Spatial pattern formation is abundant in nature and occurs in both living and non-living matter. Familiar examples include sand ripples, river deltas, zebra fur and snail shells.
In this thesis, we focus on patterns induced by mechanical stress, and develop continuum theories for three systems undergoing pattern formation on widely different length scales.
On the largest scale of several meters, we model columnar jointing of igneous rock. Using analytical calculations and numerical simulations, we derive a scaling function, which quantitatively relates the column diameter to material parameters and cooling conditions.
On the scale of micrometers, we model breast cancer tissue as a viscoelastic active fluid. The model captures experimentally observed statistical characteristics as well as the cell division process, and hints at substrate friction being important for cell speed distributions.
On the smallest scale of nanometers, we study thin films of block copolymers, which have potential applications as self-organizing templates for microelectronics. By performing a thin-shell expansion of a well-known model for block copolymers, we develop an effective model for the impact of curvature on pattern formation and ordering kinetics in a thin curved film.