Session T15: Free-Surface Flows: Instabilities

Thin films dewetting with phase separation: Dependence of surface tension and Hamaker constant on concentration

We study the instability of a thin film composed of two miscible liquids (binary fluid) on top of a solid planar surface including the fact that both the free surface and wetting energies are dependent on the mixture concentration. By assuming a linear relationship between these energies and both the bulk and surface concentrations, we analyze their effect on the eventual phase separation of the constituent fluids. The problem is formulated within the gradient dynamics formulation applied to thin film limit of the Cahn-Hilliard Navier-Stokes equations. The dependence of the free surface energy on concentration leads to a Marangoni type of effect, while the wetting energy resulting from van der Waals type of interaction between the film and the substrate is described by a concentrations dependant Hamaker constant. The linear stability analysis uncovers that both real and imaginary growth rates are possible, suggesting oscillatory and/or traveling evolution of any small perturbations of either the film thickness or the concentration field. While our problem formulation applies to any binary mixture that can be consistently modeled via the presented approach, a particular interpretation of the results is provided for the case of liquid metal alloy films on nanoscale.   

Building with blobs: harnessing liquid thread instability to assemble regular structures

3D printing has become an increasingly popular means of manufacturing thanks to the low-cost and accessible nature of Fused Deposition Modeling (FDM) machines, now common fixtures in most labs, workshops, and even homes. Under-extrusion is commonly encountered in FDM printing where too small of a volume of filament is supplied to the nozzle during a print. Reminiscent of a leaky faucet, the melted filament drips as the machine prints in a line, forming a sequence of viscous blobs and more finely drawn threads. Here we show that this instability can be harnessed to assemble regular lattices. We rationalize the physics at play in this problem and leverage our understanding of the problem to provide design guidelines to practitioners.   

Controlling the instability of the air-water interface below an impacting disc

Escape of trapped air between an approaching disc and a water surface prior to the solid-liquid impact induces instability at the air-water interface. The instability occurs at the disc edges where the exit velocity of air first exceeds the critical velocity, and a dominant unstable wavelength emerges at the onset of the instability. Here we study the effect of shear flow modulation on the growth of the interfacial instability, which is achieved by imposing axisymmetric sinusoidal perturbations of different wavelengths on the impacting discs. Using a total-internal reflection visualisation technique, we observe that the instabilities on the water surface tend to grow below the crest locations of the disc cq. near the disc edges. Small wavelengths were found to attenuate the magnitude of the dominant wavelength under the disc edges. Meanwhile, larger wavelengths induce new sites of instability and could amplify the magnitude of the dominant wavelength. The amplification is significant for the imposed wavelengths that are close to the dominant wavelength. The origin of the attenuation and amplification of the instability in modulated shear flow will be discussed, accounting for the water cavity dynamics and the local air velocity maxima development under the modulated disc.

    

`Finding the point of no return': Understanding how a free-surface in a moving contact-line problem becomes unstable and how to suppress it

The moving-contact line between a fluid, liquid and a solid is a ubiquitous phenomenon, and determining the maximum speed at which a liquid can wet/dewet a solid is a practically important problem. Using continuum models the maximum speed of wetting/dewetting can be found by calculating steady solutions of the governing equations and locating the critical capillary number, Ca_crit above which no steady-state solution can be found. Below Ca_crit, both stable and unstable steady-state solutions exist and if some appropriate measure of these solutions is plotted against Ca, a fold bifurcation appears where the stable and unstable branches meet. In this talk we develop a computational framework to describe this phenomenon and, by applying ideas from dynamical systems theory to the highly-dimensional complex system, show that, rather than just being a consequence of the fold bifurcation, the unstable solutions are `edge states' and a have profound importance on the transient behaviour of the system. Significantly, the system can become unstable when Ca<sub>crit</sub> due to finite amplitude interfacial `wobbles' are more dangerous than `stretch' perturbations.   

Ribbing instability in rotating Landau-Levich drag-out problem.

The liquid flux entrained on the outside of a rotating drum is a fundamental problem involving complex, multi-scale, multi-phase flow physics in film coating processes, tribology (journal bearings), etc. Here, we study this so-called rotating drag-out problem in the limit wherein the liquid inertia dominates meniscus surface tension forces. In both experiments and direct numerical simulations, this condition leads to multiple liquid sheets along the drum's axis at its rising side that could be interpreted as inertial analogues of the well-known axial modulation of coating film thickness in low Reynolds number meniscus coating flows. However, the liquid sheet height is proportional to U²/g, where U is the linear drum speed. By properly controlling the gap between the drum's bottom and the floor, along with the drum's immersion depth, we measure the mean rib spacing as a function of drum speed U. In order to further clarify the underlying mechanism, we also solve the incompressible two-phase Navier-Stokes equations with an explicit interface capturing (VOF method). In particular, a simplified configuration of the drag-out problem involving an inclined plate with high-speed withdrawal is investigated. Thereby, the origin of the incipient ribbing instability is elucidated.   

Author not Attending

Magnetized target fusion reactors achieve fusion conditions by confining plasma in a rotating cavity within a molten lead/lithium working fluid, contained in a cylindrical rotor drum. The cavity is forced to collapse by injecting working fluid through bore holes in the rotor walls. As the cavity collapses, the plasma is compressed and heated rapidly to fusion conditions. For fusion conditions to be attained, it is critical that the free-surface is kept uniform throughout the collapse. This presents a challenge as the interior wall of the rotor, which is perforated with bore holes, causes a non-uniform pressure field, which results in wave-like free-surface disturbances. These disturbances must be minimized as far as possible. Presently, it is not known exactly how the amplitude and shape of the disturbances is affected by the arrangement of rotor bore holes (rotor geometry). In the present work, the influence of the rotor geometry on the free-surface is investigated using a Rankine source panel method. An adjustment is made to account for the periodicity of the bores. The influence of the rotor geometry and initial fill volume of fluid in the rotor is discussed. This work is funded under the NSERC General Fusion/McGill alliance grant.

    

Optimal electrostatic control and reduced-order modelling of thick-film flow on a uniformly rotating cylinder

The dynamics of a thick, two-dimensional film of a perfectly conducting, Newtonian fluid coating a uniformly rotating, horizontal, circular cylinder are studied. The cylinder is held at a constant potential and a concentric outer electrode encloses the system, inducing electrostatic forces at the fluid-gas interface. The potential at the outer electrode is, in general, non-uniform, and can be used as a control actuation mechanism. Long-wave scalings¹ are used along with the Method of Weighted Residuals² to derive a model that is valid for both thin and thick films, and incorporates the effects of electrostatic stress, rotation, gravity, viscosity, inertia, and capillarity. Optimal control theory is applied to the model to explore the possibilities of controlling the dynamics of the film using the induced electric field.

¹ Wray, A. W. and Cimpeanu, R. (2020). Reduced-order modelling of thick inertial flows around rotating cylinders. J. Fluid Mech., 898, A1.

² Ruyer-Quil, C. and Manneville, P. (2000). Improved modeling of flows down inclined planes. Eur. Phys. J. B, 15 (2), 357–369.   

Dynamics and stability of leaky dielectric viscous films coating the inside of a vertical cylindrical tube in a radial electric field

The dynamics of an electrified viscous film coating the interior of a vertical cylindrical tube subject to a radial electrostatic field and driven by gravity is investigated theoretically. We consider the general case where both the inner gas and the annular liquid film are weakly conductive with different electrical properties. Using the Taylor-Melcher leaky dielectric model, a coupled system of evolution equations governing the interfacial position and charge distribution is derived for the axisymmetric flow with long-wave (LW) asymptotics. Linear stability analysis of the electrified film to axisymmetric disturbances is carried out. The influences of various parameters on stability characteristics have been explored in detail, including the radii of outer and inner electrodes, relative permittivities, dimensionless conductivities Σ_i, an electric Weber number E_w, and a Reynolds number. The electric field is found to be either stabilizing or destabilizing; and in the limiting case of high Σ_i in both phases, the stability conditions for the dual role of the electric field can be deduced by simply analyzing a normal-stress balance at the gas-liquid interface. In leaky dielectric cases, the dispersion curves have two branches if Σ_i was not too large and E_w is sufficiently high, in contrast to perfectly dielectric cases where only one branch is obtained. In the nonlinear regime, the temporal evolution of the instabilities is predicted by solving the LW model numerically in a wide range of parameters. Rich dynamics are revealed and classified, including steady-state and time-periodic traveling waves, finite-time touchdown singularity, and complex chaotic dynamics. Phase diagrams of the flow behaviors are constructed in the parameter space with extensive numerical simulations.   

Pattern formation instabilities on a curved shell

Pattern formation instabilities can be seen in various free surface flows. Dynamical pattern formation, that is, the initial nucleation and the consequent coarsening behaviour, is a rich source of problems that relate Marangoni effects, localised and global curvature and any external effects together. Here we look at a static case of the Turing-esq pattern formations on a curved shell as a result of the diffusion-reaction system.   

Rogue Nanowaves: A Route to Thin Film Rupture

Understanding the stability of thin films and their potential to spontaneously rupture is crucial for a wide range of fluid-based applications.  Over the last thirty years, there has been significant progress in understanding the stability of ultra thin films, where disjoining pressure competes with surface tension to dictate whether or not a hole is formed in the so-called 'spinodal regime' of dewetting.  Mathematically, this corresponds to a long-wavelength linear instability and, notably, it is known that thermal capillary nanowaves, driven by Brownian motion in the bulk, play a key role in speeding up this instability, to bring theory in line with experiments.

Here, we consider an alternative and complementary possibility - that the formation of rogue nanowaves can lead to the rupture of linearly stable films.  Experimentally, such a 'thermal regime' has been observed, but a theoretical framework and predictive capability for this regime is absent.

In this talk, it will be shown that the film's dynamics can be described by a stochastic thin film equation, which naturally captures the generation of nanowaves and, once solved computationally, predicts the rupture of linearly stable nanofilms.  Then, using large deviation theory, which has recently been applied to model rogue ocean waves, we are able to formulate a theory which describes the most likely route to rupture and the timescale associated with this process, which compare well to our direct simulations.  Finally, comparisons to experimental analyses and molecular simulations will be presented and directions of future research discussed.