## Publications

140 results found

Broadley H, Papageorgiou DT, 2022, Nonlinear gravity electro-capillary waves in two-fluid systems: solitary and periodic waves and their stability, *JOURNAL OF ENGINEERING MATHEMATICS*, Vol: 133, ISSN: 0022-0833

Alexander JP, Papageorgiou DT, 2022, Ordered and disordered dynamics in inertialess stratified three-layer shear flows, *PHYSICAL REVIEW FLUIDS*, Vol: 7, ISSN: 2469-990X

Tomlinson SD, Papageorgiou DT, 2021, Linear instability of lid- and pressure-driven flows in channels textured with longitudinal superhydrophobic grooves, *JOURNAL OF FLUID MECHANICS*, Vol: 932, ISSN: 0022-1120

Papageorgiou DT, Tanveer S, 2021, Mathematical study of a system of multi-dimensional non-local evolution equations describing surfactant-laden two-fluid shear flows, *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 477, Pages: 1-15, ISSN: 1364-5021

This article studies a coupled system of model multi-dimensional partial differential equations (PDEs) that arise in the nonlinear dynamics of two-fluid Couette flow when insoluble surfactants are present on the interface. The equations have been derived previously, but a rigorous study of local and global existence of their solutions, or indeed solutions of analogous systems, has not been considered previously. The evolution PDEs are two-dimensional in space and contain novel pseudo-differential terms that emerge from asymptotic analysis and matching in the multi-scale problem at hand. The one-dimensional surfactant-free case was studied previously, where travelling wave solutions were constructed numerically and their stability investigated; in addition, the travelling wave solutions were justified mathematically. The present study is concerned with some rigorous results of the multi-dimensional surfactant system, including local well posedness and smoothing results when there is full coupling between surfactant dynamics and interfacial motion, and global existence results when such coupling is absent. As far as we know such results are new for non-local thin film equations in either one or two dimensions.

Cimpeanu R, Gomes SN, Papageorgiou DT, 2021, Active control of liquid film flows: beyond reduced-order models, *NONLINEAR DYNAMICS*, Vol: 104, Pages: 267-287, ISSN: 0924-090X

Alexander JP, Kirk TL, Papageorgiou DT, 2020, Stability of falling liquid films on flexible substrates, *Journal of Fluid Mechanics*, Vol: 900, Pages: A40-1-A40-33, ISSN: 0022-1120

The linear stability of a liquid film falling down an inclined flexible plane under the influence of gravity is investigated using analytical and computational techniques. A general model for the flexible substrate is used leading to a modified Orr–Sommerfeld problem addressed numerically using a Chebyshev tau decomposition. Asymptotic limits of long waves and small Reynolds numbers are addressed analytically and linked to the computations. For long waves, the flexibility has a destabilising effect, where the critical Reynolds number decreases with decreasing stiffness, even destabilising Stokes flow for sufficiently small stiffness. To pursue this further, a Stokes flow approximation was considered, which confirmed the long-wave results, but also revealed a short wave instability not captured by the long-wave expansions. Increasing the surface tension has little effect on these instabilities and so they were characterised as wall modes. Wider exploration revealed mode switching in the dispersion relation, with the wall and surface mode swapping characteristics for higher wavenumbers. The zero-Reynolds-number results demonstrate that the long-wave limit is not sufficient to determine instabilities so the numerical solution for arbitrary wavenumbers was sought. A Chebyshev tau spectral method was implemented and verified against analytical solutions. Short wave wall instabilities persist at larger Reynolds numbers and destabilisation of all Reynolds numbers is achievable by increasing the wall flexibility, however increasing the stiffness reverts back to the rigid wall limit. An energy decomposition analysis is presented and used to identify the salient instability mechanisms and link them to their physical origin.

Michelin S, Game S, Lauga E,
et al., 2020, Spontaneous onset of convection in a uniform phoretic channel., *Soft Matter*, Vol: 16, Pages: 1259-1269, ISSN: 1744-683X

Phoretic mechanisms, whereby gradients of chemical solutes induce surface-driven flows, have recently been used to generate directed propulsion of patterned colloidal particles. When the chemical solutes diffuse slowly, an instability further provides active isotropic particles with a route to self-propulsion by spontaneously breaking the symmetry of the solute distribution. Here we show theoretically that, in a mechanism analogous to Bénard-Marangoni convection, phoretic phenomena can create spontaneous and self-sustained wall-driven mixing flows within a straight, chemically-uniform active channel. Such spontaneous flows do not result in any net pumping for a uniform channel but greatly modify the distribution and transport of the chemical solute. The instability is predicted to occur for a solute Péclet number above a critical value and for a band of finite perturbation wavenumbers. We solve the perturbation problem analytically to characterize the instability, and use both steady and unsteady numerical computations of the full nonlinear transport problem to capture the long-time coupled dynamics of the solute and flow within the channel.

Tomlin R, Cimpeanu R, Papageorgiou D, 2020, Instability and dripping of electrified liquid films flowing down inverted substrates, *Physical Review Fluids*, Vol: 5, Pages: 013703-1-013703-34, ISSN: 2469-990X

We consider the gravity-driven flow of a perfect dielectric, viscous, thin liquid film, wetting a flatsubstrate inclined at a non-zero angle to the horizontal. The dynamics of the thin film is influencedby an electric field which is set up parallel to the substrate surface – this nonlocal physical mechanismhas a linearly stabilizing effect on the interfacial dynamics. Our particular interest is in fluid filmsthat are hanging from the underside of the substrate; these films may drip depending on physicalparameters, and we investigate whether a sufficiently strong electric field can suppress such nonlinearphenomena. 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 presentstudy, we incorporate an electric field and analyse the absolute/convective instabilities of a hierarchyof reduced-order models to predict the dripping limit in parameter space. The spatial stability resultsfor the reduced-order models are verified by performing an impulse–response analysis with directnumerical simulations (DNS) of the Navier–Stokes equations coupled to the appropriate electricalequations. Guided by the results of the linear theory, we perform DNS on extended domains withinflow/outflow conditions (mimicking an experimental set-up) to investigate the dripping limit forboth non-electrified and electrified liquid films. For the latter, we find that the absolute instabilitythreshold provides an order-of-magnitude estimate for the electric field strength required to suppressdripping; the linear theory may thus be used to determine the feasibility of dripping suppressiongiven a set of geometrical, fluid and electrical parameters.

Game S, Hodes M, Papageorgiou D, 2019, Effects of slowly-varying meniscus curvature on internal flows in the Cassie state, *Journal of Fluid Mechanics*, Vol: 872, Pages: 272-307, ISSN: 0022-1120

The flow rate of a pressure-driven liquid through a microchannel may be enhanced by texturing its no-slip boundaries with grooves aligned with the flow. In such cases, the grooves may contain vapour and/or an inert gas and the liquid is trapped in the Cassie state, resulting in (apparent) slip. The flow rate enhancement is of benefit to different applications including the increase of throughput of a liquid in a lab-on-achip, and the reduction of thermal resistance associated with liquid metal cooling of microelectronics. At any given cross section, the meniscus takes the approximate shape of a circular arc whose curvature is determined by the pressure difference across it. Hence, it typically protrudes into the grooves near the inlet of a microchannel and is gradually drawn into the microchannel as it is traversed and the liquid pressure decreases. For sufficiently large Reynolds numbers, the variation of the meniscus shape and hence the flow geometry necessitates the inclusion of inertial (non-parallel) flow effects. We capture them for a slender microchannel, where our small parameter is the ratio of ridge pitchto-microchannel height, and order one Reynolds numbers. This is done by using a hybrid analytical-numerical method to resolve the nonlinear three-dimensional (3D) problem as a sequence of two-dimensional (2D) linear ones in the microchannel cross-section, allied with nonlocal conditions that determine the slowly-varying pressure distribution at leading and first orders. When the pressure difference across the microchannel is constrained by the advancing contact angle of the liquid on the ridges and its surface tension (which are high for liquid metals), inertial effects can significantly reduce the flow rate for realistic parameter values. For example, when the solid fraction of the ridges is 0.1, the microchannel height-to-(half) ridge pitch ratio is 6, the Reynolds number of the flow is 1 and the small parameter is 0.1, they reduce the flow rate of a liq

Sharma A, Ray PK, Papageorgiou DT, 2019, Dynamics of gravity-driven viscoelastic films on wavy walls, *Physical Review Fluids*, Vol: 4, Pages: 063305-1-063305-26, ISSN: 2469-990X

The linear stability and nonlinear dynamics of viscoelastic liquid films flowing down inclined surfaces with sinusoidal topography are investigated. The Oldroyd-B constitutive model is used and numerical solutions of a long-wave nonlinear evolution equation for the film thickness, introduced by Dávalos-Orozco [L. A. Dávalos-Orozco, Stability of thin viscoelastic films falling down wavy walls, Interfacial Phenom. Heat Transfer 1, 301 (2013)], provide insight into the influence of elasticity and wall topography on the nonlinear film dynamics, while Floquet analysis of the linearized evolution equation is used to study the onset of linear instability. Focusing initially on inertialess films (with zero Reynolds number), linear stability results are organized into three regimes based on the wall wavelength. For sufficiently short and sufficiently long wall wavelengths, the onset of instability is not tangibly affected by the topography. There is however an intermediate range of wavelengths where, as the wall wavelength is increased, the critical Deborah number for the onset of instability first decreases (topography is destabilizing) and then increases sufficiently for topography to be stabilizing (relative to the flat wall). Solutions to a perturbation amplitude equation indicate that the character of the instability changes substantially within this intermediate range; topography induces streamwise variations in the base-state velocity at the free surface which couple with perturbations and substantially influence the instability growth rate. Very similar trends are observed for Newtonian films and variations in the critical Reynolds number. Simulations of the full nonlinear evolution equation produce a broad range of nonlinear states including traveling waves, time-periodic waves, and chaos. Perturbations to the film generally saturate at higher amplitudes for cases with larger linear growth rates, e.g., with increasing Deborah number or for a destabiliz

Papageorgiou DT, 2019, Film flows in the presence of electric fields, *Annual Review of Fluid Mechanics*, Vol: 51, Pages: 155-187, ISSN: 0066-4189

The presence of electric fields in immiscible multifluid flows induces Maxwell stresses at sharp interfaces that can produce electrohydrodynamic phenomena of practical importance. Electric fields can be stabilizing or destabilizing depending on their strength and orientation. In microfluidics, fields can be used to drive systems out of equilibrium to produce hierarchical patterning, mixing, and phase separation. We describe nonlinear theories of electrohydrodynamic instabilities in immiscible multilayer flows in several geometries, including flows over or inside planar or topographically structured substrates and channels and flows in cylinders and cylindrical annuli. Matched asymptotic techniques are developed for two- and three-dimensional flows, and reduced-dimension nonlinear models are derived and studied. When all regions are slender, electrostatic extensions to lubrication or shallow-wave theories are derived. In the presence of nonslender layers, nonlocal terms emerge naturally to modify the evolution equations. Analysis and computations provide a plethora of dynamics, including nonlinear traveling waves, spatiotemporal chaos, and singularity formation. Direct numerical simulations are used to evaluate the models and go beyond their range of validity to quantify phenomena such as electric field–induced directed patterning, suppression of Rayleigh–Taylor instabilities, and electrostatically induced pumping in microchannels. Comparisons of theory and simulations with available experiments are included throughout.

Tomlin R, Gomes SN, Pavliotis G,
et al., 2019, Optimal control of thin liquid films and transverse mode effects, *SIAM Journal on Applied Dynamical Systems*, Vol: 18, Pages: 117-149, ISSN: 1536-0040

We consider the control of a three-dimensional thin liquid film on a flat substrate, inclined at a nonzero angle to the horizontal. Controls are applied via same-fluid blowing and suction through the substrate surface. The film may be either overlying or hanging, where the liquid lies above or below the substrate, respectively. We study the weakly nonlinear evolution of the fluid interface, which is governed by a forced Kuramoto--Sivashinsky equation in two space dimensions. The uncontrolled problem exhibits three ranges of dynamics depending on the incline of the substrate: stable flat film solution, bounded chaotic dynamics, or unbounded exponential growth of unstable transverse modes. We proceed with the assumption that we may actuate at every location on the substrate. The main focus is the optimal control problem, which we first study in the special case that the forcing may only vary in the spanwise direction. The structure of the Kuramoto--Sivashinsky equation allows the explicit construction of optimal controls in this case using the classical theory of linear quadratic regulators. Such controls are employed to prevent the exponential growth of transverse waves in the case of a hanging film, revealing complex dynamics for the streamwise and mixed modes. Next, we consider the optimal control problem in full generality and prove the existence of an optimal control. For numerical simulations, an iterative gradient descent algorithm is employed. Finally, we consider the effects of transverse mode forcing on the chaotic dynamics present in the streamwise and mixed modes for the case of a vertical film flow. Coupling through nonlinearity allows us to reduce the average energy in solutions without directly forcing the dominant linearly unstable modes.

Cimpeanu R, Papageorgiou DT, 2018, Three-dimensional high speed drop impact onto solid surfaces at arbitrary angles, *International Journal of Multiphase Flow*, Vol: 107, Pages: 192-207, ISSN: 0301-9322

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.

Tomlin R, Kalogirou A, Papageorgiou D, 2018, Nonlinear dynamics of a dispersive anisotropic Kuramoto–Sivashinsky equation in two space dimensions, *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 474, ISSN: 1364-5021

A Kuramoto–Sivashinsky equation in two spacedimensions arising in thin film flow is considered ondoubly periodic domains. In the absence of dispersiveeffects, this anisotropic equation admits chaoticsolutions for sufficiently large length scales withfully two-dimensional profiles; the one-dimensionaldynamics observed for thin domains are structurallyunstable as the transverse length increases. Wefind that, independent of the domain size, thecharacteristic length scale of the profiles in thestreamwise direction is about 10 space units, with thatin the transverse direction being approximately threetimes larger. Numerical computations in the chaoticregime provide an estimate for the radius of theabsorbing ball inL2in terms of the length scales, fromwhich we conclude that the system possesses a finiteenergy density. We show the property of equipartitionof energy among the low Fourier modes, and reportthe disappearance of the inertial range when solutionprofiles are two-dimensional. Consideration of thehigh frequency modes allows us to compute anestimate for the analytic extensibility of solutionsinC2. We examine the addition of a physicallyderived third-order dispersion to the problem; thishas a destabilising effect, in the sense of reducinganalyticity and increasing amplitude of solutions.However, sufficiently large dispersion may regularisethe spatiotemporal chaos to travelling waves. Wefocus on dispersion where chaotic dynamics persist,and study its effect on the interfacial structures,absorbing ball, and properties of the power spectrum.

Ray PK, Hauge J, Papageorgiou D, 2017, Nonlinear interfacial instability in two-fluid viscoelastic Couette flow, *Journal of Non-Newtonian Fluid Mechanics*, Vol: 251, Pages: 17-27, ISSN: 0377-0257

Weakly-nonlinear interfacial instabilities in two-fluid planar Couette flow are investigated for the case where one layer is thin. Taking this thin-layer thickness as a small parameter, asymptotic analysis is used to derive a nonlinear evolution equation for the interface height valid for wavelengths that scale with the channel height. Consequently, the influence of the thick layer is felt through a non-local coupling term which is obtained by solving a system of linear equations which are a simplified viscoelastic analogue to the Orr–Sommerfeld equation. The evolution equation allows for the clear identification of the influence of normal stresses at the interface on both the initial instability and the subsequent nonlinear dynamics. Results from numerical simulations illustrate: (1) an array of non-stationary states including traveling waves and chaos, (2) competition between elastic instability and instability due to viscosity stratification, and (3) the accuracy of a simplified ’localized’ evolution equation (derived using a long-wave approximation to the coupling term) when either the elasticity of the thick-layer fluid is sufficiently weak or the elasticities of the two fluids are sufficiently well-matched.

Game SE, Hodes M, Keaveny EE,
et al., 2017, Physical mechanisms relevant to flow resistance in textured microchannels, *Physical Review Fluids*, Vol: 2, ISSN: 2469-990X

Flow resistance of liquids flowing through microchannels can be reduced by replacing flat, no-slip boundaries with boundaries adjacent to longitudinal grooves containing an inert gas, resulting in apparent slip. With applications of such textured microchannels in areas such as microfluidic systems and direct liquid cooling of microelectronics, there is a need for predictive mathematical models that can be used for design and optimization. In this work, we describe a model that incorporates the physical effects of gas viscosity (interfacial shear), meniscus protrusion (into the grooves), and channel aspect ratio and show how to generate accurate solutions for the laminar flow field using Chebyshev collocation and domain decomposition numerical methods. While the coupling of these effects are often omitted from other models, we show that it plays a significant role in the behavior of such flows. We find that, for example, the presence of gas viscosity may cause meniscus protrusion to have a more negative impact on the flow rate than previously appreciated. Indeed, we show that there are channel geometries for which meniscus protrusion increases the flow rate in the absence of gas viscosity and decreases it in the presence of gas viscosity. In this work, we choose a particular definition of channel height: the distance from the base of one groove to the base of the opposite groove. Practically, such channels are used in constrained geometries and therefore are of prescribed heights consistent with this definition. This choice allows us to easily make meaningful comparisons between textured channels and no-slip channels occupying the same space.

Papageorgiou DT, Papaefthymiou ES, 2017, Nonlinear stability in three-layer channel flows, *Journal of Fluid Mechanics*, Vol: 829, ISSN: 0022-1120

The nonlinear stability of viscous, immiscible multilayer flows in plane channelsdriven both by a pressure gradient and gravity is studied. Three fluid phases arepresent with two interfaces. Weakly nonlinear models of coupled evolution equationsfor the interfacial positions are derived and studied for inertialess, stably stratifiedflows in channels at small inclination angles. Interfacial tension is demoted andhigh-wavenumber stabilisation enters due to density stratification through second-orderdissipation terms rather than the fourth-order ones found for strong interfacialtension. An asymptotic analysis is carried out to demonstrate how these models arise.The governing equations are 2 × 2 systems of second-order semi-linear parabolicpartial differential equations (PDEs) that can exhibit inertialess instabilities due tointeraction between the interfaces. Mathematically this takes place due to a transitionof the nonlinear flux function from hyperbolic to elliptic behaviour. The conceptof hyperbolic invariant regions, found in nonlinear parabolic systems, is used toanalyse this inertialess mechanism and to derive a transition criterion to predict thelarge-time nonlinear state of the system. The criterion is shown to predict nonlinearstability or instability of flows that are stable initially, i.e. the initial nonlinear fluxesare hyperbolic. Stability requires the hyperbolicity to persist at large times, whereasinstability sets in when ellipticity is encountered as the system evolves. In the formercase the solution decays asymptotically to its uniform base state, while in the lattercase nonlinear travelling waves can emerge that could not be predicted by a linearstability analysis. The nonlinear analysis predicts threshold initial disturbances abovewhich instability emerges.

Tomlin RJ, Papageorgiou DT, Pavliotis GA, 2017, Three-dimensional wave evolution on electrified falling films, *Journal of Fluid Mechanics*, Vol: 822, Pages: 54-79, ISSN: 1469-7645

We consider the full three-dimensional dynamics of a thin falling liquid film on a flat plate inclined at some non-zero angle to the horizontal. In addition to gravitational effects, the flow is driven by an electric field which is normal to the substrate far from the flow. This extends the work of Tseluiko & Papageorgiou (J. Fluid Mech., vol. 556, 2006b, pp. 361–386) by including transverse dynamics. We study both the cases of overlying and hanging films, where the liquid lies above or below the substrate, respectively. Starting with the Navier–Stokes equations coupled with electrostatics, a fully nonlinear two-dimensional Benney equation for the interfacial dynamics is derived, valid for waves that are long compared to the film thickness. The weakly nonlinear evolution is governed by a Kuramoto–Sivashinsky equation with a non-local term due to the electric field effect. The electric field term is linearly destabilising and produces growth rates proportional to $|\unicode[STIX]{x1D743}|^{3}$ , where $\unicode[STIX]{x1D743}$ is the wavenumber vector of the perturbations. It is found that transverse gravitational instabilities are always present for hanging films, and this leads to unboundedness of nonlinear solutions even in the absence of electric fields – this is due to the anisotropy of the nonlinearity. For overlying films and a restriction on the strength of the electric field, the equation is well-posed in the sense that it possesses bounded solutions. This two-dimensional equation is studied numerically for the case of periodic boundary conditions in order to assess the effects of inertia, electric field strength and the size of the periodic domain. Rich dynamical behaviours are observed and reported. For subcritical Reynolds number flows, a sufficiently strong electric field can promote non-trivial dynamics for some choices of domain size, leading to fully two-dimensional evolutions of the interface. We also observe two-dimensiona

Wray AW, Matar OK, Papageorgiou DT, 2017, Accurate low-order modeling of electrified falling films at moderate Reynolds number, *Physical Review Fluids*, Vol: 2, ISSN: 2469-990X

The two- and three-dimensional spatio-temporal dynamics of a falling, electrified leakydielectric film are studied. The method of weighted residuals is used to derive high-ordermodels that account for both inertia as well as second-order electrostatic effects. Themodels are validated against both linear theory and direct numerical simulations of theNavier-Stokes equations. It is shown that a simplified model offers a rapid computationaloption at the cost of a minimal decrease in accuracy. This model is then used to perform aparametric study in three dimensions.

Wray AW, Papageorgiou DT, Matar OK, 2017, Reduced models for thick liquid layers with inertia on highly curved substrates, *SIAM Journal on Applied Mathematics*, Vol: 77, Pages: 881-904, ISSN: 0036-1399

A method is presented for deriving reduced models for fluid flows over highly curved substrates with wider applicability and accuracy than existing models in the literature. This is done by reducing the Navier--Stokes equations to a novel system of boundary layer like equations in a general geometric setting. This is accomplished using a new, relaxed set of scalings that assert only that streamwise variations are “slow”. These equations are then solved using the method of weighted residuals, which is demonstrated to be applicable regardless of the geometry selected. A large number of results in the literature can be derived as special cases of our general formulation. A few of the more interesting cases are demonstrated. Finally, the formulation is applied to two thick annular flow systems as well as a conical system in both linear and nonlinear regimes, which traditionally has been considered inaccessible to such reduced models. Comparisons are made with direct numerical simulations of the Stokes equations. The results indicate that reduced models can now be used to model systems involving thick liquid layers.

Anderson TG, Cimpeanu R, Papageorgiou DT,
et al., 2017, Electric field stabilization of viscous liquid layers coating the underside of a surface, *PHYSICAL REVIEW FLUIDS*, Vol: 2, ISSN: 2469-990X

We investigate the electrostatic stabilization of a viscous thin film wetting the underside of a horizontal surface in the presence of an electric field applied parallel to the surface. The model includes the effect of bounding solid dielectric regions above and below the liquid-air system that are typically found in experiments. The competition between gravitational forces, surface tension, and the nonlocal effect of the applied electric field is captured analytically in the form of a nonlinear evolution equation. A semispectral solution strategy is employed to resolve the dynamics of the resulting partial differential equation. Furthermore, we conduct direct numerical simulations (DNS) of the Navier-Stokes equations using the volume-of-fluid methodology and assess the accuracy of the obtained solutions in the long-wave (thin-film) regime when varying the electric field strength from zero up to the point when complete stabilization occurs. We employ DNS to examine the limitations of the asymptotically derived behavior as the liquid layer thickness increases and find excellent agreement even beyond the regime of strict applicability of the asymptotic solution. Finally, the asymptotic and computational approaches are utilized to identify robust and efficient active control mechanisms allowing the manipulation of the fluid interface in light of engineering applications at small scales, such as mixing.

Moore MR, Mughal MS, Papageorgiou DT, 2017, Ice formation within a thin film flowing over a flat plate, *Journal of Fluid Mechanics*, Vol: 817, Pages: 455-489, ISSN: 0022-1120

We present a model for ice formation in a thin, viscous liquid film driven by aBlasius boundary layer after heating is switched off along part of the flat plate. Theflow is assumed to initially be in the Nelson et al. (J. Fluid Mech., vol. 284, 1995,pp. 159–169) steady-state configuration with a constant flux of liquid supplied atthe tip of the plate, so that the film thickness grows like x1/4in distance alongthe plate. Plate cooling is applied downstream of a point, Lx0, an O(L)-distancefrom the tip of the plate, where L is much larger than the film thickness. Thecooling is assumed to be slow enough that the flow is quasi-steady. We present athorough asymptotic derivation of the governing equations from the incompressibleNavier–Stokes equations in each fluid and the corresponding Stefan problem for icegrowth. The problem breaks down into two temporal regimes corresponding to therelative size of the temperature difference across the ice, which are analysed in detailasymptotically and numerically. In each regime, two distinct spatial regions arise, anouter region of the length scale of the plate, and an inner region close to x0 in whichthe film and air are driven over the growing ice layer. Moreover, in the early timeregime, there is an additional intermediate region in which the air–water interfacepropagates a slope discontinuity downstream due to the sudden onset of the ice atthe switch-off point. For each regime, we present ice profiles and growth rates, andshow that for large times, the film is predicted to rupture in the outer region whenthe slope discontinuity becomes sufficiently enhanced.

Noronha Moreira Antunes Gomes ST, Kalliadasis S, Papageorgiou DT,
et al., 2017, Controlling roughening processes in the stochastic Kuramoto-Sivashinsky equation, *Physica D - Nonlinear Phenomena*, Vol: 348, Pages: 33-43, ISSN: 0167-2789

We present a novel control methodology to control the roughening processes of semilinear parabolic stochastic partial differential equations in one dimension, which we exemplify with the stochastic Kuramoto-Sivashinsky equation. The original equation is split into a linear stochastic and a nonlinear deterministic equation so that we can apply linear feedback control methods. Our control strategy is then based on two steps: first, stabilize the zero solution of the deterministic part and, second, control the roughness of the stochastic linear equation. We consider both periodic controls and point actuated ones, observing in all cases that the second moment of the solution evolves in time according to a power-law until it saturates at the desired controlled value.

Dubrovina E, Craster RV, Papageorgiou DT, 2017, Two-layer electrified pressure-driven flow in topographically structured channels, *Journal of Fluid Mechanics*, Vol: 814, Pages: 222-248, ISSN: 0022-1120

The flow of two stratified viscous immiscible perfect dielectric fluids in a channel withtopographically structured walls is investigated. The flow is driven by a streamwisepressure gradient and an electric field across the channel gap. This problem isexplored in detail by deriving and studying a nonlinear evolution equation for theinterface valid for large-amplitude long waves in the Stokes flow regime. For flatwalls, the electrified flow is long-wave unstable with a critical cutoff wavenumberthat increases linearly with the magnitude of the applied voltage. In the nonlinearregime, it is found that the presence of pressure-driven flow prevents electrostaticallyinduced interface touchdown that has been observed previously – time-modulatednonlinear travelling waves emerge instead. When topography is present, linearly stableuniform flows become non-uniform spatially periodic steady states; a small-amplitudeasymptotic theory is carried out and compared with computations. In the linearlyunstable regime, intricate nonlinear structures emerge that depend, among otherthings, on the magnitude of the wall corrugations. For a low-amplitude sinusoidalboundary, time-modulated travelling waves are observed that are similar to thosefound for flat walls but are influenced by the geometry of the wall and slide over itwithout touching. The flow over a high-amplitude sinusoidal pattern is also examinedin detail and it is found that for sufficiently large voltages the interface evolves tolarge-amplitude waves that span the channel and are subharmonic relative to the wall.A type of ‘walking’ motion emerges that causes the lower fluid to wash through thetroughs and create strong vortices over the peaks of the lower boundary. Non-uniformsteady states induced by the topography are calculated numerically for moderate andlarge values of the flow rate, and their stability is analysed using Floquet theory.The effect of large flow rates is also considered asymptotically to fi

Kirk TL, Hodes M, Papageorgiou DT, 2016, Nusselt numbers for Poiseuille flow over isoflux parallel ridges accounting for meniscus curvature, *Journal of Fluid Mechanics*, Vol: 811, Pages: 315-349, ISSN: 1469-7645

We investigate forced convection in a parallel-plate-geometry microchannel with superhydrophobic walls consisting of a periodic array of ridges aligned parallel to the direction of a Poiseuille flow. In the dewetted (Cassie) state, the liquid contacts the channel walls only at the tips of the ridges, where we apply a constant-heat-flux boundary condition. The subsequent hydrodynamic and thermal problems within the liquid are then analysed accounting for curvature of the liquid–gas interface (meniscus) using boundary perturbation, assuming a small deflection from flat. The effects of this surface deformation on both the effective hydrodynamic slip length and the Nusselt number are computed analytically in the form of eigenfunction expansions, reducing the problem to a set of dual series equations for the expansion coefficients which must, in general, be solved numerically. The Nusselt number quantifies the convective heat transfer, the results for which are completely captured in a single figure, presented as a function of channel geometry at each order in the perturbation. Asymptotic solutions for channel heights large compared with the ridge period are compared with numerical solutions of the dual series equations. The asymptotic slip length expressions are shown to consist of only two terms, with all other terms exponentially small. As a result, these expressions are accurate even for heights as low as half the ridge period, and hence are useful for engineering applications.

Kalogirou A, CĂ®mpeanu R, Keaveny EE,
et al., 2016, Capturing nonlinear dynamics of two-fluid Couette flows with asymptotic models, *Journal of Fluid Mechanics*, Vol: 806, Pages: R1-R13, ISSN: 1469-7645

The nonlinear stability of two-fluid Couette flows is studied using a novel evolution equation whose dynamics is validated by direct numerical simulation (DNS). The evolution equation incorporates inertial effects at arbitrary Reynolds numbers through a non-local term arising from the coupling between the two fluid regions, and is valid when one of the layers is thin. The equation predicts asymmetric solutions and exhibits bistability, features that are essential observations in the experiments of Barthelet et al. (J. Fluid Mech., vol. 303, 1995, pp. 23–53). Related low-inertia models have been used in qualitative predictions rather than the direct comparisons carried out here, and ad hoc modifications appear to be necessary in order to predict asymmetry and bistability. Comparisons between model solutions and DNS show excellent agreement at Reynolds numbers of O(103)O(103) found in the experiments. Direct comparisons are also made with the available experimental results of Barthelet et al. (J. Fluid Mech., vol. 303, 1995, pp. 23–53) when the thin layer occupies 1/51/5 of the channel height. Pointwise comparisons of the travelling wave shapes are carried out, and once again the agreement is very good.

Kalogirou A, Papageorgiou DT, 2016, Nonlinear dynamics of surfactant-laden two-fluid Couette flows in the presence of inertia, *Journal of Fluid Mechanics*, Vol: 802, Pages: 5-36, ISSN: 1469-7645

The nonlinear stability of immiscible two-fluid Couette flows in the presence of inertia is considered. The interface between the two viscous fluids can support insoluble surfactants and the interplay between the underlying hydrodynamic instabilities and Marangoni effects is explored analytically and computationally in both two and three dimensions. Asymptotic analysis when one of the layers is thin relative to the other yields a coupled system of nonlinear equations describing the spatio-temporal evolution of the interface and its local surfactant concentration. The system is non-local and arises by appropriately matching solutions of the linearised Navier–Stokes equations in the thicker layer to the solution in the thin layer. The scaled models are used to study different physical mechanisms by varying the Reynolds number, the viscosity ratio between the two layers, the total amount of surfactant present initially and a scaled Péclet number measuring diffusion of surfactant along the interface. The linear stability of the underlying flow to two- and three-dimensional disturbances is investigated and a Squire’s type theorem is found to hold when inertia is absent. When inertia is present, three-dimensional disturbances can be more unstable than two-dimensional ones and so Squire’s theorem does not hold. The linear instabilities are followed into the nonlinear regime by solving the evolution equations numerically; this is achieved by implementing highly accurate linearly implicit schemes in time with spectral discretisations in space. Numerical experiments for finite Reynolds numbers indicate that for two-dimensional flows the solutions are mostly nonlinear travelling waves of permanent form, even though these can lose stability via Hopf bifurcations to time-periodic travelling waves. As the length of the system (that is the wavelength of periodic waves) increases, the dynamics becomes more complex and includes time-periodic, quasi-periodic

Gomes SN, Papageorgiou DT, Pavliotis GA, 2016, Stabilizing non-trivial solutions of the generalized Kuramoto-Sivashinsky equation using feedback and optimal control, *IMA JOURNAL OF APPLIED MATHEMATICS*, Vol: 82, Pages: 158-194, ISSN: 0272-4960

Papageorgiou DT, Wang Q, 2016, Using electric fields to induce patterning in leaky dielectric fluids in a rod-annular geometry, *IMA Journal of Applied Mathematics*, ISSN: 0272-4960

The stability and axisymmetric deformation of two immiscible, viscous, perfect or leaky dielectric fluids confined in the annulus between two concentric cylinders are studied in the presence of radial electricfields. The fields are set up by imposing a constant voltage potential difference between the inner andouter cylinders. We derive a set of equations for the interface in the long-wavelength approximation which retains the essential physics of the system and allows for interfacial deformations to be as large as the annular gap hence accounting for possible touchdown at the inner or outer electrode. The effects of the electric parameters are evaluated initially by performing a linear stability analysis which shows excellent agreement with the linear theory of the full axisymmetric problem in the appropriate long wavelengthregime. The nonlinear interfacial dynamics are investigated by carrying out direct numerical simulations of the derived long wave models, both in the absence and presence of electric fields. For non-electrified thin layer flows (i.e. one of the layers thin relative to the other) the long-time dynamics agree with thelubrication approximation results found in literature. When the liquid layers have comparable thickness our results demonstrate the existence of both finite time and infinite time singularities (asymptotic touching solutions) in the system. It is shown that a two-side touching solution is possible for both the non-electrified and perfect dielectric cases, while only one-side touching is found in the case of leaky dielectric liquids, where the flattened interface shape resembles the pattern solutions found in literature.Meanwhile the finite-time singular solution agrees qualitatively with the experiments of Reynolds (196

Thompson AB, Gomes SN, Pavliotis,
et al., 2016, Stabilising falling liquid film flows using feedback control, *Physics of Fluids*, Vol: 28, ISSN: 1089-7666

Falling liquid films become unstable due to inertial effects when the fluid layer is sufficiently thick or the slopesufficiently steep. This free surface flow of a single fluid layer has industrial applications including coating andheat transfer, which benefit from smooth and wavy interfaces, respectively. Here we discuss how the dynamicsof the system are altered by feedback controls based on observations of the interface height, and supplied tothe system via the perpendicular injection and suction of fluid through the wall. In this study, we modelthe system using both Benney and weighted-residual models that account for the fluid injection throughthe wall. We find that feedback using injection and suction is a remarkably effective control mechanism:the controls can be used to drive the system towards arbitrary steady states and travelling waves, and thequalitative effects are independent of the details of the flow modelling. Furthermore, we show that the systemcan still be successfully controlled when the feedback is applied via a set of localised actuators and only asmall number of system observations are available, and that this is possible using both static (where thecontrols are based on only the most recent set of observations) and dynamic (where the controls are based onan approximation of the system which evolves over time) control schemes. This study thus provides a solidtheoretical foundation for future experimental realisations of the active feedback control of falling liquid films.

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