Table Of ContentLecture Notes in Mathematical Fluid Mechanics
Alexander Khapalov
Bio-Mimetic
Swimmers in
Incompressible
Fluids
Modeling, Well-Posedness, and
Controllability
Advances in Mathematical Fluid Mechanics
Lecture Notes in Mathematical Fluid Mechanics
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Alexander Khapalov
Bio-Mimetic Swimmers
in Incompressible Fluids
Modeling, Well-Posedness,
and Controllability
AlexanderKhapalov
DepartmentofMathematicsandStatistics
WashingtonStateUniversity
Pullman,WA,USA
ISSN2297-0320 ISSN2297-0339 (electronic)
AdvancesinMathematicalFluidMechanics
ISSN2510-1374 ISSN2510-1382 (electronic)
LectureNotesinMathematicalFluidMechanics
ISBN978-3-030-85284-9 ISBN978-3-030-85285-6 (eBook)
https://doi.org/10.1007/978-3-030-85285-6
MathematicsSubjectClassification:35xx,76xx,93xx,49xx
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ToLuke,Zoe,Dasha,Elena,Irina,andPeter
Preface
Theswimmingandflyingphenomenainnaturehavebeenasourceofgreatinterest
andinspirationformanyresearchersinmathematicsandnaturalsciencesforalong
time. The bibliography of formal publications in this area is very diverse in terms
ofapproachesandmethodology,andcanbetracedasfarbackastotheworksofG.
Borelliintheseventeenthcentury.
Along these lines, the goal of this monograph is to present an original concise
mathematical theory for so-called “bio-mimetic swimmers” in the framework of
coupled system of partial differential equations (PDE) and ordinary differential
equations(ODE).Thistheoryincludes:
(cid:129) Anoriginalmodelingapproach(PartI),
(cid:129) An associated well-posedness results for the proposed models for swimmers
(PartII)
(cid:129) A controllability theory, studying the steering potential of the proposed swim-
mers(PartsIII–V).
We will be focusing on the bio-mimetic swimmers (viewed as possible artificial
mechanical devices) created to imitate the swimming locomotion of the existing
“swimmers” in nature such as fish, eels, clams, aquatic frogs, turtles, sea snakes,
andspermatozoa.
Thistheorywasinitiallydevelopedintheseriesofworks[3,19–33],published
in2005–2018,andinthismonograph,itisextendedtoaprincipallywiderclassof
swimmingmodelswithsignificantlyimprovedsteeringcapabilities.
The proposed theory is based on the original “immerse body” modeling
approach, introduced in [19] in 2005. More precisely, we model the interaction
betweenaswimmerandanincompressiblefluidsurroundingitbymakinguseofa
coupledsystemoftwosetsofpartialandordinarydifferentialequations:
(cid:129) Afluid(Navier–Stokesornon-stationaryStokes)equationforthefluiddynamics
(cid:129) Anordinarydifferentialequationforthepositionofaswimmerinthefluid
Itisassumedthattheswimmer’sbodyconsistsoffinitelymanyelements,identified
with the fluid they occupy, that are subsequently linked by the rotational and
vii
viii Preface
structural forces, which are explicitly described and serve as the means to change
thegeometricshapeoftheswimmer’sbody.Thesumofsuchforces,beinginternal
to the swimmer at hand, is equal to zero, and hence, they cannot move its center
ofmassinaspacecontainingnomedium.However,inanincompressiblefluid,the
originalswimmer’sinternalforces,ingeneral,willchange,bothintheirmagnitude
and direction, according to the instantaneous spatial orientation of the swimmer’s
bodypartsandoftheseforces.Respectively,thesumoftheactualforcesthatwill
act on the swimmer’s body parts in an incompressible fluid can become non-zero,
whichwillresultintheswimmer’slocomotion(aself-propellingmotion).
Acknowledgments
The author’s research presented in this monograph was supported in part by the
NSF Grants DMS-0504093 and DMS-10007981 and Simmons Foundation award
number317297.
The author wishes to express his gratitude to the Department of Mathematics
at the University of Rome II “Tor Vergata” and to the Instituto Nazionale di Alta
Matematica(Italy)forhospitalitythroughouthissabbaticalintheSpringsemester
of2019duringwhichpartsofthismonographwerewritten.
The author also wishes to thank the reviewers for their comments and sugges-
tions.
Pullman,WA,USA AlexanderKhapalov
Fall2020–Winter2021
Contents
1 Introduction................................................................. 1
1.1 Modeling:MimickingtheNature.................................... 2
1.2 MathematicalApproachtoSwimmingModeling .................. 6
1.3 SwimmingControllability ........................................... 7
1.4 RelatedSelectedBibliography....................................... 8
PartI Modeling of Bio-Mimetic Swimmers in 2D and 3D
IncompressibleFluids
2 Bio-MimeticFish-LikeSwimmersina2DIncompressible
Fluid:EmpiricModeling .................................................. 15
2.1 Swimmer’sBodyasaCollectionofSeparateSets ................. 15
2.2 Bio-MimeticFish-andSnake-LikeSwimmers .................... 16
2.3 Swimmer’sInternalForces .......................................... 16
2.3.1 RotationalInternalForces.................................. 17
2.3.2 ElasticInternalForces...................................... 18
2.4 Swimmer’sGeometricControls..................................... 19
2.5 InternalForcesandConservationofMomenta ..................... 20
2.5.1 AboutSwimmerswithBodyPartsDifferentinMass..... 20
2.6 Fluid Equations: Non-stationary Stokes and
Navier–StokesEquationsin2D ..................................... 21
2.7 AModelofa2DFish-LikeBio-MimeticSwimmer:The
CaseofStokesEquations............................................ 22
2.8 AModelofa2DFish-LikeBio-MimeticSwimmer:The
CaseofNavier–StokesEquations ................................... 24
3 Bio-MimeticAquaticFrog-andClam-LikeSwimmersina
2DFluid:EmpiricModeling .............................................. 25
3.1 ABio-MimeticAquaticFrog-LikeSwimmerina2D
IncompressibleFluid................................................. 25
3.2 A Bio-Mimetic Clam-Like Swimmer in a 2D
IncompressibleFluid................................................. 28
ix
x Contents
4 Bio-Mimetic Swimmers in a 3D Incompressible Fluid:
EmpiricModeling .......................................................... 31
4.1 RotationalForcesin3D.............................................. 31
4.2 AModelofa3DFish-LikeBio-MimeticSwimmer:The
CaseofStokesEquations............................................ 33
4.3 A Bio-Mimetic Frog-Like Swimmer in a 3D
IncompressibleFluid................................................. 34
4.4 A Bio-Mimetic Clam-Like Swimmer in a 3D
IncompressibleFluid................................................. 36
PartII Well-PosednessofModelsforBio-MimeticSwimmersin
2Dand3DIncompressibleFluids
5 Well-Posednessof2Dor3DBio-MimeticSwimmers:The
CaseofStokesEquations .................................................. 41
5.1 Notations.............................................................. 41
5.2 Swimmer’sBody..................................................... 42
5.3 Initial-andBoundary-ValueProblemSetup ........................ 45
5.3.1 EstimatesforInternalForces............................... 47
5.4 MainResult:ExistenceandUniquenessofSolutions.............. 48
5.5 ProofofTheorem5.1 ................................................ 49
5.5.1 PreliminaryResults:DecoupledEquationforz (t)’s..... 49
i
5.5.2 ThreeDecoupledSolutionMappingsfor(5.3.1).......... 52
5.5.3 ProofofTheorem5.1....................................... 56
6 Well-Posednessof2Dor3DBio-MimeticSwimmers.................. 59
6.1 ProblemSetupandMainResults.................................... 59
6.1.1 ProblemSetting............................................. 59
6.1.2 MainResults................................................ 60
6.2 ProofsoftheMainResults........................................... 60
6.2.1 SolutionMappingforDecoupledNavier–Stokes
Equations.................................................... 61
6.2.2 PreliminaryResults......................................... 61
6.2.3 ContinuityofB .......................................... 63
NS
6.2.4 ProofofTheorems6.1and6.2............................. 68
PartIII MicromotionsandLocalControllabilityforBio-Mimetic
Swimmersin2Dand3DIncompressibleFluids
7 LocalControllabilityof2Dand3DSwimmers:TheCaseof
Non-stationaryStokesEquations......................................... 71
7.1 DefinitionsofControllabilityforBio-MimeticSwimmers......... 71
7.2 MainResults.......................................................... 73
7.2.1 MainResults................................................ 75
