We present a two-dimensional micro-scale model for crystal dissolution and precipitation in a porous medium. The local geometry of the pore is represented as a thin strip and the model allows for changes in the pore volume. A formal limiting argument, for the limit of the width of the strip going to zero, leads to a system of one-dimensional effective upscaled equations. We show that the effective equations allow for travelling-wave solutions and prove the existence and uniqueness of these. Numerical solutions of the effective equations are compared with numerical solutions of the original equations on the thin strip and with analytical results. We also show that a model from the literature that does not allow changes in the pore volume can be obtained from the present model as a formal limit.

A configuration of near-equilibrium liquid droplets sitting on a precursor film which wets the entire substrate can coarsen in time by two different mechanisms: collapse or collision of droplets. The collapse mechanism, i.e., a larger droplet grows at the expense of a smaller one by mass exchange through the precursor film, is also known as Ostwald ripening. As was shown by K. B. Glasner and T. P. Witelski (‘Collision versus collapse of droplets in coarsening of dewetting thin films’, Phys. D209 (1–4), 2005, 80–104) in case of a one-dimensional substrate, the migration of droplets may interfere with Ostwald ripening: The configuration can coarsen by collision rather than by collapse. We study the role of migration in a two-dimensional substrate for a whole range of mobilities. We characterize the velocity of a single droplet immersed into an environment with constant flux field far away. This allows us to describe the dynamics of a droplet configuration on a two-dimensional substrate by a system of ODEs. In particular, we find by heuristic arguments that collision can be a relevant coarsening mechanism.

Adhesion of cells to one another and their environment is an important regulator of many biological processes but has proved difficult to incorporate into continuum mathematical models. This paper develops further the new modelling approach proposed by Armstrong et al. (A continuum approach to modelling cell–cell adhesion, J. Theor. Biol. 243: 98–113, 2006). The models studied in the present paper use an integro-partial differential equation for cell behaviour, in which the integral represents the sensing by cells of their local environment. This enables an effective representation of cell–cell adhesion, as well as random cell movement, and cell proliferation. The authors use this modelling approach to investigate the ability of cell–cell adhesion to generate spatial patterns during cell aggregation. The model is also extended to give a new representation of cancer growth, whose solutions reflect the balance between cell–cell and cell–matrix adhesion in regulating cancer invasion. The non-local term in these models means that there is no standard theory from which one can deduce the boundedness required for biological realism: specifically, solutions for cell density must lie between zero and a positive density corresponding to close cell packing. Here the authors derive a number of conditions, each of which is sufficient for the required boundedness, and they demonstrate numerically that cell density increases above the upper bound for some parameter sets not satisfying these conditions. Finally the authors outline what they regard as the main mathematical challenges for future work on boundedness in models of this type.

We consider a solution of a mono-component oil and wax. The latter is dissolved in the oil if the temperature is above the so-called cloud point (which depends on the concentration) and it segregates in the form of solid crystals if temperature is below the cloud point. As the solid fraction of wax increases, the diffusivity of liquid wax in the oil decreases (gelification), eventually vanishing. We study a one-dimensional model where temperature is initially above the cloud point and then it is lowered to induce diffusion and gelification. We formulate the relevant mathematical problem (a free boundary problem), studying its well-posedness and showing some qualitative results.