Sedimenting sphere in a variable-gap Hele-Shaw cell

Abstract

We have measured the trajectory and visualized the wake of a single sphere falling in a fluid confined between two closely spaced glass plates (a Hele-Shaw cell). The position of a sedimenting sphere was measured to better than 0.001d, where d is the sphere diameter, for Reynolds numbers (based on the terminal velocity) between 20 and 330, for gaps between the plates ranging from 1.014d to 1.4d. For gaps in the range 1.01d-1.05d, the behaviour of the sedimenting sphere is found to be qualitatively similar to that of an unconfined cylinder in a uniform flow, but our sedimenting sphere begins to oscillate and shed von Kármán vortices for Re > 200, which is far greater than the Re = 49 for the onset of vortex shedding behind cylinders in an open flow. When the gap is increased to 1.10d-1.40d, the vortices behind the sphere are different -they are qualitatively similar to those behind a sphere sedimenting in the absence of confining walls. Our precision measurements of the velocity of a sedimenting sphere and the amplitude and frequency of the oscillations provide a benchmark for numerical simulations of the sedimentation of particles in fluids. IntroductionThe motion of blunt bodies in a fluid has been studied since the early days of the development of fluid mechanics. Even now sedimenting bodies yield surprising discoveries and challenges for theoretical analyses, but there have been few experiments using modern imaging techniques to examine a sedimenting body.We have examined the sedimentation of a single sphere in a fluid contained between vertical plates separated by a distance only slightly greater than the sphere diameter. We focus on the flow dependence on the separation between the plates. We make measurements for Reynolds numbers ranging from about 30 to 300, but we do not examine in detail the values of the Reynolds numbers corresponding to successive bifurcations. We measure the sphere position in a co-moving reference frame and obtain high-precision results for the sphere velocity and the properties of the wake. Our results for the sphere terminal velocity should long serve as a benchmark for algorithms designed for the difficult general problem of numerical simulation of the Navier-Stokes equation in a system with moving boundaries.Our observations reveal some qualitative features that are common to bodies sedimenting in the absence of sidewalls, and to flow past a fixed sphere or a fixed cylinder in unconfined geometries. Hence we review these situations in the following section.

Hele-Shaw beach creation by breaking waves: a mathematics-inspired experiment

“…Given the results in [7] and the values of r = p ρ s /ρ 0 and St, our system straddles the boundary of the viscous and inertial regimes, in which the particles either reach the Stokes velocity or the inertial velocity, respectively. The Stokes velocity in our Hele-Shaw system is 0.75∆ρgd 2 sin θ/(18νρ 0 ) ⇡ 0.6m/s with the adjustment factor 0.75 from Lee et al [19] for the current Hele-Shaw cell of width 1.14d. (Wall effects are avoided in Courrech du Pont et al [7] because the tank width is larger than 15d.)…”

On Wave-Driven “Shingle” Beach Dynamics in a Table-Top Hele-Shaw Cell

“…Hence, we chose the Hele-Shaw cell to have a gap width of 2l = 2 mm given available Gamma Alumina particles with diameter d = 1.75 ± 0.05 mm and effective density 2.08 ± 0.2g/cm 3 with water-filled pores. We note that Lee et al (2007) considered one particle settling in a Hele-Shaw cell, and observed quasi-two-dimensional behaviour for gap widths 2l < 1.05d, and three-dimensional behaviour for 2l > 1.1d. With our choice of 2l = 2 mm and d = 1.75 ± 0.05 mm, we are in a quasi-three-dimensional regime of particle settling.…”