| The study of pattern forming systems has been of growing interest to biologists, chemists and physicists in recent years. Generally, these pattern forming systems involve competing interactions that lead to instabilities, driving the system to form a pattern. In this project, we look at two such pattern forming systems. The first is a reaction-diffusion system, where the competition is between the activator and the inhibitor, and the second is a thin layer of ferrofluid which exhibits pattern formation due to a competition between magnetic and surface energies.; Numerical simulation of the Gierer-Meinhardt model for reaction and diffusion is used to study the sequence of transitions from islands of high activator concentration to stripes of high activator concentration to wells of depleted activator. This sequence can occur by activator saturation or by inhibitor depletion. Four quantitative measures are introduced which display different trends depending upon whether the transition is driven by activator saturation or inhibitor depletion. These four measures characterize the transitions, and enhance understanding of the system.; A model for the Helmholtz free energy is derived to predict aggregate spacing in thin layers of ferrofluid. When a drop of ferrofluid is confined between two glass plates and subjected to an external magnetic field, the particles in the ferrofluid aggregate, forming a hexagonal array. This theoretical model, once fully developed, is used to predict aggregate spacing for this hexagonal pattern as a function of external magnetic field, the ramping rate of the external magnetic field, and plate separation. The results of this model are then compared to experimental data, demonstrating excellent agreement. |
