Volume-preserving Flow Research Articles

Reduction of Volume-preserving Flows on an n-dimensional Manifold

A geometric reduction procedure for volume-preserving flows with a volume-preserving symmetry on an n-dimensional manifold is obtained. Instead of the coordinate-dependent theory and the concrete coordinate transformation, we show that a volume-preserving flow with a one-parameter volume-preserving symmetry on an n-dimensional manifold can be reduced to a volume-preserving flow on the corresponding (n-1)-dimensional quotient space. More generally, if it admits an r-parameter volume-preserving commutable symmetry, then the reduced flow preserves the corresponding (n – r)-dimensional volume form.

Zygmund strong foliations

We show that for a volume-preserving Anosov flow on a 3-manifold the strong stable and unstable foliations are Zygmund-regular. We also exhibit an obstruction to higher regularity, which admits a direct geometric interpretation. Vanishing of this obstruction implies high smoothness of the joint strong subbundle and that the flow is either a suspension or a contact flow.

General volume-preserving mechanical systems

We present the general form of equations that generate a volume-preserving flow on a symplectic manifold (M, omega) via the highest Euler-Lagrange cohomology. It is shown that for every volume-preserving flow there are some 2-forms that play a similar role to the Hamiltonian in Hamilton mechanics. The ordinary canonical equations are included as a special case with a 2-form 1/(n - 1)Homega, where H is the corresponding Hamiltonian.

Persistence of invariant tori in generalized Hamiltonian systems

We present some results of KAM type, comparable to the KAM theory for volume-preserving maps and flows, for generalized Hamiltonian systems which may admit a distinct number of action and angle variables. In particular, systems under consideration can be odd dimensional. Applications to the perturbation of three-dimensional steady Euler fluid particle path flows are considered with respect to the existence problem of barriers to fluid transport and mixing.

Break-up of invariant surfaces in action–angle–angle maps and flows

An important class of three-dimensional volume-preserving maps and flows arises as a perturbation from integrable action–action–angle maps and flows. The properties of this class of maps and flows are discussed. While action–angle–angle volume-preserving maps admit an analog of the KAM theorem, general results on nonexistence of two-dimensional invariant manifolds of action–action–angle maps are proven here. Nonexistence of such two-dimensional invariant manifolds means possibility of global transport and a mechanism for such transport — the local mechanism of resonance-induced dispersion [Phys. Rev. Lett. 75 (1995) 3669] — is studied perturbatively. Resonance-induced dispersion is shown to arise from the existence of periodic orbits that survive perturbation at places where two-dimensional invariant manifolds break down.

Volume-Preserving Great Circle Flows on the 3-Sphere

In this paper, we study volume-preserving great circle flows on the 3-sphere S3, and introduce a moduli space for these, consisting of distance-decreasing holomorphic maps from connected open subsets of S2 to S2; one consequence is that the only volume-preserving great circle flows defined on the entire 3-sphere are those tangent to Hopf fibrations.

Chaotic mixing in a bounded three-dimensional flow

Even though the first theoretical example of chaotic advection was a three-dimensional flow (Hénon 1966), the number of theoretical studies addressing chaos and mixing in three-dimensional flows is small. One problem is that an experimentally tractable three-dimensional system that allows detailed experimental and computational investigation had not been available. A prototypical, bounded, three-dimensional, moderate-Reynolds-number flow is presented; this system lends itself to detailed experimental observation and allows high-precision computational inspection of geometrical and dynamical effects. The flow structure, captured by means of cuts with a laser sheet (experimental Poincaré section), is visualized via continuously injected fluorescent dye streams, and reveals detailed chaotic structures and chains of high-period islands. Numerical experiments are performed and compared with particle image velocimetry (PIV) and flow visualization results. Predictions of existing theories for chaotic advection in three-dimensional volume-preserving flows are tested. The ratio of two frequencies of particle motion – the frequency of motion around the vertical axis and the frequency of recirculation in the plane containing the axis – is identified as the crucial parameter. Using this parameter, the number of islands in the chain can be predicted. The same parameter – using as a base-case the integrable motion – allows the identification of operating conditions where small perturbations lead to nearly complete mixing.

The evolution of material lines curvature in deterministic chaotic flows

It is by now well established that curvature plays a fundamental role in the description of the topology emerging from the partially mixed structures advected by chaotic flows. This article focuses on the dynamics of curvature in volume-preserving time-periodic flows. Previous work on the subject dealt with the evolution of curvature in the time-continuous framework. Here we derive the dynamical equations for the time-discrete dynamical system associated with the Poincaré return map of the flow. We show that this approach allows one to gain more insight into understanding the mechanisms of folding of material lines as they are passively stirred by the mixing process. By exploiting the incompressibility assumption, we analyze dependence on initial conditions (i.e. on the initial curvature and tangent vectors), and discuss under which circumstances the dependence on the initial curvature vector becomes immaterial as time increases. This analysis is closely connected with the properties of an invariant geometric structure referred to as the global unstable manifold associated with the flow system. Direct numerical simulations for physically realizable systems are used to provide concrete examples of the results that arise from theoretical considerations. The impact of this information on the prediction of the behavior of diffusing-reacting mixing processes (e.g. pattern formation and generation of lamellar structures) is also addressed.

Stable ergodicity and Anosov flows

In this note we prove that if M is a 3-manifold and ϕ t : M→M is a C 2, volume-preserving Anosov flow, then the time-1 map ϕ 1 is stably ergodic if and only if ϕ t is not a suspension of an Anosov diffeomorphism.

Reduction of three-dimensional, volume-preserving flows with symmetry

We develop a general, coordinate-free theory for the reduction of volume-preserving flows with a volume-preserving symmetry on three-manifolds. The reduced flow is generated by a one-degree-of-freedom Hamiltonian which is the generalization of the Bernoulli invariant from hydrodynamics. The reduction procedure also provides global coordinates for the study of symmetry-breaking perturbations. Our theory gives a unified geometric treatment of the integrability of three-dimensional, steady Euler flows and two-dimensional, unsteady Euler flows, as well as quasigeostrophic and magnetohydrodynamic flows.

Fine Scale Structure of the Local Lyapunov Exponent

Direct numerial calculations reveal the fine-scale structures of the local Lyapunov exponent as a spatial function in the chaotic region of a three-dimensional volume-preserving flow.

A volume-preserving counterexample to the Seifert conjecture

We prove that every 3-manifold possesses aC1, volume-preserving flow with no fixed points and no closed trajectories. The main construction is a volume-preserving version of the Schweitzer plug. We also prove that every 3-manifold possesses a volume-preserving,C∞ flow with discrete closed trajectories and no fixed points (as well as a PL flow with the same geometry), which is needed for the first result. The proof uses a Dehn-twisted Wilson-type plug which also preserves volume.

Transport in 3D volume-preserving flows

The idea of surfaces of locally minimal flux is introduced as a key concept for understanding transport in steady three-dimensional, volume-preserving flows. Particular attention is paid to the role of the skeleton formed by the equilibrium points, selected hyperbolic periodic orbits and cantori and connecting orbits, to which many surfaces of locally minimal flux can be attached. Applications are given to spheromaks (spherical vortices) and eccentric Taylor-Couette Flow.

Equivalent spatial and material descriptions of finite deformation elastoplasticity in principal axes

A spatial description of the theory of rate-independent finite deformation elastoplasticity, in which the stress tensor is defined through the strain energy function, is discussed. The main assumption of isotropic elastic response and invariance requirements under superposed rigid body motion restrict the acceptable forms of the strain energy function to those given in terms of principal values of the strain measure of elastic distortion. The formulation is developed on a manifold and the corresponding material description is obtained simply by pull-back of the derived spatial form, by appealing to the notion of covariance. The method of principal axes is systematically exploited to derive the explicit expression for the stress tensor computation for an arbitrary form of the strain energy function and the explicit form of the evolution equation for an arbitrary form of the yield function. A model problem of volume-preserving plastic flow is discussed in the closure.

Gauge invariance and current algebra in nonrelativistic many-body theory

The main purpose of this paper is to further our theoretical understanding of the fractional quantum Hall effect, in particular of spin effects, in two-dimensional incompressible electron fluids subject to a strong, transverse magnetic field. As a prerequisite for an analysis of the quantum Hall effect, the authors develop a general formulation of the many-body theory of spinning particles coupled to external electromagnetic fields and moving through a general, geometrically nontrivial background. Their formulation is based on a Lagrangian path-integral quantization and is valid in arbitrary coordinates, including coordinates moving according to a volume-preserving flow. It is found that nonrelativistic quantum theory exhibits a fundamental, local U(1)\ifmmode\times\else\texttimes\fi{}SU(2) gauge invariance, and the corresponding gauge fields are identified with physical, external fields. To illustrate the usefulness of their formalism, the authors prove a general form of the quantum-mechanical Larmor theorem and discuss some well-known effects, including the Barnett-Einstein-de Haas effect and superconductivity, emphasizing the implications of U(1)\ifmmode\times\else\texttimes\fi{}SU(2) gauge invariance. They then consider two-dimensional, incompressible quantum fluids in more detail. Exploiting U(1)\ifmmode\times\else\texttimes\fi{}SU(2) gauge invariance, they calculate the leading terms in the effective actions of such systems as functionals of the U(1) and SU(2) gauge fields, on large-distance and low-frequency scales. Among the applications of these results are a simple proof of the Goldstone theorem for spin waves and the linearresponse theory of two-dimensional, incompressible Hall fluids, including a Hall effect for spin currents and sum rules for the response coefficients. For two-dimensional, incompressible systems with broken parity and time-reversal symmetry, a particularly significant implication of U(1)\ifmmode\times\else\texttimes\fi{}SU(2) gauge invariance is a duality between the physics inside the bulk of the system and the physics of gapless, chiral modes propagating along the boundary of the system. These modes form chiral $\stackrel{^}{\mathrm{u}}(1)$ and $\mathrm{s}\stackrel{^}{\mathrm{u}}(2)$ current algebras. The representation theory of these current algebras, combined with natural physical constraints, permits one to derive the quantization of the response coefficients, such as the Hall conductivity. A classification of incompressible Hall fluids is outlined, and many examples, including one concerning a superfluid $^{3}\mathrm{He}$-$\frac{A}{B}$ interface, are discussed.

Variational and projection methods for the volume constraint in finite deformation elasto-plasticity

This paper focuses on the treatment of volume constraints which in the context of elasto-plasticity typically arise as a result of assuming volume-preserving plastic flow. Projection methods based on the modification of the discrete gradient operator B , often proposed on an ad-hoc basis, are systematically obtained in the variational context furnished by a three-field Hu-Washizu principle. The fully nonlinear formulation proposed here is based on a local multiplicative split of the deformation gradient into volume-preserving and dilatational parts, without relying on rate forms of the weak form of momentum balance. This approach fits naturally in a formulation of plasticity based on the multiplicative decomposition of the deformation gradient, and enables one to exactly enforce the condition of volume-preserving plastic flow. Within the framework proposed in this paper, rate forms and incrementally objective algorithms are entirely bypassed.

Anosov flows on new three manifolds

Throughout this paper, Anosov flows ~t:M ~M are volume preserving (a nontrivial assumption in light of Franks and Williams' recent example), and their ambient manifolds M are compact, orientable and three dimensional. Our purpose is to present a simple construction that yields a family of C ~ smooth, nonalgebraic (definitions below) Anosov flows. The existence of a volume-preserving Anosov flow ~ on a given manifold M 3 has immediate corollaries. The stable and unstable foliations are codimension one and minimal. By reparameterizing �9 if necessary, the flows (called horocycle flows) generated by the strong stable and strong unstable foliations are also minimal. We therefore have the relations

Volume-preserving flow by powers of the $m\textrm{-th}$ mean curvature in the hyperbolic space

This paper concerns closed hypersurfaces of dimension $n(\geq 2)$ in the hyperbolic space ${\mathbb{H}}_{\kappa}^{n+1}$ of constant sectional curvature $\kappa$ evolving in direction of its normal vector, where the speed is given by a power $\beta (\geq 1/m)$ of the $m$th mean curvature plus a volume preserving term, including the case of powers of the mean curvature and of the $\mbox{Gaus}$ curvature. The main result is that if the initial hypersurface satisfies that the ratio of the biggest and smallest principal curvature is close enough to 1 everywhere, depending only on $n$, $m$, $\beta$ and $\kappa$, then under the flow this is maintained, there exists a unique, smooth solution of the flow for all times, and the evolving hypersurfaces exponentially converge to a geodesic sphere of ${\mathbb{H}}_{\kappa}^{n+1}$, enclosing the same volume as the initial hypersurface.

An Optimal Loewner-type Systolic Inequality and Harmonic One-forms of Constant Norm

We present a new optimal systolic inequality for a closed Riemannian manifold X, which generalizes a number of earlier inequalities, including that of C. Loewner. We characterize the boundary case of equality in terms of the geometry of the Abel-Jacobi map, A_X, of X. For an extremal metric, the map A_X turns out to be a Riemannian submersion with minimal fibers, onto a flat torus. We characterize the base of J_X in terms of an extremal problem for Euclidean lattices, studied by A.-M. Berg\'e and J. Martinet. Given a closed manifold X that admits a submersion F to its Jacobi torus T^{b_1(X)}, we construct all metrics on X that realize equality in our inequality. While one can choose arbitrary metrics of fixed volume on the fibers of F, the horizontal space is chosen using a multi-parameter version of J. Moser's method of constructing volume-preserving flows.