Analytical stress and velocity fields for gravity flows of highly frictional granular materials

For granular materials the Coulomb-Mohr yield condition characterizes the two physical processes of inter-particle cohesion and inter-particle friction. The latter effect is quantified by the so-called angle of internal friction, denoted here by . The special case arising from zero angle of internal friction corresponds to the standard Tresca yield condition of metal plasticity. For certain materials such as coal, alumina filter cake, waste rock and silica, angles of internal friction occur in the vicinity 70°–80°, and therefore the study of an idealized granular theory with an angle of internal friction equal to ninety degrees has real practical significance. Here for the special case of =90°, the governing second-order nonlinear partial differential equations for the non-dilatant double-shearing model of granular flow are presented for both plane and axially symmetric flows, and a number of simple analytical solutions of these novel equations are determined. Some of these solutions are illustrated graphically by showing the orthogonal grids which give the maximum and minimum principal stress directions and by showing the streamlines which give the particle paths.     

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°.     

Modelling peptide nanotubes for artificial ion channels

We investigate the van der Waals interaction of D,L-Ala cyclopeptide nanotubes and various ions, ion-water clusters and C(60) fullerenes, using the Lennard-Jones potential and a continuum approach which assumes that the atoms are smeared over the peptide nanotube providing an average atomic density. Our results predict that Li(+), Na(+), Rb(+) and Cl(-) ions and ion-water clusters are accepted into peptide nanotubes of 8.5 ? internal diameter whereas the C(60) molecule is rejected. The model indicates that the C(60) molecule is accepted into peptide nanotubes of 13 ? internal diameter, suggesting that the interaction energy depends on the size of the molecule and the internal diameter of the peptide nanotube. This result may be useful for the design of peptide nanotubes for drug delivery applications. Further, we also find that the ions prefer a position inside the peptide ring where the energy is minimum. In contrast, Li(+)-water clusters prefer to be in the space between each peptide ring.     

A formal exact mathematical solution for a sloping rat-hole in a highly frictional granular solid

Summary. This paper provides a formal exact analytical solution to a rat-hole with a sloping base in two and three dimensions for a highly frictional granular material. A rat-hole is the general term used to describe those stable cavities, which frequently occur in storage hoppers and stock piles, whose formation prevents further material falling through the outlet. Figure 1a depicts the typical geometric configuration, comprising upper and lower sloping surfaces that form a channel or cylindrical cavity. In granular industries this is a commonly occurring situation, for example, where the flow of material from a hopper ceases due to the formation of a stable almost cylindrical vertical cavity. Despite their practical importance, the only analytical solution applies to the perfectly cylindrical cavity, assumed infinite in length with no upper sloping surface. In order to determine analytical solutions to more realistic situations, it is necessary to make compromises with regard to both geometric and constitutive considerations. Here, for both two and three-dimensional rat-holes, we present analytical parametric solutions for the special case of a highly frictional granular material, where the angle of internal friction is equal to ninety degrees. In addition, we assume that the highly frictional granular material is at the point of yield on a sloping rigid base, and with an infinitesimal central outlet as shown in Fig. 1b. The solutions given here are bona fide exact solutions of the governing equations for a Coulomb-Mohr granular solid, and satisfy exactly the free surface conditions on the sloping upper surface and a frictional condition along the sloping rigid base. We emphasize that while all zero-stress boundary conditions are correctly satisfied, and the solutions constitute the only known exact analytical solutions for a realistic rat-hole geometry, the solutions for both geometries exhibit infinite values of the other stress component on the free surface. This feature arises as a consequence of assuming an angle of internal friction equal to ninety degrees, and throws doubt on the physical applicability of the formal exact solution.     

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°.     

Encapsulation of a benzene molecule into a carbon nanotube

Aromatic hydrocarbon molecules encapsulated in carbon nanotubes have been proposed for applications as semiconductors. They can be formed by exploiting the van der Waals interaction as a simple method to incorporate molecules into carbon nanotubes. However, the existence of energy barriers near the open ends of carbon nanotubes may be an obstacle for molecules entering carbon nanotubes. In this paper, we investigate the encapsulation mechanism of a typical aromatic hydrocarbon, namely a benzene molecule, into a carbon nanotube in order to determine the dependence on radius of the tube. A continuous approach which assumes that the molecular interactions can be approximated using average atomic densities together with the semi-empirical Lennard–Jones potential function is adopted, and an analytical expression for the interaction energy is obtained which may be readily evaluated by algebraic computer packages. In particular, we determine the threshold radius of the carbon nanotube for which the benzene molecule will enter the carbon nanotube. The analytical approach adopted here provides a computationally rapid procedure for the determination of critical numerical values.