Model for coiling and meandering instability of viscous threads

A numerical model is presented to describe both the transient and steady-state dynamics of viscous threads falling onto a plane. The steady-state coiling frequency w is calculated as a function of fall height H. In the case of weak gravity, w ~ H^{-1} and w ~ H are obtained for lower and higher fall heights respectively. When the effect of gravity is significant, the relation w ~ H^2 is observed. These results agree with the scaling laws previously predicted. The critical Reynolds number for coil-uncoil transition is discussed. When the gravity is weak, the transition occurs with hysteresis effects. If the plane moves horizontally at a constant speed, a variety of meandering oscillation modes can be observed experimentally. The present model also can describe this phenomenon. The numerically obtained state diagram for the meandering modes qualitatively agrees with the experiment.Comment: 12 pages, 10 figure

Stability of Liquid Rope Coiling

International audienceA thin ‘rope' of viscous fluid falling from a sufficient height coils as it approaches a rigid surface. Here we perform a linear stability analysis of steady coiling, with particular attention to the ‘inertiogravitational' regime in which multiple states with different frequencies exist at a fixed fall height. The basic states analyzed are numerical solutions of asymptotic ‘thin-rope' equations that describe steady coiling. To analyze their stability, we first derive in detail a set of more general equations for the arbitrary time-dependent motion of a thin viscous rope. Linearization of these equations about the steady coiling solutions yields a boundary-eigenvalue problem of order twenty-one which we solve numerically to determine the complex growth rate. The multivalued portion of the curve of steady coiling frequency vs. height comprises alternating stable and unstable segments whose distribution agrees closely with high-resolution laboratory experiments. The dominant balance of (perturbation) forces in the instability is between gravity and the viscous resistance to bending of the rope; inertia is not essential, although it significantly influences the growth rate

A discrete geometric approach for simulating the dynamics of thin viscous threads

We present a numerical model for the dynamics of thin viscous threads based on a discrete, Lagrangian formulation of the smooth equations. The model makes use of a condensed set of coordinates, called the centerline/spin representation: the kinematical constraints linking the centerline's tangent to the orientation of the material frame is used to eliminate two out of three degrees of freedom associated with rotations. Based on a description of twist inspired from discrete differential geometry and from variational principles, we build a full-fledged discrete viscous thread model, which includes in particular a discrete representation of the internal viscous stress. Consistency of the discrete model with the classical, smooth equations is established formally in the limit of a vanishing discretization length. The discrete models lends itself naturally to numerical implementation. Our numerical method is validated against reference solutions for steady coiling. The method makes it possible to simulate the unsteady behavior of thin viscous jets in a robust and efficient way, including the combined effects of inertia, stretching, bending, twisting, large rotations and surface tension

Channelized melt flow in downwelling mantle: Implications for 226Ra-210Pb disequilibria in arc magmas

We present the results of an analytical model of porous flow of viscous melt into a steadily dilating ‘‘channel’’ (defined as a cluster of smaller veins) in downwelling subarc mantle. The model predicts the pressure drop in the mantle wedge matrix surrounding the channel needed to drive melt flow as a function of position and time. Melt is sucked toward the dilatant region at a near-constant velocity (105 s1) until veins comprising the channel stop opening (t = t). Fluid elements that complete their journey within the time span t < t arrive at a channel. Our results make it possible to calculate the region of influence sampled by melt that surrounds the channel. This region is large compared to the model size of the channelized region driving flow. For a baseline dilation time of 1 year and channel half width of 2 m, melt can be sampled over an 80-m radius and has the opportunity to sample matrix material with potentially contrasting chemistry on geologically short timescales. Our mechanical results are consistent with a downgoing arc mantle wedge source region where melting and melt extraction by porous flow to a channel network are sufficiently rapid to preserve source-derived 238U-230Th-226Ra, and potentially also 226 Ra-210Pb, disequilibria, prior to magma ascent to the surface. Since this is the rate-determining step in the overall process, it allows the possibility that such short-lived disequilibria measured in arc rocks at the surface are derived from deep in the mantle wedge. Stresses due to partial melting do not appear capable of producing the desired sucking effect, while the order of magnitude rate of shear required to drive dilation of 107 s1 is much larger than values resulting from steady state subduction. We conclude that local deformation rates in excess of background plate tectonic rates are needed to ‘‘switch on’’ the dilatant channel network and to initiate the sucking effect

Delayed Capillary Breakup of Falling Viscous Jets

Thin jets of viscous fluid like honey falling from capillary nozzles can attain lengths exceeding 10 m before breaking up into droplets via the Rayleigh-Plateau (surface tension) instability. Using a combination of laboratory experiments and WKB analysis of the growth of shape perturbations on a jet being stretched by gravity, we determine how the jet's intact length lb depends on the flow rate Q, the viscosity η, and the surface tension coefficient γ. In the asymptotic limit of a high-viscosity jet, lb∼(gQ2η4/γ4)1/3, where g is the gravitational acceleration. The agreement between theory and experiment is good, except for very long jets.</p

Random field sampling for a simplified model of melt-blowing considering turbulent velocity fluctuations

In melt-blowing very thin liquid fiber jets are spun due to high-velocity air streams. In literature there is a clear, unsolved discrepancy between the measured and computed jet attenuation. In this paper we will verify numerically that the turbulent velocity fluctuations causing a random aerodynamic drag on the fiber jets -- that has been neglected so far -- are the crucial effect to close this gap. For this purpose, we model the velocity fluctuations as vector Gaussian random fields on top of a k-epsilon turbulence description and develop an efficient sampling procedure. Taking advantage of the special covariance structure the effort of the sampling is linear in the discretization and makes the realization possible

Multiple mantle upwellings in the transition zone beneath the northern East-African Rift system from relative P-wave travel-time tomography

Mantle plumes and consequent plate extension have been invoked as the likely cause of East African Rift volcanism. However, the nature of mantle upwelling is debated, with proposed configurations ranging from a single broad plume connected to the large low-shear-velocity province beneath Southern Africa, the so-called African Superplume, to multiple lower-mantle sources along the rift. We present a new P-wave travel-time tomography model below the northern East-African, Red Sea, and Gulf of Aden rifts and surrounding areas. Data are from stations that span an area from Madagascar to Saudi Arabia. The aperture of the integrated data set allows us to image structures of 100 km length-scale down to depths of 700– 800 km beneath the study region. Our images provide evidence of two clusters of low-velocity structures consisting of features with diameter of 100–200 km that extend through the transition zone, the first beneath Afar and a second just west of the Main Ethiopian Rift, a region with off-rift volcanism. Considering seismic sensitivity to temperature, we interpret these features as upwellings with excess temperatures of 100 6 50 K. The scale of the upwellings is smaller than expected for lower mantle plume sources. This, together with the change in pattern of the low-velocity anomalies across the base of the transition zone, suggests that ponding or flow of deep-plume material below the transition zone may be spawning these upper mantle upwellings