scholarly journals On the stability of the μ(I) rheology for granular flow

2017 ◽  
Vol 833 ◽  
pp. 302-331 ◽  
Author(s):  
J. D. Goddard ◽  
Jaesung Lee
Keyword(s):  
Linear Stability ◽  
Granular Flow ◽  
Convective Instability ◽  
Continuum Model ◽  
Shear Bands ◽  
Closed Form Solution ◽  
Static Field ◽  
Form Solution ◽  
Base Flow ◽  
Field Equations

This article deals with the Hadamard instability of the so-called$\unicode[STIX]{x1D707}(I)$model of dense rapidly sheared granular flow, as reported recently by Barkeret al.(J. Fluid Mech., vol. 779, 2015, pp. 794–818). The present paper presents a more comprehensive study of the linear stability of planar simple shearing and pure shearing flows, with account taken of convective Kelvin wavevector stretching by the base flow. We provide a closed-form solution for the linear-stability problem and show that wavevector stretching leads to asymptotic stabilization of the non-convective instability found by Barkeret al.(J. Fluid Mech., vol. 779, 2015, pp. 794–818). We also explore the stabilizing effects of higher velocity gradients achieved by an enhanced-continuum model based on a dissipative analogue of the van der Waals–Cahn–Hilliard equation of equilibrium thermodynamics. This model involves a dissipative hyperstress, as the analogue of a special Korteweg stress, with surface viscosity representing the counterpart of elastic surface tension. Based on the enhanced-continuum model, we also present a model of steady shear bands and their nonlinear stability against parallel shearing. Finally, we propose a theoretical connection between the non-convective instability of Barkeret al.(J. Fluid Mech., vol. 779, 2015, pp. 794–818) and the loss of generalized ellipticity in the quasi-static field equations. Apart from the theoretical interest, the present work may suggest stratagems for the numerical simulation of continuum field equations involving the$\unicode[STIX]{x1D707}(I)$rheology and variants thereof.

Stability and Imperfections in Quasistatic Viscoplastic Solutions

1990 ◽  
Vol 43 (5S) ◽  
pp. S251-S255 ◽  
Author(s):  
T. Belytschko ◽  
B. Moran ◽  
M. Kulkarni
Keyword(s):  
Strain Field ◽  
Strain Softening ◽  
Fourier Spectrum ◽  
Shear Bands ◽  
Closed Form Solution ◽  
Step Function ◽  
Form Solution ◽  
Viscoplastic Material ◽  
Strain Fields ◽  
The Stability

The effect of imperfections on the structure of shear bands in strain-softening viscoplasticity is studied via a closed form solution. The stability of various solutions is then examined by varying the data through imperfections. It is shown that a step-function imperfection, such as commonly used in finite element solutions, leads to a step-function shear strain field, which is an unstable solution. Arbitrary C0 and C1 imperfections lead to C0 and C1 strain fields, respectively. Fourier analyses show that the imperfection scales the response of the viscoplastic material: the Fourier spectrum of the strain field is strongly influenced by the Fourier spectrum of the imperfection.

scholarly journals Non-Abelian firewall

2020 ◽  
Vol 29 (14) ◽  
pp. 2043003
Author(s):  
Douglas Singleton
Keyword(s):  
Energy Density ◽  
Equivalence Principle ◽  
Closed Form Solution ◽  
Information Loss ◽  
Form Solution ◽  
Field Equations ◽  
Yang Mills ◽  
Information Loss Paradox ◽  
The Cost ◽  
Mills Field

A simple, closed-form solution to the Yang-Mills field equations is presented which has a non-Abelian firewall — a spherical “horizon” where the energy density diverges. By the gravity/gauge duality, this non-Abelian firewall implies the existence of a gravitational firewall. Gravitational firewalls have been proposed as a way of resolving the information loss paradox, but at the cost of violating the equivalence principle.

A mathematical model for localization in granular flow

1992 ◽  
Vol 436 (1897) ◽  
pp. 217-250 ◽  
Keyword(s):  
Granular Flow ◽  
Continuum Model ◽  
Shear Banding ◽  
Shear Bands ◽  
Two Dimensional ◽  
Open Problems ◽  
Static Approximation ◽  
Particular Solution ◽  
Research Questions ◽  
Expository Paper

The phrase ‘localization of flow’ refers to a phenomenon in plasticity in which deformation becomes concentrated in thin bands of intense shearing. This paper proposes a two-dimensional continuum model which isolates some mathematical issues relevant to such shear banding. The challenge is to follow the solution after ill- posedness appears in the equations. A particular solution in this régime is constructed in the quasi-static approximation. The paper also lists several open problems suggested by the model. Besides addressing research questions, the paper is partly intended as an expository paper for mathematicians interested in shear bands.

Rapid granular flows down inclined planar chutes. Part 2. Linear stability analysis of steady flow solutions

2010 ◽  
Vol 652 ◽  
pp. 461-488 ◽  
Author(s):  
MARK J. WOODHOUSE ◽  
ANDREW J. HOGG
Keyword(s):  
Linear Stability ◽  
Granular Flow ◽  
Continuum Model ◽  
Three Dimensional ◽  
Linear Instability ◽  
Steady Flows ◽  
Density Inversion ◽  
Instability Mechanism ◽  
Flow Curves ◽  
Steady Solutions

The linear stability of steady solutions for a rapid granular flow down an inclined chute, modelled using a kinetic theory continuum model, is analysed. The previous studies of Forterre & Pouliquen (J. Fluid Mech., vol. 467, 2002, p. 361) and Mitarai & Nakanishi (J. Fluid Mech., vol. 507, 2004, p. 309) are extended by considering fully three-dimensional perturbations, allowing variations in both the cross-slope and downslope directions, as well as normal to the base. Our results demonstrate the existence of three qualitatively different unstable perturbations, each of which can be the most rapidly growing instability for different steady flows. By considering the linear stability of many steady solutions along macroscopic flow curves, we show that linear stability occurs in only a small part of parameter space, and furthermore the regions of linear instability do not correlate with density inversion of the underlying steady solutions. Our results suggest that inelastic clustering is the dominant instability mechanism.

On a Closed-Form Solution of Prandtl's System of Equations

2013 ◽  
Vol 40 (2) ◽  
pp. 106-114
Author(s):  
J. Venetis ◽  
Aimilios (Preferred name Emilios) Sideridis
Keyword(s):  
Closed Form ◽  
Closed Form Solution ◽  
Form Solution ◽  
System Of Equations

Plane Wave Resonance in the Tire Air Cavity as a Vehicle Interior Noise Source

1995 ◽  
Vol 23 (1) ◽  
pp. 2-10 ◽  
Author(s):  
J. K. Thompson
Keyword(s):  
Closed Form Solution ◽  
Form Solution ◽  
Interior Noise ◽  
Cavity Resonance ◽  
Test Results ◽  
Vehicle Interior ◽  
Air Cavity ◽  
Vehicle Interior Noise ◽  
Wave Resonance ◽  
Resonance Frequencies

Abstract Vehicle interior noise is the result of numerous sources of excitation. One source involving tire pavement interaction is the tire air cavity resonance and the forcing it provides to the vehicle spindle: This paper applies fundamental principles combined with experimental verification to describe the tire cavity resonance. A closed form solution is developed to predict the resonance frequencies from geometric data. Tire test results are used to examine the accuracy of predictions of undeflected and deflected tire resonances. Errors in predicted and actual frequencies are shown to be less than 2%. The nature of the forcing this resonance as it applies to the vehicle spindle is also examined.

Capacity Analysis for Correlated Multi-Hop MIMO Channels under Colored Noise

2015 ◽  
Vol 1 ◽  
pp. 41
Author(s):  
Nguyen N. Tran ◽  
Ha X. Nguyen
Keyword(s):  
Computational Complexity ◽  
Mutual Information ◽  
Closed Form Solution ◽  
Optimal Solution ◽  
Form Solution ◽  
Maximization Problem ◽  
Capacity Analysis ◽  
Mimo Channels ◽  
Average Mutual Information ◽  
Channel Output

A capacity analysis for generally correlated wireless multi-hop multi-input multi-output (MIMO) channels is presented in this paper. The channel at each hop is spatially correlated, the source symbols are mutually correlated, and the additive Gaussian noises are colored. First, by invoking Karush-Kuhn-Tucker condition for the optimality of convex programming, we derive the optimal source symbol covariance for the maximum mutual information between the channel input and the channel output when having the full knowledge of channel at the transmitter. Secondly, we formulate the average mutual information maximization problem when having only the channel statistics at the transmitter. Since this problem is almost impossible to be solved analytically, the numerical interior-point-method is employed to obtain the optimal solution. Furthermore, to reduce the computational complexity, an asymptotic closed-form solution is derived by maximizing an upper bound of the objective function. Simulation results show that the average mutual information obtained by the asymptotic design is very closed to that obtained by the optimal design, while saving a huge computational complexity.

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