Fluid-Structure Interactions in Low-Reynolds-Number Flows (Rsc Soft Matter) - Hardcover

Hardcover

ISBN 10: 1849738130 ISBN 13: 9781849738132

Verlag: Royal Society of Chemistry, 2015

Inhaltsangabe

Fluid-structure interactions have been well studied over the years but most of the focus has been on high Reynolds number flows, inertially dominated flows where the drag force from the fluid typically varies as the square of the local fluid speed. There are though a large number of fluid-structure interaction problems at low values of the Reynolds number, where the fluid effects are dominated by viscosity and the drag force from the fluid typically varies linearly with the local fluid speed, which are applicable to many current research areas including hydrodynamics, microfluidics and hemodynamics. Edited by experts in complex fluids, Fluid-Structure Interactions in Low-Reynolds-Number Flows is the first book to bring together topics on this subject including elasticity of beams, flow in tubes, mechanical instabilities induced by complex liquids drying, blood flow, theoretical models for low-Reynolds number locomotion and capsules in flow. The book includes introductory chapters highlighting important background ideas about low Reynolds number flows and elasticity to make the subject matter more approachable to those new to the area across engineering, physics, chemistry and biology.

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Fluid–Structure Interactions in Low-Reynolds-Number Flows

By Camille Duprat, Howard A. Stone

The Royal Society of Chemistry

Copyright © 2016 The Royal Society of Chemistry
All rights reserved.

ISBN: 978-1-84973-813-2

Contents

Chapter 1 Introduction to the Elasticity of Rods Basile Audoly, 1, 
Chapter 2 Low-Reynolds-Number Flows Howard A. Stone and Camille Duprat, 25, 
Chapter 3 Model Problems Coupling Elastic Boundaries and Viscous Flows Howard A. Stone and Camille Duprat, 78, 
Chapter 4 Theoretical Models of Low-Reynolds-Number Locomotion On Shun Pak and Eric Lauga, 100, 
Chapter 5 Elastic Fibers in Flows Anke Lindner and Michael Shelley, 168, 
Chapter 6 Elastocapillarity Camille Duprat and Howard A. Stone, 193, 
Chapter 7 Mechanical Instabilities Induced by the Drying of Complex Liquids Ludovic Pauchard and Frédérique Giorgiutti-Dauphiné, 247, 
Chapter 8 Flow in Flexible/Collapsible Tubes Matthias Heil and Andrew L. Hazel, 280, 
Chapter 9 Dynamics of Membrane-Bound Particles: Capsules and Vesicles Petia M. Vlahovska, 313, 
Chapter 10 On the Importance of the Deformability of Red Blood Cells in Blood Flow Manouk Abkarian and Annie Viallat, 347, 
Subject Index, 463, 

CHAPTER 1

Introduction to the Elasticity of Rods

BASILE AUDOLY

In this chapter, we introduce the equations governing the equilibrium of slender elastic structures and provide some examples of applications, including buckling. The only prerequisites are the calculus of variations, the elementary geometry of curves and the elasticity of linear springs. By a slender structure, we mean a quasi-one-dimensional elastic body whose extent in one direction is much larger than in the two perpendicular (cross-sectional) directions. The equilibrium of slender structures is governed by ordinary differential equations, and these are considerably easier to derive and analyze than the partial differential equations governing the equilibrium of three-dimensional (3D) elastic bodies: this chapter could serve as an introduction to the elasticity of deformable bodies in general. Throughout this chapter, the forces acting on the elastic structure are prescribed, and no coupling with a fluid is considered.

The chapter is organized as follows. We start by deriving one-dimensional (1D) models governing slender structures in two steps: first we analyze a discrete truss network in two dimensions, i.e. a collection of linear springs connected by perfect hinges (Section 1.1), and next we take the continuous limit to derive the energy of an elastic rod (Section 1.2). Readers not interested in the details of this dimensional reduction can start reading directly from Section 1.2.3, where we introduce the continuous, planar elastica model, which is a 1D elastic object immersed in a two-dimensional (2D) space. In Section 1.3, we use the calculus of variations to derive the equilibrium equations for an elastica in two dimensions, and derive two famous analogies, with the oscillations of a nonlinear pendulum and with the shape of a meniscus at a fluid interface. In Section 1.4, we focus on the linear response of the elastica relevant to small applied forces, which is the classical linear beam model. In Section 1.5, we extend the elastica model to three dimensions, and illustrate the interplay of the bending and twisting modes with an analysis of helical buckling.

1.1 Discrete Setting: A Periodic Truss Network

We start by analyzing the discrete structure shown in Figure 1.1, made up of linear elastic springs connected by perfect (frictionless) hinges. This discrete structure is called a truss network. The continuous model that we shall ultimately derive is largely independent of the specific truss geometry, but for the sake of definiteness we consider a truss made up of square cells in its undeformed configuration. Let a denote the length of the sides of the squares. We pick an orthonormal Cartesian frame (ex, ey), with the corresponding axes x and y oriented parallel and perpendicular to the long dimension of the network.

In the present section, the modes of deformation and the elastic energy of the truss network are derived. This sets the stage for the dimensional reduction carried out in the following section by taking the limit of a large number of cells, a [much less than] L, where L is the length of the truss as a whole.

1.1.1 Geometric Description of a Single Cell

We start by focusing on a single cell of the truss, as shown in Figure 1.2. Let (x1, x2, x'1, x'2) denote the positions of the vertices. In terms of these, we define the centroid of the cell,

r = 1/4 (x1 + x2 + x'1 + x'2), (1.1)

and two materials vectors,

u = 1/a (x2 + x'2/2 - x1 + x'1/2), (1.2)

v = 1/a (x'1 + x'2/2 - x1 + x2/2). (1.3)

The endpoints of the vectors (au) and (av) are tied to the midpoints of the vertical and horizontal truss elements, respectively (see Figure 1.2b). As a result, they follow the truss as it deforms, hence the name material vectors. In view of the orientation of the complete truss, we call u the tangent material vector, and v the transverse material vector.

We also define the vector w, as

w = + x1 - x2 - x'1 + x'2/2a. (1.4)

If the deformed vertices in Figure 1.2b are obtained from the undeformed vertices in Figure 1.2a by an affine transformation, the vector w is zero, as can be checked. Therefore, w is a measure of the nonaffine character of the transformation: w is nonzero when the centroid r does not match the point of intersection of the diagonal truss elements.

We observe that the vertices can be reconstructed in terms of the set of four vectors defined above, namely the centroid r, the material vectors u and v, and the vector w:

x1 = r + a/2 ( - u - v + w), (1.5a)

x2 = r + a/2 ( + u - v - w), (1.5b)

x'1 = r + a/2 ( - u + v - w), (1.5c)

x'2 = r + a/2 ( + u + v + w). (1.5d)

We shall use the variables (r, u, v, w) as the main unknowns, and use the above formulas to reconstruct the vertex positions.

1.1.2 Small-Displacement Approximation, Modes of Deformation

We consider the situation where each cell of the structure deforms by a small amount. Then, the displacements of the vertices are small and one can expand r = δr, u = ex + δu, v = ey + δv and w = δw. All quantities relevant to the elastic analysis of the truss will be systematically expanded to first order with respect to δr, δu, δv and δw.

The corresponding modes of deformation of a cell are shown in Figure 1.2c: δrx and δry are rigid-body translations, δux and δvy correspond to longitudinal and transverse dilations, δuy and δvx correspond to shear, and δwx and δwy correspond to bending modes. Here, we denote the components of vectors using a subscript notation, as in δrx = δr • ex.

Note that a rigid-body rotation through an infinitesimal angle δθ is obtained by combining the two shear modes to obtain (δr, δu, δv, δw) = (0, δ θey, - δθex, 0). The shear strain is defined as γ/2, where γ =...

Details zur Ausgabe

• Verlag: Royal Society of Chemistry
• Erscheinungsdatum: 2015
• Sprache: Englisch
• ISBN 10: 1849738130
• ISBN 13: 9781849738132
• Einband: Gebundene Ausgabe
• Anzahl der Seiten: 483
• Herausgeber: Duprat Camille, Stone Howard A
• Kontakt zum Hersteller: Nicht verfügbar
• Verantwortliche Person: gpsr@libri.de; gpsr@libri.de; ; Europaallee 1; Bad Hersfeld; 36244; Deutschland