A first-principle predictive theory for a sphere falling through sharply stratified fluid at low Reynolds number

Abstract

A sphere exhibits a prolonged residence time when settling through a stable stratification of miscible fluids due to the deformation of the fluid-density field. Using a Green's function formulation, a first-principles numerically assisted theoretical model for the sphere-fluid coupled dynamics at low Reynolds number is derived. Predictions of the model, which uses no adjustable parameters, are compared with data from an experimental investigation with spheres of varying sizes and densities settling in stratified corn syrup. The velocity of the sphere as well as the deformation of the density field are tracked using time-lapse images, then compared with the theoretical predictions. A settling rate comparison with spheres in dense homogeneous fluid additionally quantifies the effect of the enhanced residence time. Analysis of our theory identifies parametric trends, which are also partially explored in the experiments, further confirming the predictive capability of the theoretical model. The limit of infinite fluid domain is considered, showing evidence that the Stokes paradox of infinite fluid volume dragged by a moving sphere can be regularized by density stratifications. Comparisons with other possible models under a hierarchy of additional simplifying assumptions are also presented.

Delayed settling of marine snow at sharp density transitions driven by fluid entrainment and diffusion-limited retention

“…This delayed settling due to entrainment of lighter fluid can be seen in Fig. 1A, a sequence of previously published images from experiments with solid spheres along with matching theory (Camassa et al 2010). However, marine aggregates are extremely porous (usually containing > 99.5% water by volume; Ploug & Passow 2007), providing for another potential mechanism for delayed settling.…”

“…The different curves shown in (C) and (D) demonstrate that delayed settling due to diffusion limited retention results in a positive (quadratic) relationship between DST and aggregate size while entrainment of lighter fluid results in a negative relationship between DST and aggregate size having a minor decrease in the magnitude of the DST (<10% for all particle diameters tested). The second model -hereafter referred to as the solid particle entrainment model -gives the settling behavior of a solid particle as it passes through a sharp density transition (Camassa et al 2009(Camassa et al , 2010. This previously developed model illustrates decreased settling velocity at the density transition due to entrainment of lighter fluid; diffusion plays no role in the delayed settling of the particle since the particle is not porous.…”

Retention and entrainment effects: Experiments and theory for porous spheres settling in sharply stratified fluids

“…These studies have shown how fluid entrained from the upper regions causes a particle to slow down significantly when entering lower regions of denser fluid. [5][6][7][8][9] With porous particles, diffusion of the stratifying agent (e.g., salt) can add an extra physical mechanism affecting settling. 10 In some cases, the initial combined density of the solid porous medium and buoyant upper fluid may even be less than that of the lower fluid.…”

“…[5][6][7][8][9] In our experiments this leads to rather complex dynamics; to make progress, we parametrize it here in the simplest possible way. To this extent, we assume that the entrained fluid is effectively enclosed in a shell region around the settling sphere, thus augmenting the radius of the fluid portion of the porous region to a cumulative effective radius a * ≥ a, see Figure 1(c).…”

“…For instance, leakage of entrained fluid through a trailing stem is observed in experiments with solid spheres. 6,8,9 On the other hand, a * is the only adjustable parameter in a theory that has thus far been based on first principles. We stress that our model of the entrained shell affects the diffusion and buoyancy components of the dynamics, leaving the viscous drag untouched.…”