20 results
Stability of a photosurfactant-laden viscous liquid thread under illumination
- Michael D. Mayer, Toby L. Kirk, Demetrios T. Papageorgiou
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- Journal:
- Journal of Fluid Mechanics / Volume 983 / 25 March 2024
- Published online by Cambridge University Press:
- 14 March 2024, A10
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This paper investigates the effects of a light-actuated photosurfactant on the canonical problem of the linear stability of a viscous thread surrounded by a dynamically passive fluid. A model consisting of the Navier–Stokes equations and a set of molar concentration equations is presented that capture light-induced switching between two stable surfactant isomer states, trans and cis. These two states display significantly different interfacial properties, allowing for some external control of the stability behaviour of the thread via incident light. Normal modes are used to generate a generalized eigenvalue problem for the growth rate which is solved with a hybrid analytical and numerical method. The results are validated with appropriate analytical solutions of increasing complexity, beginning with a solution to a clean interface, then analytical solutions for one insoluble surfactant, one soluble surfactant and a special case of two photosurfactants with a spatially uniform undisturbed state. Presenting each of these cases allows for a holistic discussion of the effect of surfactants in general on the stability of a liquid thread. Finally, the numerical solutions in the presence of two photosurfactants that display radially non-uniform undisturbed states are presented, and details of the impact of the illumination on the linear stability of the thread are discussed.
Fast reaction of soluble surfactant can remobilize a stagnant cap
- Darren G. Crowdy, Anna E. Curran, Demetrios T. Papageorgiou
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- Journal of Fluid Mechanics / Volume 969 / 25 August 2023
- Published online by Cambridge University Press:
- 11 August 2023, A8
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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.
Linear instability of lid- and pressure-driven flows in channels textured with longitudinal superhydrophobic grooves
- Samuel D. Tomlinson, Demetrios T. Papageorgiou
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- Journal:
- Journal of Fluid Mechanics / Volume 932 / 10 February 2022
- Published online by Cambridge University Press:
- 02 December 2021, A12
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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.
Stability of falling liquid films on flexible substrates
- J. Paul Alexander, Toby L. Kirk, Demetrios T. Papageorgiou
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- Journal:
- Journal of Fluid Mechanics / Volume 900 / 10 October 2020
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- 13 August 2020, A40
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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.
Effects of slowly varying meniscus curvature on internal flows in the Cassie state
- Simon E. Game, Marc Hodes, Demetrios T. Papageorgiou
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- Journal of Fluid Mechanics / Volume 872 / 10 August 2019
- Published online by Cambridge University Press:
- 10 June 2019, pp. 272-307
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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-a-chip, 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 pitch-to-microchannel height, and order-one Reynolds numbers. This is done by using a hybrid analytical–numerical method to resolve the nonlinear three-dimensional (3-D) problem as a sequence of two-dimensional (2-D) linear ones in the microchannel cross-section, allied with non-local 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 is 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 liquid metal (Galinstan) by approximately 50 %. Conversely, for sufficiently large microchannel heights, they enhance it. Physical explanations of both of these phenomena are given.
Two-layer electrified pressure-driven flow in topographically structured channels
- Elizaveta Dubrovina, Richard V. Craster, Demetrios T. Papageorgiou
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- Journal:
- Journal of Fluid Mechanics / Volume 814 / 10 March 2017
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- 02 February 2017, pp. 222-248
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The flow of two stratified viscous immiscible perfect dielectric fluids in a channel with topographically structured walls is investigated. The flow is driven by a streamwise pressure gradient and an electric field across the channel gap. This problem is explored in detail by deriving and studying a nonlinear evolution equation for the interface valid for large-amplitude long waves in the Stokes flow regime. For flat walls, the electrified flow is long-wave unstable with a critical cutoff wavenumber that increases linearly with the magnitude of the applied voltage. In the nonlinear regime, it is found that the presence of pressure-driven flow prevents electrostatically induced interface touchdown that has been observed previously – time-modulated nonlinear travelling waves emerge instead. When topography is present, linearly stable uniform flows become non-uniform spatially periodic steady states; a small-amplitude asymptotic theory is carried out and compared with computations. In the linearly unstable regime, intricate nonlinear structures emerge that depend, among other things, on the magnitude of the wall corrugations. For a low-amplitude sinusoidal boundary, time-modulated travelling waves are observed that are similar to those found for flat walls but are influenced by the geometry of the wall and slide over it without touching. The flow over a high-amplitude sinusoidal pattern is also examined in detail and it is found that for sufficiently large voltages the interface evolves to large-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 the troughs and create strong vortices over the peaks of the lower boundary. Non-uniform steady states induced by the topography are calculated numerically for moderate and large values of the flow rate, and their stability is analysed using Floquet theory. The effect of large flow rates is also considered asymptotically to find solutions that compare very well with numerical computations.
Nusselt numbers for Poiseuille flow over isoflux parallel ridges accounting for meniscus curvature
- Toby L. Kirk, Marc Hodes, Demetrios T. Papageorgiou
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- Journal:
- Journal of Fluid Mechanics / Volume 811 / 25 January 2017
- Published online by Cambridge University Press:
- 07 December 2016, pp. 315-349
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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.
Falling liquid films with blowing and suction
- Alice B. Thompson, Dmitri Tseluiko, Demetrios T. Papageorgiou
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- Journal:
- Journal of Fluid Mechanics / Volume 787 / 25 January 2016
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- 15 December 2015, pp. 292-330
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Flow of a thin viscous film down a flat inclined plane becomes unstable to long-wave interfacial fluctuations when the Reynolds number based on the mean film thickness becomes larger than a critical value (this value decreases as the angle of inclination to the horizontal increases, and in particular becomes zero when the plate is vertical). Control of these interfacial instabilities is relevant to a wide range of industrial applications including coating processes and heat or mass transfer systems. This study considers the effect of blowing and suction through the substrate in order to construct from first principles physically realistic models that can be used for detailed passive and active control studies of direct relevance to possible experiments. Two different long-wave, thin-film equations are derived to describe this system; these include the imposed blowing/suction as well as inertia, surface tension, gravity and viscosity. The case of spatially periodic blowing and suction is considered in detail and the bifurcation structure of forced steady states is explored numerically to predict that steady states cease to exist for sufficiently large suction speeds since the film locally thins to zero thickness, giving way to dry patches on the substrate. The linear stability of the resulting non-uniform steady states is investigated for perturbations of arbitrary wavelength, and any instabilities are followed into the fully nonlinear regime using time-dependent computations. The case of small amplitude blowing/suction is studied analytically both for steady states and their stability. Finally, the transition between travelling waves and non-uniform steady states is explored as the amplitude of blowing and suction is increased.
Interfacial instability in electrified plane Couette flow
- STEFAN MÄHLMANN, DEMETRIOS T. PAPAGEORGIOU
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- Journal of Fluid Mechanics / Volume 666 / 10 January 2011
- Published online by Cambridge University Press:
- 06 January 2011, pp. 155-188
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The dynamics of a plane interface separating two sheared, density and viscosity matched fluids in the vertical gap between parallel plate electrodes are studied computationally. A Couette profile is imposed onto the fluids by moving the rigid plates at equal speeds in opposite directions. In addition, a vertical electric field is applied to the shear flow by impressing a constant voltage difference on the electrodes. The stability of the initially flat interface is a very subtle balance between surface tension, inertia, viscosity and electric field effects. Under unstable conditions, the potential difference in the fluid results in an electrostatic pressure that amplifies disturbance waves on the two-fluid interface at characteristic wave lengths. Various mechanisms determining the growth rate of the most unstable mode are addressed in a systematic parameter study. The applied methodology involves a combination of numerical simulation and analytical work. Linear stability theory is employed to identify unstable parametric conditions of the perturbed Couette flow. Particular attention is given to the effect of the applied electric field on the instability of the perturbed two-fluid interface. The normal mode analyses are followed up by numerical simulations. The applied method relies on solving the governing equations for the fluid mechanics and the electrostatics in a one-fluid approximation by using a finite-volume technique combined with explicit tracking of the evolving interface. The numerical results confirm those of linear theory and, furthermore, reveal a rich array of dynamical behaviour. The elementary fluid instabilities are finger-like structures of interpenetrating fluids. For weakly unstable situations a single fingering instability emerges on the interface. Increasing the growth rates causes the finger to form a drop-like tip region connected by a long thinning fluids neck. Even more striking fluid motion occurs at higher values of the electric field parameter for which multiple fluid branches develop on the interface. For a pair of perfect dielectrics the vertical electric field was found to enhance interfacial motion irrespective of the permittivity ratio, while in leaky dielectrics the electric field can either stabilize or destabilize the interface, depending on the conductivity and permittivity ratio between the fluids.
Numerical study of electric field effects on the deformation of two-dimensional liquid drops in simple shear flow at arbitrary Reynolds number
- STEFAN MÄHLMANN, DEMETRIOS T. PAPAGEORGIOU
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- Journal of Fluid Mechanics / Volume 626 / 10 May 2009
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- 10 May 2009, pp. 367-393
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The effect of an electric field on a periodic array of two-dimensional liquid drops suspended in simple shear flow is studied numerically. The shear is produced by moving the parallel walls of the channel containing the fluids at equal speeds but in opposite directions and an electric field is generated by imposing a constant voltage difference across the channel walls. The level set method is adapted to electrohydrodynamics problems that include a background flow in order to compute the effects of permittivity and conductivity differences between the two phases on the dynamics and drop configurations. The electric field introduces additional interfacial stresses at the drop interface and we perform extensive computations to assess the combined effects of electric fields, surface tension and inertia. Our computations for perfect dielectric systems indicate that the electric field increases the drop deformation to generate elongated drops at steady state, and at the same time alters the drop orientation by increasing alignment with the vertical, which is the direction of the underlying electric field. These phenomena are observed for a range of values of Reynolds and capillary numbers. Computations using the leaky dielectric model also indicate that for certain combinations of electric properties the drop can undergo enhanced alignment with the vertical or the horizontal, as compared to perfect dielectric systems. For cases of enhanced elongation and alignment with the vertical, the flow positions the droplets closer to the channel walls where they cause larger wall shear stresses. We also establish that a sufficiently strong electric field can be used to destabilize the flow in the sense that steady-state droplets that can exist in its absence for a set of physical parameters, become increasingly and indefinitely elongated until additional mechanisms can lead to rupture. It is suggested that electric fields can be used to enhance such phenomena.
A global attracting set for nonlocal Kuramoto–Sivashinsky equations arising in interfacial electrohydrodynamics
- DMITRI TSELUIKO, DEMETRIOS T. PAPAGEORGIOU
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- European Journal of Applied Mathematics / Volume 17 / Issue 6 / December 2006
- Published online by Cambridge University Press:
- 05 February 2007, pp. 677-703
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We study a generalized class of nonlocal evolution equations which includes those arising in the modelling of electrified film flow down an inclined plane, with applications in enhanced heat or mass transfer through interfacial turbulence. Global existence and uniqueness results are proved and refined estimates of the radius of the absorbing ball in $L^2$ are obtained in terms of the parameters of the equations (the length of the system and the dimensionless electric field-measuring parameter multiplying the nonlocal term). The established estimates are compared with numerical solutions of the equations which in turn suggest an optimal upper bound for the radius of the absorbing ball. A scaling argument is used to explain this and a general conjecture is made based on extensive computations.
Theory and experiments on the stagnant cap regime in the motion of spherical surfactant-laden bubbles
- RAVICHANDRA PALAPARTHI, DEMETRIOS T. PAPAGEORGIOU, CHARLES MALDARELLI
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- Journal of Fluid Mechanics / Volume 559 / 25 July 2006
- Published online by Cambridge University Press:
- 19 July 2006, pp. 1-44
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The buoyant motion of a bubble rising through a continuous liquid phase can be retarded by the adsorption onto the bubble surface of surfactant dissolved in the liquid phase. The reason for this retardation is that adsorbed surfactant is swept to the trailing pole of the bubble where it accumulates and lowers the surface tension relative to the front end. The difference in tension creates a Marangoni force which opposes the surface flow, rigidifies the interface and increases the drag coefficient. Surfactant molecules adsorb onto the bubble surface by diffusing from the bulk to the sublayer of liquid adjoining the surface, and kinetically adsorbing from the sublayer onto the surface. The surface surfactant distribution which defines the Marangoni force is determined by the rate of kinetic adsorption and bulk diffusion relative to the rate of surface convection. In the limit in which the rate of either kinetic or diffusive transport of surfactant to the bubble surface is slow relative to surface convection and surface diffusion is also slow, surfactant collects in a stagnant cap at the back end of the bubble while the front end is stress free and mobile. The size of the cap and correspondingly the drag coefficient increases with the bulk concentration of surfactant until the cap covers the entire surface and the drag coefficient is that of a bubble with a completely tangentially immobile surface. Previous theoretical research on the stagnant cap regime has not studied in detail the competing roles of bulk diffusion and kinetic adsorption in determining the size of the stagnant cap angle, and there have been only a few studies which have attempted to quantitatively correlate simulations with measurements.
This paper provides a more complete theoretical study of and a validating set of experiments on the stagnant cap regime. We solve numerically for the cap angle and drag coefficient as a function of the bulk concentration of surfactant for a spherical bubble rising steadily with inertia in a Newtonian fluid, including both bulk diffusion and kinetic adsorption. For the case of diffusion-limited transport (infinite adsorption kinetics), we show clearly that very small bulk concentrations can immobilize the entire surface, and we calculate the critical concentrations which immobilize the surface as a function of the surfactant parameters. We demonstrate that the effect of kinetics is to reduce the cap angle (hence reduce the drag coefficient) for a given bulk concentration of surfactant. We also present experimental results on the drag of a bubble rising in a glycerol–water mixture, as a function of the dissolved concentration of a polyethoxylated non-ionic surfactant whose bulk diffusion coefficient and a lower bound on the kinetic rate constants have been obtained separately by measuring the reduction in dynamic tension as surfactant adsorbs onto a clean interface. For low concentrations of surfactant, the experiments measure drag coefficients which are intermediate between the drag coefficient of a bubble whose surface is tangentially mobile and one whose surface is completely immobilized. Using the separately obtained value for the diffusion coefficient of the polyethoxylate, we undertake simulations which provide, upon comparison with the measured drag coefficients, a tighter bound on the kinetic rate constants than were otherwise obtained using dynamic surface tension measurement, and this suggests a new method for the measurement of kinetic rate constants.
Wave evolution on electrified falling films
- DMITRI TSELUIKO, DEMETRIOS T. PAPAGEORGIOU
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- Journal of Fluid Mechanics / Volume 556 / 10 June 2006
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- 24 May 2006, pp. 361-386
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The nonlinear stability of falling film flow down an inclined flat plane is investigated when an electric field acts normal to the plane. A systematic asymptotic expansion is used to derive a fully nonlinear long-wave model equation for the scaled interface, where higher-order terms must be retained to make the long-wave approximation valid for long times. The effect of the electric field is to introduce a non-local term which comes from the potential region above the liquid film. This term is always linearly destabilizing and produces growth rates proportional to the cubic power of the wavenumber – surface tension is included and provides a short wavelength cutoff. Even in the absence of an electric field, the fully nonlinear equation can produce singular solutions after a finite time. This difficulty is avoided at smaller amplitudes where the weakly nonlinear evolution is governed by an extension of the Kuramoto–Sivashinsky equation. This equation has solutions which exist for all time and allows for a complete study of the nonlinear behaviour of competing physical mechanisms: long-wave instability above a critical Reynolds number, short-wave damping due to surface tension and intermediate growth due to the electric field. Through a combination of analysis and extensive numerical experiments, we find parameter ranges that support non-uniform travelling waves, time-periodic travelling waves and complex nonlinear dynamics including chaotic interfacial oscillations. It is established that a sufficiently high electric field will drive the system to chaotic oscillations, even when the Reynolds number is smaller than the critical value below which the non-electrified problem is linearly stable. A particular case of this is Stokes flow.
Analytical description of the breakup of liquid jets
- Demetrios T. Papageorgiou
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- Journal of Fluid Mechanics / Volume 301 / 25 October 1995
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- 26 April 2006, pp. 109-132
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A viscous or inviscid cylindrical jet with surface tension in a surrounding medium of negligible density tends to pinch owing to the mechanism of capillary instability. We construct similarity solutions which describe this phenomenon as a critical time is encountered, for three distinct cases: (i) inviscid jets governed by the Euler equations, (ii) highly viscous jets governed by the Stokes equations, and (iii) viscous jets governed by the Navier-Stokes equations. We look for singular solutions of the governing equations directly rather than by analysis of simplified models arising from slender-jet theories. For Stokes jets implicitly defined closed-form solutions are constructed which allow the scaling exponents to be fixed. Navier-Stokes pinching solutions follow rationally from the Stokes ones by bringing unsteady and nonlinear terms into the momentum equations to leading order. This balance fixes a set of universal scaling functions for the phenomenon. Finally we show how the pinching solutions can be used to provide an analytical description of the dynamics beyond breakup.
The absolute instability of an inviscid compound jet
- ANUJ CHAUHAN, CHARLES MALDARELLI, DEMETRIOS T. PAPAGEORGIOU, DAVID S. RUMSCHITZKI
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- Journal of Fluid Mechanics / Volume 549 / 25 February 2006
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- 08 February 2006, pp. 81-98
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This paper examines the emergence of the absolute instability from convectively unstable states of an inviscid compound jet. A compound jet is composed of a cylindrical jet of one fluid surrounded by a concentric annulus of a second, immiscible fluid. For all jet velocities $v$, there are two convectively unstable modes. As in the single-fluid jet, the compound jet becomes absolutely unstable below a critical dimensionless velocity or Weber number $V ({:=}\,\sqrt{v^2\,{\rho_1 R_1}/\sigma_1}$ where $\rho_{1}$, $R_{1}$ and $\sigma _{1}$ are the core density, radius and core–annular interfacial tension), which is a function of the annular/core ratios of densities $\beta$, surface tensions $\gamma$ and radii $a$. At $V\,{=}\,0$, the absolutely unstable modes and growth recover the fastest growing temporal waves. We focus specifically on the effect of $\gamma$ at $a\,{=}\,2$ and $\beta\,{=}\,1$ and find that when the outer tension is significantly less than the inner $(0.1\,{<}\,\gamma\,{<}\,0.3)$, the critical Weber number $V_{\hbox{\scriptsize{\it crit}}}$ decreases with <$\gamma$, whereas for higher ratios $(0.3\,{<}\,\gamma\,{<}\,3)$ it increases. The values (1.2–2.3) of $V_{\hbox{\scriptsize{\it crit}}}$ for the compound jet include the parameter-independent critical value of 1.77 for the single jet. Therefore, increasing the outer tension can access the absolute instability at higher dimensional velocities than for a single jet with the same radius and density as the core and a surface tension equal to the compound jet's liquid–liquid tension. We argue that this potentially facilitates distinguishing experimentally between absolute and convective instabilities because higher velocities and surface tension ratios higher than 1 extend the breakup length of the convective instability. In addition, for $0.3\,{<}\,\gamma\,{<}\,1.16$, the wavelength for the absolute instability is roughly half that of the fastest growing convectively unstable wave. Thus choosing $\gamma$ in this range exaggerates its distinction from the convective instability and further aids the potential observation of absolute instability.
Large-amplitude capillary waves in electrified fluid sheets
- DEMETRIOS T. PAPAGEORGIOU, JEAN-MARC VANDEN-BROECK
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- Journal of Fluid Mechanics / Volume 508 / 10 June 2004
- Published online by Cambridge University Press:
- 03 June 2004, pp. 71-88
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Large-amplitude capillary waves on fluid sheets are computed in the presence of a uniform electric field acting in a direction parallel to the undisturbed configuration. The fluid is taken to be inviscid, incompressible and non-conducting. Travelling waves of arbitrary amplitudes and wavelengths are calculated and the effect of the electric field is studied. The solutions found generalize the exact symmetric solutions of Kinnersley (1976) to include electric fields, for which no exact solutions have been found. Long-wave nonlinear waves are also constructed using asymptotic methods. The asymptotic solutions are compared with the full computations as the wavelength increases, and agreement is found to be excellent.
Chaotic flows in pulsating cylindrical tubes: a class of exact Navier–Stokes solutions
- MARK G. BLYTH, PHILIP HALL, DEMETRIOS T. PAPAGEORGIOU
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- Journal of Fluid Mechanics / Volume 481 / 25 April 2003
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- 28 April 2003, pp. 187-213
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We consider the unsteady motion of a viscous incompressible fluid inside a cylindrical tube whose radius is changing in a prescribed manner. We construct a class of exact solutions of the Navier–Stokes equations in the case when the vessel radius is a function of time alone so that the cross-section is circular and uniform along the pipe axis. The Navier–Stokes equations admit solutions which are governed by nonlinear partial differential equations depending on the radial coordinate and time alone, and forced by the wall motion. These solutions correspond to a wide class of bounded radial stagnation-point flows and are of practical importance. In dimensionless terms, the flow is characterized by two parameters: $\D$, the amplitude of the oscillation, and $R$, the Reynolds number for the flow. We study flows driven by a time-periodic wall motion, and find that at small $R$ the flow is synchronous with the forcing and as $R$ increases a Hopf bifurcation takes place. Subsequent dynamics, as $R$ increases, depend on the value of $\Delta$. For small $\Delta$ the Hopf bifurcation leads to quasi-periodic solutions in time, with no further bifurcations occurring – this is supported by an asymptotic high-Reynolds-number boundary layer theory. At intermediate $\Delta$, the Hopf bifurcation is either quasi-periodic (for the smaller $\Delta$) or subharmonic (for larger $\Delta$), and the solutions tend to a chaotic attractor at sufficiently large $R$; the route to chaos is found not to follow a Feigenbaum scenario. At larger values of $\Delta$, we find that the solution remains time periodic as $R$ increases, with solution branches supporting periods of successive integer multiples of the driving period emerging. On a given branch the flow exhibits a self-similarity in both time and space and these features are elucidated by careful numerics and an asymptotic analysis. In contrast to the two-dimensional case (see Hall & Papageorgiou 1999) chaos is not found at either small or comparatively large $\Delta$.
Using surfactants to control the formation and size of wakes behind moving bubbles at order-one Reynolds numbers
- YANPING WANG, DEMETRIOS T. PAPAGEORGIOU, CHARLES MALDARELLI
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- Journal:
- Journal of Fluid Mechanics / Volume 453 / 25 February 2002
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- 06 March 2002, pp. 1-19
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A bubble translating through a continuous liquid (i.e. Newtonian) phase moves as a sphere when inertial and viscous forces are small relative to capillary forces. Spherical bubbles with stress-free interfaces do not retain wakes at their trailing ends as inertial forces become important (increasing Reynolds number). This is in contrast to translating spheres with immobile interfaces in which flow separation and wake formation occurs at order-one Reynolds number. Surfactants present in the continuous phase adsorb onto a bubble surface as it translates, and affect the interfacial mobility by creating tension gradient forces. Adsorbed surfactant is convected to the trailing end of the bubble, lowers the tension there relative to the front, and creates a tension gradient which reduces the surface flow. For low bulk concentrations of surfactant (or if kinetic exchange between bulk and surface is slow relative to convection), diffusion towards the surface is much slower than convection, and surfactant is swept into an immobile cap at the trailing end. As with solid spheres, these caps entrain wakes at order-one Reynolds number. In adsorptive bubble technologies where solutes transfer between the bubble and the continuous phase, usually through thin boundary layers around the bubble surface (high Péclet number), these wakes generally form owing to the presence of surfactant impurities. The wake presence retards the interphase transfer displacing the thin boundary layer towards the front end of the bubble; as mass transfer through the wake is much slower than through the boundary layer, the mass transfer is reduced.
Our recent theoretical research has demonstrated that at low Reynolds numbers, the mobility of a surfactant-retarded bubble interface can be increased by raising the bulk concentration of a surfactant which kinetically rapidly exchanges between the surface and the bulk. At high bulk concentrations the interface saturates with surfactant, effectively removing the tension gradient. In this paper, we demonstrate theoretically that this interfacial control is still realized at order-one Reynolds numbers, and, more importantly, we show that the control can be used to manipulate the formation, size and ultimately the disappearance of a wake. This wake removal mechanism has the potential to dramatically increase the interphase transfer in adsorptive bubble technologies.
The onset of chaos in a class of Navier–Stokes solutions
- PHILIP HALL, DEMETRIOS T. PAPAGEORGIOU
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- Journal:
- Journal of Fluid Mechanics / Volume 393 / 25 August 1999
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- 25 August 1999, pp. 59-87
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The flow between parallel walls driven by the time-periodic oscillation of one of the walls is investigated. The flow is characterized by a non-dimensional amplitude Δ and a Reynolds number R. At small values of the Reynolds number the flow is synchronous with the wall motion and is stable. If the amplitude of oscillation is held fixed and the Reynolds number is increased there is a symmetry-breaking bifurcation at a finite value of R. When R is further increased, additional bifurcations take place, but the structure which develops, essentially chaotic flow resulting from a Feigenbaum cascade or a quasi-periodic flow, depends on the amplitude of oscillation. The flow in the different regimes is investigated by a combination of asymptotic and numerical methods. In the small-amplitude high-Reynolds-number limit we show that the flow structure develops on two time scales with chaos occurring on the longer time scale. The chaos in that case is shown to be associated with the unsteady breakdown of a steady streaming flow. The chaotic flows which we describe are of particular interest because they correspond to Navier–Stokes solutions of stagnation-point form. These flows are relevant to a wide variety of flows of practical importance.
Increased mobility of a surfactant-retarded bubble at high bulk concentrations
- YANPING WANG, DEMETRIOS T. PAPAGEORGIOU, CHARLES MALDARELLI
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- Journal:
- Journal of Fluid Mechanics / Volume 390 / 10 July 1999
- Published online by Cambridge University Press:
- 10 July 1999, pp. 251-270
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We study theoretically the adsorption of surfactant onto the interface of gas bubbles in creeping flow rising steadily in an infinite liquid phase containing surface-active agents. When a bubble rises in the fluid, surfactant adsorbs onto the surface at the leading edge, is convected by the surface flow to the trailing edge and accumulates and desorbs off the back end. This transport creates a surfactant concentration gradient on the surface that causes the surface tension at the back end to be lower than that at the front end, thus retarding the bubble velocity by the creation of a Marangoni force. In this paper, we demonstrate numerically that the mobility of the surfactant-retarded bubble interface can be increased by raising the bulk concentration of surfactant. At high bulk concentrations, the interface saturates with surfactant, and this saturation acts against the convective partitioning to decrease the surface surfactant gradient. We show that as the Péclet number (which scales the convective effect) increases, larger concentrations are necessary to remobilize the surface completely. These results lead to a technologically useful paradigm where the drag and interfacial mobility of a bubble can be controlled by the level of the bulk concentration of surfactant.