scholarly journals Constitutive relations for compressible granular flow in the inertial regime

2019 ◽  
Vol 874 ◽  
pp. 926-951 ◽  
Author(s):  
D. G. Schaeffer ◽  
T. Barker ◽  
D. Tsuji ◽  
P. Gremaud ◽  
M. Shearer ◽  
...  
Keyword(s):  
Steady State ◽  
Granular Flow ◽  
Numerical Solutions ◽  
Volume Fraction ◽  
Constitutive Relations ◽  
Practical Interest ◽  
Time Dependent ◽  
Direct Result ◽  
Wide Range ◽  
Inertial Regime

Granular flows occur in a wide range of situations of practical interest to industry, in our natural environment and in our everyday lives. This paper focuses on granular flow in the so-called inertial regime, when the rheology is independent of the very large particle stiffness. Such flows have been modelled with the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$-rheology, which postulates that the bulk friction coefficient $\unicode[STIX]{x1D707}$ (i.e. the ratio of the shear stress to the pressure) and the solids volume fraction $\unicode[STIX]{x1D719}$ are functions of the inertial number $I$ only. Although the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$-rheology has been validated in steady state against both experiments and discrete particle simulations in several different geometries, it has recently been shown that this theory is mathematically ill-posed in time-dependent problems. As a direct result, computations using this rheology may blow up exponentially, with a growth rate that tends to infinity as the discretization length tends to zero, as explicitly demonstrated in this paper for the first time. Such catastrophic instability due to ill-posedness is a common issue when developing new mathematical models and implies that either some important physics is missing or the model has not been properly formulated. In this paper an alternative to the $\unicode[STIX]{x1D707}(I),\unicode[STIX]{x1D6F7}(I)$-rheology that does not suffer from such defects is proposed. In the framework of compressible $I$-dependent rheology (CIDR), new constitutive laws for the inertial regime are introduced; these match the well-established $\unicode[STIX]{x1D707}(I)$ and $\unicode[STIX]{x1D6F7}(I)$ relations in the steady-state limit and at the same time are well-posed for all deformations and all packing densities. Time-dependent numerical solutions of the resultant equations are performed to demonstrate that the new inertial CIDR model leads to numerical convergence towards physically realistic solutions that are supported by discrete element method simulations.

Fourier-Bessel Series Model for the Stefan Problem With Internal Heat Generation in Cylindrical Coordinates

2018 ◽  
Author(s):  
Lyudmyla Barannyk ◽  
John Crepeau ◽  
Patrick Paulus ◽  
Ali Siahpush
Keyword(s):  
Differential Equation ◽  
Steady State ◽  
Heat Generation ◽  
Numerical Solutions ◽  
Internal Heat ◽  
Internal Heat Generation ◽  
Time Dependent ◽  
Cylindrical Coordinates ◽  
System Approaches ◽  
Melting And Solidification

A nonlinear, first-order ordinary differential equation that involves Fourier-Bessel series terms has been derived to model the time-dependent motion of the solid-liquid interface during melting and solidification of a material with constant internal heat generation in cylindrical coordinates. The model is valid for all Stefan numbers. One of the primary applications of this problem is for a nuclear fuel rod during meltdown. The numerical solutions to this differential equation are compared to the solutions of a previously derived model that was based on the quasi-steady approximation, which is valid only for Stefan numbers less than one. The model presented in this paper contains exponentially decaying terms in the form of Fourier-Bessel series for the temperature gradients in both the solid and liquid phases. The agreement between the two models is excellent in the low Stefan number regime. For higher Stefan numbers, where the quasi-steady model is not accurate, the new model differs from the approximate model since it incorporates the time-dependent terms for small times, and as the system approaches steady-state, the curves converge. At higher Stefan numbers, the system approaches steady-state faster than for lower Stefan numbers. During the transient process for both melting and solidification, the temperature profiles become parabolic.

scholarly journals Alternating direction implicit type preconditioners for the steady state inhomogeneous Vlasov equation

2017 ◽  
Vol 83 (1) ◽  
Author(s):  
Markus Gasteiger ◽  
Lukas Einkemmer ◽  
Alexander Ostermann ◽  
David Tskhakaya
Keyword(s):  
Steady State ◽  
Numerical Simulations ◽  
Vlasov Equation ◽  
Numerical Solutions ◽  
Splitting Method ◽  
Time Integration ◽  
Computational Effort ◽  
Alternating Direction Implicit ◽  
Alternating Direction ◽  
Wide Range

The purpose of the current work is to find numerical solutions of the steady state inhomogeneous Vlasov equation. This problem has a wide range of applications in the kinetic simulation of non-thermal plasmas. However, the direct application of either time stepping schemes or iterative methods (such as Krylov-based methods such as the generalized minimal residual method (GMRES) or relaxation schemes) is computationally expensive. In the former case the slowest time scale in the system forces us to perform a long time integration while in the latter case a large number of iterations is required. In this paper we propose a preconditioner based on an alternating direction implicit type splitting method. This preconditioner is then combined with both GMRES and Richardson iteration. The resulting numerical schemes scale almost ideally (i.e. the computational effort is proportional to the number of grid points). Numerical simulations conducted show that this can result in a speed-up of close to two orders of magnitude (even for intermediate grid sizes) with respect to the unpreconditioned case. In addition, we discuss the characteristics of these numerical methods and show the results for a number of numerical simulations.

scholarly journals Growth Rate of White Dwarf Mass in Binaries

1989 ◽  
Vol 114 ◽  
pp. 507-510
Author(s):  
Mariko Kato ◽  
Hideyuki Saio ◽  
Izumi Hachisu
Keyword(s):  
Growth Rate ◽  
Steady State ◽  
Star Formation ◽  
Neutron Star ◽  
Mass Loss ◽  
White Dwarf ◽  
Accretion Rate ◽  
Time Dependent ◽  
Type I ◽  
Wide Range

AbstractThe growth rate of a white dwarf which accretes hydrogen-rich or helium matter is studied. If the accretion rate is relatively small, unstable shell flash occurs and during which the envelope mass is lost. We have followed the evolutions of shell flashes by steady state approach with wind mass loss solutions to determined the mass lost from the system for wide range of binary parameters. The time-dependent models are also calculated in some cases. The mass loss due to the Roche lobe overflow are taken into account. This results seriously affects the existing scenarios on the origin of the type I supernova or on the neutron star formation induced by accretion.

scholarly journals Systematic derivation of a surface polarisation model for planar perovskite solar cells

2018 ◽  
Vol 30 (3) ◽  
pp. 427-457 ◽  
Author(s):  
N. E. COURTIER ◽  
J. M. FOSTER ◽  
S. E. J. O'KANE ◽  
A. B. WALKER ◽  
G. RICHARDSON
Keyword(s):  
Solar Cells ◽  
Numerical Solutions ◽  
Perovskite Solar Cells ◽  
Practical Interest ◽  
Operating Conditions ◽  
Asymptotic Approach ◽  
Perovskite Layer ◽  
Current Voltage ◽  
Wide Range ◽  
Vacancy Density

Increasing evidence suggests that the presence of mobile ions in perovskite solar cells (PSCs) can cause a current–voltage curve hysteresis. Steady state and transient current–voltage characteristics of a planar metal halide CH3NH3PbI3PSC are analysed with a drift-diffusion model that accounts for both charge transport and ion vacancy motion. The high ion vacancy density within the perovskite layer gives rise to narrow Debye layers (typical width ~2 nm), adjacent to the interfaces with the transport layers, over which large drops in the electric potential occur and in which significant charge is stored. Large disparities between (I) the width of the Debye layers and that of the perovskite layer (~600 nm) and (II) the ion vacancy density and the charge carrier densities motivate an asymptotic approach to solving the model, while the stiffness of the equations renders standard solution methods unreliable. We derive a simplifiedsurface polarisationmodel in which the slow ion dynamics are replaced by interfacial (non-linear) capacitances at the perovskite interfaces. Favourable comparison is made between the results of the asymptotic approach and numerical solutions for a realistic cell over a wide range of operating conditions of practical interest.

scholarly journals Steady state rheology of homogeneous and inhomogeneous cohesive granular materials

2019 ◽  
Vol 22 (1) ◽  
Author(s):  
Hao Shi ◽  
Sudeshna Roy ◽  
Thomas Weinhart ◽  
Vanessa Magnanimo ◽  
Stefan Luding
Keyword(s):  
Steady State ◽  
Granular Materials ◽  
Volume Fraction ◽  
Bond Number ◽  
Contact Friction ◽  
Volume Fractions ◽  
Wide Range ◽  
Homogeneous Stress ◽  
Shear Box ◽  
Opposing Effects

AbstractThis paper aims to understand the effect of different particle/contact properties like friction, softness and cohesion on the compression/dilation of sheared granular materials. We focus on the local volume fraction in steady state of various non-cohesive, dry cohesive and moderate to strong wet cohesive, frictionless-to-frictional soft granular materials. The results from (1) an inhomogeneous, slowly sheared split-bottom ring shear cell and (2) a homogeneous, stress-controlled simple shear box with periodic boundaries are compared. The steady state volume fractions agree between the two geometries for a wide range of particle properties. While increasing inter-particle friction systematically leads to decreasing volume fractions, the inter-particle cohesion causes two opposing effects. With increasing strength of cohesion, we report an enhancement of the effect of contact friction i.e. even smaller volume fraction. However, for soft granular materials, strong cohesion causes an increase in volume fraction due to significant attractive forces causing larger deformations, not visible for stiff particles. This behaviour is condensed into a particle friction—Bond number phase diagram, which can be used to predict non-monotonic relative sample dilation/compression.

A theoretical study on the response of a point elasto-hydrodynamic lubrication contact to a normal harmonic vibration under thermal and non-Newtonian conditions

2007 ◽  
Vol 221 (9) ◽  
pp. 1089-1110 ◽  
Author(s):  
P Yang ◽  
J Cui ◽  
J M Jin ◽  
D Dowson
Keyword(s):  
Steady State ◽  
Numerical Solutions ◽  
Hydrodynamic Lubrication ◽  
Computing Time ◽  
Harmonic Vibration ◽  
Time Dependent ◽  
Processing Unit ◽  
Infinite Plane ◽  
Shear Strain Rate ◽  
Newtonian Flow

Time-dependent thermal and non-Newtonian elastohydrodynamic lubrication of an elliptical point contact subjected to a normal harmonic vibration was studied numerically in this work. The contact was idealized as between an infinite plane and a spherical roller. The normal vibration of the roller was described by specifying the centre of the spherical roller to the infinite plane (without deformation) as a cyclic function of time. The shear-thinning rheological property of the lubricant was described by the Eyring model. The time-dependent numerical solution was achieved instant after instant in each period of a vibration. The periodic errors were checked at the end of each vibration cycle until the responses of variables such as pressure, film thickness, and temperature were all periodic functions with the same frequency of the roller's vibration. At each instant, the pressure field was solved with a multi-grid method, the surface deflection produced by pressure was determined with a multi-level multi-integration technique, the non-Newtonian flow of the lubricant was considered by using the equivalent viscosity calculated according to the shear-strain rate along the entrainment direction only, and the temperature field was evaluated with a finite-difference scheme through a column-by-column relaxation process. The computing time for a cyclic solution was 12–15 h on a personal computer with a 3.0 GHz central processing unit. The effects of both the amplitude and the frequency of the vibration were investigated. It was shown that the time-dependent solution is significantly different from the steady-state solution, especially when the amplitude of vibration is large and the frequency of vibration is high. Corresponding to a typical thermal and non-Newtonian case, numerical solutions were also obtained under isothermal and Newtonian, isothermal and non-Newtonian, and thermal and Newtonian conditions. Comparisons between these solutions indicate that, under time-dependent conditions, the effects of thermal and non-Newtonian flow are similar to those under steady-state conditions.

scholarly journals Research on the Homogenized Postirradiation Elastoplastic Constitutive Relations for Composite Nuclear Fuels

2021 ◽  
Vol 8 ◽  
Author(s):  
Jing Zhang ◽  
Jingyu Zhang ◽  
Haoyu Wang ◽  
Hongyang Wei ◽  
Changbing Tang ◽  
...  
Keyword(s):  
Metal Matrix ◽  
Volume Fraction ◽  
Constitutive Relations ◽  
Particle Volume Fraction ◽  
Tensile Tests ◽  
Uniaxial Tensile ◽  
Particle Volume ◽  
Plastic Deformations ◽  
Wide Range ◽  
Irradiation Process

A multi-scale finite element method is developed to simulate the irradiation process and postirradiation uniaxial tensile tests for metal-matrix composite fuels with representative volume elements (RVEs). The simulations of irradiation process are implemented under a wide range of burnup levels, with the irradiation effects on the mechanical constitutive relations of fuel particles and matrix taken into account comprehensively. The simulation results for the macroscopic postirradiation true stress/strain curves are obtained, excluding the irradiation-induced macroscopic deformations. The effects of particle fission density, temperature, and initial particle volume fraction are investigated and analyzed. The research results indicate that 1) a quasi-elastic stage appears during the postirradiation tension, which is mainly induced by the creation of high residual compressive stresses in the particles and matrix after irradiation; 2) with the increase of effective strains, new plastic deformations increase in the particles and matrix to result in the macroscale plastic stage; 3) the macroscale irradiation softening and hardening phenomena appear, which mainly stem from the weakened deformation resistance by the irradiation-induced plastic deformations in the matrix, the enlarged particle volume fraction after irradiation, and the irradiation hardening effects of metal matrix.

Computational Study of Scission Neutrons in Low-Energy Fission: Stationary and Time-Dependent Approaches

2011 ◽  
Vol 9 (4) ◽  
pp. 917-936 ◽  
Author(s):  
M. Rizea ◽  
N. Carjan
Keyword(s):  
Eigenvalue Problems ◽  
Numerical Solutions ◽  
Computational Study ◽  
Sparse Matrices ◽  
Practical Interest ◽  
Time Dependent ◽  
Axially Symmetric ◽  
Finite Difference Approximations ◽  
Sudden Approximation ◽  
Physical Quantities

AbstractThe emission of scission neutrons from fissioning nuclei is of high practical interest. To study this process we have used the sudden approximation and also a more realistic approach that takes into account the scission dynamics. Numerically, this implies the solution of the bi-dimensional Schrödinger equation, both stationary and time-dependent. To describe axially symmetric extremely deformed nuclear shapes, we have used the Cassini parametrization. The Hamiltonian is discretized by using finite difference approximations of the derivatives. The main computational challenges are the solution of algebraic eigenvalue problems and of linear systems with large sparse matrices. We have employed appropriate procedures (Arnoldi and bi-conjugate gradients). The numerical solutions have been used to evaluate physical quantities, like the number of emitted neutrons per scission event, the primary fragments’ excitation energy and the distribution of the emission points.

scholarly journals Stability and solution of the time-dependent Bondi–Parker flow

2020 ◽  
Vol 493 (2) ◽  
pp. 2834-2840
Author(s):  
Eric Keto
Keyword(s):  
Steady State ◽  
Transonic Flow ◽  
Initial Conditions ◽  
Steady State Solution ◽  
State Solution ◽  
Time Dependent ◽  
Hamiltonian Description ◽  
Bernoulli’S Equation ◽  
Wide Range ◽  
Isothermal And Adiabatic Flows

ABSTRACT Bondi and Parker derived a steady-state solution for Bernoulli’s equation in spherical symmetry around a point mass for two cases, respectively, an inward accretion flow and an outward wind. Left unanswered were the stability of the steady-state solution, the solution itself of time-dependent flows, whether the time-dependent flows would evolve to the steady state, and under what conditions a transonic flow would develop. In a Hamiltonian description, we find that the steady-state solution is equivalent to the Lagrangian implying that time-dependent flows evolve to the steady state. We find that the second variation is definite in sign for isothermal and adiabatic flows, implying at least linear stability. We solve the partial differential equation for the time-dependent flow as an initial-value problem and find that a transonic flow develops under a wide range of realistic initial conditions. We present some examples of time-dependent solutions.

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