The vertical transport of solid material in a stratified medium is fundamental to a number of environmental applications, with implications for the carbon cycle and nutrient transport in marine ecosystems. In this work, we study the diffusion-limited settling of highly porous particles in a density-stratified fluid through a combination of experiment, analysis, and numerical simulation. By delineating and appealing to the diffusion-limited regime wherein buoyancy effects due to mass adaptation dominate hydrodynamic drag, we derive a simple expression for the steady settling velocity of a sphere as a function of the density, size, and diffusivity of the solid, as well as the density gradient of the background fluid. In this regime, smaller particles settle faster, in contrast with most conventional hydrodynamic drag mechanisms. Furthermore, we outline a general mathematical framework for computing the steady settling speed of a body of arbitrary shape in this regime and compute exact results for the case of general ellipsoids. Using hydrogels as a highly porous model system, we validate the predictions with laboratory experiments in linear stratification for a wide range of parameters. Lastly, we show how the predictions can be applied to arbitrary slowly varying background density profiles and demonstrate how a measured particle position over time can be used to reconstruct the background density profile.
We initiate the study of stability of solutions of the 2D inviscid incompressible porous medium equation (IPM). We begin by classifying all stationary solutions of the inviscid IPM under mild conditions. We then prove some linear stability results. We then study solutions of the IPM equation which are sufficiently regular perturbations of linearly stable steady states. We prove that sufficiently regular perturbations which are also small must be globally regular and strongly converge to a steady state. The mechanism behind the stability is \emph{stratification} as opposed to previous stability…
We show that when a non-wetting fluid drains a stratified porous medium at sufficiently small capillary numbers Ca, it flows only through the coarsest stratum of the medium; by contrast, above a threshold Ca, the non-wetting fluid is also forced laterally, into part of the adjacent, finer strata. The spatial extent of this partial invasion increases with Ca. We quantitatively understand this behavior by balancing the stratum-scale viscous pressure driving the flow with the capillary pressure required to invade individual pores. Because geological formations are frequently stratified, we antici…
Imbibition plays a central role in diverse energy, environmental, and industrial processes. In many cases, the medium has multiple parallel strata of different permeabilities; however, how this stratification impacts imbibition is poorly understood. We address this gap in knowledge by directly visualizing forced imbibition in three-dimensional (3D) porous media with two parallel strata. We find that imbibition is spatially heterogeneous: for small capillary number Ca, the wetting fluid preferentially invades the fine stratum, while for Ca above a threshold value, the fluid instead preferential…
Many key environmental, industrial, and energy processes rely on controlling fluid transport within subsurface porous media. These media are typically structurally heterogeneous, often with vertically-layered strata of distinct permeabilities -- leading to uneven partitioning of flow across strata, which can be undesirable. Here, using direct in situ visualization, we demonstrate that polymer additives can homogenize this flow by inducing a purely-elastic flow instability that generates random spatiotemporal fluctuations and excess flow resistance in individual strata. In particular, we find t…
Imbibition, the displacement of a nonwetting fluid by a wetting fluid, plays a central role in diverse energy, environmental, and industrial processes. While this process is typically studied in homogeneous porous media with uniform permeabilities, in many cases, the media have multiple parallel strata of different permeabilities. How such stratification impacts the fluid dynamics of imbibition, as well as the fluid saturation after the wetting fluid breaks through to the end of a given medium, is poorly understood. We address this gap in knowledge by developing an analytical model of imbibiti…
We derive the torque on a spheroid of an arbitrary aspect ratio $κ$ sedimenting in a linearly stratified ambient. The analysis demarcates regions in parameter space corresponding to broadside-on and edgewise (longside-on) settling in the limit $Re, Ri_v \ll 1$, where $Re = ρ_0UL/μ$ and $Ri_v =γL^3g/μU$, the Reynolds and viscous Richardson numbers, respectively, are dimensionless measures of the importance of inertial and buoyancy forces relative to viscous ones. Here, $L$ is the spheroid semi-major axis, $U$ an appropriate settling velocity scale, $μ$ the fluid viscosity, and $γ\,(>0)$ the (co…
An extremely broad and important class of phenomena in nature involves the settling and aggregation of matter under gravitation in fluid systems. Some examples include: sedimenting marine snow particles in lakes and oceans (central to carbon sequestration), dense microplastics in the oceans (which impact ocean ecology and the food chain), and even iron snow on Mercury (conjectured as its magnetic field source). These fluid systems all have stable density stratification, which is known to trap particulates through upper lightweight fluid coating the sinking particles, thus providing transient b…
In this study, we present results from experiments on the retention of single oil droplets rising through a two-layer density stratification. These experiments confirm the significant slowdown observed in past literature of settling and rising particles and droplets in stratification, and are the first experiments to study single liquid droplets as opposed to solid particles. By tracking the motion of the droplets as they rise through a stratified fluid, we identify two timescales which quantitatively describe this slowdown: an entrainment timescale, and a retention timescale. The entrainment …
The hydrodynamic force experienced by a small spherical particle undergoing an arbitrary time-dependent motion in a density-stratified fluid is investigated theoretically. The study is carried out under the Oberbeck-Boussinesq approximation, and in the limit of small Reynolds and small Péclet numbers. The force acting on the particle is obtained by using matched asymptotic expansions in which the small parameter is given by a/l where a is the particle radius and l is the stratification length defined by Ardekani & Stocker (2010), which depends on the Brunt-Vaisala frequency, on the fluid kinem…
Revisiting the phenomenon of bouncing of inertial particles crossing density stratified interfaces
Turbulence Structure Laboratory, School of Mechanical Engineering, Tel Aviv University（特拉维夫大学机械工程学院湍流结构实验室）；Department of Civil and Environmental Engineering, Imperial College London（帝国理工学院土木与环境工程系）
Multi-phase reactive transport processes are ubiquitous in igneous systems. A challenging aspect of modelling igneous phenomena is that they range from solid-dominated porous to liquid-dominated suspension flows and therefore entail a wide spectrum of rheological conditions, flow speeds, and length scales. Most previous models have been restricted to the two-phase limits of porous melt transport in deforming, partially molten rock and crystal settling in convecting magma bodies. The goal of this paper is to develop a framework that can capture igneous system from source to surface at all phase…
