Accumulation of Particles Towards the Envelope of a Fluid-Infused Patch Under Evaporation

Abstract

We study the dynamics of unsaturated flows within a fluid-infused wet patch in a thin porous layer, coupled with the transport of passive particles, partly inspired by some art practice and environmental and industrial activities. The role of wetting and capillary forces and the influence of evaporation are both emphasized. Two coupled partial differential equations (PDEs) are first derived to describe the time evolution of the fluid saturation and concentration of passive particles. Three dimensionless parameters are identified after appropriate nondimensionalization of the coupled PDEs, the influence of which is demonstrated through asymptotic and numerical solutions at both the early and late times. With evaporation, in particular, the fluid front first arrives at a maximum location after an initial period of spreading. Thereafter, the front starts to recede until the fluid drains out at a finite time. Correspondingly, the concentration of particles blows up at a finite time, when the size of the wet region shrinks to zero. Meanwhile, the spatial distribution of particles evolves towards a universal curve near the front at late times, as the particles gradually accumulate at the front of the wet patch. It is recognized that, while the spreading of fluid within a porous layer behaves similar to that of a viscous gravity current subject to volume loss, the frontal structure for the concentration of particles, to some extent, is similar to that of an intruding inviscid gravity current.