3 years ago

Inhomogeneities and caustics in passive particle sedimentation in incompressible flows.

Emilio Hernández-García, Gábor Drótos, Cristóbal López, Pedro Monroy

In an incompressible flow, fluid density remains invariant along fluid element trajectories. This implies that the spatial distribution of perfect passive tracers in incompressible flows cannot develop density inhomogeneities if they are not already introduced in the initial condition. Thus, typically, tracer inhomogeneities in such flows are explained by particle interactions, or by non-ideality arising e.g. from inertial effects. However, in certain practical situations, density is measured or accumulated on (hyper-) surfaces of dimensionality lower than the full dimensionality of the flow on which motion occurs. A practical situation of this type arises when observing particle distributions sedimented on the floor of the ocean. In such cases, even if ideal tracers are distributed uniformly over 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 tracer 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 tracer advection in a simple kinematic flow, study the generic dependence on the 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.

Publisher URL: http://arxiv.org/abs/1801.10116

DOI: arXiv:1801.10116v1

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