Summary. To elucidate the dynamics of free (buoyancy-driven) subduction of oceanic lithosphere, I study a model in which a 2-D sheet of viscous fluid with thic

In liquid rope coiling, a slender jet of viscous fluid falling onto a rigid surface builds a rotating corkscrew-like structure. Here, I use a numerical continuation method to construct a complete regime diagram for liquid rope coiling. I first consider the onset of coiling, and show that a suitable onset criterion is that the radius of the rope itself be just equal to the radius of the coil. Numerical calculation of the critical surface in the space of the dimensionless fall height , flow rate and nozzle diameter shows that the surface has four distinct asymptotic limits corresponding to a viscous (V) mode, a gravitational (G) mode and two inertial modes (I1, I2) which are distinguished by how much the tail of the jet is stretched by gravity. Exact expressions for the onset frequencies in each of these four modes are determined. Finally, the regime diagram is constructed in the form of contour plots of the dimensionless coiling frequency as a function of and for several values of . The diagram exhibits a total of six modes: V, G, I1, I2, a multivalued inertio-gravitational (IG) mode and a third inertial mode I3 with viscosity-dominated stretching of the tail. The regime diagram permits prediction of the coiling frequency for given values of the fall height, flow rate, viscosity and nozzle diameter, and should therefore be useful in practical applications ranging from non-woven textile production to 3D printing.

We present numerical and analytical predictions of mucociliary clearance based on the continuum description of a viscoelastic mucus film, where momentum transfer from the beating cilia is represented via a Navier-slip boundary condition introduced by Bottier et al. (PLoS Comput. Biol., vol. 13, issue 7, 2017a, e1005552). Mucus viscoelasticity is represented via the Oldroyd-B model, where the relaxation time and the viscosity ratio have been fitted to experimental data for the storage and loss moduli of different types of real mucus, ranging from healthy to diseased conditions. We solve numerically the fully nonlinear governing equations for inertialess flow, and develop analytical solutions via asymptotic expansion in two limits: (i) weak viscoelasticity, i.e. low Deborah number; (ii) low cilia beat amplitude (CBA). All our approaches predict a drop in the mucus flow rate in relation to the Newtonian reference value, as the cilia beat frequency is increased. This relative drop increases with decreasing CBA and slip length. In diseased conditions, e.g. mucus properties characteristic of cystic fibrosis, the drop reaches 30 % in the physiological frequency range. In the case of healthy mucus, no significant drop is observed, even at very high frequency. This contrasts with the deterioration of microorganism propulsion predicted by the low-amplitude theory of Lauga (Phys. Fluids, vol. 19, issue 8, 2007, 083104), and is due to the larger beat amplitude and slip length associated with mucociliary clearance. In the physiological range of the cilia beat frequency, the low-amplitude prediction is accurate for both healthy and diseased conditions. Finally, we find that shear-thinning, modelled via a multi-mode Giesekus model, does not significantly alter our weakly viscoelastic and low-amplitude predictions based on the Oldroyd-B model.

The solutal Marangoni effect is attracting increasing interest because of its fundamental role in many isothermal directional transport processes in fluids, including the Marangoni-driven spreading on liquid surfaces or Marangoni convection within a liquid. Here we report a type of continuous Marangoni transport process resulting from Marangoni-driven spreading and Marangoni convection in an aqueous two-phase system. The interaction between a salt (CaCl₂) and an anionic surfactant (sodium dodecylbenzenesulfonate) generates surface tension gradients, which drive the transport process. This Marangoni transport consists of the upward transfer of a filament from a droplet located at the bottom of a bulk solution, coiling of the filament near the surface, and formation of Fermat’s spiral patterns on the surface. The bottom-up coiling of the filament, driven by Marangoni convection, may inspire automatic fiber fabrication.

Steady buckling (coiling) of thin falling liquid jets is sensitive to surface tension, yet an understanding of these capillary effects lags far behind what is known about surface-tension-free coiling. In experiments with submillimetric jets and ultralow flow rates, we find that the critical dispensing height $H_c$ for coiling decreases with increasing flow rate, a trend opposite to that found previously for inertia-free coiling. We resolve the apparent contradiction using nonlinear numerical simulations based on slender-jet theory which show that the trend reversal is due to the strong effect of surface tension in our experiments. We use our experiments to construct a regime diagram (coiling vs stagnation flow) in the space of capillary number $Ca$ and jet slenderness $\epsilon$ and find that it agrees well with fully nonlinear numerical simulations. However, it differs substantially from the analogous regime diagram determined experimentally by Le Merrer, Quéré, and Clanet [ Phys. Rev. Lett. 109, 064502 (2012)] for the unsteady buckling of a compressed liquid bridge. Using linear stability analysis, we show that the differences between the two regime diagrams can be explained by a combination of shape nonuniformity and the influence of gravity.

A circular jet breaks up into droplets via the Rayleigh-Plateau instability, retaining a circular cross section throughout. If, however, the nozzle from which the jet issues is elongated, the circular symmetry is broken, and the jet forms a chainlike structure with neighboring links separated by ${90}^{\circ }$. The cause of this structure is two-dimensional capillary-inertial oscillation of jet cross sections in their own plane. We perform an experimental study of chain oscillations as a function of flow rate using careful prelaminarization and 12 elliptical nozzles with different areas and eccentricities. The oscillation frequencies inferred from the observed chain-link structure do not agree with those predicted by Rayleigh's infinitesimal theory [L. Rayleigh, Proc. R. Soc. London 29A, 71 (1879)]. However, they do agree with an extended nonlinear theory of Bohr [N. Bohr, Philos. Trans. R. Soc. A 209, 281 (1909)] that accounts for finite-amplitude effects. This agreement shows that our fluid chains are nonlinear oscillations whose frequency decreases with increasing amplitude. We perform direct numerical simulations of chain oscillations using a volume-of-fluid method and find good agreement with the predictions of Bohr's theory. Finally, we generalize Bohr's theory to the case of two interacting modes with quadrupolar and octapolar azimuthal dependencies. The resulting solution explains qualitatively the “dimpled” shape of the jet's surface observed in the experiments.