Steady and Unsteady Buckling of Viscous Capillary Jets and Liquid Bridges.

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 ε 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)PRLTAO0031-900710.1103/PhysRevLett.109.064502] 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.

Liquid ropes: a geometrical model for thin viscous jet instabilities.

Thin, viscous fluid threads falling onto a moving belt behave in a way reminiscent of a sewing machine, generating a rich variety of periodic stitchlike patterns including meanders, W patterns, alternating loops, and translated coiling. These patterns form to accommodate the difference between the belt speed and the terminal velocity at which the falling thread strikes the belt. Using direct numerical simulations, we show that inertia is not required to produce the aforementioned patterns. We introduce a quasistatic geometrical model which captures the patterns, consisting of three coupled ordinary differential equations for the radial deflection, the orientation, and the curvature of the path of the thread's contact point with the belt. The geometrical model reproduces well the observed patterns and the order in which they appear as a function of the belt speed.

Insights into the dynamics of mantle plumes from uranium-series geochemistry.

The long-standing paradigm that hotspot volcanoes such as Hawaii or Iceland represent the surface expression of mantle plumes--hot, buoyant upwelling regions beneath the Earth's lithosphere--has recently been the focus of controversy. Whether mantle plumes exist or not is pivotal for our understanding of the thermal, dynamic and compositional evolution of the Earth's mantle. Here we show that uranium-series disequilibria measured in hotspot lavas indicate that hotspots are indeed associated with hot and buoyant upwellings and that weaker (low buoyancy flux) hotspots such as Iceland and the Azores are characterized by lower excess temperatures than stronger hotspots such as Hawaii. This direct link between buoyancy flux and mantle temperature is evidence for the existence of mantle plumes.

Generation of Fermat's spiral patterns by solutal Marangoni-driven coiling in an aqueous two-phase system.

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.

Plume-ridge interaction induced migration of the Hawaiian-Emperor seamounts.

The history of the Hawaiian hotspot is of enduring interest in studies of plate motion and mantle flow, and has been investigated by many researchers using the detailed history of the Hawaiian-Emperor Seamount chain. One of the unexplained aspects of this history is the apparent offset of several Emperor seamounts from the Hawaii plume track. Here we show that the volcanic migration rates of the Emperor seamounts based on existing data are inconsistent with the drifting rate of the Pacific plate, and indicate northward and then southward "absolute movements" of the seamounts. Numerical modeling suggests that attraction and capture of the upper part of the plume by a moving spreading ridge led to variation in the location of the plume's magmatic output at the surface. Flow of the plume material towards the ridge led to apparent southward movement of Meiji. Then, the upper part of the plume was carried northward until 65 Ma ago. After the ridge and the plume became sufficiently separated, magmatic output moved back to be centered over the plume stem. These changes are apparent in variations in the volume of seamounts along the plume track. Chemical and isotopic compositions of basalt from the Emperor Seamount chain changed from depleted (strong mid-ocean ridge affinity) in Meiji and Detroit to enriched (ocean island type), supporting declining influence from the ridge. Although its surface expression was modified by mantle flow and by plume-ridge interactions, the stem of the Hawaiian plume may have been essentially stationary during the Emperor period.

Meandering instability of a viscous thread.

A viscous thread falling from a nozzle onto a surface exhibits the famous rope-coiling effect, in which the thread buckles to form loops. If the surface is replaced by a belt moving with speed U , the rotational symmetry of the buckling instability is broken and a wealth of interesting states are observed [see S. Chiu-Webster and J. R. Lister, J. Fluid Mech. 569, 89 (2006)]. We experimentally studied this "fluid-mechanical sewing machine" in a more precise apparatus. As U is reduced, the steady catenary thread bifurcates into a meandering state in which the thread displacements are only transverse to the motion of the belt. We measured the amplitude and frequency omega of the meandering close to the bifurcation. For smaller U , single-frequency meandering bifurcates to a two-frequency "figure-8" state, which contains a significant 2omega component and parallel as well as transverse displacements. This eventually reverts to single-frequency coiling at still smaller U . More complex, highly hysteretic states with additional frequencies are observed for larger nozzle heights. We propose to understand this zoology in terms of the generic amplitude equations appropriate for resonant interactions between two oscillatory modes with frequencies omega and 2omega . The form of the amplitude equations captures both the axisymmetry of the U=0 coiling state and the symmetry-breaking effects induced by the moving belt.

Dynamics of liquid rope coiling.

We present a combined experimental and numerical investigation of the coiling of a liquid "rope" falling on a solid surface, focusing on three little-explored aspects of the phenomenon: The time dependence of "inertio-gravitational" coiling, the systematic dependence of the radii of the coil and the rope on the experimental parameters, and the "secondary buckling" of the columnar structure generated by high-frequency coiling. Inertio-gravitational coiling is characterized by oscillations between states with different frequencies, and we present experimental observations of four distinct branches of such states in the frequency-fall height space. The transitions between coexisting states have no characteristic period, may take place with or without a change in the sense of rotation, and usually (but not always) occur via an intermediate "figure of eight" state. We present extensive laboratory measurements of the radii of the coil and of the rope within it, and show that they agree well with the predictions of a "slender-rope" numerical model. Finally, we use dimensional analysis to reveal a systematic variation of the critical column height for secondary buckling as a function of (dimensionless) flow rate and surface tension parameters.

Periodic folding of viscous sheets.

The periodic folding of a sheet of viscous fluid falling upon a rigid surface is a common fluid mechanical instability that occurs in contexts ranging from food processing to geophysics. Asymptotic thin-layer equations for the combined stretching-bending deformation of a two-dimensional sheet are solved numerically to determine the folding frequency as a function of the sheet's initial thickness, the pouring speed, the height of fall, and the fluid properties. As the buoyancy increases, the system bifurcates from "forced" folding driven kinematically by fluid extrusion to "free" folding in which viscous resistance to bending is balanced by buoyancy. The systematics of the numerically predicted folding frequency are in good agreement with laboratory experiments.

Pattern formation in a thread falling onto a moving belt: an "elastic sewing machine".

We study the dynamics of instability and pattern formation in a slender elastic thread that is continuously fed onto a surface moving at constant speed V in its own plane. As V is decreased below a critical value V(c), the steady "dragged catenary" configuration of the thread becomes unstable to sinusoidal meanders and thence to a variety of more complex patterns including biperiodic meanders, figures of 8, "W," "two-by-one," and "two-by-two" patterns, and double coiling. Laboratory experiments are performed to determine the phase diagram of these patterns as a function of V, the thread feeding speed U, and the fall height H. The meandering state is quantified by measuring its amplitude and frequency as functions of V, which are consistent with a Hopf bifurcation. We formulate a numerical model for a slender elastic thread that predicts well the observed steady shapes but fails to predict the frequency of the onset of meandering, probably because of slippage of the thread relative to the belt. A comparison of our phase diagram with the analogous diagram for a thread of viscous fluid falling on a moving surface reveals many similarities, but each contains several patterns that are not found in the other.