Interface Motion in the Ostwald Ripening and Chemotaxis Systems

Ostwald ripening and chemotaxis are two different mechanisms that describe particle motion throughout a domain. Ostwald ripening describes the redistribution of a solid solution due to energy potentials while chemotaxis is a cellular phenomenon where organisms move based on the presence of chemical gradients in their environment. Despite the two systems coming from disparate fields, they are connected by the late-stage dynamics of interfacial motion. For the Ostwald ripening system we consider the case of N droplets in the asymptotic limit of small radii ri 1. We first derive a system of ODEs that describe the motion of the droplets and then improve this calculation by including higher order terms. Certain properties, such as area preservation and finite time extinction of certain droplets are proved and a numerical example is presented to support the claims. In the chemotaxis model we look at the asymptotic limit of diffusive forces being small compared to that of chemotactic gradients. We use a boundary-fitted coordinate system to derive an equation for the velocity of an arbitrary interface and analyze a few specific examples. The asymptotic results are also explored and confirmed using the finite element and level set methods. Our analysis reveals the mechanism of movement to be motion by curvature in Ostwald ripening and a surface diffusion law in chemotaxis. The governing rules of motion may be different in the two systems but the end result is typically characteristically similarexchange of mass and smoothing in favor of a larger and more stable configuration of drops.