Fluid-Structure Interactions in Low-Reynolds-Number Flows (Soft Matter Series, Volume 4) - Hardcover

Hardcover

ISBN 10: 1849738130 ISBN 13: 9781849738132

Publisher: Royal Society of Chemistry, 2015

Synopsis

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.

About this title

From the Back Cover

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.

Excerpt. © Reprinted by permission. All rights reserved.

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 = δuy + δvx. As expected, γ = 0 for this infinitesimal rotation — note that γ = 0 holds as well for rigid-body translations, which is the other type of rigid-body motion.

1.1.3 Scaling and Symmetry Analysis of the Energy of the Cells

The elastic energy of a cell is the sum of the elastic energy stored in each of the six springs. It is invariant with respect to rigid-body motions, and is therefore a function of (δux, δvy, γ, δwx, δwy). Since we consider linearly elastic springs and work in the limit of small displacements, the energy functional is, moreover, a quadratic form.

We use the fact that the undeformed cell has the symmetry of a rectangle: it is invariant both with respect to reflection in a vertical axis and with respect to reflection in a horizontal axis. With respect to these symmetries, the quantities δux, δvy, γ, δwx and δwy have parities (+, +), (+ , +), (-,-), (+ , -) and (- , +), respectively. This means, for instance, that δwx is unchanged by a mirror symmetry operation with a vertical axis but is changed to - δwx by a mirror symmetry operation with a horizontal axis. In order to be invariant with respect to both these symmetries, the elastic energy cannot couple quantities having different parities. We conclude that it is of the form

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII], (1.6)

where the factor a/2 in front has been pulled out by convention. Here, Q is a quadratic form depending on the stretch variables δux and δvy, and g, b and c are elastic moduli. Note that g is multiplied by the shear strain squared, (γ/2)2: g is called the shear modulus.

Let k denote the typical stiffness of the springs. Observe that δux, δvy, γ, δwx and δwy are all dimensionless: as a result, all moduli, including the quadratic form Q itself, scale like (ka):

Q ~ g ~ b ~ c ~ ka. (1.7)

Detailed expressions for these quantities are derived in the next two sections — readers not interested in these details can skip directly to Section 1.1.6.

1.1.4 Elongation of Springs

Within the small-displacement approximation, the small elongations of the horizontal springs are ε12 = |x2 - x1| a ≈ (δx2 - δx1 * ex = a(δux - δwx) and ε1'2' = |x'2 - x'1| - a [approximately equal to] a(δux + δwx). By a similar calculation, the elongations of the vertical springs are ε11' ≈ a(δvy - δwy) and ε22' ≈ a(δvy + δwy), and those of the diagonal springs are ε12' = |x'2 - x1| - a[square root of 2] ≈ (a[square root of 2]/2) ex + ey * (δu + δv) and ε1'2 = (a[square root of 2]/2 (ex - ey) · (δu - δv).

1.1.5Energy of a Single Cell

Let us assume that all horizontal springs have the same spring constant, and likewise for vertical springs, and for diagonal springs. Let kx, ky and kz denote the respective spring constants. We consider the case where these spring constants are given by

kx = kβ2, ky = k, kz = 2kβ, (1.8)

where β is a parameter and k is a typical stiffness. As we shall see later, β is related to the Poisson's ratio of the structure. In view of the equation above, the spring constants of the three types of springs are not independent. Little would be changed if they were independent, however, as the continuous model of a planar elastica involves only two independent elastic moduli.

We can then write the elastic energy of the cell as

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.9)

Inserting the expressions for the elongations derived in Section 1.1.4, we obtain an energy of the type anticipated earlier in eqn (1.6). In addition, we obtain the following explicit expressions for the moduli:

g = 8akβ, (1.10a)

b = 1/2akβ2, (1.10b)

c = !/2ak. (1.10c)

The quadratic form Q reads

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.11a)

where

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII]. (1.11b)

Note that the scaling law derived in eqn (1.7) is confirmed, as all moduli are expressed as (ak) times a function of β.

1.1.6 Assembling the Periodic Truss

The periodic truss shown in Figure 1.1 is obtained by chaining cells, as sketched in Figure 1.3. Note that the stiffness of the vertical springs is now 2ky: as shown in the figure, each vertical spring in the periodic truss is obtained by merging two springs from adjacent cells.

The energy of the truss is obtained by summing the energy of the cells:

[MATHEMATICAL EXPRESSION NOT REPRODUCIBLE IN ASCII] (1.12)

where the index i runs over all the cells.

Each cell has its own set of vectors δui, δvi, δwi. The vectors pertaining to neighboring cells are not independent, however: the vertex positions xi and x'i can be reconstructed by applying eqn (1.5) to either one of the adjacent cells. For the results to agree, the following conditions must hold:

ri+1 - ri/a = ui + ui+1/2, (1.13a)

vi+1 - vi = wi + wi+1. (1.13b)

When the applied loading is conservative, the equilibrium shape of the truss can be found by minimizing the sum of the elastic energy in eqn (1.12) and the potential energy of the external loads, subject to the constraints expressed by eqn (1.13).

1.2 Continuous Limit: String, 2D Elastica etc.

In this section, we present a continuous version of the periodic truss network. We analyze the limit of the energy of the truss Etruss in the limit of a large aspect ratio (a/L [right arrow] 0). This defines the planar elastica model.

1.2.1 A Generic Continuous Model

In the continuous limit, we assume that the periodic truss can be represented by an elastic curve, specified through a parametric equation r(S). Here, S is a Lagrangian coordinate, playing exactly the same role as the discrete vertex index i in the periodic truss. The curve r(S) is called the centerline, and is the continuous limit of the polygonal chain passing through the midpoints of the vertical bars of the discrete truss, r(S = ai) = ri. In the continuous setting, r(S) is the centroid of the cross section labeled by the Lagrangian coordinate S.

We partially relax the assumption of small displacements introduced in the analysis of the discrete truss, now allowing large rotations and large displacements (but small strains; see below). As a result, r(S) can be an arbitrary curve, even though the undeformed shape of the truss is straight.

We also consider a smooth vector field v(S) representing the transverse material vector vi of the truss network; see Figure 1.4a. In the smooth setting, like in the discrete case, the norm of v(S) captures the dilation or contraction of the material cross sections, and its direction follows their rotations.

In view of the constraints (1.13), we view the longitudinal material vector u(S) and the bending vector w(S) as secondary quantities, which we reconstruct from

u(S) = r'(S), (1.14a)

w(S) = a/2 v'(S); (1.14b)

u is called the material tangent to the centerline. In the continuous setting, we use primes to denote differentiation with respect to the arclength parameter S. Note that the small parameter a represents the microscopic size of the cross sections, and will go to zero later on.

Next, we define the strain variables by

ε = 1/2 (r'2 - 1), (1.15a)

η = 1/2(v2 - 1), (1.15b)

γ = r' · v, (1.15c)

κ = r" · v. (1.15d)

Here, ε captures the stretching of the centerline (ε = 0 when |r'| = 1), η measures the change of the norm of the vector v (which represents the stretching of transverse springs in the discrete model), γ is a measure of shear (it is nonzero only when the transverse material vector does not stay perpendicular to the tangent u to the centerline) and κ is a measure of curvature (since v is close to a unit vector perpendicular to the centerline). These definitions of the strain are not unique: other definitions are possible, which are equivalent in the limit of small strain. For instance, we could have defined the axial strain by ε = |r'| - 1 instead.

It is important to note that all these strain variables are invariant with respect to a rigid-body rotation of both the centerline r(S) and the transverse material vector v(S), as they depend only on the dot products (and norms) of r', v and their derivatives. In this sense, (ε, η, γ, κ) extend the linear measures of strain (δux, δvy, γ, δwx, δwy) which we introduced earlier in the discrete setting. This allows us now to work in the more general framework of finite displacements and finite rotations, even though we continue to assume small strains, i.e.

(Continues...)

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Edition details

• Publisher: Royal Society of Chemistry
• Publication date: 2015
• Language: English
• ISBN 10: 1849738130
• ISBN 13: 9781849738132
• Binding: Hardcover
• Edition number: 1
• Number of pages: 483
• Editor: Duprat Camille, Stone Howard A