimensional vectors that contain both force and torqueF = [F L]> and translation and rotation U = [U Ω]>.2.3 The complex reciprocal theoremStone and Samuel showed that determining the rigid-body motion of a deforming swimmer[62], U and Ω, may be greatly simplified by appealing to the Lorentz reciprocal theorem [63].They showed that by using the solution to the auxiliary rigid-body resistance problem onemay solve for the swimming kinematics of a deforming body without resolving the flow fieldit generates. Lauga later extended the use of the reciprocal theorem for swimming to non-Newtonian fluids [27] and these ideas were then subsequently further developed [36, 56] andintegral theorems have subsequently been used in a number of recent theoretical studies oflocomotion in complex fluids [34, 37, 41, 42, 64]. We present here integral theory for a generalnon-Newtonian fluid in the formalism of Elfring and Lauga [56] before showing its applicationto simple bodies.For a force- and torque-free swimmer of surface S in a non-Newtonian fluid we denotethe velocity field and its associated stress tensor by u and σ, while for a body of the sameinstantaneous shape subject to (an arbitrary) rigid-body translation and rotation Uˆ and Ωˆ ina Newtonian fluid (τˆ = ηˆ ˆ˙γ) the velocity field and its associated stress tensor are denoted byuˆ and σˆ. Each fluid is in mechanical equilibrium and hence the mixed products uˆ · (∇ · σ) =u · (∇ · σˆ) = 0, then by integrating over the volume of fluid V (t) external to the surface S(t)with normal n (positive into the fluid) and invoking the divergence theorem one obtains∫Sn · σ · uˆ dS +∫Vσ :∇uˆ dV =∫Sn · σˆ · u dS +∫Vσˆ :∇u dV = 0. (2.6)The first term on the left-hand side of Eq. (2.6) is zero because the swimmer is force- andtorque-free, and hence the second term on the left-hand side is also zero, by construction. Thefirst term on the right-hand side of Eq. (2.6) may be expanded by using the boundary motion82.3. The complex reciprocal theoremon S in (2.1) and so, assuming the fluids are incompressible, we haveFˆ · U = −∫Sn · σˆ · uS dS − ηˆ∫Vˆ˙γ :∇u dV, (2.7)0 =∫Vτ :∇uˆ dV. (2.8)Here Fˆ represents the force and torque resulting from the rigid-body motion of S.Due to the linearity of the Stokes equations we may write uˆ = Gˆ · Uˆ, σˆ = Tˆ · Uˆ whileFˆ = −Rˆ · Uˆ. We also assume a simplistic decomposition of the constitutive relation into linearand nonlinear components τ = ηγ˙ + N(u, τ ), but, as shown by Lauga[36], we expect sucha form to arise when solving the problem perturbatively, either expanding in a weakly non-Newtonian limit or a small deformation limit, in either case, the effects of nonlinearities aresmall compared to the leading order Newtonian behavior. Substituting into (2.7) and utilizing(2.8) while discarding the arbitrary vector Uˆ yieldsU = Rˆ−1 ·[∫SuS · (n · Tˆ ) dS − ηˆη∫VN :∇Gˆ dV]. (2.9)The first term in the brackets represents thrust generated in a Newtonian fluid (as we showbelow) whereas the volume integral only contributes if the fluid in the swimming problem isnon-Newtonian and hence is a measure of the modification of the swimming dynamics due tothe presence of non-Newtonian stresses.One may also obtain integral statements for drag and thrust in complex fluids. In the dragproblem, the body is simply undergoing rigid body motion, namely uD(xS) = U + Ω × rS ,whereas in the thrust problem the body is deforming but otherwise held fixed in place uT (xS) =uS . Due to the linearity of the Stokes equations, these two flow fields sum to give the flow fielddue to swimming in a Newtonian fluid, but in a complex fluid this is in general not the casedue to the nonlinearity of the constitutive relation. Following the approach above we can showthat the drag on a body of shape S under rigid body motionFD = −ηηˆRˆ · U −∫VND :∇Gˆ dV. (2.10)Whereas the thrust generated by a deforming body with uT (xS) = uS , isFT =ηηˆ∫SuS · (n · Tˆ ) dS −∫VNT :∇Gˆ dV. (2.11)The nonlinear components, ND = N(uD, τD) and NT = N(uT , τT ), depend on the respectivedrag and thrust fields. As we can see, taking FD + FT = does not, in general, lead to (2.9),unlike for a Newtonian fluid, because N 6= NT + ND.We note that the formulas derived here are independent of the choice of viscosity, ηˆ, in theauxiliary Newtonian problem. In particular, there is no requirement that ηˆ = η although such92.3. The complex reciprocal theorema choice (when sensible) will simplify the appearance of the formulas.2.3.1 Small amplitude deformationsA deforming body will have a time dependent shape, S(t), but if the amplitude of thedeformation is small, then we can, through Taylor series recast the problem onto a simpler,static reference surface, S0 with boundary velocity uS0 [65, 66]. The velocity field is expandedu(rS) = u(x0) +∆r · ∇u|x0 +O(|∆r|2), (2.12)where ∆r = rS − r0 and x0 ∈ S0. Rearranging we obtain the boundary condition on S0,u(x0) = U + Ω× r0 + uS0 , (2.13)whereuS0 =∂∆r∂t+ Ω×∆r−∆r · ∇u|x0 +O(|∆r|2). (2.14)Note that the swimming gait on S0, uS0 , depends on gradients of the (unknown) flow fieldu and the rotation rate of the swimmer Ω. However if we take ∆r = r1 where 1, is adimensionless measure of gait amplitude, then expand the velocity field in a regular perturbationseries, u =∑m mum, with boundary condition uS0() = uS01 + 2uS02 + ... whereuS01 =∂r1∂t, (2.15)uS02 = Ω1 × r1 − r1 · ∇u1|x0 , (2.16)we obtain boundary conditions for the velocity field on S0, which are known order-by-order.Furthermore, upon expanding all fields in , we see that the leading order effect of the nonlin-earity in the constitutive equation enters, at most, at quadratic order N = 2N2[u1, τ1]. Weproceed by applying the reciprocal theorem in (2.9) directly on S0, which is permissible becausethe stress in the fluid between S and S0 is divergence free [56], to obtainU = Rˆ−1 ·∫S0uS01 · (n · Tˆ ) dS + 2Rˆ−1 ·[∫S0uS02 · (n · Tˆ ) dS −ηˆη∫V0N2 :∇Gˆ dV]+O(3).(2.17)The first term typically does not contribute to steady-state swimming as for a periodic gaitwe have uS01 = 0 (where here an overbar denotes a time-average over a period 2pi/ω). If theboundary conditions have → − symmetry then we should expect〈uS01〉= 0 (where theangle brackets denote a surface average over S0), which leads to zero net velocity to leadingorder for simple bodies as we will see. When there is no net motion of the body at leadingorder, U1 = , then the boundary conditions in the swimming problem are precisely equal to102.4. Viscoelastic fluid relationsa thrust problem where the body is held fixed to leading order, u1(x0) = uS01 . In this caseit follows that the first viscoelastic correction, is equal in the swimming and thrust problems,N2 = NT,2, in other words the change in swimming speed is due entirely to the modificationof the thrust by the complex rheology. As a result, to leading order, Newtonian drag (but withviscosity η) balances the thrust generated in a viscoelastic fluidU =ηˆηRˆ−1 · FT , (2.18)where the thrust is given by the reciprocal theorem (2.11). As we shall show below, for a gen-eralized swimming sheet in a viscoelastic fluid, the thrust for any single temporal deformationmode is diminished thereby causing a reduction of swimming speed.2.4 Viscoelastic fluid relationsIn this work we consider a general constitutive relation for polymeric fluids given byAjτ (j) = η(j)0 Bjγ˙ + N(j)(u, τ (j)), (2.19)where the deviatoric stress tensor written as a sum of j relaxation modes, τ =∑j τ(j), η(j)0is the zero-shear-rate viscosity for the j-th mode, Aj and Bj are linear operators in time andN(j) is a symmetric tensor that depends nonlinearly on the velocity and stress and representsthe transport and stretching of the polymeric microstructure by the flow [1, 27].We shall neglect here the influence of a particular initial stress state in the fluid, suitablewhen determining the steady swimming speed of a microorganism, in which case we may writethe flow and stress fields as time periodic, expressed generally asu =∑pu(p)e−ipωt. (2.20)Upon summing over all relaxation modes (j) we may write the constitutive relationship for eachtemporal mode (p) asτ (p) = η∗(p)γ˙(p) + N(p), (2.21)where η∗(p) =∑j η(j)0 [Bj(p)/Aj(p)] and N(p) =∑j [1 + Aj(p)]−1N(j,p) where Aj(p) and Bj(p)are the characteristic polynomials of the differential operators (i.e eipωtAj [e−ipωt]). For eachFourier mode we thus have a linear response with complex viscosity, η∗(p), and a nonlinearterm. As an example, the Oldroyd-B equation, which has a single relaxation mode, yieldsη∗(p) = η0(1 − ipωλ2)/(1 − ipωλ1) where λ1 is the relaxation time and λ2 is the retardationtime.Subsitution of Eq. (2.21) into Eq. (2.9) and recasting onto a static shape S0 we obtain a112.5. Model Swimmersspectral decomposition of the swimming velocity in a non-Newtonian fluidU(p) = Rˆ−1 ·[ ∫S0uS0,(p) · (n · Tˆ ) dS − ηˆη∗(p)∫V0N(p) :∇Gˆ dV]. (2.22)We are often interested in only the zeroth mode, or mean swimming velocity, U(0) = U. In eithercase, we see that the non-Newtonian contribution arises directly as a result of the tensor N andtherefore, linearly viscoelastic fluids yield precisely the same swimming velocity as Newtonianfluids for a given prescribed gait.2.5 Model SwimmersIt is typical, for analytical tractability, for S0 to align with a coordinate system, in particu-lar, spherical, cylindrical and planar swimmers. For two-dimensional swimmers the resistanceproblem is ill-posed, but as shown previously for Newtonian fluids [58], use of the reciprocaltheorem for swimming is still valid in two dimensions. Here, we provide simplified reciprocaltheorem formulas for spherical, cylindrical and planar swimmers in complex fluids.2.5.1 Spherical swimmersIf the body is a sphere of radius a, then the rigid body problem leads to resistance tensorsRˆFU = 6piηˆaI, RˆLΩ = 8piηˆa3I, RˆFΩ = 0 and RˆLU = 0, while traction on the surface givesn · Tˆ = −3ηˆ2a [I 2Ξ] where Ξij = ijkrk. With these relations, the swimming speed for a sphereU = − 14pia2∫S0[I32a2Ξ>]· uS0 dS − ηˆη∫V0N :∇Gˆ · Rˆ−1 dV, (2.23)where for a sphere∇Gˆ · Rˆ−1 = 18piηˆ[(1 + a26 ∇2)∇G ∇(1|r|3Ξ)]. (2.24)and G = 1|x|(I + xx|x|2)is the Oseen tensor, as shown by Lauga [36]. Setting N = 0 one obtainsthe result for a Newtonian fluid as shown by Stone and Samuel [62].2.5.2 Cylindrical swimmersFor cylindrical swimmers, who may rotate about their axis of symmetry but translate onlyin the plane perpendicular to this, the stress may be writtenn · Tˆ = − ηˆa[αI‖ 2Ξ · I⊥], (2.25)122.5. Model Swimmerswhere α is a dimensionless constant (which is singular in Re). We use the ‖ subscript todenote in-plane components while ⊥ denotes the out of plane component. Integrating overthe perimeter we may obtain the mobilities per unit length Rˆ−1FU = (2piηˆα)−1I‖ and Rˆ−1LΩ =(4piηˆa2)−1I⊥. Combining these terms we obtain the swimming velocityU = − 12pia∫S0[I1a2Ξ>]· uS0 dS − ηˆη∫V0N :∇Gˆ · Rˆ−1 dV, (2.26)where for a cylinder∇Gˆ · Rˆ−1 = 14piηˆ[(1 + a24 ∇2)∇G 1a∇(1rΞ)], (2.27)and G = − ln(r/a)I + xxr2is the Oseen tensor in 2D. Notice that the result is independent of αand thus the singular nature of the resistance problem is avoided.2.5.3 Planar swimmersHere we consider a two-dimensional planar swimmer (sheet), which may have differentprescribed velocities, uS1 and uS2 , on each side (S0 = S1 ∪ S2), unequally spaced betweentwo rigid surfaces. We restrict the sheet to in-plane translation and so look only at the two-dimensional resistance problem, which is simply shear flow hencen · Tˆ = − ηˆhI‖, ∇Gˆ = −1hnI‖, (2.28)where now I‖ is two-dimensional. If the distances between the swimmer and the two walls areh1 and h2, we have then as the only non-zero mobility per unit area R−1FU =h1h2ηˆ(h1+h2)I‖. Keepingin mind that the unit normal n and distance to the wall h changes depending on the side wecombine with the above to obtainU =1h1 + h2(− h2〈uS1〉− h1 〈uS2〉+ 1η[∫ h10h2 〈N · n〉 dxn +∫ h20h1 〈N · n〉 dxn]).(2.29)Here again, the angle brackets denote a surface average and dxn = dx · n. In an unboundedfluid we simply take h1 = h2 →∞ to obtainU = − 〈uS0〉+ 1η〈∫ ∞0N · n dxn〉, (2.30)where the average is over both sides of the sheet.132.6. Generalized Sheet in a viscoelastic fluid2.6 Generalized Sheet in a viscoelastic fluidGiven the simple form of the integral theorem for swimming of two-dimensional objects welook to categorize the motion of a generally deforming flat sheet. We consider motions that areperiodic in time and in space∆r(r0, t) = A∑n,pcn,peink·r0e−ipωt, (2.31)where A is the characteristic amplitude of deformation, k is the wave vector, r0 is a referencepoint on a flat plane with normal n and n ·k = 0. The system is invariant in the direction n×k.The Fourier coefficients, cn,p = an,p(k/ |k|) + bn,pn, include both transverse and longitudinaldeformations. The velocity field boundary condition on the sheet is henceuS =∂∆r∂t= −ωA∑n,pipcn,peink·r0−ipωt. (2.32)We wish to describe this motion on the reference surface (S0) and so expand in powers of ∆ruS0 = uS −∆r · ∇u|r0 + ...= −ωA∑n,pipcn,peink·r0−ipωt −A∑n,peink·r0−ipωtcn,p · ∇u|r0 + ... (2.33)This expansion necessitates obtaining gradients of the flow field u unless there are no transversedeformations, cn,p ·n = 0. To determine the swimming motion when transverse deformations arepresent, one may posit a regular pertubation expansion of all fields in powers of ≡ A|k| 1.Using the generalized boundary conditions above, one can obtain the leading order (Newtonian)flow field, u1, using classical methods discussed elsewhere [67]. By the integral formula (2.30),the time-averaged (steady) swimming velocity in a Newtonian fluid is then immediately foundto beUN= −〈uS0〉= ωA2k∑pUˆN2 (p) +O(ωk2A3), (2.34)where each frequency componentUˆN2 (p) =∑nnp[an,pa†n,p − bn,pb†n,p]. (2.35)We see that all modes are decoupled. For traveling waves in the k direction n = p (or n = −pfor waves in the −k direction) and we arrive at known results for a general swimming sheetwith both transverse and longitudinal waves [58, 68, 69] whereUˆN2 (p) = p2[ap,pa†p,p − bp,pb†p,p]. (2.36)142.6. Generalized Sheet in a viscoelastic fluidNote that the compressional and transverse waves lead to oppositely signed motion even thoughboth waves are traveling in the same direction and so there is no net motion, at O(2), if∑p p2[ap,pa†p,p − bp,pb†p]= 0.Now to study the effects on viscoelasticity due to this general boundary deformation wemust select a viscoelastic constitutive equation. In general, under a small-amplitude expansionwe have at the first two ordersτ(p)1 = η∗(p)γ(p)1 , (2.37)τ(p)2 = η∗(p)γ(p)2 + N(p)2 (u1). (2.38)The leading order velocity field is Newtonian while the stress field displays both a viscous andelastic response but there is no viscoelastic effect on locomotion at this order of approximationas N1 = 0. Non-linearities arising from the constitutive relation enter at second order butbecause we are interested here only in the steady swimming speed, we need only to evaluatethe mean, N2 = N(0)2 . We use here the Oldroyd-B constitutive equation, which in the smallamplitude limit is broadly representative of nonlinear viscoelastic fluids for swimming [30]. Thenonlinear contribution is given by the tensorN2 =∑pη0ipω(η∗(p)η0− 1)[(∇u(-p)1)> · γ˙(p)1 + γ˙(p)1 ·∇u(−p)1 − u(−p)1 ·∇γ˙(p)1 ] .Armed with the solution of the leading order Newtonian flow field u1, only straightforwardintegration, via (2.30), is then needed to obtain the swimming velocity for a generally deformingflat sheet in a (Oldroyd-B) viscoelastic fluid, to leading orderU = ωA2k∑pη∗(p)η0UˆN2 (p). (2.39)We see from the above equation that the contribution of each mode is rescaled by the real partof the dimensionless complex viscosity. This factor is always less than or equal to one,Re[η∗(p)η0]=1 + p2De2β1 + p2De2≤ 1, (2.40)because the retardation time, λ2, is smaller than the relaxation time, λ1, in viscoelastic fluids,β = λ2/λ1 < 1. This means that the swimming speed due to any individual temporal mode isslower than in a Newtonian fluid. The Deborah number De = ωλ1 characterizes the responseof the fluid, when the time scale of actuation is much longer than that of the relaxation of thefluid, De→ 0, and we recover the Newtonian swimming speed. Physically, the thrust (by way ofequation (2.18)) is reduced by a factor equal to the frequency dependent viscosity, meaning thehigher frequencies are increasingly damped. Now because each individual mode is less effective152.6. Generalized Sheet in a viscoelastic fluidas a means of propulsion,Re[η∗(p)η0] ∣∣∣UˆN2 (p)∣∣∣ ≤ ∣∣∣UˆN2 (p)∣∣∣ , (2.41)slower swimming in a viscoelastic fluid is guaranteed if UˆN2 (p) does not change sign ∀p, anexample being a sheet passing only unidirectional transverse waves. However, swimming neednot be slower in general because UˆN2 (p) can certainly change sign in more general deformations.In particular, one may note that unidirectional transverse and longitudinal waves lead to motionin opposite directions and therefore may result in faster swimming in a viscoelastic fluid.A simple example of faster swimming is found by recalling that in a Newtonian fluid theswimming speed (at O(2)) is zero when a sheet is passing unidirectional transverse and com-pressional deformation waves if∑p p2[ap,pa†p,p − bp,pb†p,p]= 0. In a viscoelastic fluid the sameactuation may lead to net motion because the coefficients are rescaled by a factor that diminishesfor higher modes. For example, if we take two modes, q and m, where q2bq,qb†q,q = m2am,ma†m,mand all other modes zero, in a Newtonian fluid the thrust due to the two modes cancels oneanother and therefore there is no net motion at O(2), UN = ωA2k∑p UˆN2 (p) = 0. In a vis-coelastic fluid however, the higher frequency term is damped by a larger factor and hence thelower frequency component generates a larger thrust leading to the swimming speedU = ωA2k∑pη∗(p)η0UˆN2 (p),= ωA2km2am,ma†m,m2(q2 −m2)(1− β)De2(1 +m2De2)(1 + q2De2), (2.42)which is maximum when De = 1/√mq and decays quadratically as De→ 0. If q2 > m2 then theswimmer moves in the direction of k as the thrust due to the compressional waves is dominant,if q2 < m2 the opposite is true, while if m = q there is only a single mode and the swimmingspeed is zero.Because higher frequencies are damped by larger factors, swimmers may go faster in aviscoelastic fluid compared to a Newtonian fluid, as in the previous example, depending on theamplitudes of the modes and in general, the Deborah number. Lauga and Riley explore thiseffect with oppositely traveling transverse waves (for which UˆN2 (p) = −p2[bp,pb†p,p − b−p,pb†−p,p]can change sign) finding the necessary conditions for two such waves to yield faster swimming[39]. Generally a swimmer may go faster, slower, or even stop in a viscoelastic fluid providedthere are multiple modes and UˆN2 (p) changes sign.It may seem obvious that the complex viscosity, measureable by the linear response (2.37),determines the change in swimming speed due to a viscoelastic fluid, but if the fluid were onlylinearly viscoelastic there would be absolutely no change in the swimming speed from that ofa Newtonian fluid. It is the nonlinear response that leads to change in the swimming speed,not the linear response, the tensor N2 happens to lead to a rescaling of the Newtonian solution162.6. Generalized Sheet in a viscoelastic fluidby the viscous modulus and that result is remarkably robust. The same result holds for asheet swimming near a wall and two-dimensional pumping [56], as well as helical swimming,both in an unbound fluid and near wall [40]. The question one might ask is why should eachmode, individually, be slower? In other words, why should the thrust generated by any singlecomponent of a deforming boundary condition necessarily be diminished in a viscoelastic fluid?Ultimately we find that the nonlinear non-Newtonian response of the fluid is of the same formas the motion due to the deforming boundary in a Newtonian fluid, but oppositely signed,namely1η0∫ ∞0〈N2[u1] · n〉dxn =∑p[η∗(p)η0− 1]〈r(−p)1 ·∇u(p)1〉,= −ωA2k∑pDeG∗(p)η0ωUˆN2 (p). (2.43)Here we have written the response in terms of the elastic modulus G∗ = −ipωη∗ ,whose realpart isRe[G∗(p)η0ω]=p2De(1− β)1 + p2De2, (2.44)and so when summed with the Newtonian contribution the result for each mode is always slower.We will refer to this as an elastic reponse because the contribution of each mode is scaled bythe (real part of the) elastic modulus, which goes to zero as De→ 0.This begs the question of whether one may elicit an elastic response from the fluid byprescribing a velocity boundary condition with no associated deformation? We find that if wedirectly prescribe a slip velocity uS = uS0 , without deforming the body, ∆r = 0, for exampleby modifying surface chemistry [70], the results can be markedly different. If we prescribe ageneral time-periodic slip velocity on a flat sheet,uS0 = −ωA∑n,pipcn,peink·r0−ipωt, (2.45)to leading order this is the same boundary condition as prescribed above for a deforming sheet,but because here there is no associated deformation〈uS0〉= 0 at all orders and the contributionto the swimming speed is entirely due to viscoelastic part of the thrust, namelyU = −ωA2k∑pDeG∗(p)η0ωUˆN2 (p). (2.46)We see a quadratic decay to zero as De → 0 because, as constructed, there cannot be any netmotion for these boundary conditions in a Newtonian fluid. For large Deborah numbers theswimming speed approachs the limit U → −(1 − β)UN as De → ∞ where UN refers to theswimming speed of a deforming sheet in a Newtonian fluid as given by (2.34). Analogously, the172.7. Conclusionresponse is larger for higher modes. We also note that we were able to construct a response of asimilar form (see (2.42)) with traveling deformation waves by superposing both transverse andlongitudinal waves whose thrust is in opposition. If the thrust of these modes cancels exactlyin a Newtonian fluid, as it did in our example, the swimming speed is determined by an elasticresponse, while if the two terms do not cancel exactly, there is an additional contribution tothe swimming speed that scales as Re[η∗/η0].We often see this form of elastic mediated response, U ∝ Re[G∗/η0ω], when no net motionwould occur if the fluid were Newtonian. A similar response was found by Lauga for a squirmerwith a slip velocity prescribed so that the swimming speed in a Newtonian fluid is zero [27].Pak et al. also found the swimming velocity ∝ De(1− β)/(1 + De2) for two co-rotating spheresthat propel due to the symmetry-breaking axial flows generated by viscoelastic hoop stresses[34], and also similarly found by Yazdi and Ardekani for a reciprocal squirmer near a wall[37]. One also observes an elastic mediated synchronization of a system of two swimmers ina viscoelastic fluid that conversely displays no relative phase evolution in a Newtonian fluid[56, 71]. A similar response is also noted for force generation by the flapping motion of a rigidrod [72]. Naturally one might prescribe a slip velocity, with or without deformation, that wouldlead to swimming both in a Newtonian fluid and in a non-Newtonian fluid and therefore theswimming velocity could contain terms that vary as both the viscous and elastic modulus ofthe fluid, but if these boundary conditions are meant to emulate a deforming body then theresult may not be physically sensible.2.7 ConclusionIn this chapter, we used a reciprocal theorem formulation to study the motion of swimmingbodies in non-Newtonian fluids. We first used this formulation to show that generally, for smallamplitude deformations, the leading order change in swimming speed is due to a modificationof thrust generated by the swimmer in complex fluids. We then used this theory to studythe effect of viscoelastic stresses on the swimming velocity of a sheet undergoing both generaltransverse and longitudinal deformations. We showed that it is possible to swim faster orslower, depending on the swimming gait, not only with oppositely traveling transverse waves butalso with unidirectional traveling waves provided both transverse and longitudinal deformationmodes are present and then demonstrate why and how this occurs, thereby extending theresults of Riley and Lauga to this case [39]. What becomes clear, even in this simplified model,is that it may be quite deceptive to draw any conclusion about whether one swimmer shouldgo faster or slower based only on observations of a different (simpler) swimmer, because of theunequal damping of higher and lower frequency deformation modes in viscoelastic fluids. Inparticular, it is only with multiple (small amplitude) deformation modes that a swimming speedenhancement can be realized in viscoelastic fluids. Additionally, we found that while swimmingby deformation elicits a particular response in which all swimming modes are individually182.7. Conclusiondiminished in a viscoelastic fluid, one may prescribe slip velocities that lead to a completelydifferent (elastic) response in which all modes, individually, lead to a speed increase.It remains to be seen how these results extend to finite bodies and whether a swimminggait may be decomposed into components that yield a viscous or an elastic response fromthe fluid. For example, numerical results indicate that even finite filaments passing sinusoidaldeformation waves do not experience a speed increase in viscoelastic fluids, yet when a front-back asymmetry in amplitude (resembling flapping) is introduced an elastic response is excitedthat leads to a speed increase [44], and when that amplitude asymmetry is reversed a speeddecrease is conversely observed [47]. Experimental work by Shen and Arratia [51], with thenematode C. elegans, revealed a trend of decreasing speed with the increase in De. Accordingto Sznitmann et al.[73, 74], C. elegans passes a transverse deformation consisting of oppositetraveling waves with identical frequencies and our results indicate that this would lead to slowerspeeds, at small amplitudes, due to the lack of multiple modes required for enhancement. Thefront-back amplitude asymmetry of the swimmer also has an effect[47], but such effects may beminimal for swimmers much longer than a typical wavelength[46, 50].The picture developed here is only valid at small amplitudes and it is not clear in what wayslarge amplitude motions change the response of the fluid. There is indication that a single-modeswimming sheet is always slower, even for large amplitude deformations, as demonstrated bothanalytically [56] and numerically [49], but the opposite trend was found for helical swimmers,which are slower at small amplitude but were observed, both experimentally [50] and numer-ically [46], to swim faster at large amplitude. For large amplitude deformations one shouldalso consider the effects of finite extension as microorganisms typically reside in regimes wherethe Deborah number is not small[27, 30]. Finally, we note that the results presented here arelimited to swimmers with a prescribed gait and the picture is further complicated when fluidstresses lead to changes in the gait itself [35, 38].19Chapter 3Rheology and flow studies of dragreducing gravel packing fluids3.1 IntroductionDrag reduction was first reported in polymer solutions about 70 years ago by Toms [75](Mysels [76]) and thus, is commonly known as Toms phenomenon. In essence, it was foundthat a noticeable reduction in pressure drop occurs in turbulent pipe flow upon the addition ofa small amount of polymeric additive. Later, surfactant solutions were also observed to causea similar drag reduction [77] in turbulent flows. This drag reduction phenomenon is utilizedin a variety of petroleum industry applications [78] with the most successful large scale usein the Trans-Alaska pipeline [79]. It is also applied in district heating and cooling systemsprominently in Japan [80, 81]. The interaction of two canonical areas of interest, complexfluid rheology and turbulence, with widespread successful industrial application has made itan extensively researched interdisciplinary topic. Several numerical and experimental studies,a number of reviews and books on the effects of polymer additives have been published in theliterature[82–88], and more recently a number on surfactant additives as well [11, 89–94].In this chapter, our focus is on particular fluids used in gravel-packing operations in oiland gas wells (see [95–97] for an overview). A range of different fluids are used to transport asolid phase to its point of application (as is also the case in hydraulic fracturing) but micellarsolutions are often attractive in these applications as they combine initially high viscositiesand thus enhanced solids transport with drag reducing properties along with the ability toreform structures that degrade in flow (useful at very high shear rates in turbulence regimeobserved operationally). By forming a variety of micellar or vesicular structures such as worm-like micelles also known as cylindrical, rod-like or thread-like micelles resembling the long chainsof polymers at concentrations above the critical micelle concentration (CMC), surfactant alsolead to complex viscoelastic rheology. Operationally, these fluids experience wide variations inshear rates and flow regimes. From the surface to the cross-over port, flow ranges from inertialto weakly turbulent, but are then highly turbulent upon passing through small diameter shunttubes used in alternate path packing [98, 99]. Because the mechanical properties of thesecomplex fluids can change dramatically depending on the flow, it is necessary to understandand characterize the rheology of these fluids in order to make better process predictions (seefor example [100, 101]).203.2. Experimental workIn this work, we study a zwitterionic viscoelastic surfactant (VES) solution to understandthe structural and flow properties of the solution by performing a variety of rheological testsand then flow loop experiments. As a point of reference, we compare to the commonly usedpolymeric additive i.e. xanthan gum. We show that these solutions have qualitatively differentrheological behaviours with the surfactant solution behaving like a gel in contrast to a shear-thinning viscoelastic polymer solution at room temperature. Further experiments suggestedshear-thickening in the surfactant solutions as a result of shear induced structures (SIS). Theseobservations highlight a strong structural significance in VES solutions especially at the highershear rates of the drag reducing turbulent regime. We are able to predict the laminar flow curvesfrom our flow loop measurements using the rheological characterization within experimentalaccuracy (see sec. 3.2.3) and also, measure the drag reduction behaviour in the turbulentregime of the solutions. A novel representation scheme is used to plot the results depictingboth the physical state of the solution as well as the overall drag reduction effect of additives,useful for an engineering application. The generalized Reynolds number (Reg) is used withsolution viscosity (η) to predict the physical state of the solution whereas the solvent Reynoldsnumber (Res) with solvent viscosity (ηs) shows the overall drag reduction effect in turbulentregime by comparing the solution flow details with the initial state i.e. solvent.3.2 Experimental work3.2.1 MaterialsCommonly used viscous gravel packing fluids in the oil and gas industry include viscoelasticsurfactants (VES), xanthan and hydroxyethyl cellulose (HEC). The VES fluids are surfactantbased whereas xanthan and HEC are polymer based fluids. In this study, we consider a VESand xanthan formulation in liquid form supplied by Schlumberger, similar to those in [100, 101].CaCl2 and NaCl brine solutions were used as the solvents for hydration.The VES fluids can be formulated using different types of surfactants which include an-ionic surfactants, cationic surfactants and zwitterionic surfactants. The VES fluid used hereis a zwitterionic surfactant formulation [102], which can be hydrated in both monovalent(NaBr,NaCl,KCl) and divalent (CaBr2, CaCl2) brine solutions. The xanthan used is a het-eropolysaccharide based formulation which hydrates only in monovalent brines (NaBr,NaCl,KCl).3.2.2 Sample preparationThe solutions were prepared by batch mixing the viscosifiers in brine solutions at specifiedconcentrations. Before the flow loop tests, the batch mixing protocol was developed using prepa-ration of lab scale quantities to perform basic rheological tests ensuring viscosity developmentas per company standards.We prepared six solutions at room temperature which included two xanthan solutions (0.04ppg, 0.06 ppg) mixed in 9.2 ppg NaCl brines, two NaCl based VES solutions (4.5%, and 5.5%213.2. Experimental workby volume in 9.2 ppg brine) and two CaCl2 based VES solutions (3.5% and 5.5% by volume in10.7 ppg and 11 ppg brines respectively). A ppg means lbm per gal (US) in all the measurementswhere 1 ppg = 119.826 kg/m3. For each solution, a similar protocol was followed. The requiredamount of salts were mixed in fresh water for 1 hour and the density was measured using adensity meter (DMA 35 by Anton Paar) at room temperature. While stirring, the formulatedindustrial fluids were added to the brines and allowed to mix at 250 rpm for 14 hours.3.2.3 MethodsRheometerRheological measurements were performed on a Malvern Kinexus ultra+ rotational rheome-ter using a cone and plate geometry (CP 4/40) for xanthan and a parallel plate (PP 20) serratedgeometry (gap = 1mm) for the VES fluids. A variety of steady and small amplitude oscillatoryshear (SAOS) dynamic tests as well as sequences with both steady and dynamic tests wereperformed at room temperature under controlled stress (CS) or controlled rate (CR) conditionsto characterize the complex fluids [103–105]. Dynamic frequency sweeps were conducted in thelinear viscoelastic (LVE) region determined by strain or stress sweeps.The sampling time for a steady state test was a minimum of either 2 minutes or the timeto achieve an equilibrium value with a tolerance of 1% for 2 seconds. For dynamic tests, thenumber of raw data points for integration was set as 32, 768 with a minimum and maximumintegration times as 5 sec and time period of the oscillation respectively. The frequency valuetogether with the lower and upper bounds of the integration time determine the integrationperiod. The nominal and maximum time to equilibrium for dynamic tests were set as 500periods and 20 seconds respectively. At each oscillation frequency, the minimum time valuebetween the nominal and maximum time to equilibrium corresponds to the waiting time beforethe start of integration time.FlowloopThe setup of the flow loop is illustrated schematically in Fig. 3.1. It consists of a 0.22m3capacity mixing tank, 15 HP AMT 4251-95 variable-frequency driven centrifugal pump, anelectromagnetic flow meter (FT, OMEGA FMG 606) with an accuracy of 0.2% of full scale (1.2m3/min) i.e. an error of ±0.0024 m3/min and two pressure transducers (PT, Omega, PX 409)with an accuracy of 0.08% of full scale (206.8 KPa) i.e. an error of ±0.16 KPa. The flow rateand pressure drop were measured across the test section: a 4.59m long straight, smooth pipewith an inner diameter of 48.4mm.We took measurements at different constant flow rates. At each pump frequency, a waitingtime of 25 s is allowed for the flow to stabilize before data recordings are made. The data isaveraged over 1000 recordings made at a frequency of 200Hz for each pump frequency. Beforemixing every new fluid sample, water was pumped in the loop to confirm the calibration. We223.3. ResultsFT PT PTFigure 3.1: A schematic of the flow loopperformed three runs for each solution with a pump frequency increasing from 5Hz up to 55Hzand then, decreasing back to 5Hz. The range of flow rates and pressure drops for our fluidswere observed as 0.1 − 0.7 m3/min and 5 − 20KPa respectively. The pump could provide amaximum flow rate of 1m3/min with water as the working fluid. A gap of 30 minutes and 3hours were allowed between the 1st - 2nd and 2nd - 3rd runs respectively. The data from theforward sequence (5− 55 Hz) of the first run was discarded for analysis to remove errors fromany inhomogeneity in the fluid as a result of any incomplete mixing.3.3 Results3.3.1 RheometryXanthan gum, a common drag reducing polymeric additive, is well characterized in theliterature [106, 107]. Its inclusion here thus provides a benchmark for our rheological experi-ments and serves to verify our measurement protocols. In Fig. 3.2, we plot the variation ofapparent viscosity. The solution displays shear thinning behaviour over a wide range of shearrates, while approaching a plateau towards both the lower and higher ends of the shear-raterange tested. A power-law model can be used to describe the shear-thinning behaviour whereasthe complete characteristics are commonly described with the Carreau [1] or Carreau-Yasuda[108] model (see Fig. 3.2 for the fit). While the qualitative rheological behaviour of xanthan issimilarly reported in the literature [109–114], differences in the solution concentration as wellas natural variations in the xanthan gum itself lead quantitative differences in the values of theconstitutive parameters.A dynamic frequency sweep in the LVE regime as shown in Fig. 3.3 depicts a viscoelasticliquid behaviour. At low frequencies, the elastic modulus (G′) is below the viscous modulus(G′′) with a crossover frequency of nearly 0.1 Hz. Beyond the crossover frequency, G′ is alwaysgreater than G′′. The high phase angle at low frequencies again indicates viscous behavior [104]233.3. Results10-2 10-1 100 101 102 10310-210-1100101102Experimental resultsCarreau-YasudaFigure 3.2: Viscosity vs shear rate for xanthan (0.04 ppg) at room temp.which implies that suspended materials, such as sand (or proppants), will sediment and settle.After applying two minutes of pre-shear at rates of 100 s−1, 750 s−1 and 1100 s−1, we ob-served simple relaxation behaviour under small amplitude oscillatory shear tests upon cessationof the pre-shear as shown in Fig. 3.4. We see that the highly pre-strained fluid relaxes back toa dynamic equilibrium within an order of 10 seconds upon ceasing the shear.10-2 10-1 100 10110-11001010102030405060708090Figure 3.3: A SAOS frequency sweep (strain = 5 %) for xanthan (0.06 ppg) at room temp.Examining the viscoelastic surfactant fluid, it displayed markedly different rheology. First,we note that it displayed minor wall slip effects at low shear rates. A comparison of apparentviscosities between smooth and serrated geometries reveals 20%−30% higher values for serratedgeometry recordings at low shear rates. Thus, we used the serrated plate geometry to perform243.3. Results0 10 20 30 40 50 6010-1100101102Figure 3.4: A SAOS sequence (1 Hz, CS: 1 Pa) to monitor structure build up after 100 s−1,750 s−1 and 1100 s−1 for xanthan (0.06 ppg) at room temp. Here t = 0 indicates the timewhen the pre-shear is stopped.all further rheological sequences. Visual inspection of samples readily suggests gel-like behavior(see Fig. 3.5 for table-top images). Using the tube inversion method of the table-top rheology,the sample d) indicates a finite yield stress by holding it’s own weight for an extended period oftime (overnight) [115]. Dynamic rheological measurements in the LVE regime were performedin order to quantify the fluid’s structural response. We see in Fig. 3.6 that indeed G′ isalways higher than G′′, with both being nearly independent of frequency. This indicates gel-like response with an equilibrium storage modulus and thus, practically infinite relaxation times(outside the experimental range) which means deformation stresses do not relax even at longtime scales.a)b)c)d)Figure 3.5: Photographs at room temperature of 20 ml VES samples for a) NaCl, 4.5%, b)NaCl, 5.5%, c) CaCl2, 3.5% d) CaCl2, 5.5%253.3. Results10-2 10-1 100 10110-11001010102030405060708090Figure 3.6: A SAOS frequency sweep (CS = 1 Pa) for VES (5.5%, 11 ppg CaCl2) at roomtemp.The gel-like behaviour of VES is again observed in measurements of apparent viscosity insteady shear. Figure 3.7 shows a shear ramp where the shear rates are increased in stepwisemanner and shear stress recordings are made to calculate the apparent viscosity. At low shearrates in the range 0.1 - 1 s−1, we see ηγ˙ ' c constant, as is common with yield stress behaviour.This is also observed in the inset plot of figure 3.7 which shows the stress ramp where insteadstress values are increased in a stepwise manner and shear rates are recorded to calculate theapparent viscosity. The sudden decrease over three decades in apparent viscosity above a criticalvalue of stress indicates yielding of the fluid. This critical value can be taken as the yield stressof the sample.10-2 10-1 100 101 102 10310-210-110010110210310-210-1100101102103100 101 10210-2100102104Figure 3.7: Shear rheology profile of VES (3.5%, 10.7 ppg CaCl2) at room temp.263.3. ResultsAnother interesting aspect observed in Fig. 3.7 was the apparent shear-thickening of the fluidat higher shear rates. The apparent viscosity approaches a plateau and then starts increasingafter a shear rate of about 750 s−1. This shear-thickening behaviour was not observed to thesame extent with smooth plates. To understand this, we performed a combination of steadyand dynamic tests. We shear the fluid at a fixed shear rate for 2 minutes and then perform aSAOS test for 5 minutes to monitor the structural changes, before moving to the next shearrate and repeating. Figure 3.8 shows changes in elastic and viscous modulus under SAOS, uponcessation of imposed pre-shear rates of 100 s−1, 750 s−1 and 1000 s−1. We see an oscillatorystructure relaxation, suggesting a strong variation in shear induced structures (SIS) with shear.The SIS persist with a higher moduli values even after ceasing the pre-shear. The time of theirpersistence increases with the pre-shear value.0 50 100 150 200 250 30010-210-1100101102Figure 3.8: A SAOS sequence (1 Hz, CS: 1 Pa) to monitor structure build up after 100 s−1,750 s−1 and 1000 s−1 for VES (3.5% 10.7 ppg CaCl2) at room temp. Here t = 0 indicates thetime when the pre-shear is stopped.3.3.2 Flow loopFlow loop tests were performed to quantify the level of drag reduction using viscoelasticsurfactant additives. The flow rate and pressure drop were measured across a section to computethe Fanning friction factor (f) and Reynolds number (Re). The Fanning friction factor is definedasf =τw12ρU2, (3.1)where ρ is the density of the fluid, U is the average flow speed calculated from the measuredflow rate and τw is the wall shear shear stress, calculated from the measured pressure drop ∆P ,τw =∆Pd4l. (3.2)273.3. ResultsHere d is the pipe diameter and l is the length of the test section. Thus, the friction factor maybe written asf =∆Pd2ρU2l. (3.3)For the viscoelastic fluids considered here, the viscosity of the solvent and the solution candiffer substantially, which means the Reynolds number can be defined in different ways basedmainly on how the viscosity is defined. Using the (Newtonian) solvent viscosity (ηs) leads to asolvent Reynolds numberRes =ρUdηs. (3.4)On the other hand, using the polymer/surfactant solution viscosity, η(γ˙) leads to a generalizedReynolds numberReg =ρUdη. (3.5)For a give flow, a reference shear rate must be used and here the nominal wall shear rate,γ˙w = 8U/d, is used.Physically, the solvent Reynolds number can be used to quantify the overall drag reduc-tion effect of the additives with respect to the solvent in the turbulent regime [82], while thegeneralized Reynolds number helps predict the physical state of the solution.To form the Reynolds numbers, the viscosities of the Newtonian solvents (NaCl and CaCl2brines) were measured while the solutions were modelled as power-law fluids using the rheo-logical data for shear rates in the 100 s−1 to 500 s−1 range (which largely represent laminarflows).In the laminar regime, the Fanning friction factor for cylindrical pipe flow isf =16Reg. (3.6)This result is compared to the measured values of the friction factor, given by (3.3), in Fig. 3.9.The results of the measured friction show good agreement with the laminar friction law notonly for the xanthan solution but also the viscoelastic surfactant solutions (with either solvent).As the Reynolds number is increased, the flow becomes turbulent. In this regime, it isfar more instructive to use a solvent Reynolds number in order to quantify the level of dragreduction. Thus in Fig. 3.9, we display all data in the turbulent regime as a function of solventReynolds number. Also shown in Fig. 3.9 is a curve corresponding to the Blasius equation forthe drag of a turbulent (Newtonian) flow in a smooth pipe [116]. We see that drag reductionis observed in the turbulent regime for both xanthan and the VES fluids. Virk’s asymptote formaximum drag reduction (MDR) is also shown Fig. 3.9 along with the more recent empiricallimit for surfactant solutions proposed by Zakin [117, 118]. The jump in the data occurs simplybecause the solvent viscosity is several times lower than the solution viscosity, while the NaClviscosity is 0.39 times the CaCl2 viscosity, explaining the bigger jump for xanthan and NaCl283.4. Discussion and ConclusionsVES fluid. We also observe a higher drag reduction for VES than xanthan as expected from[101]. Hysteresis was observed for VES fluids, meaning that the results for VES fluids do notfollow the same path on the forward (5 − 55 Hz) and backward (55 − 5 Hz) pump frequencycycles of a loop run. It is most prominent in the turbulent regime of the NaCl fluid wheredrag reduction was weaker for the backward cycle than forward cycle. In other words, a higherpressure drop was recorded for the same flow rate during backward cycle which could be aresult of shear-thickening or structural changes at higher shear rates from the end of a forwardcycle in a run.102 103 104 105 10610-410-310-210-1100Figure 3.9: Friction factor vs Reynolds number for a xanthan (0.06ppg) and VES (3.5%, 10.7ppg CaCl2 and 4.5%, 9.2 ppg NaCl) solutions3.4 Discussion and ConclusionsWe calibrated our setup with a xanthan solution, which behaved as a typical shear-thinningviscoelastic fluid, with viscous behaviour at long timescales (low frequencies) and an elasticresponse at short timescales (high frequencies). The zwitterionic VES fluids behaved like anelastic gel at room temperature with a non-zero equilibrium elastic modulus andG′ > G′′ even atlow frequencies. As is common for gels, the VES fluid displayed yield stress behaviour in steady-shear rheology experiments. Structurally, the gel-like behaviour suggests that the chains arestrongly constrained [104] and thus, sedimentation rates would be lower than in a comparablefluid with less constrained (entangled) structure such as xanthan [119]. Similar rheologicalresponses were observed for the zwitterionic VES solutions with both NaCl and CaCl2 brinesolvents. This indifference to inorganic salts is expected in zwitterionic solutions as it has293.4. Discussion and Conclusionsboth positive and negative charged head groups in the same molecule, unlike ionic surfactantsolutions and therefore, will have weak electrostatic interactions and negligible sensitivity tosalts [120]. This insensitivity has been observed earlier in zwitterionic solutions [121, 122],whereas ionic surfactants show rheological variations with salts [123, 124].Gel-like behaviour has been observed for anionic [125], cationic [126] and zwitterionic surfac-tant solutions [121, 122] below ‘gelation’ temperatures where these fluids all possess long tailedwormlike micelles (carbon 22 chains). The stress relaxation, over timescale τr, for molecularaggregates (i.e. wormlike micelles in VES fluids) happens through reptation, with characteristictimescale τrep, as well as the breaking and recombination of chains, over timescale τb [10]. Weobserved no single finite relaxation time from the dynamic tests at room temperature whichleads us to believe that the VES fluids in this work are in a regime where τb τrep and thusthe breaking/recombination is an unfavourable process [127]. The relaxation would hence bedominated by the reptation process similar to unbreakable polymers with τr ∼ τrep ∼ L¯3, whereL¯ is the average contour length [12, 13, 128]. Here, we have practically infinite relaxation timeand thus, longer average lengths for our fluid, and these long and relatively unbreakable chainsof wormlike micelles could lead to gelation at room temperature by entanglement, explainingthe gel-like behaviour [129, 130].We also observed shear-thickening in VES fluids after a critical shear rate which may bedue to shear induced structures (SIS) [131–134]. These structures impart higher elasticity,birefringence and viscosity to the fluid [135] and their buildup (induction) time decreases withincreasing shear rates [131]. The steady and dynamic test sequence results (Fig. 3.8) indicatestronger SIS with high elasticity persistent for longer times as the pre-shear value increases.It’s possible that the long induction periods for SIS become short enough (at shear rates above750s−1) to affect the macroscopic properties such as apparent viscosity for our fluid explainingthe shear-thickening. In order to better understand the SIS, we need to consider birefrin-gence studies, rheo-optical measurements and optical techniques such as NMR, Cryo-TEM andSANS, as well as exploring geometry dependent effects. These are necessary if we are to makepredictions in practical shear rate regimes.The two Reynolds numbers used here are both useful for an engineering application. 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Graham, “Drag reduction and the dynamics of turbulence in simple and complexfluids,” Physics of Fluids, vol. 26, no. 10, 2014.41Appendix AOldroyd-B modelA.1 The model equationTo analyze the viscoelastic fluid, we decompose the stress as τ = τ s + τ p where τ s and τ pare the Newtonian and polymer contributions respectively. The Newtonian contribution can bewritten as τ s = ηsγ˙ where γ˙ = ∇u +∇uT . Now, we use the upper convected Maxwell modelto express the polymer contribution asτ p + λ∇τ p= ηpγ˙, (A.1)where λ is the relaxation time and ηp is the polymer viscosity contribution with η = ηs + ηp.The upper convected derivative for any tensor A is defined as;∇A=∂A∂t+ u.∇A− (∇uT .A+ A.∇u) (A.2)which can be described as the rate of change of A calculated in the reference frame of fluid i.e.translating and deforming with the fluid. Now, to simplify we add the Newtonian contributionsto (A.1) on both sides using τ s = ηsγ˙,τ p + λ∇τ p +τs + λ∇τ s = ηpγ˙ + ηsγ˙ + ηsλ∇γ˙τ + λ∇τ = η[γ˙ +ηsλη∇γ˙]τ + λ1∇τ = η[γ˙ + λ2∇γ˙] (A.3)where λ1 = λ and λ2 < λ1.Now, (A.3) can be non dimensionalize using ηω for stresses and ω for shear rates to give;τ +De1∇τ = γ˙ +De2∇γ˙ (A.4)where De are Deborah numbers which are expressed as De1 = λ1ω and De2 = λ2ω.42A.2. Small amplitude motion : Perturbations and Fourier seriesA.2 Small amplitude motion : Perturbations and FourierseriesWe can write the u, τ and γ˙ as regular perturbation expansion in powers of for a swimmingmotion which is a result of small amplitude deformations,u = u1 + 2u2 + .......... τ = τ1 + 2τ2 + ....... γ˙ = γ˙1 + 2γ˙2 + ......., (A.5)where << 1 is a dimensionless measure of deformation amplitude. As we see that the upperconvected derivative brings a time derivative so we define all fields to be time-periodic expressedin terms of Fourier series. For example, E can expressed asE(a, b, t) =12E˜∗(0) +12n=∞Σn=−∞E˜(n)(a, b)e−nit, (A.6)where tildes & ∗ denote the complex & complex conjugate quantity respectively. We haveE˜(−n) = E˜∗(n) which results in [E˜(−n)enit]∗ = [E˜∗(n)enit]∗ = [E˜(n)e−nit] so the terms in sum-mation at n & −n add up to give a real quantity as f˜ + f˜∗ = 2Re[f ]. Hence, the (A.6) can bewritten as;E(a, b, t) =12E˜∗(0) +12n=∞Σn=−∞E˜(n)(a, b)e−nit= Re[n=∞Σn=0E˜(n)(a, b)e−nit](A.7)= Re[n=∞Σn=0E˜∗(n)(a, b)enit](A.8)So, the u, τ and γ˙ at each order can be expressed as;uj(a, b, t) =12u˜∗(0)j +12n=∞Σn=−∞u˜(n)j (a, b)e−nit = Re[n=∞Σn=0u˜(n)j (a, b)e−nit]= Re[n=∞Σn=0u˜∗(n)j (a, b)enit](A.9)τj(a, b, t) =12τ˜∗(0)j +12n=∞Σn=−∞τ˜(n)j (a, b)e−nit= Re[n=∞Σn=0τ˜(n)j (a, b)e−nit]= Re[n=∞Σn=0τ˜∗(n)j (a, b)enit](A.10)˜˙γj(a, b, t) =12˜˙γ∗(0)j +12n=∞Σn=−∞˜˙γ(n)j (a, b)e−nit= Re[n=∞Σn=0˜˙γ(n)j (a, b)e−nit]= Re[n=∞Σn=0˜˙γ∗(n)j (a, b)enit](A.11)43A.3. First orderA.3 First orderOn matching the first order of (A.4), we getτ1 +De1∂τ1∂t= γ˙1 +De2∂γ˙1∂t. (A.12)Now, using the Fourier expansions i.e. (A.10) and (A.11) we can write[12τ˜∗(0)1 +12n=∞Σn=−∞τ˜(n)1 (a, b)e−nit]+De1∂∂t(12τ˜∗(0)1 +12n=∞Σn=−∞τ˜(n)1 (a, b)e−nit)=[12˜˙γ∗(0)1 +12n=∞Σn=−∞˜˙γ(n)1 (a, b)e−nit]+De1∂∂t(12˜˙γ∗(0)1 +12n=∞Σn=−∞˜˙γ(n)1 (a, b)e−nit). (A.13)Matching the time order terms, the zeroth mode can be written as12[τ˜∗(0)1 + τ˜(0)1]=12[˜˙γ∗(0)1 +˜˙γ(0)1],→ Re[τ˜ (0)1 ] = Re[˜˙γ(0)1 ], τ˜ ∗(0)1 = ˜˙γ∗(0)1 & τ˜ (0)1 = ˜˙γ(0)1 . (A.14)Similarly, the nth mode can be written asn=∞Σn=−∞,n6=012(1− niDe1)τ˜ (n)1 (a, b)e−nit =n=∞Σn=−∞,n6=012(1− inDe2)˜˙γ(n)1 (a, b)e−nit. (A.15)Thus, we haveτ˜(n)1 =(1− inDe2)(1− niDe1)˜˙γ(n)1 , (A.16)τ˜∗(n)1 =(1 + inDe2)(1 + niDe1)˜˙γ∗(n)1 , (A.17)which even satisfies the n = 0 result. So, it can be written generally asτ(n)1 = Gγ˙(n)1 for G =(1− inDe2)(1− niDe1) (A.18)A.4 General Relationship & Second orderAt higher orders following the (A.4) and (A.2), the general relationship can be written as(1 +De1∂∂t)τj −(1 +De2∂∂t)γ˙j =j−1Σn=1[De1(∇uTj−n.τn + τn.∇uj−n − u1−n.∇τn)−De2(∇uTj−n.γ˙n + γ˙n.∇uj−n − uj−n.∇γ˙n)]. (A.19)44A.4. General Relationship & Second orderSo, we can write the second order as(1 +De1∂∂t)τ2 −(1 +De2∂∂t)γ˙2 =De1(∇uT1 .τ1 + τ1.∇u1 − u1.∇τ1)−De2(∇uT1 .γ˙1 + γ˙1.∇u1 − u1.∇γ˙1). (A.20)Now, we describe the above equation using Fourier components with their time modes andwrite ∇uT in a similar way using (A.9). Thus, the left hand side is written as=(1 +De1∂∂t)τ2 −(1 +De2∂∂t)γ˙2=(1 +De1∂∂t)(12τ˜∗(0)2 +12n=∞Σn=−∞τ˜(n)2 (a, b)e−nit)−(1 +De2∂∂t)(12˜˙γ∗(0)2 +12n=∞Σn=−∞˜˙γ(n)2 (a, b)e−nit)=12(τ˜∗(0)2 + τ˜(0)2 − ˜˙γ∗(0)2 − ˜˙γ(0)2)+12n=∞Σn=−∞,n 6=0[(1− niDe1)τ˜ (n)2 − (1− niDe2)˜˙γ(n)2]e−nit.(A.21)Similarly, we will solve below only the first term just for ease and then extrapolate the resultsto the other terms;∇uT1 .τ1 =(12∇u˜T∗(0)1 +12n=∞Σn=−∞∇u˜T (n)1 e−nit)·(12τ˜∗(0)1 +12n=∞Σn=−∞τ˜(n)1 e−nit)=12Re[∇u˜T (0)1 · τ˜ (0)1 +∇u˜T (0)1 · τ˜ ∗(0)1]+14(n=∞Σn=−∞,n6=0∇u˜T (n)1 e−nit ·n=∞Σn=−∞,n6=0τ˜(n)1 e−nit+n=∞Σn=−∞,n6=0∇u˜T (n)1 e−nit · τ˜ ∗(0)1 + ∇u˜T∗(0)1 ·n=∞Σn=−∞τ˜(n)1 e−nit)= Re[∇u˜T (0)1 · τ˜ (0)1]+14(n=∞Σn=−∞,n 6=0∇u˜T (n)1 e−nit ·n=∞Σn=−∞,n 6=0τ˜(n)1 e−nit +n=∞Σn=−∞,n 6=0∇u˜T (n)1 e−nit · τ˜ ∗(0)1 + ∇u˜T∗(0)1 ·n=∞Σn=−∞τ˜(n)1 e−nit). (A.22)Now, the second term will also give the zeroth mode of time when there is an inner product ofnth mode of ∇u˜T and -nth mode of τ˜ or vice-versa. So, the zeroth mode of first term of R.H.Scan be written as= Re[∇u˜T (0)1 .τ˜ (0)1]+14n=∞Σn=1(∇u˜T (n)1 .τ˜ (−n)1 +∇u˜T (−n)1 .τ˜ (n)1)= Re[∇u˜T (0)1 .τ˜ (0)1]+14n=∞Σn=1(∇u˜T (n)1 .τ˜ ∗(n)1 +∇u˜T∗(n)1 .τ˜ (n)1)= Re[∇u˜T (0)1 .τ˜ (0)1]+12n=∞Σn=1Re(∇u˜T (n)1 .τ˜ ∗(n)1), (A.23)45A.4. General Relationship & Second orderusing (A.17) we get,= Re[∇u˜T (0)1 . ˜˙γ(0)1]+12n=∞Σn=1Re((1 + inDe2)(1 + niDe1)∇u˜T (n)1 . ˜˙γ∗(n)1)(A.24)After extrapolating these results to other terms on R.H.S for a zeroth mode of time, we get= (De1 −De2)(∇u˜T (0)1 . ˜˙γ(0)1 + ˜˙γ(0)1 .∇u˜(0)1 − u˜(0)1 .∇ ˜˙γ(0)1)+12n=∞Σn=1Re[(De1 −De2)(1 + niDe1)(∇u˜T (n)1 . ˜˙γ∗(n)1 + ˜˙γ∗(n)1 .∇u˜(n)1 − u˜(n)1 .∇ ˜˙γ∗(n)1)]. (A.25)So, we write zeroth mode at second order using (A.21) and (A.25);Re[τ˜(0)2 ]− Re[˜˙γ(0)2 ] = τ (0)2 − γ˙(0)2 = (De1 −De2)(∇u˜T (0)1 . ˜˙γ(0)1 + ˜˙γ(0)1 .∇u˜(0)1 − u˜(0)1 .∇ ˜˙γ(0)1)+12n=∞Σn=1Re[(De1 −De2)(1 + niDe1)(∇u˜T (n)1 . ˜˙γ∗(n)1 + ˜˙γ∗(n)1 .∇u˜(n)1 − u˜(n)1 .∇ ˜˙γ∗(n)1)].(A.26)Now, let’s see the second mode of second order. From (A.21),LHS =12[(1− 2iDe1)τ˜ (2)2 − (1− 2iDe2)˜˙γ(2)2](A.27)Using (A.22), the first term of R.H.S. for second mode can be written as=14[∇u˜T (1)1 .τ˜ (1)1 +∇u˜T (2)1 .τ˜ ∗(0)1 +∇u˜T∗(0)1 .τ˜ (2)1]=14[1− iDe21− iDe1∇u˜T (1)1 .˜˙γ(1)1 +∇u˜T (2)1 . ˜˙γ∗(0)1 +1− 2iDe21− 2iDe1∇u˜T∗(0)1 .˜˙γ(2)1]. (A.28)Thus, the R.H.S after extending this to other terms is expressed below;R.H.S =14[1− iDe21− iDe1(∇u˜T (1)1 . ˜˙γ(1)1 + ˜˙γ(1)1 .∇u˜(1)1 − u˜(1)1 .∇ ˜˙γ(1)1)+(∇u˜T (2)1 . ˜˙γ∗(0)1 + ˜˙γ∗(0)1 .∇u˜(2)1 − u˜(2)1 .∇ ˜˙γ∗(0)1)1− 2iDe21− 2iDe1(∇u˜T∗(0)1 . ˜˙γ(2)1 + ˜˙γ(2)1 .∇u˜∗(0)1 − u˜∗(0)1 .∇ ˜˙γ(2)1)]. (A.29)46A.4. General Relationship & Second orderHence, the second mode of second order can be written as(1− 2iDe1)τ˜ (2)2 − (1− 2iDe2)˜˙γ(2)2 =12[1− iDe21− iDe1(∇u˜T (1)1 . ˜˙γ(1)1 + ˜˙γ(1)1 .∇u˜(1)1 − u˜(1)1 .∇ ˜˙γ(1)1)+(∇u˜T (2)1 . ˜˙γ∗(0)1 + ˜˙γ∗(0)1 .∇u˜(2)1 − u˜(2)1 .∇ ˜˙γ∗(0)1)1− 2iDe21− 2iDe1(∇u˜T∗(0)1 . ˜˙γ(2)1 + ˜˙γ(2)1 .∇u˜∗(0)1 − u˜∗(0)1 .∇ ˜˙γ(2)1)].(A.30)Now, the right hand sides can be described as A(n)j and thus in general,τ(0)j − γ˙(0)j = A(0)j ,(1− niDe1)τ˜ (n)j − (1− niDe2)˜˙γ(n)j = A(n)j . (A.31)For a motion with deformation of the form y = Σn=∞n=−∞,n6=0 a˜ne−nit, we don’t have a zerothmode at order 1 as it means just a shift of y with a˜o which is not relevant. So, we areinterested in such kind of motions where there are no zeroth modes at first order. Hence, ourresults modifies asFirst Orderτ(n)1 =(1− inDe2)(1− niDe1) γ˙(n)1 ∀ n ≥ 1, (A.32)Second order, zeroth modeτ(0)2 − γ˙(0)2 =12n=∞Σn=1Re[(De1 −De2)(1 + niDe1)(∇u˜T (n)1 . ˜˙γ∗(n)1 +˜˙γ∗(n)1 .∇u˜(n)1 − u˜(n)1 .∇ ˜˙γ∗(n)1)], (A.33)Second order, second mode(1− 2iDe1)τ˜ (2)2 − (1− 2iDe2)˜˙γ(2)2 =12[1− iDe21− iDe1(∇u˜T (1)1 . ˜˙γ(1)1 + ˜˙γ(1)1 .∇u˜(1)1 − u˜(1)1 .∇ ˜˙γ(1)1)].(A.34)47Appendix BThe complex reciprocal theorem [56]We take u and σ as the velocity field and stress tensor for a force free swimmer of surfaceS, while uˆ and σˆ are the velocity field and stress tensor for a body with the same instanta-neous shape subject to rigid body translation Uˆ and rotation Ωˆ. Each fluid is in mechanicalequilibrium so ∇ · σ =∇ · σˆ = 0 and thus, using equality of virtual powers;∫V∇ · σ · uˆ dV =∫V∇ · σˆ · u dV = 0 (B.1)where the V is external to surface S with normal n into the fluid. So, using divergence theoremwhile noting the definition of normal n into the fluid and force free conditions we have;∫Sn · σ · uˆ dS +∫Vσ :∇uˆ dV =∫Sn · σˆ · u dS +∫Vσˆ :∇u dV = 0, (B.2)⇒ 0 =∫Sn · σˆ · u dS +∫Vσˆ :∇u dV. (B.3)The first term on the left hand side is zero because the swimmer is force free and thus, thesecond terms also goes to zero. Now, u(x ∈ ∂ S ) = U + Ω× rS + uS whereas the stress tensoris σ = −pI + τ . Taking the fluids to be incompressible, we expand the first term and rewritethe second term to getFˆ ·U + Lˆ ·Ω = −∫Sn · σˆ · uS dS −∫Vτˆ :∇u dV (B.4)Now, we use the 6 dimensional vectors: U = [U Ω]> and FˆD = [Fˆ Lˆ]> (where the subscriptd indicates the rigid-body drag) and write the equation compactly asFˆD · U = −∫Sn · σˆ · uS dS −∫Vτˆ :∇u dV (B.5)Now, taking the rigid body motion in a Newtonian fluid we have τˆ = ηˆ ˆ˙γ and using the identityˆ˙γ :∇u = γ˙ :∇uˆ while rearranging the dot products on vectors,U · FˆD = −∫SuS · (n · σˆ) dS − ηˆ∫Vγ˙ :∇uˆ dV. (B.6)48Appendix B. The complex reciprocal theoremAs Stokes equations are linear so we can write uˆ = Gˆ · Uˆ, σˆ = Tˆ · Uˆ and FˆD = −Rˆ · Uˆ where Rˆis the symmetric resistance tensor which can be written asRˆ = −∫S[n · Tˆ r× (n · Tˆ )]>dS. (B.7)Substituting while cancelling the arbitrary Uˆ factor, we getU = Rˆ−1 ·[∫SuS ·(n · Tˆ)dS + ηˆ∫Vγ˙ :∇Gˆ dV]. (B.8)Now, we have the second term of the left hand side from (B.2) and using the incompressibility,uˆ = Gˆ · Uˆ and τ = ηγ˙ +N for a viscoelastic fluid governed by Oldroyd-B model, we write∫Vσ :∇uˆ dV = η∫Vγ˙ :∇uˆ dV +∫VN :∇uˆ dV = 0⇒∫Vγ˙ :∇Gˆ dV = −1η∫VN :∇Gˆ dV (B.9)We substitute this in (B.8) to getU = Rˆ−1 ·[∫SuS ·(n · Tˆ)dS − ηˆη∫VN :∇Gˆ dV](B.10)Similarly, we could use the appropriate conditions on the swimmer body as mentioned insection 2.3 to simplify the equation B.2 and compute integral statements for drag and thrustin complex fluids as (2.10) and (2.11).49Appendix CRheology tablesWe used a variety methods to perform rheology sequences and test the characteristics astabulated in C.1.Methods Shear ramp, stress ramp, time sweep, single frequencySAOS, strain sweep, stress sweep, frequency sweepCharacteristics Shear profile, thixotropy/rheopexy, stress relaxation,creep recovery, LVE, Structural responseTable C.1: Methods and characteristics tested in RheologyAn example of data displaying wall slip on comparison of smooth and serrated plates geom-etry is tabulated in C.2. The serrated plate viscosities are as high as 30% indicating wall slipfor the sample. The results were repeatable qualitatively observing higher values for serratedgeometry.Smooth PP Serrated PP Differenceγ˙ η γ˙ η %0.10 37.53 0.10 47.50 26.570.22 21.34 0.22 25.37 18.880.46 11.95 0.46 15.26 27.701 7.00 1 9.13 30.382.15 4.74 2.15 5.25 10.654.64 2.67 4.64 2.72 1.6810 1.28 10.00 1.20 −6.7921.54 0.63 21.55 0.60 −3.5646.42 0.33 46.41 0.32 −3.73100 0.18 100 0.18 −2.66215.50 0.10 215.50 9.83× 10−2 −5.26464.20 5.67× 10−2 464.20 5.46× 10−2 −3.811000 5.15× 10−2 1000 6.70× 10−2 29.91Table C.2: Apparent viscosity comparison between smooth and serrated parallel plate geometryfor VES (3.5%, 10.7 ppg CaCl2)We also performed initial rheology tests to choose the gap value for the given roughness ofthe parallel plate geometry. The lower and higher values pose different challenges of wall slipand edge fracture respectively. We performed low shear rheology for the VES (3.5%, 10.7 ppgCaCl2) sample at different gap values and select the gap size with consistent higher viscosity50Appendix C. Rheology tablesrecordings. The rheology results are tabulated in C.3 for a shear ramp with γ˙ (s−1) variationand τ (Pa)recordingsmadebytherheometer.d → 0.5 mm 0.75 mm 1 mm 1.25 mmγ˙ τ η τ η τ η τ η0.02 2.29 106.30 2.53 117.50 2.92 136 2.27 105.500.05 3.13 67.44 3.14 67.70 4.26 91.84 2.79 60.130.10 3.93 39.28 3.91 39.09 5.30 53.05 3.31 33.130.22 3.03 14.07 5.33 24.73 6.56 30.46 4.42 20.510.46 3.16 6.80 6.48 13.96 6.12 13.19 4.84 10.421.00 4.05 4.05 5.49 5.49 7.38 7.38 5.41 5.402.15 5.70 2.65 6.99 3.24 7.60 3.53 6.24 2.904.64 8.03 1.73 8.66 1.87 9.02 1.94 7.99 1.7210.00 10.45 1.04 10.02 1.00 9.61 0.96 9.18 0.92Table C.3: Rheological results for different gap values of serrated PP geometry with VES (3.5%,10.7 ppg CaCl2)Thus, we chose 1 mm and observed gap dependency for the sample. The lower value of 0.75mm of gap size showed results with similar values to smooth plates geometry tests performedright after gap variation tests, strongly indicating wall slip for our VES fluid. Therefore,detailed gap and roughness dependency studies need to be performed in the future to findoptimum values tackling wall slip and edge fracture for the fluid.51Appendix DFlow loop dataIn this appendix, we present flow loop results data of 3 samples for reference. The tableD.1 presents Xanthan (0.06 ppg) data whereas tables D.2 and D.3 for the VES (3.5%, 10.7ppg CaCl2) and VES (4.5%, 9.2 ppg NaCl) respectively. The first two columns representthe measured parameters namely, flow rate and pressure drop respectively whereas the lastthree columns (solvent Re, generalized Re and friction factor) are the computed dimensionlessnumbers used in the fig. 3.9. It should be noted that generalized Re is not computed for higherflow rates in NaCl based sample because of the edge fracture in the rheometery at higher shearrates (above 550− 650 s−1) resulting in viscosity measurements only at lower end of the shearrates.52Appendix D. Flow loop dataQ (l/min) ∆P (Pa) Ren Reg f620.50 17 966.85 254 742.09 10 370.11 2.76× 10−3608.16 17 293.36 249 679.00 10 031.64 2.77× 10−3595.47 16 891.66 244 466.13 9687.82 2.82× 10−3577.96 16 220.88 237 280.63 9221.69 2.87× 10−3554.39 15 389.14 227 601.44 8608.23 2.96× 10−3531.87 14 567.43 218 357.79 8038.06 3.05× 10−3507.95 13 921.47 208 537.30 7449.35 3.19× 10−3490.04 13 107.77 201 182.27 7020.09 3.23× 10−3465.94 12 251.16 191 289.22 6458.71 3.34× 10−3437.25 11 425.91 179 510.88 5814.68 3.54× 10−3409.80 10 793.14 168 242.51 5223.82 3.80× 10−3377.58 9784.82 155 015.61 4562.59 4.06× 10−3345.18 8904.19 141 713.13 3933.74 4.42× 10−398.17 4954.14 40 302.78 500.48 3.04× 10−2177.89 5819.64 73 030.90 1343.40 1.09× 10−2220.54 6427.08 90 541.06 1919.74 7.81× 10−3257.46 7116.19 105 699.69 2482.61 6.34× 10−3293.86 7858.97 120 644.85 3092.49 5.38× 10−3328.01 8633.71 134 664.28 3712.02 4.74× 10−3363.31 9394.36 149 157.15 4398.90 4.21× 10−3395.40 10 461.99 162 330.73 5062.88 3.95× 10−3429.36 11 212.34 176 270.16 5805.30 3.59× 10−3457.97 12 137.84 188 015.86 6461.88 3.42× 10−3482.68 12 656.93 198 163.02 7051.41 3.21× 10−3500.79 13 497.66 205 596.46 7496.19 3.18× 10−3521.26 14 112.22 214 000.98 8012.02 3.07× 10−3535.58 14 377.12 219 879.72 8380.91 2.96× 10−3544.63 15 062.44 223 596.23 8617.52 3.00× 10−3558.62 15 505.97 229 341.08 8988.39 2.94× 10−3561.46 15 118.33 230 506.51 9064.39 2.83× 10−3556.53 15 271.50 228 481.31 8932.49 2.91× 10−3565.60 15 606.83 232 203.91 9175.52 2.88× 10−3562.20 15 358.82 230 810.88 9084.28 2.87× 10−3542.32 14 659.96 222 646.50 8556.80 2.95× 10−3513.66 13 823.99 210 882.40 7819.03 3.10× 10−3509.68 13 631.39 209 247.91 7718.62 3.10× 10−3497.77 13 467.44 204 356.53 7421.25 3.21× 10−3484.65 12 665.34 198 972.47 7099.32 3.19× 10−3465.14 11 982.22 190 960.30 6630.83 3.27× 10−3439.87 11 299.59 180 587.45 6043.38 3.45× 10−3417.24 10 735.29 171 296.04 5535.74 3.64× 10−3389.34 9758.07 159 841.43 4934.57 3.80× 10−353Appendix D. Flow loop dataQ (l/min) ∆P (Pa) Ren Reg f358.71 9081.70 147 265.50 4306.63 4.17× 10−3324.21 8152.77 133 101.65 3640.75 4.58× 10−3287.79 7388.32 118 152.92 2987.11 5.27× 10−3255.12 6733.49 104 736.76 2445.15 6.11× 10−3223.59 6090.51 91 795.60 1964.12 7.20× 10−3190.22 5515.32 78 094.30 1501.63 9.01× 10−3145.74 4875.07 59 834.74 964.81 1.36× 10−295.81 4297.23 39 334.30 480.66 2.77× 10−2125.38 5219.26 51 475.61 757.09 1.99× 10−2110.86 4974.10 45 512.25 617.39 2.43× 10−2136.36 5268.20 55 983.77 870.05 1.70× 10−2169.70 5785.54 69 670.28 1249.97 1.20× 10−2201.96 6422.81 82 915.45 1667.70 9.44× 10−3236.22 7175.51 96 980.40 2161.95 7.71× 10−3274.18 7822.90 112 565.14 2767.33 6.24× 10−3318.84 8722.99 130 900.55 3553.29 5.14× 10−3383.06 10 287.59 157 263.10 4815.46 4.20× 10−3425.89 11 378.24 174 848.11 5739.82 3.76× 10−3458.33 12 186.37 188 165.59 6482.00 3.48× 10−3476.21 12 903.98 195 504.58 6906.16 3.41× 10−3487.29 12 955.57 200 053.34 7174.37 3.27× 10−3505.60 13 160.95 207 570.46 7626.45 3.09× 10−3489.14 12 781.62 200 813.13 7219.57 3.20× 10−3503.22 13 577.97 206 593.81 7567.09 3.21× 10−3515.55 13 803.15 211 657.26 7876.80 3.11× 10−3534.09 14 321.15 219 267.07 8351.46 3.01× 10−3541.99 14 806.57 222 510.15 8557.08 3.02× 10−3511.54 13 927.83 210 009.33 7775.46 3.19× 10−3489.25 13 084.11 200 860.30 7222.38 3.28× 10−3471.31 12 341.38 193 494.02 6788.90 3.33× 10−3451.48 11 658.34 185 353.89 6322.33 3.43× 10−3430.97 10 997.36 176 934.60 5853.73 3.55× 10−3411.00 10 631.57 168 733.52 5411.13 3.77× 10−3388.85 9955.31 159 639.94 4936.63 3.95× 10−3391.27 10 011.08 160 632.71 4987.59 3.92× 10−3370.44 9384.11 152 084.46 4555.62 4.10× 10−3337.52 8520.71 138 567.67 3904.66 4.48× 10−3304.40 7811.49 124 969.35 3290.57 5.05× 10−3269.61 7070.81 110 687.31 2691.28 5.83× 10−3227.81 6111.29 93 528.52 2035.97 7.06× 10−3166.51 5234.36 68 361.87 1211.32 1.13× 10−2131.69 4732.42 54 065.21 821.22 1.64× 10−2Table D.1: Flow loop measured and computed data for Xanthan (0.06 ppg) solution54Appendix D. Flow loop dataQ (l/min) ∆P (Pa) Ren Reg f598.77 14 628.10 112 680.96 5788.05 2.08× 10−3559.55 13 100.20 105 300.34 5807.49 2.13× 10−3527.26 11 435.58 99 224.45 5824.60 2.09× 10−3492.48 10 310.86 92 677.73 5844.31 2.16× 10−3453.00 9305.69 85 248.53 5795.65 2.31× 10−3414.04 8522.51 77 917.07 5297.22 2.53× 10−3373.52 7777.61 70 292.64 4778.87 2.84× 10−3332.37 6932.95 62 548.20 4371.79 3.19× 10−3291.83 6310.67 54 918.21 3547.87 3.77× 10−3245.35 6012.10 46 171.57 2685.52 5.08× 10−3201.98 5757.51 38 009.20 1965.22 7.18× 10−3151.60 5576.13 28 528.77 1239.92 1.23× 10−2102.74 5120.09 19 334.27 664.03 2.47× 10−2107.10 5045.63 20 155.13 576.45 2.24× 10−2158.61 4923.58 29 848.18 1130.34 9.95× 10−3201.47 5221.38 37 914.08 1703.56 6.54× 10−3235.32 5570.17 44 284.73 2223.49 5.11× 10−3268.80 6353.98 50 584.70 2793.16 4.47× 10−3310.05 6701.94 58 348.14 3568.07 3.54× 10−3344.49 7245.36 64 829.65 4274.49 3.10× 10−3382.80 7850.33 72 038.03 5121.61 2.72× 10−3414.86 8411.20 78 070.76 5610.41 2.48× 10−3447.67 9271.59 84 246.38 6054.21 2.35× 10−3482.42 10 145.11 90 785.26 6524.11 2.22× 10−3513.31 11 215.54 96 598.25 6670.07 2.16× 10−3545.51 12 106.07 102 658.93 6580.13 2.07× 10−3578.52 13 374.45 108 870.06 6494.46 2.03× 10−3603.60 14 873.90 113 589.31 6433.26 2.07× 10−3632.16 16 220.24 118 964.37 6367.25 2.06× 10−3660.85 17 548.28 124 364.53 6304.50 2.04× 10−3676.04 18 607.15 127 222.61 6272.62 2.07× 10−3674.56 18 857.79 126 943.77 6275.69 2.11× 10−3677.93 19 278.90 127 578.74 6268.71 2.13× 10−3688.10 19 962.19 129 491.42 6247.93 2.14× 10−3693.69 20 364.24 130 544.01 6236.66 2.15× 10−3697.54 20 588.73 131 268.75 6228.96 2.15× 10−3696.64 20 312.42 131 099.91 6230.75 2.13× 10−3690.88 20 035.64 130 015.84 6242.30 2.13× 10−3690.42 19 341.19 129 929.20 6243.23 2.06× 10−3688.73 18 377.85 129 609.77 6246.66 1.97× 10−3684.69 17 867.81 128 849.37 6254.86 1.94× 10−355Appendix D. Flow loop dataQ (l/min) ∆P (Pa) Ren Reg f671.50 16 852.48 126 367.48 6282.06 1.90× 10−3658.76 15 855.87 123 970.35 6308.96 1.86× 10−3642.01 15 108.67 120 817.81 6345.32 1.86× 10−3621.30 14 287.33 116 921.47 6391.90 1.88× 10−3597.89 13 488.30 112 515.81 6446.91 1.92× 10−3573.53 12 739.02 107 930.70 6507.02 1.97× 10−3542.25 11 958.09 102 045.02 6588.94 2.07× 10−3504.97 10 967.12 95 028.89 6694.49 2.19× 10−3460.31 10 408.51 86 625.45 6225.17 2.50× 10−3420.03 9824.66 79 044.01 5680.35 2.83× 10−3374.99 9291.27 70 568.89 4943.80 3.36× 10−3330.88 8683.43 62 267.82 3988.93 4.03× 10−3283.15 8525.67 53 284.86 3053.70 5.40× 10−3237.75 8095.11 44 740.97 2262.92 7.28× 10−3192.76 7533.13 36 274.43 1579.18 1.03× 10−2141.92 6896.00 26 707.11 934.11 1.74× 10−296.04 6370.56 18 073.65 478.17 3.51× 10−298.29 4977.79 18 497.16 499.70 2.62× 10−2156.73 4922.74 29 495.07 1169.08 1.02× 10−2197.33 5315.28 37 134.53 1778.51 6.94× 10−3230.22 5856.33 43 324.77 2355.19 5.62× 10−3268.10 6407.53 50 452.46 3326.91 4.53× 10−3308.50 6783.95 58 056.48 4014.01 3.62× 10−3344.99 7201.89 64 922.74 4698.77 3.08× 10−3379.22 7766.70 71 365.05 5165.03 2.74× 10−3415.69 8395.82 78 228.62 5661.78 2.47× 10−3449.69 9106.82 84 626.63 6124.83 2.29× 10−3480.99 10 053.77 90 516.85 6551.14 2.21× 10−3513.26 11 090.88 96 588.38 6486.75 2.14× 10−3544.85 12 129.06 102 533.33 6518.44 2.08× 10−3575.28 13 304.21 108 260.28 6547.42 2.04× 10−3602.57 14 298.53 113 396.95 6572.23 2.00× 10−3616.06 14 786.28 115 934.45 6584.11 1.98× 10−3615.49 15 515.74 115 826.87 6583.61 2.08× 10−3628.57 16 308.77 118 288.30 6594.92 2.10× 10−3644.75 17 044.28 121 334.06 6608.61 2.08× 10−3662.17 18 159.59 124 613.06 6623.01 2.10× 10−3676.23 18 857.84 127 257.34 6634.36 2.10× 10−3684.25 19 480.21 128 768.11 6640.76 2.11× 10−3692.89 20 005.07 130 392.60 6647.55 2.12× 10−3689.11 19 484.45 129 682.60 6644.59 2.09× 10−3686.55 18 890.43 129 199.42 6642.57 2.04× 10−356Appendix D. Flow loop dataQ (l/min) ∆P (Pa) Ren Reg f680.38 18 081.37 128 038.30 6637.68 1.98× 10−3669.81 17 011.13 126 050.57 6629.21 1.93× 10−3656.73 16 063.64 123 587.93 6618.54 1.89× 10−3639.96 15 129.90 120 432.72 6604.59 1.88× 10−3618.99 14 132.87 116 486.88 6586.66 1.87× 10−3596.52 13 420.95 112 258.26 6566.82 1.92× 10−3574.03 12 670.83 108 024.51 6546.25 1.95× 10−3549.54 11 883.82 103 416.05 6523.00 2.00× 10−3515.77 11 061.40 97 061.46 6489.34 2.11× 10−3477.99 10 172.58 89 951.85 6510.25 2.26× 10−3437.16 9445.00 82 268.48 5954.16 2.51× 10−3392.97 8942.40 73 951.98 5352.26 2.94× 10−3347.27 8504.24 65 351.54 4729.80 3.58× 10−3303.19 7881.47 57 055.78 3888.87 4.36× 10−3255.76 7643.67 48 130.14 2852.59 5.94× 10−3211.33 7379.01 39 770.00 2015.11 8.40× 10−3163.28 6695.57 30 727.19 1259.56 1.28× 10−2111.23 6173.74 20 931.80 625.93 2.54× 10−2Table D.2: Flow loop measured and computed data for VES (3.5%, 10.7 ppg CaCl2)Q (l/min) ∆P (Pa) Ren Reg f745.02 16 209.32 306 285.52 1.77× 10−3740.99 15 483.48 304 625.30 1.71× 10−3729.67 15 020.29 299 973.57 1.71× 10−3728.92 14 780.91 299 663.64 1.69× 10−3723.44 14 724.97 297 410.77 1.71× 10−3706.96 14 463.63 290 637.60 1.75× 10−3693.41 14 148.03 285 067.25 1.78× 10−3669.27 13 941.44 275 142.47 1.89× 10−3634.72 13 279.59 260 938.77 2.00× 10−3597.77 12 683.55 245 747.82 2.15× 10−3562.80 12 112.82 231 371.41 2.32× 10−3524.03 11 504.89 215 435.17 2.54× 10−3483.68 11 143.25 198 845.46 2.89× 10−3443.54 10 536.18 182 343.41 3.25× 10−3402.52 9937.03 165 479.03 3.72× 10−3362.09 9370.77 148 858.08 3280.56 4.33× 10−3317.11 8799.27 130 368.39 2635.78 5.30× 10−3273.25 8088.38 112 337.33 2061.76 6.57× 10−3230.81 7319.49 94 886.59 1560.49 8.33× 10−3186.02 6447.01 76 472.98 1093.11 1.13× 10−2138.48 5659.87 56 928.80 671.69 1.79× 10−257Appendix D. Flow loop dataQ (l/min) ∆P (Pa) Ren Reg f91.52 5062.88 37 626.58 339.19 3.66× 10−297.94 5477.70 40 262.48 411.11 3.50× 10−2151.02 5853.98 62 084.56 820.01 1.57× 10−2197.10 6477.67 81 028.47 1253.73 1.02× 10−2234.94 7131.79 96 584.66 1658.83 7.92× 10−3277.24 7564.04 113 976.49 2159.95 6.03× 10−3321.16 7917.24 132 032.56 2730.64 4.71× 10−3357.14 8335.76 146 821.90 3234.28 4.01× 10−3392.94 8806.24 161 539.48 3.50× 10−3428.18 9223.09 176 026.69 3.08× 10−3464.14 9781.22 190 811.66 2.78× 10−3499.07 10 293.83 205 172.44 2.53× 10−3530.88 11 028.54 218 247.78 2.40× 10−3564.29 11 462.13 231 985.48 2.21× 10−3596.58 12 157.71 245 260.38 2.09× 10−3626.82 12 799.32 257 689.38 2.00× 10−3657.61 13 539.24 270 349.08 1.92× 10−3677.31 14 022.64 278 449.83 1.87× 10−3673.05 14 263.93 276 695.96 1.93× 10−3685.38 14 647.89 281 763.65 1.91× 10−3690.60 15 095.98 283 911.86 1.94× 10−3701.48 15 744.89 288 385.42 1.96× 10−3703.53 16 210.42 289 228.82 2.01× 10−3702.26 16 205.88 288 704.87 2.01× 10−3700.22 16 389.16 287 864.94 2.05× 10−3698.72 16 866.62 287 251.07 2.12× 10−3694.24 16 776.87 285 406.63 2.13× 10−3683.15 17 010.70 280 847.89 2.23× 10−3679.21 16 836.12 279 227.89 2.24× 10−3661.25 16 767.57 271 847.00 2.35× 10−3653.65 16 554.00 268 720.29 2.38× 10−3632.89 16 081.09 260 185.20 2.46× 10−3598.40 15 321.24 246 007.75 2.62× 10−3559.70 14 538.62 230 097.86 2.85× 10−3523.43 13 742.30 215 187.03 3.07× 10−3482.95 12 923.37 198 546.50 3.40× 10−3444.96 12 026.50 182 924.90 3.72× 10−3405.69 11 342.56 166 780.62 3963.05 4.22× 10−3366.41 10 510.96 150 632.70 3369.15 4.80× 10−3325.20 9682.62 133 692.97 2785.60 5.61× 10−3281.44 8750.49 115 701.77 2212.31 6.77× 10−3237.61 7868.46 97 683.27 1689.02 8.54× 10−358Appendix D. Flow loop dataQ (l/min) ∆P (Pa) Ren Reg f194.48 6908.19 79 952.17 1227.28 1.12× 10−2147.59 6003.34 60 674.28 790.51 1.69× 10−298.26 5435.38 40 395.55 413.28 3.45× 10−2121.29 8397.16 49 862.59 353.28 3.42× 10−2147.34 8150.45 60 574.35 502.28 2.25× 10−2190.50 8934.78 78 315.87 799.24 1.48× 10−2235.17 9882.35 96 682.31 1169.86 1.07× 10−2272.95 10 698.20 112 212.96 1531.52 8.61× 10−3315.05 11 582.24 129 520.22 1985.04 6.99× 10−3356.42 12 130.58 146 526.28 2481.16 5.72× 10−3396.90 12 470.31 163 167.01 3013.93 4.74× 10−3435.62 12 899.55 179 086.71 3566.52 4.07× 10−3470.63 13 307.33 193 479.25 3.60× 10−3507.15 13 703.62 208 493.76 3.19× 10−3544.30 14 155.99 223 764.78 2.86× 10−3577.15 14 657.31 237 271.06 2.64× 10−3516.41 14 095.67 212 300.58 3.17× 10−3518.62 14 380.03 213 207.43 3.20× 10−3555.01 15 171.37 228 168.31 2.95× 10−3589.10 15 898.03 242 182.34 2.75× 10−3612.26 16 643.74 251 707.10 2.66× 10−3629.04 17 266.05 258 602.93 2.62× 10−3643.34 17 909.84 264 481.28 2.59× 10−3659.13 18 376.71 270 972.39 2.53× 10−3671.29 18 863.50 275 973.94 2.51× 10−3659.54 18 892.94 271 142.80 2.60× 10−3645.78 18 776.40 265 484.31 2.70× 10−3635.77 18 613.31 261 370.56 2.76× 10−3619.29 18 324.35 254 593.96 2.86× 10−3598.98 17 876.70 246 244.10 2.99× 10−3580.75 17 334.04 238 753.06 3.08× 10−3570.37 16 853.68 234 485.97 3.10× 10−3557.58 16 186.03 229 225.41 3.12× 10−3519.59 15 163.50 213 606.94 3.37× 10−3481.98 14 163.74 198 148.04 3.65× 10−3444.28 13 243.81 182 647.57 3695.78 4.02× 10−3403.30 12 325.12 165 798.71 3102.40 4.54× 10−3362.58 11 384.51 149 061.41 2559.33 5.19× 10−3319.81 10 392.29 131 474.83 2039.54 6.09× 10−3276.12 9517.34 113 516.07 1563.83 7.48× 10−3231.90 8568.49 95 337.35 1140.59 9.55× 10−3187.57 7542.30 77 109.62 777.12 1.28× 10−2140.40 6799.48 57 720.90 460.31 2.07× 10−2Table D.3: Flow loop measured and computed data for VES (4.5%, 9.2 ppg NaCl)59
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Drag and thrust effects of Viscoelastic fluids Goyal, Gaurav 2016
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Title | Drag and thrust effects of Viscoelastic fluids |
Creator |
Goyal, Gaurav |
Publisher | University of British Columbia |
Date Issued | 2016 |
Description | Viscoelastic fluids are non-Newtonian fluids exhibiting both viscous and elastic properties. Many fluids of practical importance (polymers, surfactants, mucus, shampoos etc.) display viscoelastic effects to different degrees under a wide range of flow conditions and thus, these fluids present a variety of problems. In this work, we study two problems at very different flow conditions in viscoelastic fluids: a) the effect of swimming gait on bio-locomotion and b) characterizing the drag reducing fluids used for gravel-packing operations in the petroleum industry. For the first problem, we give formulas for the swimming of simplified two-dimensional bodies at low Reynolds numbers in complex fluids using the reciprocal theorem. By way of these formulas, we calculate the swimming velocity due to small-amplitude deformations on the simplest of these bodies, a two-dimensional sheet, to explore general conditions on the swimming gait under which the sheet may move faster, or slower, in a viscoelastic fluid compared to a Newtonian fluid. We show that in general, for small amplitude deformations, a speed increase can only be realized by multiple deformation modes in contrast to slip flows. Additionally, we show that a change in swimming speed is directly due to a change in thrust generated by the swimmer. Later, we work with viscoelastic additives (xanthan and a zwitterionic viscoelastic surfactant, VES), widely used as drag reducers for gravel-packing applications. While the behavior of xanthan is well characterized in the literature, much less is known about the VES characteristics, despite widespread use. We performed a number of rheological tests and flow-loop experiments on VES solutions to understand the structural characteristics to make better process predictions. Unlike xanthan, which displays typical viscoelastic liquid characteristics, VES displays elastic gel-like behaviour. The gel-like behaviour suggests long and relatively unbreakable chain lengths of the wormlike micelles in the VES at room temperature leading to gelation by entanglement. Also, a shear-thickening behaviour of VES samples at higher shear rates is observed, possibly as a result of shear-induced structures. Finally, we present a novel representation scheme to depict the flow-loop results relating the rheological characterization while observing drag reduction. |
Genre |
Thesis/Dissertation |
Type |
Text |
Language | eng |
Date Available | 2017-01-21 |
Provider | Vancouver : University of British Columbia Library |
Rights | Attribution-NonCommercial-NoDerivatives 4.0 International |
DOI | 10.14288/1.0340489 |
URI | http://hdl.handle.net/2429/60135 |
Degree |
Master of Applied Science - MASc |
Program |
Mechanical Engineering |
Affiliation |
Applied Science, Faculty of |
Degree Grantor | University of British Columbia |
Graduation Date | 2017-02 |
Campus |
UBCV |
Scholarly Level | Graduate |
Rights URI | http://creativecommons.org/licenses/by-nc-nd/4.0/ |
Aggregated Source Repository | DSpace |
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