New stress and velocity fields for highly frictional granular materials

The idealised theory for the quasi-static flow of granular materials which satisfy the Coulomb-Mohr hypothesis is considered. This theory arises in the limit that the angle of internal friction approaches $\pi/2$, and accordingly these materials may be referred to as being `highly frictional'. In this limit, the stress field for both two-dimensional and axially symmetric flows may be formulated in terms of a single nonlinear second order partial differential equation for the stress angle. To obtain an accompanying velocity field, a flow rule must be employed. Assuming the non-dilatant double-shearing flow rule, a further partial differential equation may be derived in each case, this time for the streamfunction. Using Lie symmetry methods, a complete set of group-invariant solutions is derived for both systems, and through this process new exact solutions are constructed. Only a limited number of exact solutions for gravity driven granular flows are known, so these results are potentially important in many practical applications. The problem of mass flow through a two-dimensional wedge hopper is examined as an illustration.

Free surface problems for static Coulomb-Mohr granular solids

This paper is concerned with some plane strain and axially symmetric free surface problems which arise in the study of static granular solids that satisfy the Coulomb-Mohr yield condition. Such problems are inherently nonlinear, and hence difficult to attack analytically. Given a Coulomb friction condition holds on a solid boundary, it is shown that the angle a free surface is allowed to attach to the boundary is dependent only on the angle of wall friction, assuming the stresses are all continuous at the attachment point, and assuming also that the coefficient of cohesion is nonzero. As a model problem, the formation of stable cohesive arches in hoppers is considered. This undesirable phenomena is an obstacle to flow, and occurs when the hopper outlet is too small. Typically, engineers are concerned with predicting the critical outlet size for a given hopper and granular solid, so that for hoppers with outlets larger than this critical value, arching cannot occur. This is a topic of considerable practical interest, with most accepted engineering methods being conservative in nature. Here, the governing equations in two limiting cases (small cohesion and high angle of internal friction) are considered directly. No information on the critical outlet size is found; however solutions for the shape of the free boundary (the arch) are presented, for both plane and axially symmetric geometries.

Perturbation solutions for flow through symmetrical hoppers with inserts and asymmetrical wedge hoppers

Under certain circumstances, an industrial hopper which operates under the "funnel-flow" regime can be converted to the "mass-flow" regime with the addition of a flow-corrective insert. This paper is concerned with calculating granular flow patterns near the outlet of hoppers that incorporate a particular type of insert, the cone-in-cone insert. The flow is considered to be quasi-static, and governed by the Coulomb-Mohr yield condition together with the non-dilatant double-shearing theory. In two dimensions, the hoppers are wedge-shaped, and as such the formulation for the wedge-in-wedge hopper also includes the case of asymmetrical hoppers. A perturbation approach, valid for high angles of internal friction, is used for both two-dimensional and axially symmetric flows, with analytic results possible for both leading order and correction terms. This perturbation scheme is compared with numerical solutions to the governing equations, and is shown to work very well for angles of internal friction in excess of 45 degree.

Optimal fluid injection strategies for in situ mineral leaching in two-dimensions

This paper examines the ground-water flow problem associated with the injection and recovery of certain corrosive fluids into mineral bearing rock. The aim is to dissolve the minerals in situ, and then recover them in solution. In general, it is not possible to recover all the injected fluid, which is of concern economically and environmentally. However, a new strategy is proposed here, that allows all the leaching fluid to be recovered. A mathematical model of the situation is solved approximately using an asymptotic solution, and exactly using a boundary integral approach. Solutions are shown for two-dimensional flow, which is of some practical interest as it is achievable in old mine tunnels, for example.

An analytical method to solve a general class of nonlinear reactive transport models

Three recent papers published in Chemical Engineering Journal studied the solution of a model of diffusion and nonlinear reaction using three different methods. Two of these studies obtained series solutions using specialized mathematical methods, known as the Adomian decomposition method and the homotopy analysis method. Subsequently it was shown that the solution of the same particular model could be written in terms of a transcendental function called Gauss’ hypergeometric function. These three previous approaches focused on one particular reactive transport model. This particular model ignored advective transport and considered one specific reaction term only. Here we generalize these previous approaches and develop an exact analytical solution for a general class of steady state reactive transport models that incorporate (i) combined advective and diffusive transport, and (ii) any sufficiently differentiable reaction term R(C). The new solution is a convergent Maclaurin series. The Maclaurin series solution can be derived without any specialized mathematical methods nor does it necessarily involve the computation of any transcendental function. Applying the Maclaurin series solution to certain case studies shows that the previously published solutions are particular cases of the more general solution outlined here. We also demonstrate the accuracy of the Maclaurin series solution by comparing with numerical solutions for particular cases.

Generalised finite volume strategies for simulating transport in strongly orthotropic porous media

In this work two different finite volume computational strategies for solving a representative two-dimensional diffusion equation in an orthotropic medium are considered. When the diffusivity tensor is treated as linear, this problem admits an analytic solution used for analysing the accuracy of the proposed numerical methods. In the first method, the gradient approximation techniques discussed by Jayantha and Turner [Numerical Heat Transfer, Part B: Fundamentals, 40, pp.367–390, 2001] are applied directly to the

Efficient simulation of unsaturated flow using exponential time integration

We assess the performance of an exponential integrator for advancing stiff, semidiscrete formulations of the unsaturated Richards equation in time. The scheme is of second order and explicit in nature but requires the action of the matrix function φ(A) where φ(z) = [exp(z) - 1]/z on a suitability defined vector v at each time step. When the matrix A is large and sparse, φ(A)v can be approximated by Krylov subspace methods that require only matrix-vector products with A. We prove that despite the use of this approximation the scheme remains second order. Furthermore, we provide a practical variable-stepsize implementation of the integrator by deriving an estimate of the local error that requires only a single additional function evaluation. Numerical experiments performed on two-dimensional test problems demonstrate that this implementation outperforms second-order, variable-stepsize implementations of the backward differentiation formulae.

Corrected mean-field models for spatially-dependent advection-diffusion-reaction phenomena

In the exclusion-process literature, mean-field models are often derived by assuming that the occupancy status of lattice sites is independent. Although this assumption is questionable, it is the foundation of many mean-field models. In this work we develop methods to relax the independence assumption for a range of discrete exclusion process-based mechanisms motivated by applications from cell biology. Previous investigations that focussed on relaxing the independence assumption have been limited to studying initially-uniform populations and ignored any spatial variations. By ignoring spatial variations these previous studies were greatly simplified due to translational invariance of the lattice. These previous corrected mean-field models could not be applied to many important problems in cell biology such as invasion waves of cells that are characterised by moving fronts. Here we propose generalised methods that relax the independence assumption for spatially inhomogeneous problems, leading to corrected mean-field descriptions of a range of exclusion process-based models that incorporate (i) unbiased motility, (ii) biased motility, and (iii) unbiased motility with agent birth and death processes. The corrected mean-field models derived here are applicable to spatially variable processes including invasion wave type problems. We show that there can be large deviations between simulation data and traditional mean-field models based on invoking the independence assumption. Furthermore, we show that the corrected mean-field models give an improved match to the simulation data in all cases considered.

Note on similarity solutions for viscous flow over an impermeable and non-linearly (quadratic) stretching sheet

Similarity solutions for flow over an impermeable, non-linearly (quadratic) stretching sheet were studied recently by Raptis and Perdikis (Int. J. Non-linear Mech. 41 (2006) 527–529) using a stream function of the form ψ=αxf(η)+βx2g(η). A fundamental error in their problem formulation is pointed out. On correction, it is shown that similarity solutions do not exist for this choice of ψ

Towards a classification of criterion referenced assessment models in mathematics courses : student and academic perspectives

We present the findings of a study into the implementation of explicitly criterion- referenced assessment in undergraduate courses in mathematics. We discuss students' concepts of criterion referencing and also the various interpretations that this concept has among mathematics educators. Our primary goal was to move towards a classification of criterion referencing models in quantitative courses. A secondary goal was to investigate whether explicitly presenting assessment criteria to students was useful to them and guided them in responding to assessment tasks. The data and feedback from students indicates that while students found the criteria easy to understand and useful in informing them as to how they would be graded, it did not alter the way the actually approached the assessment activity.

Direct instruction in mathematics : issues for schools with high Indigenous enrolments : a literature review

Direct instruction, an approach that is becoming familiar to Queensland schools that have high Aboriginal and Torres Strait Islander populations, has been gaining substantial political and popular support in the United States of America [USA], England and Australia. Recent examples include the No Child Left Behind policy in the USA, the British National Numeracy Strategy and in Australia, Effective Third Wave Intervention Strategies. Direct instruction, stems directly from the model created in the 1960s under a Project Follow Through grant. It has been defined as a comprehensive system of education involving all aspects of instruction. Now in its third decade of influencing curriculum, instruction and research, direct instruction is also into its third decade of controversy because of its focus on explicit and highly directed instruction for learning. Characteristics of direct instruction are critiqued and discussed to identify implications for teaching and learning for Indigenous students.

Continuous passive motion applied to whole joints stimulates chondrocyte blosynthesis of PRG4

Continuous passive motion (CPM) is currently a part of patient rehabilitation regimens after a variety of orthopedic surgical procedures. While CPM can enhance the joint healing process, the direct effects of CPM on cartilage metabolism remain unknown. Recent in vivo and in vitro observations suggest that mechanical stimuli can regulate articular cartilage metabolism of proteoglycan 4 (PRG4), a putative lubricating and chondroprotective molecule found in synovial fluid and at the articular cartilage surface. ----- ----- Objectives: (1) Determine the topographical variation in intrinsic cartilage PRG4 secretion. (2) Apply a CPM device to whole joints in bioreactors and assess effects of CPM on PRG4 biosynthesis.----- ----- Methods: A bioreactor was developed to apply CPM to bovine stifle joints in vitro. Effects of 24 h of CPM on PRG4 biosynthesis were determined.----- ----- Results: PRG4 secretion rate varied markedly over the joint surface. Rehabilitative joint motion applied in the form of CPM regulated PRG4 biosynthesis, in a manner dependent on the duty cycle of cartilage sliding against opposing tissues. Specifically, in certain regions of the femoral condyle that were continuously or intermittently sliding against meniscus and tibial cartilage during CPM, chondrocyte PRG4 synthesis was higher with CPM than without.----- ----- Conclusions: Rehabilitative joint motion, applied in the form of CPM, stimulates chondrocyte PRG4 metabolism. The stimulation of PRG4 synthesis is one mechanism by which CPM may benefit cartilage and joint health in post-operative rehabilitation. (C) 2006 Osteoarthritis Research Society International. Published by Elsevier Ltd. All rights reserved.

Real-time evaluation of driver’s alertness on highways

Suburbanisation has been internationally a major phenomenon in the last decades. Suburb-to-suburb routes are nowadays the most widespread road journeys; and this resulted in an increment of distances travelled, particularly on faster suburban highways. The design of highways tends to over-simplify the driving task and this can result in decreased alertness. Driving behaviour is consequently impaired and drivers are then more likely to be involved in road crashes. This is particularly dangerous on highways where the speed limit is high. While effective countermeasures to this decrement in alertness do not currently exist, the development of in-vehicle sensors opens avenues for monitoring driving behaviour in real-time. The aim of this study is to evaluate in real-time the level of alertness of the driver through surrogate measures that can be collected from in-vehicle sensors. Slow EEG activity is used as a reference to evaluate driver's alertness. Data are collected in a driving simulator instrumented with an eye tracking system, a heart rate monitor and an electrodermal activity device (N=25 participants). Four different types of highways (driving scenario of 40 minutes each) are implemented through the variation of the road design (amount of curves and hills) and the roadside environment (amount of buildings and traffic). We show with Neural Networks that reduced alertness can be detected in real-time with an accuracy of 92% using lane positioning, steering wheel movement, head rotation, blink frequency, heart rate variability and skin conductance level. Such results show that it is possible to assess driver's alertness with surrogate measures. Such methodology could be used to warn drivers of their alertness level through the development of an in-vehicle device monitoring in real-time drivers' behaviour on highways, and therefore it could result in improved road safety.

Driving performance impairments due to hypovigilance on monotonous roads

Drivers' ability to react to unpredictable events deteriorates when exposed to highly predictable and uneventful driving tasks. Highway design reduces the driving task mainly to a lane-keeping manoeuvre. Such a task is monotonous, providing little stimulation and this contributes to crashes due to inattention. Research has shown that driver's hypovigilance can be assessed with EEG measurements and that driving performance is impaired during prolonged monotonous driving tasks. This paper aims to show that two dimensions of monotony - namely road design and road side variability - decrease vigilance and impair driving performance. This is the first study correlating hypovigilance and driver performance in varied monotonous conditions, particularly on a short time scale (a few seconds). We induced vigilance decrement as assessed with an EEG during a monotonous driving simulator experiment. Road monotony was varied through both road design and road side variability. The driver's decrease in vigilance occurred due to both road design and road scenery monotony and almost independently of the driver's sensation seeking level. Such impairment was also correlated to observable measurements from the driver, the car and the environment. During periods of hypovigilance, the driving performance impairment affected lane positioning, time to lane crossing, blink frequency, heart rate variability and non-specific electrodermal response rates. This work lays the foundation for the development of an in-vehicle device preventing hypovigilance crashes on monotonous roads.