The thesis is divided into two parts. In the first part, we look at shape transitions in elastic systems that are understood using the paradigm of confinement vs bendability. The phase diagram of morphological transitions of an infinitely wide sheet wrapping a 2D droplet and coiling of an elastic filament confined to a spherical bubble are described using this framework.

In the second part, we investigate the role of history force on the dynamics of spherical particles in fluid flows. The non-local particle evolution equation becomes a local modified Robin boundary condition to the heat equation, which is solved for various flow scenarios exactly. A spectrally accurate numerical method arising out of this approach can be used to solve particle evolution in general flow scenarios.