Session M16: Free-Surface Flows III

Numerical Simulation of the ``Fluid Mechanical Sewing Machine''

A thin thread of viscous fluid falling onto a moving conveyor belt generates a wealth of complex ``stitch'' patterns depending on the belt speed and the fall height. To understand the rich nonlinear dynamics of this system, we have developed a new numerical code for simulating unsteady viscous threads, based on a discrete description of the geometry and a variational formulation for the viscous stresses. The code successfully reproduces all major features of the experimental state diagram of Morris et al. (Phys. Rev. E 2008). Fourier analysis of the motion of the thread's contact point with the belt suggests a new classification of the observed patterns, and reveals that the system behaves as a nonlinear oscillator coupling the pendulum modes of the thread.   

Frequency structure of the nonlinear instability of a dragged viscous thread

A thread of viscous fluid falling onto a moving belt exhibits a spectacular variety of modes of motion as the belt speed and nozzle height are varied [1]. For modest nozzle heights, four clear regimes are observed. For large belt speed, the thread is dragged into a stretched centenary configuration which is confined to a plane. As the belt speed is lowered, this exhibits a supercritical Hopf bifurcation to a meandering mode [2]. At very low belt speeds, the motion resembles the usual coiling motion of a viscous thread falling on a stationary surface. In between the meandering and coiling regimes, a window of novel multifrequency motion, previously called ``figures of eight" is found. We examined the longitudinal and transverse motion of the thread in all these states, using an automated apparatus that allows a detailed exploration of the parameter space. We found that the multifrequency window is characterized by a complex pattern of motion whose main frequencies are locked in a 3:2 ratio. This motion appears and disappears with finite amplitude at sharp bifurcations, without measurable hysteresis. \\[0pt] [1] S. Chiu-Webster and J. R. Lister, J. Fluid Mech., 569, 89 (2006).\\[0pt] [2] S. W. Morris, J. H. P. Dawes, N. M. Ribe and J. R. Lister, Phys. Rev. E, 77, 066218 (2008).   

Liquid supercoiling

Supercoiling is defined as the large-scale secondary coiling of a slender body that is already coiled at a smaller scale (e.g., telephone cords and DNA strands). We demonstrate experimentally a novel fluid-mechanical form of supercoiling that occurs in the context of the familiar ``liquid rope coiling'' instability of a thin thread of viscous fluid falling onto a rigid surface. Under appropriate conditions, the coiling instability generates a tall pile of coils in the form of a hollow cylindrical column, which in turn becomes unstable to a secondary coiling instability with a frequency $\approx 10\%$ of the primary one. To place this phenomenon in a broader context, we determine experimentally the phase diagram for the different possible behaviors of the thread (stagnation flow, simple coiling, rotatory folding, periodic column collapse, supercoiling) in the space of the fluid viscosity, the flow rate, and the fall height. We formulate a mathematical model for supercoiling by combining a thin-shell description of the column wall with a slender-thread description of the column as a whole. This leads to a set of coupled ordinary differential equations in one space dimension (the arclength along the axis of the coiling column) that we solve numerically using a continuation method. A comparison of the predicted and observed frequencies of secondary coiling will be shown.   

Shape of a viscous curtain

When a viscous liquid springs out of a horizontal circular tube, at large enough flow rate the jet progressively takes the shape of a thin vertical curtain limited by two approximately parabolic rims. We will describe the shapes observed depending on the experimental conditions.   

On the the Contact Lens Problem: Modeling Rigid and Elastic Beams on Thin Films

Generally, contact lenses are prescribed by the practitioner to fit each individual patient's eye, but these fitting-philosophies are based on empirical studies and a certain degree of trial-and-error. A badly fitted lens can cause a range of afflictions, which varies from mild dry-eye-discomfort, to more serious corneal diseases. Thus, at this heart of this problem, is the question of \emph{how a rigid or elastic plate interacts with the free-surface of a thin viscous film}. In this talk, we present several mathematical models for the study of these plate-and-fluid problems. Asymptotic and numerical results are described, and we explain the role of elasticity, surface tension, viscosity, and pressure in determining the equilibrium solutions. Finally, we discuss the implications of our work on the contact lens problem, as well as on other coating processes which involve elastic substrates.   

Numerical evidence for formation of ``resonant'' coherent structures by an ensemble of small rigid particles in thermocapillary flows

The effect of formation of coherent particulate accumulation structures (PAS) in a thermocapillary flow was discovered more than a decade ago. It happens in regimes of the flow that are characterized by a hydrothermal wave travelling in the azimuthal direction. Those structures are dynamic and rotate azimuthally together with the travelling wave. They are appearing as a result of ubiquitous in nature nonlinear phenomenon of phase locking, when the turnover particle motion is synchronized with the rotating wave. Synchronization of the particles with the wave most commonly ends up in resonance leading to the PAS mode coinciding with that of the wave. Other resonance modes of PAS, however, may occur at the same time. We present numerical evidence for such structures.   

Holy balls!

Why can some balls walk on water while others cannot? We investigate the rebound dynamics of elastic spheres impacting on a free-surface. Several variables determine whether or not a sphere will bounce when impacting a free-surface including velocity, impact angle, size and elasticity. Stiff elastic spheres, such as a racquetball, successfully skip at low impact angles and high velocities, but tend not to bounce when the impact angle becomes too large. However, the more compliant Waboba$^{\textregistered}$ (WAter BOuncing BAll) bounces marvelously even at very high impact angles. Unlike a stiffer ball, the Waboba$^{\textregistered}$ flattens out quickly as it is forming a cavity. The cavity lip forms a ramp and the flattened ball then skips off the water surface. We demonstrate how this phenomenon surprisingly resembles a skipping stone. Using high-speed video we explore the rebound dynamics for various values of elasticity, velocity, angle and size and determine when an object will bounce off the water surface.   

The 2D Selective Withdrawal Transition: Analogies with MEM Systems

Free boundaries in selective withdrawal systems have been observed undergoing topological transitions through apparently-singular steady states as the withdrawal rate is increased. We transfer the study of this transition to a simpler class of differential equations that are analogous to those found in the study of microelectromechanical (MEM) systems. In true electrostatic free boundary problems, the electrostatic pressure varies as the square of the electric field. Artificially replacing this quadratic dependence with a linear dependence that matches the linear relationship between a Newtonian fluid stress field and velocity gradient components, we obtain a good model for the fluid system with fewer degrees of freedom. By first considering a restricted family of one dimensional boundaries related to two-parameter conformal maps, we aim to identify the mechanisms that control the boundary breakdown in two dimensions.   

Convection, evaporation, and condensation of binary fluids in confined geometries

Phase change has a major effect on convection in liquid layers with a free surface. Significant latent heat generated at the free surface as a result of phase change can dramatically alter the interfacial temperature, inducing thermocapillary stresses. For binary fluids, differential evaporation leads to a variation in the concentration, and hence, induces solutocapillary stresses. This talk describes numerical and experimental studies of convection in alcohol-water mixtures due to a horizontal temperature gradient in the presence of phase change. Evaporation and condensation is known to be a notoriously difficult problem to model due to a poorly defined vapor transport problem which is strongly influenced by the presence/absence and flows of non-condensable gases (e.g., air). This issue is addressed by using a sealed cuvette heated at one end and cooled at the other. Both numerics and experiments show that, by adding or removing air from the cuvette, the direction of flow in a liquid layer covering the bottom of the cell can be reversed by emphasizing either thermocapillary or solutocapillary stresses.   

A new mechanism for atomization and the primary instability in liquid-gas mixing layers

We investigate numerically and experimentally the appearance of instabilities in two planar coflowing liquids sheets. As a function of the momentum ratio $M= \rho_G U_G^2/\rho_L U_L^2$, two different regimes are distinguished. At low momentum ratios the frequency of the waves appearing in the primary atomization region is influenced by the liquid velocity, whereas another asymptotic regime is obtained for large momentum ratios. In this regime, the gas velocity and the ratio between the gas boundary layer and the thickness of the separator plate influence the observed frequency. The low $M$, liquid-dominated regime appears to be a noise amplifier, while the gas-dominated, large $M$ regime displays the characteristics of a global mode. Current computational results are in agreement with experimental observations. These results are compared to the predictions of linearized stability theory. We discuss both the inviscid linearized stability theory and the viscous, Orr-Sommerfeld stability theory, and conjecture that the viscous stability theory is valid in the low $M$ regime.