602 research outputs found
Landau damping in the multiscale Vlasov theory
Vlasov kinetic theory is extended by adopting an extra one particle
distribution function as an additional state variable characterizing the
micro-turbulence internal structure. The extended Vlasov equation keeps the
reversibility, the Hamiltonian structure, and the entropy conservation of the
original Vlasov equation. In the setting of the extended Vlasov theory we then
argue that the Fokker-Planck type damping in the velocity dependence of the
extra distribution function induces the Landau damping. The same type of
extension is made also in the setting of fluid mechanics
One and two-fiber orientation kinetic theories of fiber suspensions
http://dx.doi.org/10.1016/j.jnnfm.2012.10.009The morphology influencing rheological properties of suspensions of rigid spheres constitutes the flow induced collective ordering of the spheres characterized by two or more sphere distribution functions. When the rigid spheres are replaced by rigid fibers, the collective order in the position of the spheres is replaced by the flow induced orientation of the fibers that suffices to be characterized by one-fiber orientation distribution function. A flow induced collective ordering of fibers (both in position and orientation), that can only be characterized by two or more fiber distribution functions, can still however constitute an important part of the morphology. We show that two types of interaction among fibers, one being the Onsager-type topological interaction entering the free energy and the other the hydrodynamics interaction entering the dissipative part of the time evolution, give indeed rise to a collective order in the orientation influencing the rheology of fiber suspensions
Solid-fluid dynamics of yield-stress fluids
On the example of two-phase continua experiencing stress induced solid-fluid
phase transitions we explore the use of the Euler structure in the formulation
of the governing equations. The Euler structure guarantees that solutions of
the time evolution equations possessing it are compatible with mechanics and
with thermodynamics. The former compatibility means that the equations are
local conservation laws of the Godunov type and the latter compatibility means
that the entropy does not decrease during the time evolution. In numerical
illustrations, in which the one-dimensional Riemann problem is explored, we
require that the Euler structure is also preserved in the discretization.Comment: 51 pages, 7 figure
Extra mass flux in fluid mechanics
The conditions of existence of extra mass flux in single component
dissipative non-relativistic fluids are clarified. By considering Galilean
invariance we show that if total mass flux is equal to total momentum density,
then mass, momentum, angular momentum and booster (center-of-mass) are
conserved. However, these conservation laws may be fulfilled also by other
means. We show an example of weakly non-local hydrodynamics where the
conservation laws are satisfied as well although the total mass flux is
different from momentum density
Hamiltonian Coupling of Electromagnetic Field and Matter
Reversible part of evolution equations of physical systems is often generated
by a Poisson bracket. We discuss geometric means of construction of Poisson
brackets and their mutual coupling (direct, semidirect and matched-pair
products) as well as projections of Poisson brackets to less detailed Poisson
brackets. This way the Hamiltonian coupling between transport of mixtures and
electrodynamics is elucidated
Ehrenfest regularization of Hamiltonian systems
Imagine a freely rotating rigid body. The body has three principal axes of
rotation. It follows from mathematical analysis of the evolution equations that
pure rotations around the major and minor axes are stable while rotation around
the middle axis is unstable. However, only rotation around the major axis (with
highest moment of inertia) is stable in physical reality (as demonstrated by
the unexpected change of rotation of the Explorer 1 probe). We propose a
general method of Ehrenfest regularization of Hamiltonian equations by which
the reversible Hamiltonian equations are equipped with irreversible terms
constructed from the Hamiltonian dynamics itself. The method is demonstrated on
harmonic oscillator, rigid body motion (solving the problem of stable minor
axis rotation), ideal fluid mechanics and kinetic theory. In particular, the
regularization can be seen as a birth of irreversibility and dissipation. In
addition, we discuss and propose discretizations of the Ehrenfest regularized
evolution equations such that key model characteristics (behavior of energy and
entropy) are valid in the numerical scheme as well
Continuum Mechanics and Thermodynamics in the Hamilton and the Godunov-type Formulations
Continuum mechanics with dislocations, with the Cattaneo type heat
conduction, with mass transfer, and with electromagnetic fields is put into the
Hamiltonian form and into the form of the Godunov type system of the first
order, symmetric hyperbolic partial differential equations (SHTC equations).
The compatibility with thermodynamics of the time reversible part of the
governing equations is mathematically expressed in the former formulation as
degeneracy of the Hamiltonian structure and in the latter formulation as the
existence of a companion conservation law. In both formulations the time
irreversible part represents gradient dynamics. The Godunov type formulation
brings the mathematical rigor (the well-posedness of the Cauchy initial value
problem) and the possibility to discretize while keeping the physical content
of the governing equations (the Godunov finite volume discretization)
Deterministic solution of the kinetic theory model of colloidal suspensions of structureless particles
A direct modeling of colloidal suspensions consists of calculating trajectories of all suspended objects. Due to the large time computing and the large cost involved in such calculations, we consider in this paper another route. Colloidal suspensions are described on a mesoscopic level by a distribution function whose time evolution is governed by a Fokker–Plancklike equation. The difficulty encountered on this route is the high dimensionality of the space in which the distribution function is defined. A novel strategy is used to solve numerically the Fokker–Planck equation circumventing the curse of dimensionality issue. Rheological and morphological predictions of the model that includes both direct and hydrodynamic interactions are presented in different flows
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