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[...]... introductory graduate course on vorticity and incompressible flow Chapters 1 and 2 contain elementary material on incompressible flow, emphasizing the role of vorticity and vortex dynamics together with a review of concepts from partial differential equations that are useful elsewhere in the book These formulations of the equations of motion for incompressible flow are utilized in Chaps 3 and 4 to study the existence... proofreading and help with the figures and typesetting: Michael Brenner, Richard Clelland, Diego Cordoba, Weinan E, Pedro Embid, Andrew Ferrari, Judy Horowitz, Benjamin Jones, Phyllis Kronhaus, Monika Nitsche, Mary Pugh, Philip Riley, Thomas Witelski, and Yuxi Zheng We thank Robert Krasny for providing us with Figures 9.4 and 9.5 in Chap 9 1 An Introduction to Vortex Dynamics for Incompressible Fluid Flows... Properties of the Vorticity in Ideal Fluid Flows For inviscid fluids (ν = 0) vorticity equation (1.33) reduces to (Dω/Dt) = Dω or, equivalently, to Dω = ω · ∇v Dt (1.50) because the matrix ∇v = D + , where D and are the symmetric and the antisymmetric parts, respectively, and ω × ω ≡ 0 First, we derive the vorticity- transport formula, an important result that proves that the inviscid vorticity equation... Now we take the superposition of a jet and a vortex, with D = diag(−γ1 , −γ2 , γ1 + γ2 ), γ j > 0, and ω0 = (0, 0, ω0 )t Then Eq (1.23) reduces to the scalar vorticity equation, and ω(t) = ω0 e(γ1 +γ2 )t Observe that in this case the vorticity ω aligns with the eigenvector e3 = (0, 0, 1)t corresponding to the positive eigenvalue λ3 = γ1 + γ2 of D and that the vorticity ω(t) increases exponentially... be recovered from the vorticity ω by a certain nonlocal operator, and Eq (1.36) provides a simple illustration of this fact Using Eqs (1.34)–(1.36), first we illustrate the effects of convection To simplify notation we suppress superscripts in the scalar velocity v 3 and the vorticity ω2 and subscripts in the space variable x1 We have Example 1.5 Example 1.5 A Basic Shear-Layer Flow Taking γ = ν = 0,... exact solutions with shear, vorticity, convection, and diffusion We show that although deformation can increase vorticity, diffusion can balance this effect Inviscid fluids have the remarkable property that vorticity is transported (and sometimes stretched) along streamlines We discuss this in detail in Section 1.6, including the fact that vortex lines move with the fluid and circulation over a closed... (1.51) leads to an interpretation of vorticity amplification (or decay) Recall that, because the fluid is incompressible, by Proposition 1.2 det(∇α X (α, t)) = 1 Thus for any fixed α and t, ∇α X (α, t) can be an arbitrary 3 × 3 (real) matrix with determinant 1 and three (complex) eigenvalues λ, λ−1 , and 1, with |λ| ≥ 1 When |λ| > 1, formula (1.51) shows that the vorticity increases when ω0 aligns roughly... order in (|x–x0 |), every incompressible velocity field v(x, t) is the sum of infinitesimal translation, rotation, and deformation velocities A large part of this book addresses the interactions among these three contributions to the velocity field To illustrate the interaction between a vorticity and a deformation, we now derive a large class of exact solutions for both the Euler and the Navier–Stokes equations... space and time Also, because the velocity is linear in x, the effects of viscosity do not alter these solutions Nevertheless, these solutions model the typical local behavior of incompressible flows Before proving this proposition, first we give some examples of the exact solutions in Eqs (1.24) that illustrate the interactions between a rotation and a deformation Example 1.1 Jet Flows Taking ω0 = 0 and. .. Contents 12 Reduced Hausdorff Dimension, Oscillations, and Measure-Valued Solutions of the Euler Equations in Two and Three Dimensions The Reduced Hausdorff Dimension Oscillations for Approximate-Solution Sequences without L 1 Vorticity Control Young Measures and Measure-Valued Solutions of the Euler Equations Measure-Valued Solutions with Oscillations and Concentrations Notes for Chapter 12 References . w0 h1" alt="" Vorticity and Incompressible Flow This book is a comprehensive introduction to the mathematical theory of vorticity and incompressible flow. CANTWELL This page intentionally left blank Vorticity and Incompressible Flow ANDREW J. MAJDA New York University ANDREA L. BERTOZZI Duke University PUBLISHED