Instability and dripping of electrified liquid films flowing down inverted substrates

We consider the gravity-driven flow of a perfect dielectric, viscous, thin liquid film, wetting a flat substrate inclined at a nonzero angle to the horizontal. The dynamics of the thin film is influenced by an electric field which is set up parallel to the substrate surface—this nonlocal physical mechanism has a linearly stabilizing effect on the interfacial dynamics. Our particular interest is in fluid films that are hanging from the underside of the substrate; these films may drip depending on physical parameters, and we investigate whether a sufficiently strong electric field can suppress such nonlinear phenomena. For a non-electrified flow, it was observed by Brun et al. [Phys. Fluids 27, 084107 (2015)] that the thresholds of linear absolute instability and dripping are reasonably close. In the present study, we incorporate an electric field and analyze the absolute and convective instabilities of a hierarchy of reduced-order models to predict the dripping limit in parameter space. The spatial stability results for the reduced-order models are verified by performing an impulse-response analysis with direct numerical simulations (DNS) of the Navier–Stokes equations coupled to the appropriate electrical equations. Guided by the results of the linear theory, we perform DNS on extended domains with inflow and outflow conditions (mimicking an experimental setup) to investigate the dripping limit for both non-electrified and electrified liquid films. For the latter, we find that the absolute instability threshold provides an order-of-magnitude estimate for the electric-field strength required to suppress dripping; the linear theory may thus be used to determine the feasibility of dripping suppression given a set of geometrical, fluid, and electrical parameters

Deformation of a liquid film by an impinging gas jet: Modelling and experiments

© 2019, Avestia Publishing. We consider liquid in a cylindrical beaker and study the deformation of its surface under the influence of an impinging gas jet. Analyzing such a system not only is of fundamental theoretical interest, but also of industrial importance, e.g., in metallurgical applications. The solution of the full set of governing equations is computationally expensive. Therefore, to obtain initial insight into relevant regimes and timescales of the system, we first derive a reduced-order model (a thin-film equation) based on the long-wave assumption and on appropriate decoupling the gas problem from that for the liquid and taking into account a disjoining pressure. We also perform direct numerical simulations (DNS) of the full governing equations using two different approaches, the Computational Fluid Dynamics (CFD) package in COMSOL and the volume-of-fluid Gerris package. The DNS are used to validate the results for the thinfilm equation and also to investigate the regimes that are beyond the range of validity of this equation. We additionally compare the computational results with experiments and find good agreement

Three-dimensional high speed drop impact onto solid surfaces at arbitrary angles

The rich structures arising from the impingement dynamics of water drops onto solid substrates at high velocities are investigated numerically. Current methodologies in the aircraft industry estimating water collection on aircraft surfaces are based on particle trajectory calculations and empirical extensions thereof in order to approximate the complex fluid-structure interactions. We perform direct numerical simulations (DNS) using the volume-of-fluid method in three dimensions, for a collection of drop sizes and impingement angles. The high speed background air flow is coupled with the motion of the liquid in the framework of oblique stagnation-point flow. Qualitative and quantitative features are studied in both pre- and post-impact stages. One-to-one comparisons are made with experimental data available from the investigations of Sor and García-Magariño (2015), while the main body of results is created using parameters relevant to flight conditions with droplet sizes in the ranges from tens to several hundreds of microns, as presented by Papadakis et al. (2004). Drop deformation, collision, coalescence and microdrop ejection and dynamics, all typically neglected or empirically modelled, are accurately accounted for. In particular, we identify new morphological features in regimes below the splashing threshold in the modelled conditions. We then expand on the variation in the number and distribution of ejected microdrops as a function of the impacting drop size beyond this threshold. The presented drop impact model addresses key questions at a fundamental level, however the conclusions of the study extend towards the advancement of understanding of water dynamics on aircraft surfaces, which has important implications in terms of compliance to aircraft safety regulations. The proposed methodology may also be utilised and extended in the context of related industrial applications involving high speed drop impact such as inkjet printing and combustion

Early-time jet formation in liquid–liquid impact problems: theory and simulations

We perform a thorough qualitative and quantitative comparison of theoretical predictions and direct numerical simulations for the two-dimensional, vertical impact of two droplets of the same fluid. In particular, we show that the theoretical predictions for the location and velocity of the jet root are excellent in the early stages of the impact, while the predicted jet velocity and thickness profiles are also in good agreement with the computations before the jet begins to bend. By neglecting the role of the surrounding gas both before and after impact, we are able to use Wagner theory to describe the early-time structure of the impact. We derive the model for general droplet velocities and radii, which encompasses a wide range of impact scenarios from the symmetric impact of identical drops to liquid drops impacting a deep pool. The leading-order solution is sufficient to predict the curve along which the root of the high-speed jet travels. After moving into a frame fixed in this curve, we are able to derive the zero-gravity shallow-water equations governing the leading-order thickness and velocity of the jet. Our numerical simulations are performed in the open-source software Gerris, which allows for the level of local grid refinement necessary for a problem with such a wide variety of length scales. The numerical simulations incorporate more of the physics of the problem, in particular the surrounding gas, the fluid viscosities, gravity and surface tension. We compare the computed and predicted solutions for a range of droplet radii and velocities, finding excellent agreement in the early stage. In light of these successful comparisons, we discuss the tangible benefits of using Wagner theory to confidently track properties such as the jet-root location, jet thickness and jet velocity in future studies of splash jet/ejecta evolution

Optimisation of bulk carrier loading and discharge

This report summarises progress made towards the problem submitted by Rusal Aughinish at the 93rd European Study Group with Industry. Rusal Aughinish is a company that refines alumina from bauxite. The problem presented to the study group was to review the percentage of time that the company’s inner berth was occupied and how to minimise this percent- age. A number of different approaches were taken with this aim in mind. Firstly, data supplied by Rusal Aughinish was analysed. This analysis found that there is an optimal loading rate (with respect to eliminating demurrage costs) and suggested bands of optimal ship sizes. Further to these studies, two models of Rusal Aughinish’s shipping process were developed by the group: a simulation model and an analytical model. Both models were found to replicate the shipping process reasonably well and were, hence, used to study alumina output, berth occupancy and demurrage costs

Mathematical Modelling of the Impact of Liquid Properties on Droplet Size from Flat Fan Nozzles

Flat fan nozzles atomize crop protection products, breaking them into droplets. Droplet size matters - smaller droplets give better perfor- mance, but very small droplets drift. We want to use mathematical models to better understand how liquid properties affect droplet size. There are three types of breakup: wavy sheet, perforation, and rim. In wavy sheet breakup, increasing viscosity or surface tension increases droplet size. To investigate further, we carry out direct numerical simulations of jet breakup, which show that suface tension has little effect, but increasing viscosity leads to fewer droplets. Decreasing the jet velocity also results in fewer droplets, with a wider size distribution. Each type of breakup involves primary breakup into cylinders of fluid, then secondary breakup into droplets. We thus consider the breakup of a cylinder of fluid. Direct numerical simulations suggest that within the tested parameter range viscosity has little impact on droplet size, however it does influence the timescale on which the instability evolves considerably. Linear stability analysis suggests that increasing viscosity increases the wavelength of the most unstable mode, which we expect leads to larger droplets, and that it reduces the rate of breakup. Perforations - holes in the sheet - also lead to breakup. We find how the length fraction of the sheet that is void changes with time. After breakup, the droplets continue to evolve. We develop a model, based on a transport equation, for this process. A key parameter is the breakup rate constant - larger values lead to more breakup, fewer large droplets, and a narrower size distribution. Together, these mathematical approaches improve our understanding of how droplets form, and can be used to guide experimental work

Electrostatically induced mixing in confined stratified multi-fluid systems

Electrostatic control mechanisms underpin a wide range of modern industrial processes, from lab-on-a-chip devices to microfluidic sensors for security applications. During the last decades, the striking impact of fluid interface manipulation in contexts such as polymer self-assembly, micromanufacturing and mixing in viscous media has established the field of electrically driven interfacial flows as invaluable. This work investigates electrostatically induced interfacial instabilities and subsequent generation of nonlinear coherent structures in immiscible, viscous, dielectric multi-layer stratified flows confined in channels with plane walls. The present study demonstrates theoretically that interfacial instabilities can be utilized to achieve efficient mixing in different immiscible fluid regions. This is accomplished by electrostatically driving stable flows far from their equilibrium states to attain time-oscillatory and highly nonlinear flows producing mixing. The nonlinear electrohydrodynamic instabilities play the role of imposed background velocity fields or moving device parts in more traditional mixing protocols. Initially, simple yet efficient on–off voltage protocols are investigated and subsequently symmetry-breaking voltage distributions are considered and shown to considerably enhance the achieved level of mixing. Both two- and three-dimensional flows, containing realistic fluid configurations (water and oils), are computed using direct numerical simulations based on the Navier–Stokes equations. Such numerical investigations facilitate the quantitative study of the flow into the fully nonlinear regime and constitute the basis of optimization methods in the context of microfluidic mixing applications in two- and three-dimensional geometries

A parameter-free perfectly matched layer formulation for the finite-element-based solution of the Helmholtz equation

This paper presents a parameter-free perfectly matched layer (PML) method for the finite-element-based solution of the Helmholtz equation. We employ one of Bermúdez et al.'s unbounded absorbing functions for the complex coordinate mapping underlying the PML. With this choice, the only free parameter that controls the accuracy of the numerical solution for a fixed numerical cost (characterised by the number of elements in the bulk and the PML regions) is the thickness of the perfectly matched layer, δPML. We show that, for the case of planar waves, the absorbing function performs best for PMLs whose thickness is much smaller than the wavelength. We then perform extensive numerical experiments to explore its performance for non-planar waves, considering domain shapes with smooth and polygonal boundaries, different solution types (smooth and singular), and a wide range of wavenumbers, k , to identify an optimal range for the normalised PML thickness, kδPML, such that, within this range, the error introduced by the presence of the PML is consistently small and insensitive to change. This implies that if the PML thickness is chosen from within this range no further PML optimisation is required, i.e. the method is parameter-free. We characterise the dependence of the error on the discretisation parameters and establish the conditions under which the convergence of the solution under mesh refinement is controlled exclusively by the discretisation of the bulk mesh

Electrohydrodynamically induced mixing and pumping of multifluid systems in microchannels

We investigate electrostatically induced hydrodynamics in stratified flows. Vertical electric fields are used to destabilise stably stratified systems in channel geometries and generate interfacial motion. Efficient electrohydrodynamically actuated control processes are studied theoretically and shown to induce time dependent flows in small scale confined geometries without requiring an imposed velocity field or moving parts. Using linear stability theory, the most unstable wavenumbers for a given microscale geometry are identified in order to deduce electric field strengths that can be utilised to produce a required wave pattern. Starting from simple mechanisms, such as uniform field on-off protocols, promising results are presented in this context. Two-dimensional computations using the volume-of-fluid (VOF) method are conducted to fully validate the linear stability theory. Practical optimisation possibilities such as distributions of field strengths and time intervals between on and off positions are examined numerically in the nonlinear regime. We also propose a mechanism to induce pumping by generating a travelling wave voltage distribution on one or both of the electrodes. The generated flux allows for further improvement of the microfluidic mixing process and could have numerous other relevant ramifications. The analytical and numerical tools constructed enable the study of competitive alternatives in a broad spectrum of applications, from microfluidic mixing to electrostatically induced soft lithography

Electrohydrodynamically induced mixing and pumping of multifluid systems in microchannels

We investigate electrostatically induced hydrodynamics in stratified flows. Vertical electric fields are used to destabilise stably stratified systems in channel geometries and generate interfacial motion. Efficient electrohydrodynamically actuated control processes are studied theoretically and shown to induce time dependent flows in small scale confined geometries without requiring an imposed velocity field or moving parts. Using linear stability theory, the most unstable wavenumbers for a given microscale geometry are identified in order to deduce electric field strengths that can be utilised to produce a required wave pattern. Starting from simple mechanisms, such as uniform field on-off protocols, promising results are presented in this context. Two-dimensional computations using the volume-of-fluid (VOF) method are conducted to fully validate the linear stability theory. Practical optimisation possibilities such as distributions of field strengths and time intervals between on and off positions are examined numerically in the nonlinear regime. We also propose a mechanism to induce pumping by generating a travelling wave voltage distribution on one or both of the electrodes. The generated flux allows for further improvement of the microfluidic mixing process and could have numerous other relevant ramifications. The analytical and numerical tools constructed enable the study of competitive alternatives in a broad spectrum of applications, from microfluidic mixing to electrostatically induced soft lithography