Search for: All records

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. Last, 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. 

In this paper we study the self-induced low-Reynolds-number flow generated by a cylinder immersed in a stratified fluid. In the low Péclet limit, where the Péclet number is the ratio of the radius of the cylinder and the Phillips length scale, the flow is captured by a set of linear equations obtained by linearising the governing equations with respect to the prescribed far field conditions. We specifically focus on the low Péclet regime and develop a Green's function approach to solve the linearised equations governing the flow over the cylinder. We cross check our analytical solution against numerical solution of the nonlinear equations to obtain the range of the Péclet numbers for which the linear solution is valid. We then take advantage of the analytical solution to find explicit far-field decay rates of the flow. Our detailed analysis points out that the streamfunction and the velocity field decays algebraically in the far field. Intriguingly, this algebraic decay of the flow is much slower when compared with the exponential decay of the flow generated by a slow moving cylinder in the homogeneous Stokes regime, in the absence of stratification. Consequently, the flow generated by a cylinder in the stratified Stokes regime will have a larger domain of influence when compared with the flow generated by a cylinder in the homogeneous Stokes regime. 

Abstract We present two accurate and efficient algorithms for solving the incompressible, irrotational Euler equations with a free surface in two dimensions with background flow over a periodic, multiply connected fluid domain that includes stationary obstacles and variable bottom topography. One approach is formulated in terms of the surface velocity potential while the other evolves the vortex sheet strength. Both methods employ layer potentials in the form of periodized Cauchy integrals to compute the normal velocity of the free surface, are compatible with arbitrary parameterizations of the free surface and boundaries, and allow for circulation around each obstacle, which leads to multiple-valued velocity potentials but single-valued stream functions. We prove that the resulting second-kind Fredholm integral equations are invertible, possibly after a physically motivated finite-rank correction. In an angle-arclength setting, we show how to avoid curve reconstruction errors that are incompatible with spatial periodicity. We use the proposed methods to study gravity-capillary waves generated by flow around several elliptical obstacles above a flat or variable bottom boundary. In each case, the free surface eventually self-intersects in a splash singularity or collides with a boundary. We also show how to evaluate the velocity and pressure with spectral accuracy throughout the fluid, including near the free surface and solid boundaries. To assess the accuracy of the time evolution, we monitor energy conservation and the decay of Fourier modes and compare the numerical results of the two methods to each other. We implement several solvers for the discretized linear systems and compare their performance. The fastest approach employs a graphics processing unit (GPU) to construct the matrices and carry out iterations of the generalized minimal residual method (GMRES). 

Abstract An extremely broad and important class of phenomena in nature involves the settling and aggregation of matter under gravitation in fluid systems. Here, we observe and model mathematically an unexpected fundamental mechanism by which particles suspended within stratification may self-assemble and form large aggregates without adhesion. This phenomenon arises through a complex interplay involving solute diffusion, impermeable boundaries, and aggregate geometry, which produces toroidal flows. We show that these flows yield attractive horizontal forces between particles at the same heights. We observe that many particles demonstrate a collective motion revealing a system which appears to solve jigsaw-like puzzles on its way to organizing into a large-scale disc-like shape, with the effective force increasing as the collective disc radius grows. Control experiments isolate the individual dynamics, which are quantitatively predicted by simulations. Numerical force calculations with two spheres are used to build many-body simulations which capture observed features of self-assembly. 

Abstract Alpine regions are changing rapidly due to loss of snow and ice in response to ongoing climate change. While studies have documented ecological responses in alpine lakes and streams to these changes, our ability to predict such outcomes is limited. We propose that the application of fundamental rules of life can help develop necessary predictive frameworks. We focus on four key rules of life and their interactions: the temperature dependence of biotic processes from enzymes to evolution; the wavelength dependence of the effects of solar radiation on biological and ecological processes; the ramifications of the non‐arbitrary elemental stoichiometry of life; and maximization of limiting resource use efficiency across scales. As the cryosphere melts and thaws, alpine lakes and streams will experience major changes in temperature regimes, absolute and relative inputs of solar radiation in ultraviolet and photosynthetically active radiation, and relative supplies of resources (e.g., carbon, nitrogen, and phosphorus), leading to nonlinear and interactive effects on particular biota, as well as on community and ecosystem properties. We propose that applying these key rules of life to cryosphere‐influenced ecosystems will reduce uncertainties about the impacts of global change and help develop an integrated global view of rapidly changing alpine environments. However, doing so will require intensive interdisciplinary collaboration and international cooperation. More broadly, the alpine cryosphere is an example of a system where improving our understanding of mechanistic underpinnings of living systems might transform our ability to predict and mitigate the impacts of ongoing global change across the daunting scope of diversity in Earth's biota and environments.