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Stretching and Folding Versus Cutting and Shuffling: an Illustrated

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Stretching and Folding Versus Cutting and Shuffling: an Illustrated Stretching and folding versus cutting and shuffling: An illustrated perspective on mixing and deformations of continua Ivan C. Christov,1, a) Richard M. Lueptow,2, b) and Julio M. Ottino3, c) 1)Department of Engineering Sciences and Applied Mathematics, Northwestern University, Evanston, Illinois 60208, USA 2)Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208, USA 3)Department of Chemical and Biological Engineering, Northwestern Institute on Complex Systems, and Department of Mechanical Engineering, Northwestern University, Evanston, Illinois 60208, USA We compare and contrast two types of deformations inspired by mixing applications – one from the mixing of fluids (stretching and folding), the other from the mixing of granular matter (cutting and shuffling). The connection between mechanics and dynamical systems is discussed in the context of the kinematics of deformation, emphasizing the equiv- alence between stretches and Lyapunov exponents. The stretching and folding motion exemplified by the baker’s map is shown to give rise to a dynamical system with a positive Lyapunov exponent, the hallmark of chaotic mixing. On the other hand, cutting and shuffling does not stretch. When an interval exchange transformation is used as the basis for cutting and shuffling, we establish that all of the map’s Lyapunov exponents are zero. Mixing, as quantified by the interfacial area per unit volume, is shown to be exponential when there is stretching and folding, but linear when there is only cutting and shuffling. We also discuss how a simple computational approach can discern stretching in discrete data. I. INTRODUCTION layer and the underlying static bed of granular material in an avalanche.5 This newaspect ofthe flow leads todifferent mod- The essence of mixing of a fluid with itself can be under- els for the kinematics. In particular, mixing in granular flows 6 stood in terms of an array of striations of, say, two different in rotating containers (“tumblers”) can be thought of as “cut- 7 colors of the same fluid (or two different fluids such as cof- ting and shuffling,” a process different from stretching and 8 fee and cream) undergoing stretching and folding. On top of folding. More on the fascinating behavior of granular matter stretching and folding we may superimpose diffusion, reac- can be found in Ref. 9. tion, and, in special circumstances, breakup processes lead- Stretching is a fundamental concept in mechanics and is ing to droplet formation.1,2 This approach is the backbone of covered in every continuum mechanics textbook in the con- lamellar models of mixing. A fundamental measure of the text of kinematics (see, for example, the classic volumes of 10 11 quality of mixing is aV, the interfacial area per unit volume of Truesdell and Gurtin ), where it is identified with shear or the striations (lamella or layers). Let S be the interfacial area extensional strain. Cutting and shuffling, in contrast, has been between fluid layers within a volume V enclosing the point x explored only recently. To illustrate the fundamental differ- at time t, then the interfacial area per unit volume is given by2 ence between the mixing mechanisms of stretching and fold- ing versus cutting and shuffling, we comparetwo types of sim- S ple idealized mixing protocols: the well-known baker’s map aV (x, t) = lim . (1) V 0 V ffl → and cutting and shu ing maps based on interval exchange transformations. Along the way we discuss the elegant con- A larger aV corresponds to better mixing. We can imagine many iterative mixing protocols that gen- nection between continuum mechanics and dynamical sys- tems and show how some of the calculations of the relevant erate large values of aV and create striations of the material of continually decreasing thickness in time. For fluids, a mul- kinematic quantities, such as the deformation gradient and the titude of clever mixing designs can lead to the thinning of principal stretches, are performed. Pertinent concepts from lamella, many inspired by a direct correspondence between the theory of mixing are also reviewed within this context. arXiv:1010.2256v2 [physics.flu-dyn] 21 Mar 2011 the kinematics of mixing and chaotic dynamical systems.3 We then discuss how to discern and measure stretching (or The simplest representation of mixing in terms of stretching lack thereof) in practice. and folding is the Smale horseshoe map, which stretches out a piece of material and folds it onto itself to form the shape of a horseshoe. A limiting case of this procedure is a map II. KINEMATICS OF DEFORMATION that stretches, cuts and re-stacks to generate interfacial area, the baker’s transformation, named after the process by which We restrict our discussiuon to two spatial dimensions for a baker kneads dough. simplicity.12 Consider a motion Φ from the undeformed (ref- Granular mixing has been studied as well, but less exten- erence or initial) configuration of the continuum (body) to B0 sively than fluid mixing. In many respects, the ideas applied the deformed (current or final) configuration t at time t, both to fluids carry over to granular matter.4 A key difference be- of which are regions in the Euclidean plane. KeepingB with the tween the two is that granular flows may present surfaces of mathematics notation, in this paper, we denote sets by capi- discontinuity, such as the interface between a flowing surface tal calligraphic letters. Typically, we concern ourselves with 2 motions that map the body back into itself, that is, t = 0, stretchσ ˆ = max σ1, σ2 , which is a scalar fieldσ ˆ = σˆ (X; n) but this restriction is not important for what follows.B WeB are that describes the{ stretching} experiencedby the body due to its interested in discrete-time motions such as a repetitive mixing motion. In practice, it is convenient to calculate the eigenval- protocol. That is, we allow only t = nT, where T is the dura- ues κ1, κ2 of the (right) Cauchy–Green strain tensor C F⊤F tion (period) of the motion and n is a positive integer. Then, instead.{ These} are just the squares of the principal stretch≡ es. we may write the motion as a map In this wayσ ˆ can be computed without explicitly finding the polar decomposition of F. Φ: 0 t (2) Problem 2. Show that κ = σ2 using the definition of C, the B →B i i polar decomposition theorem, and the properties of eigenval- such that after one iteration every X in is mapped to an B0 ues. x = Φ(X) in t, where by x (x, y)⊤ we shall denote the po- sition vector inB the deformed≡ configuration and X (X, Y) ≡ ⊤ represents the coordinates in the undeformed configuration. III. DYNAMICAL SYSTEMS FRAMEWORK OF KINEMATICS The superscript denotes the transpose, meaning that (x, y) ⊤ is a row vector and (x, y)⊤ is a column vector. Unless other- Equation (2) also defines a dynamical system, which in wise noted, all vectors not written out in component form are the most general sense is defined as a rule of evolution on considered to be column vectors. a state space (the body).16 This connection between the kine- We can now define the deformation gradient (or Jacobian matics of continua and dynamical systems has been success- matrix of the map) as fully exploited in the study of both fluid mixing2 and granular 4 ∂x ∂x mixing. There is a direct correspondence between the languages of x ∂(x, y) ∂X ∂Y F = ( )⊤ ∂y ∂y . (3) dynamical systems and continuum mechanics, the most im- ∇ ≡ ∂(X, Y) ≡ ∂X ∂Y portant of which, for the present purposes, is the correspon- dence between stretches and Lyapunov (characteristic) expo- It is typically assumed that the map is invertible and differ- nents. For a discrete-time map, these exponents are defined entiable a sufficient number of times so that F exists and (see, for example, Ref. 17, Sec. 5.3.1) as 0 < det F < .10,11 In this paper we relax the differentiability ∞ assumption to consider a wider (and, arguably, more interest- 1 n λ(X, v) = lim ln ( Φ )⊤v , (6) n n ing) class of motion. →∞ k ∇ k 2,10,11,13 The polar decomposition theorem allows us to write where Φn Φ Φ (n compositions of the map). The Lyapunov exponents≡ ◦···◦ depend on the position X and on the F = RU, (4) direction v ( v = 1). The quantity Φn can be calculated k k ∇ X where U is a symmetric positive definite matrix14 (due to by the chain rule along a trajectory starting at a given in 0. If F ( Φ)⊤ happens to be independent of X, we have the assumption that det F > 0) and R is a proper-orthogonal B n ≡ ∇n 15 ( Φ ) = F . Lyapunov exponents are important in the con- matrix. That is, we can locally decompose the deformation ∇ ⊤ into a rotation R and a stretch U. Because the matrix U is sym- text of the asymptotic stability of infinitesimal perturbations metric positive definite, it has an orthonormal basis of eigen- and, provided certain conditions are satisfied, can be inter- preted as the growth (or decay) rates of these perturbations vectors e1, e2 and strictly positive real eigenvalues σ1, σ2 satisfying{ } { } along a trajectory. A related concept is the finite-time Lyapunov exponents de- Uei = σiei. (5) fined as 1 n The eigenvalues σi and eigenvectors ei are called, respec- γ(X, v; n) = ln ( Φ )⊤v . (7) tively, the principal stretches and the principal directions be- n k ∇ k cause an infinitesimal line segment of length dℓ oriented in the Note that if ( Φn) = Fn, we have ∇ ⊤ direction ei has length σidℓ after undergoing the motion (that 1 is, after the map Φ is applied).
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