  Patterns are quotidian in nature. Distinct multiscale patterns are generally
a consequence of nonequilibrium dynamical processes associated with mechanical
or hydrodynamic instabilities. In this thesis, I report experimental
investigations on pattern formation in a few examples of fluid and granular
systems, and uncover the underlying mechanisms that give rise to those
patterns.
  Leidenfrost drops are known to experience star-shaped oscillations with
little damping. However, the underlying mechanism remains unclear. Here I
report that the hydrodynamic coupling between the rapid evaporated vapor flow
and vapor-liquid interface excites the star-shaped oscillations, suggesting a
purely hydrodynamic origin.
  Polygonal desiccation crack patterns are commonly observed in natural
systems. However, it is unclear whether similar crack patterns spanning
multiple length scales share the same underlying physics. I report experimental
results on polygonal cracks in drying suspensions of micron-sized particles. In
cornstarch-water mixtures, multi-scale crack patterns were observed due to two
distinct desiccation mechanisms. We also find the characteristic area of the
polygonal cracks, and film thickness obey a universal power law.
  Finally, I report sedimentations of non-Brownian particles in viscous fluids.
We observed an effective repulsion between particles with nonuniform density in
both two-body and many-body systems in two and three dimensions, in contrast to
particles with uniform density. We also characterize the statistical properties
of the sedimentation patterns of particles in three dimensions.
  The patterns I report in this thesis represent typical examples in fluid and
granular systems that are driven by nonequilibrium dynamics, and the underlying
mechanisms we uncover are expected to enhance our understanding of how these
seemingly simple patterns arise in natural systems.
