Experimental evidence for granular shear-flow instability in the Epstein regime

Abstract

Stability analysis of two-fluid protoplanetary disc models has enriched our understanding of how solids can grow into larger bodies called planetesimals. Dust particles entrained in a gas stream modify the flow, creating shear layers prone to instability. In such environments, drag occurs in the free-molecular (Epstein) regime. Recreating these two-phase flows on Earth is difficult due to gravity-driven buoyancy. Here, we use particle image velocimetry to study a low-pressure dust-gas mixture at Knudsen numbers up to 10 in microgravity. We observe a granular shear flow instability, characterized by a periodic velocity field, which can be modeled to first order as a Kelvin-Helmholtz (KH) instability. This behavior resembles a Kelvin-Helmholtz instability and provides a benchmark for two-fluid theories relevant to planet formation.

Figures (11)

• Figure 1: Coordinate system used in this work. (a): overlay of cylindrical and Cartesian coordinates onto cylindrical pipe flow. The $r-\theta$ plane corresponds to the cylinder cross section. The direction of flow in this system is in the positive x-direction. (b): cut-away view of the cylinder cross section, indicating the imaging plane with respect to the camera placement.
• Figure 2: Basic principle of the system, shown as a slice on the $x-r$ plane. (a): the background laminar gas flow has uniform velocity; (b): dust particles start from rest when injection begins; (c): as the dust and gas act upon one another at the flow's midline, the mixture assumes a slower velocity than the original gas speed i.e. the center of mass dust-gas velocity, $v_{\rm com} < v_g$.; (d): it is expected that the presence of a differential velocity and density will lead to the onset of an instability; (e): as the instability develops, vortices may form; (f): in a fully developed state, the dust and gas will be mixed by turbulence.
• Figure 3: Line profiles of the velocity in the y-direction, taken during the apparent onset phase of the fluid instability. Solid blue lines are the extracted data and the dashed red lines represent sinusoidal fits to the data. Each profile derives from a single image in the time series, for clarity only a few exemplary fits are shown.
• Figure 4: Wavelet frequency spectrum at select time intervals. From $t$ = 4 s to $t$ = 5 s of the measurement, the correlation with the wave function is maximal. The dominant frequencies are concentrated above 300 Hz and the correlation with the wave function is very strong. There is a very strong feature occurring around $t$ = 4.36 s, shown in closeup. At two later times, $t$ = 10 s to $t$ = 11 s and $t$ = 17 s to $t$ = 18 s, the range of frequencies gradually tends towards lower values and the magnitude of the coefficients decreases over time.
• Figure 5: (a): light-sheet optic mounted on top of the viewing port of the shear-flow chamber of the TEMPus VoLA microgravity experiment Capelo:2022. (b): The entire shear-flow chamber with dimensions $\sim1$ m and inner tube diameter 8 cm.