Self-diffusion in dense granular shear flows

Diffusivity is a key quantity in describing velocity fluctuations in granular materials. These fluctuations are the basis of many thermodynamic and hydrodynamic models which aim to provide a statistical description of granular systems. We present experimental results on diffusivity in dense, granular shear in a 2D Couette geometry. We find that self-diffusivities are proportional to the local shear rate with diffusivities along the mean flow approximately twice as large as those in the perpendicular direction. The magnitude of the diffusivity is D \approx \dot\gamma a^2 where a is the particle radius. However, the gradient in shear rate, coupling to the mean flow, and drag at the moving boundary lead to particle displacements that can appear sub- or super-diffusive. In particular, diffusion appears superdiffusive along the mean flow direction due to Taylor dispersion effects and subdiffusive along the perpendicular direction due to the gradient in shear rate. The anisotropic force network leads to an additional anisotropy in the diffusivity that is a property of dense systems with no obvious analog in rapid flows. Specifically, the diffusivity is supressed along the direction of the strong force network. A simple random walk simulation reproduces the key features of the data, such as the apparent superdiffusive and subdiffusive behavior arising from the mean flow, confirming the underlying diffusive motion. The additional anisotropy is not observed in the simulation since the strong force network is not included. Examples of correlated motion, such as transient vortices, and Levy flights are also observed. Although correlated motion creates velocity fields qualitatively different from Brownian motion and can introduce non-diffusive effects, on average the system appears simply diffusive.Comment: 13 pages, 20 figures (accepted to Phys. Rev. E

Computationally Efficient 3D Eddy Current Loss Prediction in Magnets of Interior Permanent Magnet Machines

This paper proposes a computationally efficient method based on imaging technique, for accurate prediction of 3- dimensional (3D) eddy current loss in the rotor magnets of interior permanent magnet (IPM) machines. 2D time-stepped finite element analysis is employed to generate the radial and the tangential 2D magnetic field information within the magnet for application of the 3D imaging technique. The method is validated with 3D time-stepped finite element analysis (FEA) for an 8 pole-18 slot IPM machine evaluating its resistance limited magnet loss with increase in axial and tangential segmentation. Magnet loss considering eddy current reaction at high frequencies is evaluated from the proposed method by employing the diffusion of the 2D magnetic field variation along the axial plane. The loss associated with all the frequencies together in the armature currents is evaluated by considering each of the harmonics separately in the proposed method employing the frozen permeability to account for magnetic saturation. The results obtained are verified with 3D FEA evaluating the magnet loss at fundamental, 10 and 20 kHz time harmonics in armature currents

Regularity and uniqueness in quasilinear parabolic systems

Inspired by a problem in steel metallurgy, we prove the existence, regularity, uniqueness, and continuous data dependence of solutions to a coupled parabolic system in a smooth bounded 3D domain, with nonlinear and nonhomogeneous boundary conditions. The nonlinear coupling takes place in the diffusion coefficient. The proofs are based on anisotropic estimates in tangential and normal directions, and on a refined variant of the Gronwall lemma

Regularity and uniqueness in quasilinear parabolic systems

Inspired by a problem in steel metallurgy, we prove the existence, regularity, uniqueness, and continuous data dependence of solutions to a coupled parabolic system in a smooth bounded 3D domain, with nonlinear and nonhomogeneous boundary conditions. The nonlinear coupling takes place in the diffusion coefficient. The proofs are based on anisotropic estimates in tangential and normal directions, and on a refined variant of the Gronwall lemma

A hybridizable discontinuous Galerkin method for electromagnetics with a view on subsurface applications

Two Hybridizable Discontinuous Galerkin (HDG) schemes for the solution of Maxwell's equations in the time domain are presented. The first method is based on an electromagnetic diffusion equation, while the second is based on Faraday's and Maxwell--Amp\`ere's laws. Both formulations include the diffusive term depending on the conductivity of the medium. The three-dimensional formulation of the electromagnetic diffusion equation in the framework of HDG methods, the introduction of the conduction current term and the choice of the electric field as hybrid variable in a mixed formulation are the key points of the current study. Numerical results are provided for validation purposes and convergence studies of spatial and temporal discretizations are carried out. The test cases include both simulation in dielectric and conductive media