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Record Nr. |
UNISA996466398103316 |
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Autore |
Khapalov Alexander Y. |
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Titolo |
Bio-mimetic swimmers in incompressible fluids : modeling, well-posedness, and controllability / / Alexander Khapalov |
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Pubbl/distr/stampa |
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Cham, Switzerland : , : Birkhäuser, , [2021] |
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©2021 |
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ISBN |
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Descrizione fisica |
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1 online resource (177 pages) |
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Collana |
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Advances in Mathematical Fluid Mechanics |
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Disciplina |
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Soggetti |
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Fluid mechanics - Mathematical models |
Mecànica de fluids |
Models matemàtics |
Biomimètica |
Llibres electrònics |
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Lingua di pubblicazione |
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Formato |
Materiale a stampa |
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Livello bibliografico |
Monografia |
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Nota di contenuto |
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Intro -- Preface -- Acknowledgments -- Contents -- 1 Introduction -- 1.1 Modeling: Mimicking the Nature -- 1.2 Mathematical Approach to Swimming Modeling -- 1.3 Swimming Controllability -- 1.4 Related Selected Bibliography -- Part I Modeling of Bio-Mimetic Swimmers in 2D and 3D Incompressible Fluids -- 2 Bio-Mimetic Fish-Like Swimmers in a 2D Incompressible Fluid: Empiric Modeling -- 2.1 Swimmer's Body as a Collection of Separate Sets -- 2.2 Bio-Mimetic Fish- and Snake-Like Swimmers -- 2.3 Swimmer's Internal Forces -- 2.3.1 Rotational Internal Forces -- 2.3.2 Elastic Internal Forces -- 2.4 Swimmer's Geometric Controls -- 2.5 Internal Forces and Conservation of Momenta -- 2.5.1 About Swimmers with Body Parts Different in Mass -- 2.6 Fluid Equations: Non-stationary Stokes and Navier-Stokes Equations in 2D -- 2.7 A Model of a 2D Fish-Like Bio-Mimetic Swimmer: The Case of Stokes Equations -- 2.8 A Model of a 2D Fish-Like Bio-Mimetic Swimmer: The Case of Navier-Stokes Equations -- 3 Bio-Mimetic Aquatic Frog- and Clam-Like Swimmers in a 2D Fluid: Empiric Modeling -- 3.1 A Bio-Mimetic Aquatic Frog-Like Swimmer in a 2D Incompressible Fluid -- 3.2 A Bio-Mimetic Clam-Like Swimmer in a |
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2D Incompressible Fluid -- 4 Bio-Mimetic Swimmers in a 3D Incompressible Fluid: Empiric Modeling -- 4.1 Rotational Forces in 3D -- 4.2 A Model of a 3D Fish-Like Bio-Mimetic Swimmer: The Case of Stokes Equations -- 4.3 A Bio-Mimetic Frog-Like Swimmer in a 3D Incompressible Fluid -- 4.4 A Bio-Mimetic Clam-Like Swimmer in a 3D Incompressible Fluid -- Part II Well-Posedness of Models for Bio-Mimetic Swimmers in 2D and 3D Incompressible Fluids -- 5 Well-Posedness of 2D or 3D Bio-Mimetic Swimmers: The Case of Stokes Equations -- 5.1 Notations -- 5.2 Swimmer's Body -- 5.3 Initial- and Boundary-Value Problem Setup -- 5.3.1 Estimates for Internal Forces. |
5.4 Main Result: Existence and Uniqueness of Solutions -- 5.5 Proof of Theorem 5.1 -- 5.5.1 Preliminary Results: Decoupled Equation for zi(t)'s -- 5.5.2 Three Decoupled Solution Mappings for (5.3.1) -- 5.5.2.1 Solution Mapping A for zi(t), i = 1, …, n -- 5.5.2.2 Solution Mapping for Decoupled Non-stationary Stokes Equations -- 5.5.2.3 The Force Term -- 5.5.3 Proof of Theorem 5.1 -- 5.5.3.1 Proof of Existence: A Fixed Point Argument -- 5.5.3.2 Proof of Uniqueness -- 6 Well-Posedness of 2D or 3D Bio-Mimetic Swimmers… -- 6.1 Problem Setup and Main Results -- 6.1.1 Problem Setting -- 6.1.2 Main Results -- 6.2 Proofs of the Main Results -- 6.2.1 Solution Mapping for Decoupled Navier-Stokes Equations -- 6.2.2 Preliminary Results -- 6.2.3 Continuity of BNS -- 6.2.4 Proof of Theorems 6.1 and 6.2 -- Part III Micromotions and Local Controllability for Bio-Mimetic Swimmers in 2D and 3D Incompressible Fluids -- 7 Local Controllability of 2D and 3D Swimmers: The Case of Non-stationary Stokes Equations -- 7.1 Definitions of Controllability for Bio-Mimetic Swimmers -- 7.2 Main Results -- 7.2.1 Main Results -- 7.2.2 Main Results in Terms of Projections of Swimmers' Forces on the Fluid Velocity Space -- 7.3 Preliminary Results -- 7.3.1 Implicit Solution Formula -- 7.3.2 Differentiation with Respect to vj's and wk's -- 7.4 Volterra Equations for d zi (τ) d vj's -- 7.5 Auxiliary Estimates -- 7.6 Proof of Theorem 7.2 -- 7.7 Proof of Theorem 7.1 -- 7.7.1 Step 1 -- 7.7.2 Step 2 -- 8 Local Controllability of 2D and 3D Swimmers: The Case of Navier-Stokes Equations -- 8.1 Problem Setting -- 8.2 Main Results -- 8.2.1 Main Results: Micromotions in 2D and 3D -- 8.2.2 Main Results: Local Controllability in 2D -- 8.2.3 Main Results: Local Controllability in 3D -- 8.2.4 Methodology of Controllability Proofs -- 8.3 Derivatives ∂u∂vj |vjs=0 : 2D Case. |
8.3.1 Auxiliary Notations -- 8.3.2 Equation for wh and its Well-Posedness -- 8.3.3 Auxiliary Regularity Results for Parabolic Systems from Lad2 -- 8.3.4 Auxiliary System of Linear Equations Systems -- 8.3.5 Derivatives ∂u∂vj |vjs=0 -- 8.4 Derivatives ∂zi∂vj | vjs = 0 as Solutions to Volterra Equations: 2D Case -- 8.4.1 Expression for zi(t -- h) - zi(t -- 0)h -- 8.4.2 Evaluation of the Integrand in the 1st Term on the Right in (8.4.2) -- 8.4.3 Volterra Equations -- 8.5 Proofs of Theorems 8.1 and 8.2 -- 8.5.1 Further Modification of (8.4.12) -- 8.5.2 Proofs of Theorem 8.1 and of Theorem 8.2 in the Case of Local Controllability Near Equilibrium (i.e., When u0 = 0) -- 8.5.2.1 Step 1 -- 8.5.2.2 Step 2 -- 8.5.2.3 Step 3: Proof of Theorem 8.1 when u0 = 0 -- 8.5.2.4 Step 4 -- 8.5.3 Proof of Theorems 8.2 and 8.1 -- 8.5.3.1 Step 1 -- 8.5.3.2 Step 2 -- 8.6 Proofs of Theorems 8.1 and 8.4 -- 8.6.1 Adjustments in Sects.8.3 and 8.4 -- 8.6.2 Adjustments in Sect.8.5 -- 8.6.2.1 Section 8.5.2.4, Step 4 in the 3D Case -- 8.6.2.2 Section 8.5.3 in the 3D Case -- Part IV Transformations of Swimmers' Internal Forces Acting in 2D and 3D Incompressible Fluids -- 9 Transformation of Swimmers' Forces Acting in a 2D Incompressible Fluid -- 9.1 Main Results -- 9.1.1 Qualitative Estimates for Forces Acting Upon Small Sets in an Incompressible 2D Fluid -- 9.1.2 Transformations of Forces |
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Acting Upon Small Rectangles in an Incompressible 2D Fluid -- 9.1.3 Transformations of Forces Acting Upon Small Discs in an Incompressible 2D Fluid -- 9.1.4 Interpretation of Theorems 9.3 and 9.4: What Shape of S Is Better for Locomotion? -- 9.2 Proof of Theorem 9.1 -- 9.2.1 Step 1 -- 9.2.2 Step 2: Green's Formula -- 9.2.3 Step 3: Evaluation of the Integral of the Gradient of the 1-st Terms on the Right in (9.2.7) Over A. |
9.2.4 Step 4: Evaluation of the Integral of the Gradient of the 2-nd Term in (9.2.7) Over A -- 9.3 Proof of Theorem 9.2 -- 9.3.1 Step 1 -- 9.3.2 Step 2 -- 9.3.3 Step 3 -- 9.3.4 Step 4 -- 9.4 Proofs of Theorems 9.3 and 9.4 -- 9.4.1 Proof of Theorem 9.3 -- 9.4.2 Step 1 -- 9.4.3 Step 2 -- 9.4.4 Step 3 -- 9.4.5 Step 4 -- 9.4.6 Step 5 -- 9.4.7 Step 6 -- 9.4.8 Step 7 -- 9.4.9 Step 8 -- 9.4.10 Step 9 -- 9.4.11 Proof of Theorem 9.4: Forces Acting Upon Small Discs in a Fluid -- 10 Transformation of Swimmers' Forces Acting in a 3D Incompressible Fluid -- 10.1 Main Results -- 10.1.1 Qualitative Estimates for Forces Acting Upon Small Sets in an Incompressible 3D Fluid -- 10.1.2 A General Formula for 1meas{S}S(PH bξ)(x)dx -- 10.1.3 The Case of Parallelepipeds -- 10.1.4 Spheres in 3D -- 10.1.5 Instrumental Observations in Relation to Controlled Steering -- 10.2 Proofs of Theorems 10.1 and 10.2 -- 10.2.1 Proof of Theorem 10.1 -- 10.2.1.1 Step 1 -- 10.2.1.2 Step 2: Green's Formula -- 10.2.1.3 Step 3: Evaluation of the First Term on the Right in (10.2.7)over A -- 10.2.1.4 Step 4 -- 10.2.1.5 Step 5 -- 10.2.2 Proof of Theorem 10.2 -- 10.2.2.1 Step 1 -- 10.2.2.2 Step 2 -- 10.2.2.3 Step 3 -- 10.2.2.4 Step 4: Calculation of the Terms in the Last Line in (10.2.26) -- 10.3 Proofs of Main Results -- 10.3.1 Proofs of Theorems 10.3-10.5 -- 10.3.1.1 Auxiliary Formulas -- 10.3.1.2 Proof of Theorem 10.3 -- 10.3.1.3 Proof of Theorem 10.4 -- 10.3.1.4 Proof of Theorem 10.5 -- Part V Global Steering for Bio-Mimetic Swimmers in 2D and 3D Incompressible Fluids -- 11 Swimming Capabilities of Swimmers in 2D and 3D Incompressible Fluids: Force Controllability -- 11.1 Discussion of Concepts for Global Swimming Locomotion -- 11.2 An Instrumental Observation -- 11.3 Illustrating Examples in 2D: A Snake- or Fish-Like and Breaststroke Locomotions. |
11.3.1 Fish- or Snake-Like Locomotion to the Left -- 11.3.2 Turning Motion of One Rectangle, While the Other Two Retain Their Position -- 11.3.3 Breaststroke Locomotion for a Swimmer Consisting of 3 Rectangles: A Bio-Mimetic Clam (Scallop) -- 11.3.4 Breaststroke Locomotion for a Swimmer Consisting of 5 Rectangles: A Bio-Mimetic Aquatic Frog -- 11.4 Breaststroke Pattern for a Swimmer Consisting of 3 Discs -- 11.5 Illustrating Examples in 3D -- 11.6 Breaststroke Locomotion of a Swimmer Consisting of 3 Balls in 3D -- References. |
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