Constitutive relations for the shear band evolution in granular matter under large strain

A so-called ＂split-bottom ring shear cell＂ leads to wide shear bands under slow, quasi-static deformation. Unlike normal cylindrical Couette shear cells or rheometers, the bottom plate is split such that the outer part of it can move with the outer wall, while the other part （inner disk） is immobile. From discrete element simulations （DEM）, several continuum fields like the density, velocity, deformation gradient and stress are computed and evaluated with the goal to formulate objective constitutive relations for the powder flow behavior. From a single simulation, by applying time- and （local） space-averaging, a non-linear yield surface is obtained with peculiar stress dependence. The anisotropy is always smaller than the macroscopic friction coefficient. However, the lower bound of anisotropy increases with the strain rate, approaching the maximum according to a stretched exponential with a specific rate that is consistent with a shear path of about one particle diameter.     

On the stability and dissipation of wall boundary conditions for compressible flows

Characteristic formulations for boundary conditions have demonstrated their effectiveness to handle inlets and outlets, especially to avoid acoustic wave reflections. At walls, however, most authors use simple Dirichlet or Neumann boundary conditions, where the normal velocity (or pressure gradient) is set to zero. This paper demonstrates that there are significant differences between characteristic and Dirichlet methods at a wall and that simulations are more stable when using walls modelled with a characteristic wave decomposition. The derivation of characteristic methods yields an additional boundary term in the continuity equation, which explains their increased stability. This term also allows to handle the two acoustic waves going towards and away from the wall in a consistent manner. Those observations are confirmed by stability matrix analysis and one‐ and two‐dimensional simulations of acoustic modes in cavities. Copyright © 2009 John Wiley & Sons, Ltd.     

Linear and non‐linear simulations of feedback control in plane Poiseuille flow

This paper examines the performance of optimal linear quadratic state and output feedback controllers in stabilizing two‐dimensional perturbations in a plane Poiseuille flow. The synthesis of the controllers is based on a linearized model of the flow using a new set of interpolating polynomials in the wall‐normal direction, which automatically satisfy the homogeneous Dirichlet and Neumann boundary conditions at the walls and eliminate spurious eigenvalues. The controllers are implemented into a non‐linear Navier–Stokes solver, which is modified to compute the evolution of the flow perturbations. Two cases are examined, one with small initial disturbances that do not violate the linearity assumptions and the other with much larger disturbances that trigger the non‐linear convection terms. For the smallest disturbances, the solver accurately reproduced the results of the linear simulations of open‐ and closed‐loop systems. The simulations for the larger disturbances without control showed a rapid initial growth but the flow soon reached a saturated state in agreement with previous findings in the literature. The large initial growth is a consequence of the non‐normal nature of the system dynamics. The state feedback and output feedback controllers were able to reduce significantly the perturbation energy. For the larger disturbances, the energy calculated from the state variables is well below the energy evaluated by direct integration of the velocity field. This is probably due to the non‐linear terms transferring energy to harmonics of the considered wavenumber, which are not sensed by the linear controller. Copyright © 2008 John Wiley & Sons, Ltd.     

High‐order boundary conditions for linearized shallow water equations with stratification,dispersion and advection

The two‐dimensional linearized shallow water equations are considered in unbounded domains with density stratification. Wave dispersion and advection effects are also taken into account. The infinite domain is truncated via a rectangular artificial boundary ??, and a high‐order open boundary condition (OBC) is imposed on ??. Then the problem is solved numerically in the finite domain bounded by ??. A recently developed boundary scheme is employed, which is based on a reformulation of the sequence of OBCs originally proposed by Higdon. The OBCs can easily be used up to any desired order. They are incorporated here in a finite difference scheme. Numerical examples are used to demonstrate the performance and advantages of the computational method, with an emphasis is on the effect of stratification. Published in 2004 by John Wiley & Sons, Ltd.     

A two-dimensional effective model describing fluid–structure interaction in blood flow: analysis, simulation and experimental validation

We derive a closed system of effective equations describing a time-dependent flow of a viscous incompressible Newtonian fluid through a long and narrow elastic tube. The 3D axially symmetric incompressible Navier–Stokes equations are used to model the flow. Two models are used to describe the tube wall: the linear membrane shell model and the linearly elastic membrane and the curved, linearly elastic Koiter shell model. We study the behavior of the coupled fluid–structure interaction problem in the limit when the ratio between the radius and the length of the tube, , tends to zero. We obtain the reduced equations that are of Biot type with memory. An interesting feature of the reduced equations is that the memory term explicitly captures the viscoelastic nature of the coupled problem. Our model provides significant improvement over the standard 1D approximations of the fluid–structure interaction problem, all of which assume an ad hoc closure assumption for the velocity profile. We performed experimental validation of the reduced model using a mock circulatory flow loop assembled at the Cardiovascular Research Laboratory at the Texas Heart Institute. Experimental results show excellent agreement with the numerically calculated solution. Major applications include blood flow through large human arteries. To cite this article: S. Čanić et al., C. R. Mecanique 333 (2005).     

A comparative study on low‐order,amplitude‐equation and perturbation approaches in thermal convection

A comparison among three weakly nonlinear approaches for thermo‐gravitational instability in a Newtonian fluid layer heated from below is presented. First, the dynamical systems describing the time evolution of the problem from different weakly nonlinear approaches, namely, the Lorenz model, the amplitude equations and the perturbation expansion approaches are obtained. Next, the steady states and their stability, as well as the transient behaviour are obtained from each dynamical system. The similarity and difference among the three models are emphasized. The role of each of the nondimensional groups, the Rayleigh number and the Prandtl number is compared for the three models. The different approaches lead to similar behaviours when the Rayleigh number is just above its critical value and Prandtl number is high. However, only the dynamical system obtained from the amplitude equations is able to reflect the role of the Prandtl number. On the other hand, the amplitude equations and perturbation expansion techniques are not suitable for predicting the uniform oscillatory behaviour observed frequently in Rayleigh–Bénard convection. The novelty of the current work lies in studying the critical differences in the findings of the three popular approaches to investigate weakly nonlinear thermal convection for the first time. Copyright © 2011 John Wiley & Sons, Ltd.     

Deformation, stress state and thermodynamic force for a growing void in an elastic–plastic material

The growth of a spherical void in an elastic–plastic body, subjected to external pressure or tension and a gas pressure as well as a surface stress at the void surface, is investigated. The deformation, strain and stress state in the full body is presented. In addition, the local and global energy terms are calculated. Finally the total thermodynamic force on the void surface as well as the total dissipation are evaluated and compared allowing the calculation of the mechanical contribution to void growth due to diffusion of vacancies generated by plastification or irradiation.     

The influence of source terms on stability,accuracy and conservation in two‐dimensional shallow flow simulation using triangular finite volumes

The two‐dimensional shallow water model is a hyperbolic system of equations considered well suited to simulate unsteady phenomena related to some surface wave propagation. The development of numerical schemes to correctly solve that system of equations finds naturally an initial step in two‐dimensional scalar equation, homogeneous or with source terms. We shall first provide a complete formulation of the second‐order finite volume scheme for this equation, paying special attention to the reduction of the method to first order as a particular case. The explicit first and second order in space upwind finite volume schemes are analysed to provide an understanding of the stability constraints, making emphasis in the numerical conservation and in the preservation of the positivity property of the solution when necessary in the presence of source terms. The time step requirements for stability are defined at the cell edges, related with the traditional Courant–Friedrichs–Lewy (CFL) condition. Copyright © 2007 John Wiley & Sons, Ltd.