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Thomas Machon - One of the best experts on this subject based on the ideXlab platform.
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the godbillon vey invariant as topological vorticity compression and obstruction to steady flow in Ideal Fluids
Proceedings of The Royal Society A: Mathematical Physical and Engineering Sciences, 2020Co-Authors: Thomas MachonAbstract:If the vorticity field of an Ideal fluid is tangent to a foliation, additional conservation laws arise. For a class of zero-helicity vorticity fields, the Godbillon-Vey (GV) invariant of foliations...
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The Godbillon–Vey invariant as a restricted Casimir of three-dimensional Ideal Fluids
Journal of Physics A: Mathematical and Theoretical, 2020Co-Authors: Thomas MachonAbstract:We show the Godbillon-Vey invariant arises as a `restricted Casimir' invariant for three-dimensional Ideal Fluids associated to a foliation. We compare to a finite-dimensional system, the rattleback, where analogous phenomena occur.
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the godbillon vey invariant as topological vorticity compression and obstruction to steady flow in Ideal Fluids
arXiv: Mathematical Physics, 2020Co-Authors: Thomas MachonAbstract:If the vorticity field of an Ideal fluid is tangent to a foliation, additional conservation laws arise. For a class of zero-helicity vorticity fields the Godbillon-Vey (GV) invariant of foliations is defined and is shown to be an invariant purely of the vorticity, becoming a higher-order helicity-type invariant of the flow. GV non-zero gives both a global topological obstruction to steady flow and, in a particular form, a local obstruction. GV is interpreted as helical compression and stretching of vortex lines. Examples are given where the value of GV is determined by a set of distinguished closed vortex lines.
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the godbillon vey invariant as a restricted casimir of three dimensional Ideal Fluids
arXiv: Mathematical Physics, 2020Co-Authors: Thomas MachonAbstract:We show the Godbillon-Vey invariant arises as a `restricted Casimir' invariant for three-dimensional Ideal Fluids associated to a foliation. We compare to a finite-dimensional system, the rattleback, where analogous phenomena occur.
Thomas C Sideris - One of the best experts on this subject based on the ideXlab platform.
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Global Existence and Asymptotic Behavior of Affine Motion of 3D Ideal Fluids Surrounded by Vacuum
Archive for Rational Mechanics and Analysis, 2017Co-Authors: Thomas C SiderisAbstract:The 3D compressible and incompressible Euler equations with a physical vacuum free boundary condition and affine initial conditions reduce to a globally solvable Hamiltonian system of ordinary differential equations for the deformation gradient in $${{\rm GL}^+(3, \mathbb{R})}$$ GL + ( 3 , R ) . The evolution of the fluid domain is described by a family of ellipsoids whose diameter grows at a rate proportional to time. Upon rescaling to a fixed diameter, the asymptotic limit of the fluid ellipsoid is determined by a positive semi-definite quadratic form of rank r = 1, 2, or 3, corresponding to the asymptotic degeneration of the ellipsoid along 3− r of its principal axes. In the compressible case, the asymptotic limit has rank r = 3, and asymptotic completeness holds, when the adiabatic index $${\gamma}$$ γ satisfies $${4/3 < \gamma < 2}$$ 4 / 3 < γ < 2 . The number of possible degeneracies, 3− r , increases with the value of the adiabatic index $${\gamma}$$ γ . In the incompressible case, affine motion reduces to geodesic flow in $${{\rm SL}(3, \mathbb{R})}$$ SL ( 3 , R ) with the Euclidean metric. For incompressible affine swirling flow, there is a structural instability. Generically, when the vorticity is nonzero, the domains degenerate along only one axis, but the physical vacuum boundary condition fails over a finite time interval. The rescaled fluid domains of irrotational motion can collapse along two axes.
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finite time singularities in Ideal Fluids with swirl
Physica D: Nonlinear Phenomena, 1995Co-Authors: Rainer Grauer, Thomas C SiderisAbstract:Three-dimensional Ideal, incompressible Fluids with swirl are studied numerically using two different methods: standard finite differences and a projection method based on upwind differencing. Both methods give quantitatively similar results, leading to the conclusion that singularities form in finite time in a manner consistent with known theoretical criteria. The effect of singularities in incompressible flows on nearby compressible flows is discussed.
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numerical computation of 3d incompressible Ideal Fluids with swirl
Physical Review Letters, 1991Co-Authors: Rainer Grauer, Thomas C SiderisAbstract:We investigate numerically the question of blowup in finite time for the «swirling flow» of the three-dimensional incompressible Euler equations. Using rotational symmetry, the Euler equations reduce to a two-dimensional problem which is numerically solved by finite differences. The elliptic equation relating vorticity to velocity is solved with the multigrid method. Calculations were performed with 896×640 mesh points
Sauro Succi - One of the best experts on this subject based on the ideXlab platform.
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Oxford Scholarship Online - Kinetic Theory of Dense Fluids
Oxford Scholarship Online, 2018Co-Authors: Sauro SucciAbstract:This chapter presents the basic elements of the kinetic theory of non-Ideal Fluids, to which both kinetic and potential energy contribute on comparable footing. Non-Ideal Fluids lie at the heart of many complex fluid-dynamic applications, such as those involving multiphase and multicomponent flows. This chapter features a degree of abstraction which may not come by handy to the reader with limited interest to the formal theory of classical many-body systems. The interested readers can safely skip the math and retain the basic bottomline. They may just skip this chapter altogether, but in this author’s opinion, this is likely to come with a toll on the full appreciation of Lattice Boltzmann theory for non-Ideal Fluids, in fact one of the most successful offsprings of Lattice Boltzmann theory.
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the emergence of supramolecular forces from lattice kinetic models of non Ideal Fluids applications to the rheology of soft glassy materials
Soft Matter, 2012Co-Authors: M Sbragaglia, R Benzi, Massimo Bernaschi, Sauro SucciAbstract:A systematic study for a single-species lattice Boltzmann model with frustrated-short range attractive and mid/long-range repulsive-interactions is presented. The equilibrium analysis, carried out along the guidelines proposed by [X. Shan, Phys. Rev. E: Stat., Nonlinear, Soft Matter Phys., 2008, 77, 066702], allows us to determine the surface tension density and the resulting disjoining pressure developing in a thin film when two interfaces overlap. Numerical simulations of confined flows are then performed with a multicomponent model and successfully tested against a recent suggestion, motivated by experimental facts, on the existence of a cooperative length underlying the non-local rheology of highly confined soft-glassy materials [Goyon et al., Nature, 2008, 454, 8487; Goyon et al., Soft Matter, 2010, 6, 2668–2678].
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mesoscopic simulation of non Ideal Fluids with self tuning of the equation of state
Soft Matter, 2012Co-Authors: Carlos E Colosqui, Giacomo Falcucci, Stefano Ubertini, Sauro SucciAbstract:A dynamic optimization strategy is presented to generate customized equations of state (EOS) for the numerical simulation of non-Ideal Fluids at high density ratio. While stable branches of the analytical EOS are preserved, the spinodal region is self-tuned during the simulation, in order to compensate for numerical errors caused by discretization in phase space. The employed EOS permits the readily setting of the sound speeds for the gas and liquid phases, thus allowing stable simulation with high density (1 : 10 to 1 : 1000) and compressibility ratios (250 : 1–25000 : 1). The present technique is demonstrated for lattice Boltzmann simulation of (free-space) multiphase systems with flat and circular interfaces.
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modern lattice boltzmann methods for multiphase microflows
Ima Journal of Applied Mathematics, 2011Co-Authors: Giacomo Falcucci, Stefano Ubertini, Daniele Chiappini, Sauro SucciAbstract:During the last decade, the Lattice Boltzmann (LB) method has captured an increasing attention as an efficient tool for the numerical simulation of complex Fluids, particularly multi-phase and multi-component flows. In this paper, we revisit the basic features of two modern variants of lattice Boltzmann models for non-Ideal Fluids, which offer promising perspectives for the numerical simulation of complex micro-flows.
Konstantin Ilin - One of the best experts on this subject based on the ideXlab platform.
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on general transformations and variational principles for the magnetohydrodynamics of Ideal Fluids part 4 generalized isovorticity principle for three dimensional flows
Journal of Fluid Mechanics, 1999Co-Authors: V A Vladimirov, H K Moffatt, Konstantin IlinAbstract:The equations of magnetohydrodynamics (MHD) of an Ideal fluid have two families of topological invariants: the magnetic helicity invariants and the cross-helicity invariants. It is first shown that these invariants define a natural foliation (described as isomagnetovortical, or imv for short) in the function space in which solutions {u(x,t), h(x,t)} of the MHD equations reside. A relaxation process is constructed whereby total energy (magnetic plus kinetic) decreases on an imv folium (all magnetic and cross-helicity invariants being thus conserved). The energy has a positive lower bound determined by the global cross-helicity, and it is thus shown that a steady state exists having the (arbitrarily) prescribed families of magnetic and cross-helicity invariants. The stability of such steady states is considered by an appropriate generalization of (Arnold) energy techniques. The first variation of energy on the imv folium is shown to vanish, and the second variation δ 2 E is constructed
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on general transformations and variational principles for the magnetohydrodynamics of Ideal Fluids part iii stability criteria for axisymmetric flows
Journal of Plasma Physics, 1997Co-Authors: V A Vladimirov, H K Moffatt, Konstantin IlinAbstract:The general theory developed in Part I of the present series is here applied to axisymmetric solutions of the equations governing the magnetohydrodynamics of Ideal incompressible Fluids. We first show a helpful analogy between axisymmetric MHD flows and flows of a stratified fluid in the Boussinesq approximation. We then construct a general Casimir as an integral of an arbitrary function of two conserved fields, namely the vector potential of the magnetic field and the scalar field associated with the ‘modified vorticity field’, the additional frozen-in field introduced in Part I. Using this Casimir, sucient conditions for linear stability to axisymmetric perturbations are obtained by standard Arnold techniques. We exploit Arnold’s method to obtain sucient conditions for nonlinear (Lyapunov) stability of the MHD flows considered. The appropriate norm is a sum of the magnetic and kinetic energies and the mean square vector potential of the magnetic field.
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on general transformations and variational principles for the magnetohydrodynamics of Ideal Fluids part 2 stability criteria for two dimensional flows
Journal of Fluid Mechanics, 1996Co-Authors: V A Vladimirov, H K Moffatt, Konstantin IlinAbstract:The techniques developed in Part 1 of the present series are here applied to two-dimensional solutions of the equations governing the magnetohydrodynamics of Ideal incompressible Fluids. We first demonstrate an isomorphism between such flows and the flow of a stratified fluid subjected to a field of force that we describe as'pseudo-gravitational'. We then construct a general Casimir as an integral of an arbitrary function of two conserved fields, namely the vector potential of the magnetic field, and the analogous potential of the 'modified vorticity field', the additional frozen field introduced in Part 1. Using this Casimir, a linear stability criterion is obtained by standard techniques. In 4, the (Arnold) techniques of nonlinear stability are developed, and bounds are placed on the second variation of the sum of the energy and the Casimir of the problem. This leads to criteria for nonlinear (Lyapunov) stability of the MHD flows considered. The appropriate norm is a sum of the magnetic and kinetic energies and the mean-square vector potential of the magnetic field.
V A Vladimirov - One of the best experts on this subject based on the ideXlab platform.
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on general transformations and variational principles for the magnetohydrodynamics of Ideal Fluids part 4 generalized isovorticity principle for three dimensional flows
Journal of Fluid Mechanics, 1999Co-Authors: V A Vladimirov, H K Moffatt, Konstantin IlinAbstract:The equations of magnetohydrodynamics (MHD) of an Ideal fluid have two families of topological invariants: the magnetic helicity invariants and the cross-helicity invariants. It is first shown that these invariants define a natural foliation (described as isomagnetovortical, or imv for short) in the function space in which solutions {u(x,t), h(x,t)} of the MHD equations reside. A relaxation process is constructed whereby total energy (magnetic plus kinetic) decreases on an imv folium (all magnetic and cross-helicity invariants being thus conserved). The energy has a positive lower bound determined by the global cross-helicity, and it is thus shown that a steady state exists having the (arbitrarily) prescribed families of magnetic and cross-helicity invariants. The stability of such steady states is considered by an appropriate generalization of (Arnold) energy techniques. The first variation of energy on the imv folium is shown to vanish, and the second variation δ 2 E is constructed
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on general transformations and variational principles for the magnetohydrodynamics of Ideal Fluids part iii stability criteria for axisymmetric flows
Journal of Plasma Physics, 1997Co-Authors: V A Vladimirov, H K Moffatt, Konstantin IlinAbstract:The general theory developed in Part I of the present series is here applied to axisymmetric solutions of the equations governing the magnetohydrodynamics of Ideal incompressible Fluids. We first show a helpful analogy between axisymmetric MHD flows and flows of a stratified fluid in the Boussinesq approximation. We then construct a general Casimir as an integral of an arbitrary function of two conserved fields, namely the vector potential of the magnetic field and the scalar field associated with the ‘modified vorticity field’, the additional frozen-in field introduced in Part I. Using this Casimir, sucient conditions for linear stability to axisymmetric perturbations are obtained by standard Arnold techniques. We exploit Arnold’s method to obtain sucient conditions for nonlinear (Lyapunov) stability of the MHD flows considered. The appropriate norm is a sum of the magnetic and kinetic energies and the mean square vector potential of the magnetic field.
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on general transformations and variational principles for the magnetohydrodynamics of Ideal Fluids part 2 stability criteria for two dimensional flows
Journal of Fluid Mechanics, 1996Co-Authors: V A Vladimirov, H K Moffatt, Konstantin IlinAbstract:The techniques developed in Part 1 of the present series are here applied to two-dimensional solutions of the equations governing the magnetohydrodynamics of Ideal incompressible Fluids. We first demonstrate an isomorphism between such flows and the flow of a stratified fluid subjected to a field of force that we describe as'pseudo-gravitational'. We then construct a general Casimir as an integral of an arbitrary function of two conserved fields, namely the vector potential of the magnetic field, and the analogous potential of the 'modified vorticity field', the additional frozen field introduced in Part 1. Using this Casimir, a linear stability criterion is obtained by standard techniques. In 4, the (Arnold) techniques of nonlinear stability are developed, and bounds are placed on the second variation of the sum of the energy and the Casimir of the problem. This leads to criteria for nonlinear (Lyapunov) stability of the MHD flows considered. The appropriate norm is a sum of the magnetic and kinetic energies and the mean-square vector potential of the magnetic field.
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on general transformations and variational principles for the magnetohydrodynamics of Ideal Fluids part 1 fundamental principles
Journal of Fluid Mechanics, 1995Co-Authors: V A Vladimirov, H K MoffattAbstract:A new frozen-in field w (generalizing vorticity) is constructed for Ideal magnetohydrodynamic flow. In conjunction with the frozen-in magnetic field h, this is used to obtain a generalized Weber transformation of the MHD equations, expressing the velocity as a bilinear form in generalized Weber variables. This expression is also obtained from Hamilton's principle of least action, and the canonically conjugate Hamiltonian variables for MHD flow are identified. Two alternative energy-type variational principles for three-dimensional steady MHD flow are established. Both involve a functional R which is the sum of the total energy and another conserved functional, the volume integral of a function Φ of Lagrangian coordinates. It is shown that the first variation δ 1 R vanishes if Φ is suitably chosen (as minus a generalized Bernoulli integral). Expressions for the second variation δ 2 R are presented
