### Shear flow in da Vinci fluids

Working in collaboration with __Prof. Rafi Blumenfeld__, this project attempted to model horizontal shear flow of granular matter with the da Vinci fluid model, which proposes that the primary mechanism of energy dissipation in a granular fluid is solid friction between infinitesimal layers. The threshold nature of this process - static friction has greater magnitude than dynamic friction - leads naturally to formation of sections of granular fluid that move together as though a solid, known as plugs. We investigated the predictions of this model by formulating the da Vinci fluid equations for a two dimensional system representing an infinite horizontal slab of fluid, bounded by surfaces that can be sheared at some velocity, as shown in the diagram opposite. Through mathematical analysis and construction of a computaitonal solver we were able to investigate how this system evolves in time.

Further to these predictions we considered how the density of the fluid may vary with fluid velocity and pressure, how the effective frictional coefficient between layers may depend on this variation in density, and how such relationships might affect formation and splitting of plugs.

### Derivation of packing fraction for packed ellipses

This project took the first steps in an effort to theoretically derive the packing fraction for random packings of identical ellipses. We separate the problem into two parts, one depending on the packing procedure and one that does not, and we outline a method for calculating the relevant probability density function for the latter. The method is based on analysis of the graph formed by connecting contact points between ellipses, as shown in the figure opposite. This graph consists of polygons that tessellate the plane fully. Some polygons, labelled "g" in the figure cover a grain, others, labelled "c" cover an inter-grain space. By deriving the distribution of areas of these polygons under certain assumptions, we will be able to compute the probability density of the packing fraction.

In this work, we derived: (i) the probability density function for the length of a chord connecting two randomly distributed neighbouring contact points on the ellipse boundary and (ii) the probability density function for the area of a triangle circumscribed by an ellipse of given aspect ration. We then outline the procedure for derivation of the area distributions of polygons of more edges similarly and relate these to the calculation of the packing fraction.