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.

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.

scholarly journals A characterisation of Hilbert spaces via orthogonality and proximinality

2005 ◽  
Vol 71 (1) ◽  
pp. 107-111
Author(s):  
Fathi B. Saidi
Keyword(s):  
Banach Space ◽  
Hilbert Space ◽  
Banach Spaces ◽  
Hilbert Spaces ◽  
Real Banach Space ◽  
Nonzero Element ◽  
Dimensional Subspace ◽  
Two Dimensional ◽  
Orthogonal Direction ◽  
Open Problems

In this paper we adopt the notion of orthogonality in Banach spaces introduced by the author in [6]. There, the author showed that in any two-dimensional subspace F of E, every nonzero element admits at most one orthogonal direction. The problem of existence of such orthogonal direction was not addressed before. Our main purpose in this paper is the investigation of this problem in the case where E is a real Banach space. As a result we obtain a characterisation of Hilbert spaces stating that, if in every two-dimensional subspace F of E every nonzero element admits an orthogonal direction, then E is isometric to a Hilbert space. We conclude by presenting some open problems.

Numerical Analysis of Bridge Expansion-Induced Rail Deformation of Ballast Truck

2014 ◽  
Vol 580-583 ◽  
pp. 3208-3214 ◽  
Author(s):  
Zhen Wei Xiong ◽  
Xin Ling Liang ◽  
Xian Xing Dai ◽  
Ping Wang
Keyword(s):  
Granular Flow ◽  
Element Model ◽  
Discrete Element Model ◽  
Board Structure ◽  
Vertical Deformation ◽  
Two Dimensional ◽  
Running Safety ◽  
Bridge Plate ◽  
Ballast Track ◽  
The Relationship

when the ballast track stretch with the bridge, ballast which is near expansion joint will move confusedly. As a result, rail produced vertical deformation. The deformation will affect the running safety and comfortability of train. At present, there are two kinds of treatments which are cover board structure and baffle structure to deal ballast’s movement. Aiming at the different modes of stretching when the two kinds of structures and different arrangement condition of bridge plate are applied, the rail-sleeper-ballast discrete element model is developed by the method of two-dimensional granular flow. The relationship between rail deformation and bridge expansion is analyzed on the foundation of the model. Results show as flows: when bridge extends or shortens, rail always produced upwarp deformation. Bridge plate should arrange asymmetrically. Like this, the rail deformation decrease by 40%. And adopting the baffle structure can effectively reduce the influence of bridge expansion in ballast truck.

A continuum model for two-dimensional dislocation distributions

1987 ◽  
Vol 55 (5) ◽  
pp. 617-629 ◽  
Author(s):  
A. K. Head ◽  
S. D. Howison ◽  
J. R. Ockendon ◽  
J. B. Titchener ◽  
P. Wilmott
Keyword(s):  
Continuum Model ◽  
Two Dimensional

scholarly journals Free boundary problems in shock reflection/diffraction and related transonic flow problems

2015 ◽  
Vol 373 (2050) ◽  
pp. 20140276 ◽  
Author(s):  
Gui-Qiang Chen ◽  
Mikhail Feldman
Keyword(s):  
Free Boundary ◽  
High Speed ◽  
Free Boundary Problems ◽  
Fluid Flows ◽  
Shock Reflection ◽  
Two Dimensional ◽  
Open Problems ◽  
Boundary Problems ◽  
Flow Problems ◽  
Concave Corner

Shock waves are steep wavefronts that are fundamental in nature, especially in high-speed fluid flows. When a shock hits an obstacle, or a flying body meets a shock, shock reflection/diffraction phenomena occur. In this paper, we show how several long-standing shock reflection/diffraction problems can be formulated as free boundary problems, discuss some recent progress in developing mathematical ideas, approaches and techniques for solving these problems, and present some further open problems in this direction. In particular, these shock problems include von Neumann's problem for shock reflection–diffraction by two-dimensional wedges with concave corner, Lighthill's problem for shock diffraction by two-dimensional wedges with convex corner, and Prandtl-Meyer's problem for supersonic flow impinging onto solid wedges, which are also fundamental in the mathematical theory of multidimensional conservation laws.

scholarly journals Effect of boundary on the two-dimensional inclined channel for a dilute granular flow distribution

2004 ◽  
Vol 53 (10) ◽  
pp. 3389
Author(s):  
Zhou Ying ◽  
Bao De-Song ◽  
Zhang Xun-Sheng ◽  
Lei Zhe-Min ◽  
Hu Guo-Qi ◽  
...  
Keyword(s):  
Granular Flow ◽  
Flow Distribution ◽  
Two Dimensional ◽  
Inclined Channel

scholarly journals Symmetric shear banding and swarming vortices in bacterial superfluids

2018 ◽  
Vol 115 (28) ◽  
pp. 7212-7217 ◽  
Author(s):  
Shuo Guo ◽  
Devranjan Samanta ◽  
Yi Peng ◽  
Xinliang Xu ◽  
Xiang Cheng
Keyword(s):  
High Speed ◽  
Shear Banding ◽  
Shear Bands ◽  
Local Stress ◽  
Transport Processes ◽  
Shear Stresses ◽  
Planar Shear ◽  
Collective Motions ◽  
Swimming Bacteria ◽  
Microscopic Origin

Bacterial suspensions—a premier example of active fluids—show an unusual response to shear stresses. Instead of increasing the viscosity of the suspending fluid, the emergent collective motions of swimming bacteria can turn a suspension into a superfluid with zero apparent viscosity. Although the existence of active superfluids has been demonstrated in bulk rheological measurements, the microscopic origin and dynamics of such an exotic phase have not been experimentally probed. Here, using high-speed confocal rheometry, we study the dynamics of concentrated bacterial suspensions under simple planar shear. We find that bacterial superfluids under shear exhibit unusual symmetric shear bands, defying the conventional wisdom on shear banding of complex fluids, where the formation of steady shear bands necessarily breaks the symmetry of unsheared samples. We propose a simple hydrodynamic model based on the local stress balance and the ergodic sampling of nonequilibrium shear configurations, which quantitatively describes the observed symmetric shear-banding structure. The model also successfully predicts various interesting features of swarming vortices in stationary bacterial suspensions. Our study provides insights into the physical properties of collective swarming in active fluids and illustrates their profound influences on transport processes.

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