## Publications

151 results found

Tomlinson SD, Mayer MD, Kirk TL,
et al., 2024, Thermal resistance of heated superhydrophobic channels with thermocapillary stress, *ASME Journal of Heat and Mass Transfer*, Vol: 146, ISSN: 0017-9310

A pressure-driven channel flow between a longitudinally ridged superhydrophobic surface (SHS) and solid wall is studied, where a constant heat flux enters the channel from either the SHS or solid wall. First, a model is developed which neglects thermocapillary stresses (TCS) in the transverse direction. The caloric, convective, and total thermal resistance are evaluated, and their dependence on the shape of the liquid–gas interface (meniscus), gas ridge width, texture period, channel height, streamwise TCS, Péclet number, and channel length is established. The caloric resistance is minimized with menisci that protrude into the gas cavity, large slip fractions, small channel heights, and small streamwise TCSs. When heating from the SHS, the convective resistance increases, and therefore, a design compromise exists between caloric and convective resistances. However, when heating from the solid wall, the convective resistance remains the same and SHSs that minimize caloric resistance are optimal. We investigate both water and Galinstan for microchannel applications and find that both configurations can have a lower total thermal resistance than a smooth-walled channel. Heating from the solid wall is shown to always have the lowest total thermal resistance. Numerical simulations are used to analyze the effect of transverse TCSs. Our model captures much of the physics in heated superhydrophobic channels but is computationally inexpensive when compared to the numerical simulations.

Crowdy D, Curran A, Papageorgiou D, 2023, Fast reaction of soluble surfactant can remobilize a stagnant cap, *Journal of Fluid Mechanics*, Vol: 969, Pages: 1-29, ISSN: 0022-1120

Analytical solutions are derived showing that a stagnant cap of surfactant at the interface between two viscous fluids caused by a linear extensional flow can be remobilized by fast kinetic exchange of surfactant with one of the fluids. Using a complex variable formulation of this multiphysics problem at zero capillary number, zero Reynolds number and zero bulk Péclet number, and assuming a linear equation of state, it is shown that the system is governed by a forced complex Burgers equation at arbitrary surface Péclet number. Consequently, this nonlinear system is shown to be linearizable using a complex analogue of the Cole–Hopf transformation. Steady equilibria of the system at any finite value of the surface Péclet number are found explicitly in terms of parabolic cylinder functions. While surface diffusion is naturally expected to mollify sharp gradients associated with stagnant caps and to remobilize the interface, this work gives an analytical demonstration of the less intuitive result that fast kinetic exchange has a similar effect. Indeed, the analytical approach here imposes no limit on the surface Péclet number, which can be taken to be infinitely large so that surface diffusion is completely absent. Mathematically, the solution structure is then very rich allowing a theoretical investigation of this extreme case where it is seen that fast surfactant exchange with the bulk can alone remobilize a stagnant cap. Remarkably, it is also possible to track explicitly the time evolution of the system to these remobilized equilibria by finding time-evolving exact solutions.

Papageorgiou DT, Tanveer S, 2023, Singular effect of interfacial slip for an otherwise stable two-layer shear flow: analysis and computations, *Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences*, Vol: 479, ISSN: 1364-5021

We consider instability of the flat interface in a two-layer Couette flow model developed earlier (Kalogirou & Papageorgiou, 2016, J. Fluid Mech. 802, 5–36; Katsiavria & Papageorgiou, 2022, Wave Motion 114, 103018. (doi:10.1016/j.wavemoti.2022.103018)) for a thin layer near one of the walls. For the case when the less viscous fluid resides next to the moving wall, we find that even a small slip effect at the interface can destabilize an otherwise highly stable flow to the Turing-type instability. The singular effect of small slip in an otherwise very stable configuration may have important ramifications in physical and technological applications. The neutral points of the dispersion relation give rise to travelling wave solutions that are continued to finite amplitude numerically and their linear stability properties identified for a set of parameter values for disturbances that include subharmonic modes with twice the wavelength of the nonlinear travelling wave. We determined Hopf and regular bifurcation points of travelling waves and rigorously justified their existence for some set of parameter values. Weakly nonlinear analysis close to bifurcation from a flat state is also presented for small amplitude waves in general. We also present global existence and regularity results for periodic initial conditions without any restriction on parameters.

Papageorgiou DT, Smyrlis Y-S, Tomlin RJ, 2022, Optimal analyticity estimates for non-linear active-dissipative evolution equations, *IMA Journal of Applied Mathematics*, Vol: 87, Pages: 964-984, ISSN: 0272-4960

Active–dissipative evolution equations emerge in a variety of physical and technological applications including liquid film flows, flame propagation, epitaxial film growth in materials manufacturing, to mention a few. They are characterized by three main ingredients: a term producing growth (active), a term providing damping at short length scales (dissipative) and a nonlinear term that transfers energy between modes and crucially produces a nonlinear saturation. The manifestation of these three mechanisms can produce large-time spatiotemporal chaos as evidenced by the Kuramoto-Sivashinsky equation (negative diffusion, fourth-order dissipation and a Burgers nonlinearity), which is arguably the simplest partial differential equation to produce chaos. The exact form of the terms (and in particular their Fourier symbol) determines the type of attractors that the equations possess. The present study considers the spatial analyticity of solutions under the assumption that the equations possess a global attractor. In particular, we investigate the spatial analyticity of solutions of a class of one-dimensional evolutionary pseudo-differential equations with Burgers nonlinearity, which are periodic in space, thus generalizing the Kuramoto-Sivashinsky equation motivated by both applications and their fundamental mathematical properties. Analyticity is examined by utilizing a criterion involving the rate of growth of suitable norms of the nth spatial derivative of the solution, with respect to the spatial variable, as n tends to infinity. An estimate of the rate of growth of the nth spatial derivative is obtained by fine-tuning the spectral method, developed elsewhere. We prove that the solutions are analytic if γ, the order of dissipation of the pseudo-differential operator, is higher than one. We also present numerical evidence suggesting that this is optimal, i.e. if γ is not larger that one, then the solution is not in general analytic. Extensive numeri

Katsiavria A, Papageorgiou DT, 2022, Nonlinear waves in viscous multilayer shear flows in the presence of interfacial slip, *Wave Motion*, Vol: 114, Pages: 1-14, ISSN: 0165-2125

The stability of immiscible two-fluid Couette flows is considered when slip is present at the liquid–liquid interface. The phenomena are modelled by incorporating a Navier slip condition at the interface to replace that of no slip. A nonlinear asymptotic theory is developed when the flow geometry consists of a thin layer slipping over a thick fluid layer that scales with the channel height. A nonlocal, nonlinear evolution equation is derived that is valid at finite Reynolds numbers, slip lengths, viscosity and density ratios. The nonlocal term arises from the coupling between the phases and its Fourier symbol is calculated in closed form in terms of Airy functions, thus generalising past results by the inclusion of slip. The linear spectrum is calculated and it is shown that in geometries containing a thin layer, the role of slip is to introduce dispersion and reduce instability or enhance stability.

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, Pages: 1-22, ISSN: 0022-0833

Starting from the Euler equations governing the flow of two immiscible incompressible fluids in a horizontal channel, allowing gravity and surface tension, and imposing an electric field across the channel, a nonlinear long-wave analysis is used to derive a 2×2 system of evolution equations describing the interface position and a modified tangential velocity jump across it. Travelling waves of permanent form are shown to exist and are constructed in the periodic case producing wave trains and the infinite case yielding novel gravity electro-capillary solitary waves. Various regimes are analysed including a hydrodynamically passive but electrically active upper layer, pairs of perfect dielectric fluids and a perfectly conducting lower fluid. In all cases, the presence of the field produces both depression and elevation waves travelling at the same speed, for given sets of parameters. The stability of the non-uniform travelling waves is investigated by numerically solving appropriate linearised eigenvalue problems. It is found that depression waves are neutrally stable whereas elevation ones are unstable unless the surface tension is large. Stability or instability is shown to be linked mathematically to the type of local eigenvalues of the nonlinear flux matrix used to obtain travelling and solitary waves; if these are real (hyperbolic flux matrix), the system is stable, and if they are complex (elliptic), the system is unstable. The latter is a manifestation of Kelvin–Helmholtz instability in electrified flows.

Tomlinson SD, Papageorgiou DT, 2022, Linear instability of lid- and pressure-driven flows in channels textured with longitudinal superhydrophobic grooves, *Journal of Fluid Mechanics*, Vol: 932, Pages: 1-37, ISSN: 0022-1120

It is known that an increased flow rate can be achieved in channel flows when smooth walls are replaced by superhydrophobic surfaces. This reduces friction and increases the flux for a given driving force. Applications include thermal management in microelectronics, where a competition between convective and conductive resistance must be accounted for in order to evaluate any advantages of these surfaces. Of particular interest is the hydrodynamic stability of the underlying basic flows, something that has been largely overlooked in the literature, but is of key relevance to applications that typically base design on steady states or apparent-slip models that approximate them. We consider the global stability problem in the case where the longitudinal grooves are periodic in the spanwise direction. The flow is driven along the grooves by either the motion of a smooth upper lid or a constant pressure gradient. In the case of smooth walls, the former problem (plane Couette flow) is linearly stable at all Reynolds numbers whereas the latter (plane Poiseuille flow) becomes unstable above a relatively large Reynolds number. When grooves are present our work shows that additional instabilities arise in both cases, with critical Reynolds numbers small enough to be achievable in applications. Generally, for lid-driven flows one unstable mode is found that becomes neutral as the Reynolds number increases, indicating that the flows are inviscidly stable. For pressure-driven flows, two modes can coexist and exchange stability depending on the channel height and slip fraction. The first mode remains unstable as the Reynolds number increases and corresponds to an unstable mode of the two-dimensional Rayleigh equation, while the second mode becomes neutrally stable at infinite Reynolds numbers. Comparisons of critical Reynolds numbers with the experimental observations for pressure-driven flows of Daniello et al. (Phys. Fluids, vol. 21, issue 8, 2009, p. 085103) are encouraging.

Alexander JP, Papageorgiou DT, 2022, Ordered and disordered dynamics in inertialess stratified three-layer shear flows, *Physical Review Fluids*, Vol: 7, Pages: 1-30, ISSN: 2469-990X

Unlike inertialess two-layer shear flows, three-layer ones can become unstable to long-wave interfacial instabilities due to a resonance mechanism between the interfaces. This interaction is codified in this paper through a set of coupled nonlinear evolution equations derived here in the limit of strong surface tension. A number of parameters are employed to cover a fairly general range of three-layer shear flows driven by a constant pressure gradient. The equations are analyzed using a combination of linear and computational techniques, identifying two linear instability mechanisms noted in the literature previously. The first is a kinematic instability due to the viscosity jumps across fluid phases and the second is a counterintuitive diffusion-derived instability, known in the literature as the Majda-Pego instability and mostly studied for second order diffusion. In the present work it is fourth order, due to surface tension, making the problem mathematically much more challenging. Three unstable parameter regimes of interest are identified linearly and are explored nonlinearly via pseudospectral numerical simulations. For thin middle layers we find steady-state traveling waves or states with asymptotically thinning regions leading to interfacial contact. However, for thin upper or lower layers, complex spatiotemporal dynamics emerge at large times that are characterized by fast time oscillations of the near-wall interface and slow oscillations of that farther away. Data analysis suggests that the dynamics is quasiperiodic in time and additionally coarsening phenomena are observed for large domain sizes leading to modulated traveling wave trains. The kinematic instability mechanism is shown to be triggered nonlinearly via the Majda-Pego mechanism. It can also be triggered by sufficiently large amplitude initial disturbances where linear instabilities are absent, although the transition is not necessarily self-sustaining in all cases.

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

The ability to robustly and efficiently control the dynamics of nonlinear systems lies at the heart of many current technological challenges, ranging from drug delivery systems to ensuring flight safety. Most such scenarios are too complex to tackle directly, and reduced-order modelling is used in order to create viable representations of the target systems. The simplified setting allows for the development of rigorous control theoretical approaches, but the propagation of their effects back up the hierarchy and into real-world systems remains a significant challenge. Using the canonical set-up of a liquid film falling down an inclined plane under the action of active feedback controls in the form of blowing and suction, we develop a multi-level modelling framework containing both analytical models and direct numerical simulations acting as an in silico experimental platform. Constructing strategies at the inexpensive lower levels in the hierarchy, we find that offline control transfer is not viable; however, analytically informed feedback strategies show excellent potential, even far beyond the anticipated range of applicability of the models. The detailed effects of the controls in terms of stability and treatment of nonlinearity are examined in detail in order to gain understanding of the information transfer inside the flows, which can aid transition towards other control-rich frameworks and applications.

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.

Papageorgiou DT, Tanveer S, 2019, Analysis and computations of a non-local thin-film model for two-fluid shear driven flows, *PROCEEDINGS OF THE ROYAL SOCIETY A-MATHEMATICAL PHYSICAL AND ENGINEERING SCIENCES*, Vol: 475, ISSN: 1364-5021

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- Citations: 4

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.

Game S, Hodes M, Kirk T,
et al., 2018, Nusselt Numbers for Poiseuille Flow Over Isoflux Parallel Ridges for Arbitrary Meniscus Curvature, *JOURNAL OF HEAT TRANSFER-TRANSACTIONS OF THE ASME*, Vol: 140, ISSN: 0022-1481

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Karamanis G, Hodes M, Kirk T,
et al., 2018, Solution of the extended Graetz-Nusselt problem for liquid flow over isothermal parallel ridges, *Journal of Heat Transfer: Transactions of the ASME*, Vol: 140, Pages: 1-15, ISSN: 0022-1481

We consider convective heat transfer for laminar flow of liquid between parallel plates. The configurations analyzed are both plates textured with symmetrically aligned isothermal ridges oriented parallel to the flow, and one plate textured as such and the other one smooth and adiabatic. The liquid is assumed to be in the Cassie state on the textured surface(s) to which a mixed boundary condition of no-slip on the ridges and no-shear along flat menisci applies. The thermal energy equation is subjected to a mixed isothermal-ridge and adiabatic-meniscus boundary condition on the textured surface(s). We solve for the developing three-dimensional temperature profile resulting from a step change of the ridge temperature in the streamwise direction assuming a hydrodynamically developed flow. Axial conduction is accounted for, i.e., we consider the extended Graetz–Nusselt problem; therefore, the domain is of infinite length. The effects of viscous dissipation and (uniform) volumetric heat generation are also captured. Using the method of separation of variables, the homogeneous part of the thermal problem is reduced to a nonlinear eigenvalue problem in the transverse coordinates which is solved numerically. Expressions derived for the local and the fully developed Nusselt number along the ridge and that averaged over the composite interface in terms of the eigenvalues, eigenfunctions, Brinkman number, and dimensionless volumetric heat generation rate. Estimates are provided for the streamwise location where viscous dissipation effects become important.

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.

Karamanis G, Hodes M, Kirk T,
et al., 2017, Solution of the Graetz-Nusselt problem for liquid flow over isothermal parallel ridges, *Journal of Heat Transfer: Transactions of the ASME*, Vol: 139, Pages: 1-12, ISSN: 0022-1481

We consider convective heat transfer for laminar flow of liquid between parallel plates that are textured with isothermal ridges oriented parallel to the flow. Three different flow configurations are analyzed: one plate textured and the other one smooth; both plates textured and the ridges aligned; and both plates textured, but the ridges staggered by half a pitch. The liquid is assumed to be in the Cassie state on the textured surface(s), to which a mixed boundary condition of no-slip on the ridges and no-shear along flat menisci applies. Heat is exchanged with the liquid either through the ridges of one plate with the other plate adiabatic, or through the ridges of both plates. The thermal energy equation is subjected to a mixed isothermal-ridge and adiabatic-meniscus boundary condition on the textured surface(s). Axial conduction is neglected and the inlet temperature profile is arbitrary. We solve for the three-dimensional developing temperature profile assuming a hydrodynamically developed flow, i.e., we consider the Graetz–Nusselt problem. Using the method of separation of variables, the thermal problem is essentially reduced to a two-dimensional eigenvalue problem in the transverse coordinates, which is solved numerically. Expressions for the local Nusselt number and those averaged over the period of the ridges in the developing and fully developed regions are provided. Nusselt numbers averaged over the period and length of the domain are also provided. Our approach enables the aforementioned quantities to be computed in a small fraction of the time required by a general computational fluid dynamics (CFD) solver.

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.

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