We study the stability of density stratified flow between co-rotating vertical cylinders with rotation rates Ωo < Ωi and radius ratio ri/ro = 0.877, where subscripts o and i refer to the outer and inner cylinders. Just as in stellar and planetary accretion disks, the flow has rotation, anticyclonic shear, and a stabilizing density gradient parallel to the rotation axis. The primary instability of the laminar state leads not to axisymmetric Taylor vortex flow but to the non-axisymmetric stratorotational instability (SRI), so named by Shalybkov and Rüdiger (2005). The present work extends the range of Reynolds numbers and buoyancy frequencies (N = (−g/ρ)(∂ρ/∂z)) examined in the previous experiments by Boubnov and Hopfinger (1997) and Le Bars and Le Gal (2007). Our observations reveal that the axial wavelength of the SRI instability increases nearly linearly with Froude number, F r = Ωi/N . For small outer cylinder Reynolds number, the SRI occurs for inner inner Reynolds number larger than for the axisymmetric Taylor vortex flow (i.e., the SRI is more stable). For somewhat larger outer Reynolds numbers the SRI occurs for smaller inner Reynolds numbers than Taylor vortex flow and even below the Rayleigh stability line for an inviscid fluid. Shalybkov and Rüdiger (2005) proposed that the laminar state of a stably stratified rotating shear flow should be stable for Ωo/Ωi > ri/ro, but we find that this stability criterion is violated for N sufficiently large; however, the destabilizing effect of the density stratification diminishes as the Reynolds number increases. At large Reynolds number the primary instability leads not to the SRI but to a previously unreported nonperiodic state that mixes the fluid.
Beginning from the shallow water equations (SWEs), a nonlinear self-similar analytic solution is derived for barotropic flow over varying topography. We study conditions relevant to the ocean slope where the flow is dominated by Earth's rotation and topography. The solution is found to extend the topographic β-plume solution of Kuehl (2014) in two ways. (1) The solution is valid for intensifying jets.(2) The influence of nonlinear advection is included. The SWEs are scaled to the case of a topographically controlled jet, and then solved by introducing a similarity variable, η = cx n x y n y . The nonlinear solution, valid for topographies h = h 0 − αxy 3 , takes the form of the Lambert W -function for pseudo velocity. The linear solution, valid for topographies h = h 0 − αxy −γ , takes the form of the error function for transport. Kuehl's results considered the case −1 ≤ γ < 1 which admits expanding jets, while the new result considers the case γ < −1 which admits intensifying jets and a nonlinear case with γ = −3.
In this study, we present a novel experimental apparatus, its construction, data acquisition methodology, and validation for the study of peristaltic flows. The apparatus consists of a series of stepper motor actuators, which deflect a deformable membrane to produce peristaltic flows. We show that this apparatus design has significant advantages over previous designs that have been used to study peristaltic flows by offering a much wider range of modeling capabilities. Comparisons between the capabilities of our apparatus and previous ones show our apparatus spanning a larger range of wavelength λ, wave speed c, amplitude A, and wave shapes (i.e., the apparatus is not constrained to nondispersive waves or to a sinusoidal shape). We provide details on the experimental design and construction for ease of reconstruction to the reader. The apparatus capabilities are validated for a large parameter range by comparing experimental measurements to analytic results from Ibanez et al. [1] for high Reynolds number (Re > 1), and Jaffrin and Shapiro [2] for low Reynolds number (Re < 1) applications. We show that the apparatus is useful for biophysical peristaltic studies and has potential applications in other types of studies.
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