A general metriplectic framework with application to dissipative extended magnetohydrodynamics

General equations for conservative yet dissipative (entropy producing) extended magnetohydrodynamics are derived from two-fluid theory. Keeping all terms generates unusual cross-effects, such as thermophoresis and a current viscosity that mixes with the usual velocity viscosity. While the Poisson bracket of the ideal version of this model has already been discovered, we determine its metriplectic counterpart that describes the dissipation. This is done using a new and general thermodynamic point of view to derive dissipative brackets, a means of derivation that is natural for understanding and creating dissipative dynamics without appealing to underlying kinetic theory orderings. Finally, the formalism is used to study dissipation in the Lagrangian variable picture where, in the context of extended magnetohydrodynamics, non-local dissipative brackets naturally emerge.

Lagrangian and Dirac constraints for the ideal incompressible fluid and magnetohydrodynamics

The incompressibility constraint for fluid flow was imposed by Lagrange in the so-called Lagrangian variable description using his method of multipliers in the Lagrangian (variational) formulation. An alternative is the imposition of incompressibility in the Eulerian variable description by a generalization of Dirac’s constraint method using noncanonical Poisson brackets. Here it is shown how to impose the incompressibility constraint using Dirac’s method in terms of both the canonical Poisson brackets in the Lagrangian variable description and the noncanonical Poisson brackets in the Eulerian description, allowing for the advection of density. Both cases give the dynamics of infinite-dimensional geodesic flow on the group of volume preserving diffeomorphisms and explicit expressions for this dynamics in terms of the constraints and original variables is given. Because Lagrangian and Eulerian conservation laws are not identical, comparison of the various methods is made.

LAGRANGIAN DYNAMICS OF CABLE-DRIVEN PARALLEL MANIPULATORS: A VARIABLE MASS FORMULATION

In this paper, dynamic analysis of cable-driven parallel manipulators (CDPMs) is performed using the Lagrangian variable mass formulation. This formulation is used to treat the effect of a mass stream entering into the system caused by elongation of the cables. In this way, a complete dynamic model of the system is derived, while preserving the compact and tractable closed-form dynamics formulation. First, a general formulation for a CDPM is given, and the effect of change of mass in the cables is integrated into its dynamics. The significance of such a treatment is that a complete analysis of the dynamics of the system is achieved, including vibrations, stability, and any robust control synthesis of the manipulator. The formulation obtained is applied to a typical planar CDPM. Through numerical simulations, the validity and integrity of the formulations are verified, and the significance of the variable mass treatment in the analysis is examined. For this example, it is shown that the effect of introducing a mass stream into the system is not negligible. Moreover, it is non linear and strongly dependent on the geometric and inertial parameters of the robot, as well as the maneuvering trajectory.

A NEW LAGRANGIAN FORMULATION OF IDEAL MAGNETOHYDRODYNAMICS

We propose a reformulation of ideal magnetohydrodynamics written in the Lagrangian variable as an enlarged system of hyperelastic type, with a specific potential. We study the hyperbolicity of the model and prove that the acoustic tensor is positive for all directions which are non orthogonal to the magnetic field. The consequences for Eulerian ideal magnetohydrodynamics and for numerical discretization are briefly discussed.

Constructing Poisson and Dissipative Brackets of Mixtures by using Lagrangian-to-Eulerian Transformation

This paper aims to construct the bracket formalism of mixture continua by using the method of Lagrangian- to-Eulerian (LE) transformation. The LE approach first builds up the transformation relations between the Eulerian state variables and the Lagrangian canonical variables, and then transforms the bracket in Lagrangian form to the bracket in Eulerian form. For the conservative part of the bracket formalism, this study systematically generates the noncanonical Poisson brackets of a two-component mixture. For the dissipative part, we deduce the Eulerian-variable-based dissipative brackets for viscous and diffusive mechanisms from their Lagrangian-variable-based counterparts. Finally, the evolution equations of a micromorphic fluid, which can be treated as a multi-component mixture, are derived by constructing its Poisson and dissipative brackets.

Self-similar strong shocks in an exponential medium

The self-similar one-dimensional propagation of a strong shock wave in a medium with exponentially varying density and ray-tube area is studied, using the Eulerian approach of Sedov. Conservation integrals analogous to Sedov's are obtained, with the expression for the Lagrangian variable. Calculated results are compared with the predictions of the CCW (Chisnell, Chester and Whitham) approximation. It was found that, in contrast to the implosion case, the propagation parameter from the CCW approximation is in error by 15% or more.