A compressible Hamiltonian electromagnetic gyrofluid model

A Lie-Poisson bracket is presented for a four-field gyrofluid model with magnetic field curvature and compressible ions, thereby showing the model to be Hamiltonian. The corresponding Casimir invariants are presented, and shown to be associated to four Lagrangian invariants advected by distinct velocity fields. This differs from a cold ion limit, in which the Lie-Poisson bracket transforms into the sum of direct and semidirect products, leading to only three Lagrangian invariants.     

Structure of magnetic field lines

In this paper the Hamiltonian structure of magnetic lines is studied in many ways. First it is used vector analysis for defining the Poisson bracket and Casimir variable for this system. Second it is derived Pfaffian equations for magnetic field lines. Third, Lie derivative and derivative of Poisson bracket is used to show structure of this system. Finally, it is shown Nambu structure of the magnetic field lines.     

Dispersive deformations of the Hamiltonian structure of Euler’s equations

Euler’s equations for a two-dimensional fluid can be written in the Hamiltonian form, where the Poisson bracket is the Lie–Poisson bracket associated with the Lie algebra of divergence-free vector fields. For the two-dimensional hydrodynamics of ideal fluids, we propose a derivation of the Poisson brackets using a reduction from the bracket associated with the full algebra of vector fields. Taking the results of some recent studies of the deformations of Lie–Poisson brackets of vector fields into account, we investigate the dispersive deformations of the Poisson brackets of Euler’s equation: we show that they are trivial up to the second order.     

Non-integrability of the semiclassical Jaynes–Cummings models without the rotating-wave approximation

Two versions of the semiclassical Jaynes–Cummings model without the rotating wave approximation are investigated. It is shown that for a non-zero value of the coupling constant the version introduced by Belobrov, Zaslavsky, and Tartakovsky is Hamiltonian with respect to a certain degenerated Poisson bracket. Moreover, it is proved that both models are not integrable.     

Poisson brackets associated to the conformal geometry of curves

In this paper we present an invariant moving frame, in the group theoretical sense, along curves in the Möbius sphere. This moving frame will describe the relationship between all conformal differential invariants for curves that appear in the literature. Using this frame we first show that the Kac-Moody Poisson bracket on can be Poisson reduced to the space of conformal differential invariants of curves. The resulting bracket will be the conformal analogue of the Adler-Gel'fand-Dikii bracket. Secondly, a conformally invariant flow of curves induces naturally an evolution on the differential invariants of the flow. We give the conditions on the invariant flow ensuring that the induced evolution is Hamiltonian with respect to the reduced Poisson bracket. Because of a certain parallelism with the Euclidean case we study what we call Frenet and natural cases. We comment on the implications for completely integrable systems, and describe conformal analogues of the Hasimoto transformation.

     

Dissipative N-point-vortex Models in the Plane

A method is presented for constructing point vortex models in the plane that dissipate the Hamiltonian function at any prescribed rate and yet conserve the level sets of the invariants of the Hamiltonian model arising from the SE (2) symmetries. The method is purely geometric in that it uses the level sets of the Hamiltonian and the invariants to construct the dissipative field and is based on elementary classical geometry in ℝ³. Extension to higher-dimensional spaces, such as the point vortex phase space, is done using exterior algebra. The method is in fact general enough to apply to any smooth finite-dimensional system with conserved quantities, and, for certain special cases, the dissipative vector field constructed can be associated with an appropriately defined double Nambu–Poisson bracket. The most interesting feature of this method is that it allows for an infinite sequence of such dissipative vector fields to be constructed by repeated application of a symmetric linear operator (matrix) at each point of the intersection of the level sets.     

Effect of the chemical reaction and radiation absorption on the unsteady MHD free convection flow past a semi infinite vertical permeable moving plate with heat source and suction

Analytical solutions for heat and mass transfer by laminar flow of a Newtonian, viscous, electrically conducting and heat generation/absorbing fluid on a continuously vertical permeable surface in the presence of a radiation, a first-order homogeneous chemical reaction and the mass flux are reported. The plate is assumed to move with a constant velocity in the direction of fluid flow. A uniform magnetic field acts perpendicular to the porous surface, which absorbs the fluid with a suction velocity varying with time. The dimensionless governing equations for this investigation are solved analytically using two-term harmonic and non-harmonic functions. Graphical results for velocity, temperature and concentration profiles of both phases based on the analytical solutions are presented and discussed.     

On the three-dimensional finite Larmor radius approximation: The case of electrons in a fixed background of ions

This paper is concerned with the analysis of a mathematical model arising in plasma physics, more specifically in fusion research. It directly follows, Han-Kwan (2010) [18], where the three-dimensional analysis of a Vlasov–Poisson equation with finite Larmor radius scaling was led, corresponding to the case of ions with massless electrons whose density follows a linearized Maxwell–Boltzmann law. We now consider the case of electrons in a background of fixed ions, which was only sketched in Han-Kwan (2010) [18]. Unfortunately, there is evidence that the formal limit is false in general. Nevertheless, we formally derive from the Vlasov–Poisson equation a fluid system for particular monokinetic data. We prove the local in time existence of analytic solutions and rigorously study the limit (when the inverse of the intensity of the magnetic field and the Debye length vanish) to a new anisotropic fluid system. This is achieved thanks to Cauchy–Kovalevskaya type techniques, as introduced by Caflisch (1990) [7] and Grenier (1996) [14]. We finally show that this approach fails in Sobolev regularity, due to multi-fluid instabilities.     

Poisson structures for geometric curve flows in semi-simple homogeneous spaces

We apply the equivariant method of moving frames to investigate the existence of Poisson structures for geometric curve flows in semi-simple homogeneous spaces. We derive explicit compatibility conditions that ensure that a geometric flow induces a Hamiltonian evolution of the associated differential invariants. Our results are illustrated by several examples of geometric interest.     

Existence for the stationary MHD-equations coupled to heat transfer with nonlocal radiation effects

We consider the problem of influencing the motion of an electrically conducting fluid with an applied steady magnetic field. Since the flow is originating from buoyancy, heat transfer has to be included in the model. The stationary system of magnetohydrodynamics is considered, and an approximation of Boussinesq type is used to describe the buoyancy. The heat sources given by the dissipation of current and the viscous friction are not neglected in the fluid. The vessel containing the fluid is embedded in a larger domain, relevant for the global temperature- and magnetic field- distributions. Material inhomogeneities in this larger region lead to transmission relations for the electromagnetic fields and the heat flux on inner boundaries. In the presence of transparent materials, the radiative heat transfer is important and leads to a nonlocal and nonlinear jump relation for the heat flux. We prove the existence of weak solutions, under the assumption that the imposed velocity at the boundary of the fluid remains sufficiently small.     

Existence of weak solutions to the time-dependent MHD equations coupled to the heat equation with nonlocal radiation boundary conditions

We study the coupling of the system of magnetohydrodynamics to the heat equation in the context of an application to crystal growth. The heat sources are given by the dissipation of current that occurs in the fixed conductors of the system. According to Boussinesq’s model, the dissipative heating is neglected in the fluid. We take into account the natural interface conditions for the magnetic field, and nonlocal radiation boundary conditions for the heat flux. We prove the existence of a weak solution with a defect measure, concentrated in a singular set.     

Hall effect on unsteady MHD Couette flow and heat transfer of a Bingham fluid with suction and injection

The unsteady magnetohydrodynamic flow of an electrically conducting viscous incompressible non-Newtonian Bingham fluid bounded by two parallel non-conducting porous plates is studied with heat transfer considering the Hall effect. An external uniform magnetic field is applied perpendicular to the plates and the fluid motion is subjected to a uniform suction and injection. The lower plate is stationary and the upper plate moves with a constant velocity and the two plates are kept at different but constant temperatures. Numerical solutions are obtained for the governing momentum and energy equations taking the Joule and viscous dissipations into consideration. The effect of the Hall term, the parameter describing the non-Newtonian behavior, and the velocity of suction and injection on both the velocity and temperature distributions are studied.     

Existence of resonances in magnetic scattering

The Schrödinger operator with a compactly supported magnetic field is shown to produce infinitely many resonances, in any odd dimension⩾3. The proof is based on the Poisson formula for resonances and properties of the magnetic heat invariants.     

Cattaneo–Christov heat flux model for three-dimensional magnetohydrodynamic flow of an Eyring Powell fluid over an exponentially stretching surface with convective boundary condition

The present model concentrates on three-dimensional steady incompressible flow of an Eyring-Powell nanofluid past an exponentially stretching sheet with magnetic field. The Cattaneo–Christov heat flux with convective boundary condition is accounted for. Shooting method is the instrumental for obtaining numerical solution of the transformed-converted system of the mathematical models. Behavior of the determining thermo-physical parameters on the velocity, temperature, skin friction, heat transfer rate, and finally isotherms are considered. The major relevant outcomes of the current investigation are that increment in Eyring-Powell parameter uplifts flow velocity, while that peters out the fluid temperature. Enhanced values of the mixed convection parameter weakened the skin friction coefficient while it slightly strengthened the rate of heat transfer.     

Spin-boson model through a Poisson-driven stochastic process

We give a functional integral representation of the semigroup generated by the spin-boson Hamiltonian by making use of a Poisson point process and a Euclidean field. We present a method of constructing Gibbs path measures indexed by the full real line which can be applied also to more general stochastic processes with jump discontinuities. Using these tools we then show existence and uniqueness of the ground state of the spin-boson, and analyze ground state properties. In particular, we prove super-exponential decay of the number of bosons, Gaussian decay of the field operators, derive expressions for the positive integer, fractional and exponential moments of the field operator, and discuss the field fluctuations in the ground state.     

Effects of partial slip,viscous dissipation and Joule heating on Von Kármán flow and heat transfer of an electrically conducting non-Newtonian fluid

The steady Von Kármán flow and heat transfer of an electrically conducting non-Newtonian fluid is extended to the case where the disk surface admits partial slip. The fluid is subjected to an external uniform magnetic field perpendicular to the plane of the disk. The constitutive equation of the non-Newtonian fluid is modeled by that for a Reiner–Rivlin fluid. The momentum equations give rise to highly non-linear boundary value problem. Numerical solutions for the governing non-linear equations are obtained over the entire range of the physical parameters. The effects of slip, magnetic parameter and non-Newtonian fluid characteristics on the velocity and temperature fields are discussed in detail and shown graphically. Emphasis has been laid to study the effects of viscous dissipation and Joule heating on the thermal boundary layer. It is interesting to find that the non-Newtonian cross-viscous parameter has an opposite effect to that of the slip and the magnetic parameter on the velocity and the temperature fields.     

Nonlinear transparency regimes for three-component acoustic pulses in a system of electron and nuclear spins

We study acoustic solitons consisting of one longitudinal and two transverse components and propagating in the direction perpendicular to an external magnetic field in a crystal containing paramagnetic impurities of electron and nuclear spins. The coupling of the electron spin subsystem to the longitudinal sound allows making the velocity of the latter close to that of the transverse acoustic waves, which provides an effective interaction between all components of the elastic field by means of the nuclear spin subsystem. We derive a three-component system of material and reduced wave equations describing this process and construct its soliton solutions in the form of stationary and breather pulses. Based on them, we study the peculiarities of the dynamics of the elastic field components and reveal the differences from the two-component model. The existence of two families of breathers is an important distinctive feature of the considered case.     

Effects of slip on steady Bödewadt flow and heat transfer of an electrically conducting non-Newtonian fluid

The steady flow and heat transfer arising due to the rotation of a non-Newtonian fluid at a larger distance from a stationary disk is extended to the case where the disk surface admits partial slip. The constitutive equation of the non-Newtonian fluid is modeled by that for a Reiner–Rivlin fluid. The fluid is subjected to an external uniform magnetic field perpendicular to the plane of the disk. The momentum equation gives rise to a highly nonlinear boundary value problem. Numerical solution of the governing nonlinear equations are obtained over the entire range of the physical parameters. The effects of slip, non-Newtonian fluid characteristics and the magnetic interaction parameter on the momentum boundary layer and thermal boundary layer are discussed in detail and shown graphically. It is observed that slip has prominent effects on the velocity and temperature fields.     

Global well-posedness and existence of uniform attractor for magnetohydrodynamic equations

We study the global well-posedness and existence of uniform attractor for magnetohydrodynamic (MHD) equations. The hydrodynamic system consists of the Navier–Stokes equations for the fluid velocity and pressure coupled with a reduced from of the Maxwell equations for the magnetic field. The fluid velocity is assumed to satisfy a no-slip boundary condition, while the magnetic field is subject to a time-dependent Dirichlet boundary condition. We first establish the global existence of weak and strong solutions to Equations (1.1)-(1.4). And at this stage, we further derive the existence of a uniform attractor for Equations (1.1)-(1.4).     

The effect of variable properties on the unsteady Couette flow with heat transfer considering the Hall effect

The influence of temperature dependent viscosity and thermal conductivity on the transient Couette flow with heat transfer is studied. An external uniform magnetic field is applied perpendicular to the parallel plates and the Hall effect is taken into consideration. The fluid is acted upon by a constant pressure gradient. The two plates are kept at two constant but different temperatures and the viscous and Joule dissipations are considered in the energy equation. A numerical solution for the governing non-linear equations of motion and the energy equation is obtained. The effect of the Hall term and the temperature dependent viscosity and thermal conductivity on both the velocity and temperature distributions is examined.