  In an incompressible flow, fluid density remains invariant along fluid
element trajectories. This implies that the spatial distribution of
non-interacting noninertial particles in such flows cannot develop density
inhomogeneities beyond those that are already introduced in the initial
condition. However, in certain practical situations, density is measured or
accumulated on (hyper-) surfaces of dimensionality lower than the full
dimensionality of the flow in which the particles move. An example is the
observation of particle distributions sedimented on the floor of the ocean. In
such cases, even if the initial distribution of noninertial particles is
uniform within a finite support in an incompressible flow, advection in the
flow will give rise to inhomogeneities in the observed density. In this paper
we analytically derive, in the framework of an initially homogeneous particle
sheet sedimenting towards a bottom surface, the relationship between the
geometry of the flow and the emerging distribution. From a physical point of
view, we identify the two processes that generate inhomogeneities to be the
stretching within the sheet, and the projection of the deformed sheet onto the
target surface. We point out that an extreme form of inhomogeneity, caustics,
can develop for sheets. We exemplify our geometrical results with simulations
of particle advection in a simple kinematic flow, study the dependence on
various parameters involved, and illustrate that the basic mechanisms work
similarly if the initial (homogeneous) distribution occupies a more general
region of finite extension rather than a sheet.
