A sphere settling in a stratified fluid at small Reynolds number

Abstract

We present a combined theoretical, experimental and numerical investigation of a sphere settling in a linearly stratified fluid at low Reynolds number (0.01 </= Re </= 2.1). We developed the microscale Synthetic Schlieren technique to study the wake of a microscale sphere settling through a density stratification. A video-microscope was used to magnify and image apparent displacements of a micron-sized random-dot pattern. Due to the nature of the wake, density gradient perturbations in the horizontal direction greatly exceed those in the vertical, requiring modification of a previously developed axisymmetric technique. We demonstrated that Schlieren could be extended to microscale (100 [mu]m) and obtained the first quantitative measurement of the density field in the wake of a sphere settling in a stratified fluid. As stratification breaks directional symmetry, the direction of motion strongly influences the dynamics, unlike in the homogeneous case. Previous work primarily focused on particles moving parallel to isopycnals. Here we investigate motion perpendicular to isopycnals. As the sphere settles, the particle draws lighter fluid downwards, generating buoyancy forces: this results in a long density wake, extending many particle diameters downstream. Using time-lapse photography, the drag on the sphere was measured and we have obtained the first experimental quantification of the added drag on a sphere due to stratification. We found that stratification increases the hydrodynamic drag, and that the added drag coefficient scales with the Richardson number Ri = a3N2/(vU) as Ri1/2, where a is the particle radius, U its speed, v the kinematic fluid viscosity and N the buoyancy frequency. These observations are confirmed by numerical simulations, and are in contrast with earlier results for higher Re.

(cont.) By analyzing the numerical velocity, pressure, density and vorticity fields around the sphere, we found that the pressure and viscous drags both increased with stratification. Combining these analyses with the investigations on isopycnal perturbations around the sphere and the buoyancy force in the wake, we conclude that the bulk of the wake does not contribute to the drag. Based on the experimental and numerical results, we derived a scaling argument which suggests that the added drag results from the buoyancy of the fluid in a small region of width (v/N)1/2 around the sphere. Here the physical mechanism responsible for the added drag in a stratified fluid at low Re is drastically different from mechanisms proposed at higher Re. The observed increase in drag could enhance retention time of particles at density interfaces as the parameter regime studied here applies to small particles in the ocean and affects the ecology of marine microorganisms by influencing particle-organism interactions.