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We analyze the instability of a vortex column in a dilute polymer solution at large $textit{Re}$ and $textit{De}$ with $textit{El} = textit{De}/textit{Re}$, the elasticity number, being finite. Here, $textit{Re} = Omega_0 a^2/ u_s$ and $textit{De} = Omega_0 tau$ are, respectively, the Reynolds and Deborah numbers based on the core angular velocity ($Omega_0$), the radius of the column ($a$), the solvent-based kinematic viscosity ($ u_s = mu_s/rho$), and the polymeric relaxation time ($tau$). The stability of small-amplitude perturbations in this distinguished limit is governed by the elastic Rayleigh equation whose spectrum is parameterized by $ textrm{E} = textit{El}(1-beta)$, $beta$ being the ratio of the solvent to the solution viscosity. The neglect of the relaxation terms, in the said limit, implies that the polymer solution supports undamped elastic shear waves propagating relative to the base-state flow. The existence of these shear waves leads to multiple (three) continuous spectra associated with the elastic Rayleigh equation in contrast to just one for the original Rayleigh equation. Further, unlike the neutrally stable inviscid case, an instability of the vortex column arises for finite E due to a pair of elastic shear waves being driven into a resonant interaction under the differential convection by the irrotational shearing flow outside the core. An asymptotic analysis for the Rankine profile shows the absence of an elastic threshold; although, for small E, the growth rate of the unstable discrete mode is transcendentally small, being O$(textrm{E}^2e^{-1/textrm{E}^{frac{1}{2}}})$. An accompanying numerical investigation shows that the instability persists for smooth vorticity profiles, provided the radial extent of the transition region (from the rotational core to the irrotational exterior) is less than a certain $textrm{E}$-dependent threshold.
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Cilia and flagella are highly conserved slender organelles that exhibit a variety of rhythmic beating patterns from non-planar cone-like motions to planar wave-like deformations. Although their internal structure, composed of a microtubule-based axoneme driven by dynein motors, is known, the mechanism responsible for these beating patterns remains elusive. Existing theories suggest that the dynein activity is dynamically regulated, via a geometric feedback from the ciliums mechanical deformation to the dynein force. An alternative, open-loop mechanism based on a flutter instability was recently proven to lead to planar oscillations of elastic filaments under follower forces. Here, we show that an elastic filament in viscous fluid, clamped at one end and acted on by an external distribution of compressive axial forces, exhibits a Hopf bifurcation that leads to non-planar spinning of the buckled filament at a locked curvature. We also show the existence of a second bifurcation, at larger force values, that induces a transition from non-planar spinning to planar wave-like oscillations. We elucidate the nature of these instabilities using a combination of nonlinear numerical analysis, linear stability theory, and low-order bead-spring models. Our results show that away from the transition thresholds, these beating patterns are robust to perturbations in the distribution of axial forces and in the filament configuration. These findings support the theory that an open-loop, instability-driven mechanism could explain both the sustained oscillations and the wide variety of periodic beating patterns observed in cilia and flagella.
We revisit the somewhat classical problem of the linear stability of a rigidly rotating liquid column in this communication. Although literature pertaining to this problem dates back to 1959, the relation between inviscid and viscous stability criteria has not yet been clarified. While the viscous criterion for stability, given by $We = n^2+k^2-1$, is both necessary and sufficient, this relation has only been shown to be sufficient in the inviscid case. Here, $We = rho Omega^2 a^3/gamma$ is the Weber number and measures the relative magnitudes of the centrifugal and surface tension forces, with $Omega$ being the angular velocity of the rigidly rotating column, $a$ the column radius, $rho$ the density of the fluid, and $gamma$ the surface tension coefficient; $k$ and $n$ denote the axial and azimuthal wavenumbers of the imposed perturbation. We show that the subtle difference between the inviscid and viscous criteria arises from the surprisingly complicated picture of inviscid stability in the $We-k$ plane. For all $n >1$, the viscously unstable region, corresponding to $We > n^2+k^2-1$, contains an infinite hierarchy of inviscidly stable islands ending in cusps, with a dominant leading island. Only the dominant island, now infinite in extent along the $We$ axis, persists for $n= 1$. This picture may be understood, based on the underlying eigenspectrum, as arising from the cascade of coalescences between a retrograde mode, that is the continuation of the cograde surface-tension-driven mode across the zero Doppler frequency point, and successive retrograde Coriolis modes constituting an infinite hierarchy.
The Tayler instability is a kink-type flow instability which occurs when the electrical current through a conducting fluid exceeds a certain critical value. Originally studied in the astrophysical context, the instability was recently shown to be also a limiting factor for the upward scalability of liquid metal batteries. In this paper, we continue our efforts to simulate this instability for liquid metals within the framework of an integro-differential equation approach. The original solver is enhanced by multi-domain support with Dirichlet-Neumann partitioning for the static boundaries. Particular focus is laid on the detailed influence of the axial electrical boundary conditions on the characteristic features of the Tayler instability, and, secondly, on the occurrence of electro-vortex flows and their relevance for liquid metal batteries.
The flow of quantized vortex lines in superfluid 3He-B is laminar at high temperatures, but below 0.6 Tc turbulence becomes possible, owing to the rapidly decreasing mutual friction damping. In the turbulent regime a vortex evolving in applied flow may become unstable, create new vortices, and start turbulence. We monitor this single-vortex instability with NMR techniques in a rotating cylinder. Close to the onset temperature of turbulence, an oscillating component in NMR absorption has been observed, while the instability generates new vortices at a low rate ~ 1 vortex/s, before turbulence sets in. By comparison to numerical calculations, we associate the oscillations with spiral vortex motion, when evolving vortices expand to rectilinear lines.
Extremely small amounts of surface-active contaminants are known to drastically modify the hydrodynamic response of the water-air interface. Surfactant concentrations as low as a few thousand molecules per square micron are sufficient to eventually induce complete stiffening. In order to probe the shear response of a water-air interface, we design a radial flow experiment that consists in an upward water jet directed to the interface. We observe that the standard no-slip effect is often circumvented by an azimuthal instability with the occurence of a vortex pair. Supported by numerical simulations, we highlight that the instability occurs in the (inertia-less) Stokes regime and is driven by surfactant advection by the flow. The latter mechanism is suggested as a general feature in a wide variety of reported and yet unexplained observations.
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