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Phagocytosis refers to a process in which one cell type fully encloses and consumes unwanted cells, debris or particulate matter. It has an important role in immune systems through the destruction of pathogens and the inhibiting of cancerous cells. In this paper, we combine cell–cell adhesion and predator–prey modelling to generate a new model for phagocytosis that can relate the interaction between cells in both space and time. Stability analysis for both homogeneous and non-homogeneous steady states is provided for one-dimensional model indicating the range of parameters that leads to phagocytosis. Finally, the paper presents numerical results for both one and two-dimensional models, which show excellent agreement with a real phenomenon of bacteria phagocytized by neutrophil cell.

Dye-sensitized solar cells (DSSCs) are a novel approach to the renewable energy problem, allowing sunlight conversion while reducing production costs. Electricity generation is achieved through a series of chemical reactions designed to transport electrons as a means of creating a circuit. Current challenges facing their development is to reduce the impact of recombination reactions, whilst enhancing their efficiency. In this paper, we investigate the behaviour of DSSCs by solving a set of partial differential equations which are used to model the density of electrons in the conduction band of a DSSC's semiconductor and in the electrolyte couple. Previously, approaches to solve these equations are by assuming steady-state models and then making use of numerical methods. In this paper, we obtain full analytical solutions to these equations and based on values of parameters associated with DSSCs available in the literature, we derive results which can be used to determine the performance of DSSCs.

Coarse aggregates increase inhomogeneity and alter rheological properties of self-compacting concrete. In this work, the effect of coarse aggregates on self-compacting concrete is studied by considering its pipe flow. Self-compacting concrete is modelled as suspension of coarse aggregates in cement mortar and a two-phase Smoothed Particle Hydrodynamics (SPH) model is employed. Cement mortar, a yield-stress fluid, is modelled by fluid SPH particles, while each coarse aggregate is represented by a group of solid SPH particles, which moves as a rigid body. Simulation results show that, at the same volume fractions, the flow rate is higher for larger coarse aggregates, that is up to 24% in the investigated cases. Both effective yield stress and effective plastic viscosity of the concretes increase with the coarse aggregate content. Effective yield stress is smaller for bigger coarse aggregates. The reduction in effective yield stress is a consequence of bigger averaged gap between larger coarse aggregates through which the cement mortar flows.

A mathematical model is proposed to describe the interaction between graphite, polyester and silica solid substrates whose surfaces are functionalized by hydroxyl group, carboxyl group and fluorine atoms. Assuming a relaxed form of these solid substrates, they are modelled as rectangular boxes, while the functional groups are modelled as boundary surfaces. Using the 6–12 Lennard–Jones potential together with a continuum approximation, the total van der Waals interaction energy is obtained as a function of the interfacial separation distance between two surfaces of the solid substrates. Here, the adhesion energy is the minimum energy calculated at the equilibrium separation distance. Our results reveal that among the substrates studied, graphite has the potential to be used as a base material in paints because of its strong adhesion when it is paired with either polyester or silica surfaces.

Lipid nanotubes are of particular interest for use as a template to create various one-dimensional nanostructures and as a carrier for drug and gene delivery. Understanding the encapsulation process is therefore crucial for such development. This paper models the interactions between lipid nanotubes and spheres of gold and silver nanoparticles and determines the critical dimension of lipid nanotubes that maximises the interaction with the nanoparticles. Our results confirm the acceptance of gold and silver nanoparticles inside lipid nanotubes. Further, we find that the lipid nanotube of radius approximately 10.23 nm is most favourable to encapsulate both types of nanoparticles.

The selective separation of biomolecules is a critical process in food, biomedical and pharmaceutical industries. Due to its size and properties, mesoporous silica offers many advantages as a separation media for biomolecules such as proteins and enzymes. In this paper, we investigate mathematically the separation of proteins trypsin and lysozyme using mesoporous silica materials. These proteins are modelled as densely packed spheres, while the silica pore is assumed to have a cylindrical structure. The Lennard–Jones potential together with a continuum approximation is employed to determine the interaction among the proteins and the interaction between a protein and a silica pore. For these systems, the total interaction energies are obtained analytically as functions of the protein size and the pore dimensions. We find that the pore radii which give rise to the maximum adsorption energies for trypsin and lysozyme are 21.74 A and 17.74 A, respectively. Since the binding energy between any two protein molecules is found to be three orders of magnitude lower than the adsorption energy of the protein into the silica pore, proteins prefer to be separated and stay inside the pore. Further, we find that using silica pores with radii in the range between 17.23 A and 21.24 A allows the entrance of only lysozyme, as such separating lysozyme from trypsin. These results agree with previous experimental study, confirming that mesoporous silica pores may be used to separate trypsin–lysozyme mixture.

In this paper, we investigate the oscillation of a fullerene that is moving within the centre of a bundle of nanotubes. In particular, certain fullerene–nanotube bundle oscillators, namely C 60 -carbon nanotube bundle, C 60 -boron nitride nanotube bundle, B 36 N 36 -carbon nanotube bundle and B 36 N 36 -boron nitride nanotube bundle are studied using the Lennard–Jones potential and the continuum approach which assumes a uniform distribution of atoms on the surface of each molecule. We address issues regarding the maximal suction energies of the fullerenes which lead to the generation of the maximum oscillation frequency. Since bundles are also found to comprise double-walled nanotubes, this paper also examines the oscillation of a fullerene inside a double-walled nanotube bundle. Our results show that the frequencies obtained for the oscillation within double-walled nanotube bundles are slightly higher compared to those of single-walled nanotube bundle oscillators. Our primary purpose here is to extend a number of established results for carbon to the boron nitride nanostructures.

Nanostructures such as carbon nanotubes and fullerenes offer the means to create new mechanical devices operating at the nanoscale. Such devices include oscillators constructed from an inner carbon nanotube sliding inside another carbon nanotube. The resultant oscillatory frequency is found to be in the gigahertz range and they have applications in the computing industry for signalling devices, such as an ultra-fast optical filter. While most research in the area is dominated by molecular dynamics simulations, our approach here is to use elementary mechanical principles and classical applied mathematical modelling techniques to formulate explicit analytical criteria and ideal model behaviour. In this paper, we first investigate the suction force experienced by a single-walled carbon nanotube located near an open end of a semi-infinite single-walled carbon nanotube, using the Lennard-Jones potential and the continuum approximation. Second the equilibrium position of an offset inner tube with reference to the cross-section of the outer tube is determined Refereed/Peer-reviewed

Fundamental to the development of electrorheological nanofluids is a precise knowledge of the electrostatic force of attraction between uncharged particles subject to an applied external electric field. In order to design more effective materials, more complicated geometries have been proposed containing both conducting and dielectric materials. Accordingly, solution methods are needed which address the issues of accurately calculating the force of attraction for closely spaced, polarised, dielectric particles; while at the same time are amenable to the material boundaries of various particle geometries. In this paper the method of multipole re-expansion is employed for the problem of two spherical particles consisting of a core, coated with a conducting layer which in turn is surrounded by an outer dielectric coating of arbitrary size. These particles are imbedded in a dielectric medium and are subject to an externally applied electric field. An exact solution is given which has not appeared in the literature previously, and which can be exploited to efficiently determine accurate numerical values to be used for the design of new electrorheological fluids. Also the solution in various limits is analysed to recover firstly the solution for homogeneous dielectric particles, secondly to produce an exact solution for conducting media and thirdly to determine an accurate approximation for conducting spheres with very thin coatings, and which provides the opportunity to make use of existing conducting media theory. Quite remarkably, the solution for thin coatings is shown to differ from the conducting solution only by a factor of 1 - 1 / k , where k is the dielectric constant. As might be expected, the force would approach zero for coatings with very low dielectric constants, or approach the conducting solution when the dielectric constant is large. It is conjectured that this result may be applicable to bodies of any shape.

For axially symmetric flows of dilatant granular materials, the velocity equations uncouple from the stress equations in certain plastic regimes, and assuming dilatant double shearing a closed set of three first-order partial differential equations are obtained. These supposedly simple equations are deceptive, because although they are simple in appearance, the determination of exact solutions is non-trivial. For one of the known families of solutions which has not been studied previously, the authors present the non-linear ordinary differential equation for the stress angle ψ and determine two small ψ approximations. Furthermore, the stream function and streamlines are obtained for ψ determined numerically and from the two small ψ approximations. For purposes of comparison, the streamlines for three further known exact solutions are also presented. In addition, we briefly examine the circumstances for which solutions of the velocity equations satisfy the principle of non-negative plastic work. For example, we are able to establish that in the case when the velocity equations are derived from a plastic potential, the solutions always satisfy the principle when the material has no cohesion.

New exact analytical solutions are presented for both stress and velocity fields for a Coulomb–Mohr granular solid assuming non-dilatant double-shearing theory. The solutions determined apply to highly frictional materials for which the angle of internal friction φ is assumed equal to 90°. This major assumption is made primarily to facilitate exact analytical solutions, and it is discussed at length in the Introduction, both in the context of real materials which exhibit large angles of internal friction, and in the context of using the solutions derived here as the leading term in a regular perturbation solution involving powers of 1−sinφ. The analytical velocity fields so obtained are illustrated graphically by showing the direction of the principal stress as compared to the streamlines. The stress solutions are also exploited to determine the static stress distribution for a granular material contained within vertical boundaries and a horizontal base, which is assumed to have an infinitesimal central outlet through which material flows until a rat-hole of parabolic or cubic profile is obtained, and no further flow takes place. A rat-hole is a stable structure that may form in storage hoppers and stock-piles, preventing any further flow of material. Here we consider the important problems of two-dimensional parabolic rat-holes of profile y=ax2, and three-dimensional cubic rat-holes of profile z=ar3, which are both physically realistic in practice. Analytical solutions are presented for both two and three-dimensional rat-holes for the case of a highly frictional granular solid, which is stored at rest between vertical walls and a horizontal rigid plane, and which has an infinitesimal central outlet. These solutions are bona fide exact solutions of the governing equations for a Coulomb–Mohr granular solid, and satisfy exactly the free surface condition along the rat-hole surface, but approximate frictional conditions along the containing boundaries. The analytical solutions presented here constitute the only known solutions for any realistic rat-hole geometry, other than the classical solution which applies to a perfectly vertical cylindrical cavity.