Absolute and convective instabilities in non-local active-dissipative equations arising in the modelling of thin liquid films

This paper was presented at the 4th Micro and Nano Flows Conference (MNF2014), which was held at University College, London, UK. The conference was organised by Brunel University and supported by the Italian Union of Thermofluiddynamics, IPEM, the Process Intensification Network, the Institution of Mechanical Engineers, the Heat Transfer Society, HEXAG - the Heat Exchange Action Group, and the Energy Institute, ASME Press, LCN London Centre for Nanotechnology, UCL University College London, UCL Engineering, the International NanoScience Community, www.nanopaprika.eu.Absolute and convective instabilities in a non-local model that arises in the analysis of thin-film flows over flat or corrugated walls in the presence of an applied electric field are discussed. Electrified liquid films arise, for example, in coating processes where liquid films are deposited onto a target surfaces with a view to producing an evenly coating layer. In practice, the target surface, or substrate, may be irregular in shape and feature corrugations or indentations. This may lead to non-uniformities in the thickness of the coating layer. Attempts to mitigate film-surface irregularities can be made using, for example, electric fields. We analyse the stability of such thin-film flows and show that if the amplitude of the wall corrugations and/or the strength of the applied electric field is increased the convectively unstable flow undergoes a transition to an absolutely unstable flow

Absolute and convective instabilities in non-local active-dissipative equations arising in the modelling of thin liquid films

This paper was presented at the 4th Micro and Nano Flows Conference (MNF2014), which was held at University College, London, UK. The conference was organised by Brunel University and supported by the Italian Union of Thermofluiddynamics, IPEM, the Process Intensification Network, the Institution of Mechanical Engineers, the Heat Transfer Society, HEXAG - the Heat Exchange Action Group, and the Energy Institute, ASME Press, LCN London Centre for Nanotechnology, UCL University College London, UCL Engineering, the International NanoScience Community, www.nanopaprika.eu.Absolute and convective instabilities in a non-local model that arises in the analysis of thin-film flows over flat or corrugated walls in the presence of an applied electric field are discussed. Electrified liquid films arise, for example, in coating processes where liquid films are deposited onto a target surfaces with a view to producing an evenly coating layer. In practice, the target surface, or substrate, may be irregular in shape and feature corrugations or indentations. This may lead to non-uniformities in the thickness of the coating layer. Attempts to mitigate film-surface irregularities can be made using, for example, electric fields. We analyse the stability of such thin-film flows and show that if the amplitude of the wall corrugations and/or the strength of the applied electric field is increased the convectively unstable flow undergoes a transition to an absolutely unstable flow

Electric fields as a means of controlling thin film flow over topography

This paper was presented at the 2nd Micro and Nano Flows Conference (MNF2009), which was held at Brunel University, West London, UK. The conference was organised by Brunel University and supported by the Institution of Mechanical Engineers, IPEM, the Italian Union of Thermofluid dynamics, the Process Intensification Network, HEXAG - the Heat Exchange Action Group and the Institute of Mathematics and its Applications.Gravity-driven, steady-state flow of a thin liquid film over a substrate containing topography in the presence of a normal electric field is investigated. The liquid is assumed to be a perfect conductor and the air above it an ideal dielectric. The Navier-Stokes equations are solved using a new depth-averaged approximation that is capable of analysing film flows with inertia, with the flow coupled to the electric field via a Maxwell normal stress term that results from the solution of Laplace’s equation for the electric potential above the film. The latter is solved analytically using separation of variables and Fourier series. The coupled solver is used to analyse the interplay between inertia and electric field effects for flow over onedimensional step and trench topographies and to predict the effect of an electric field on three-dimensional Stokes flow over a two-dimensional trench topography. Sample results are given which investigate the magnitude of the electric fields needed to suppress free surface disturbances induced by topography in each of the cases considered.This study is funded by the European Union via Marie Curie Action Contract MEST-CT-2005-020599

Stability of a horizontal viscous fluid layer in a vertical time periodic electric field

The stability of a horizontal interface between two viscous fluids, one of which is conducting and the other is dielectric, acted upon by a vertical time-periodic electric field is considered. The two fluids are bounded by electrodes separated by a finite distance. By means of Floquet theory, the marginal stability curves are obtained, thereby elucidating the dependency of the critical voltage and wavenumber upon the fluid viscosities. The limit of vanishing viscosities is shown to be in excellent agreement with the marginal stability curves predicted by means of a Mathieu equation. The methodology to obtain the marginal stability curves developed here is applicable to any arbitrary but time periodic-signal, as demonstrated for the case of a signal with two different frequencies. As a special case, the marginal stability curves for an applied ac voltage biased by a dc voltage are depicted. It is shown that the mode coupling caused by the normal stress at the interface due to the electric field leads to appearance of harmonic modes and subharmonic modes. This is in contrast to the application of a voltage with a single frequency which always leads to a harmonic mode. Whether a harmonic or subharmonic mode is the most unstable one depends on details of the excitation signal. It is also shown that the electrode spacing has a distinct effect on the stability bahavior of the system

Absolute and convective instabilities in non-local active-dissipative equations arising in the modelling of thin liquid films

Absolute and convective instabilities in a non-local model that arises in the analysis of thin-film flows over flat or corrugated walls in the presence of an applied electric field are discussed. Electrified liquid films arise, for example, in coating processes where liquid films are deposited onto a target surfaces with a view to producing an evenly coating layer. In practice, the target surface, or substrate, may be irregular in shape and feature corrugations or indentations. This may lead to non-uniformities in the thickness of the coating layer. Attempts to mitigate film-surface irregularities can be made using, for example, electric fields. We analyse the stability of such thin-film flows and show that if the amplitude of the wall corrugations and/or the strength of the applied electric field is increased the convectively unstable flow undergoes a transition to an absolutely unstable flo

Electrified thin film flow at finite Reynolds number on planar substrates featuring topography

The flow of a gravity-driven thin liquid film over a substrate containing topography, in the presence of a normal electric field, is investigated. The liquid is assumed to be a perfect conductor and the air above it a perfect dielectric. Of particular interest is the interplay between inertia, for finite values of the Reynolds number, Re, and electric field strength, expressed in terms of the Weber number, We, on the resultant free-surface disturbance away from planarity. The hydrodynamics of the film are modelled via a depth-averaged form of the Navier–Stokes equations which is coupled to a Fourier series separable solution of Laplace’s equation for the electric potential: detailed steady-state solutions of the coupled equation set are obtained numerically. The case of two-dimensional flow over different forms of discrete and periodically varying spanwise topography is explored. In the case of the familiar free-surface capillary peaks and depressions that arise for steep topography, and become more pronounced with increasing Re, greater electric field strength affects them differently. In particular, it is found that for topography heights commensurate with the long-wave approximation: (i) the capillary ridge associated with a step-down topography at first increases before decreasing, both monotonically, with increasing We and (ii) the free-surface hump which arises at a step-up topography continues to increase monotonically with increasing We, the increase achieved being smaller the larger the value of Re. A series of results for the more practically relevant problem of three-dimensional film flow over substrate containing a localised square trench topography is provided. These exhibit behaviour and features consistent with those observed for two-dimensional flow, in that as We is increased the primary free-surface capillary ridges and depressions are at first enhanced, with a corresponding narrowing, before becoming suppressed. In addition, it is found that, while the well-known horse-shoe shaped disturbance characteristic of such flows continues to persist with increasing Re in the absence of an electric field, when the latter is present and We increased in value the associated comet tail disappears as does the related downstream surge. The phenomenon is explained with reference to the competition between the corresponding capillary pressure and Maxwell stress distributions

Electrified thin-film flow over inclined topography

We consider both a long-wave model and a first-order weighted-residual integral boundary layer (WIBL) model in the investigation of thin film flow down a topographical incline whilst under the effects of a normal electric field. The liquid is assumed to be a perfect dielectric, although is trivially extended to the case of a perfect conductor. The perfect dielectric case with no topography includes a simple modified electric Weber number which incorporates the relative electrical permittivity constant into itself. Linear stability analysis is carried out for both models, and critical Reynolds numbers which depend on the electric Weber number and the capillary number are produced. Regions of stability, convective instability and absolute instability are then determined for both models in terms of our electric Weber number and Reynolds number parameters in the case of no topography. Time-dependent simulations are produced to corroborate the aforementioned regions and investigate the effect of normal electric field strength in addition to sinusoidal and rectangular topographical amplitude on our system for various domain sizes. For the time-dependent simulations we find strong agreement with the linear stability analysis, and the results suggest that the inclusion of a normal electric field may have some stabilising properties in the long-wave model which are absent in the case of a flat wall, for which the electric field is always linearly destabilising. This stabilising effect is not observed for the same parameters in the WIBL model with a sinusoidal wall, although a similar effect is noticed in the WIBL model with a rectangular wall. We also investigate the simultaneous effect of domain size, wall amplitude and electric field strength on the critical Reynolds numbers for both models, and find that increasing the electric field strength can make large-amplitude sinusoidal topography stabilising rather than destabilising for the long-wave model. Continuation curves of steady solutions and bifurcation diagrams are also produced, and comparisons between the two models are made for various parameter values, which show excellent agreement with the literature. Subharmonic branches and time-periodic solutions are additionally included, similarly showing very good agreement with the literature

Electrokinetic and electrohydrodynamic problems in multifluid flows

The present thesis deals with microfluidic systems under the influence of electric fields. The purpose of this research is to identify key behaviours which are highly relevant for applications in lab-on-chip devices such as pH control, patterning, mixing and cell trapping. In the first part of the thesis we consider the electrokinetics of charged, porous membranes and present a mathematical model for the ionic transport under the effects of a horizontal electric field. First, we investigate the behaviour of a system that consists of one anion membrane with two reservoirs and produce numerical solutions with the aim to gain a better understanding of the mechanisms that lead to overlimiting current. We then analyse the features of a system where a bipolar membrane is held in an electrolyte bath with water and a salt. We use our model to confirm findings from experiments such as the hysteretic behaviour of the IV curve and the water splitting phenomenon. In the second part of the thesis we examine the behaviour of interfaces between two fluids that are sandwiched between two electrodes. We find that introducing a constant flow rate into the system leads to time modulated travelling waves. In the case of flat channel walls these look like moving strips which are reminiscent of the patterns found in the no flow case. Adding corrugations on one or both electrodes leads to a rich variety of dynamics. We develop a Floquet stability analysis which takes into account the fact that the base state of the system is nonuniform. This is a very useful tool for identifying the different types of behaviours which arise as we change the applied voltage and the overall flow rate. We examine the streamlines of the fluid to explore the advantages of the different regimes: just by changing the applied voltage it is possible to transition from an environment which is favourable to efficient mixing to one which could enhance cell trapping.Open Acces

Numerical-asymptotic models for the manipulation of viscous films via dielectrophoresis

The effect of an externally applied electric field on the motion of an interface between two viscous dielectric fluids is investigated. We first develop a powerful, efficient and widely applicable boundary integral method to compute the interface dynamics in a general multiphysics model comprising coupled Laplace and Stokes flow problems in a periodic half-space. In particular, we exploit the relevant Stokes and Laplace Green's functions to reduce the problem to one defined on the interfacial part of the domain alone. Secondly, motivated by recent experimental work that seeks to underpin the development of switchable liquid optical devices, we concentrate on a fluid–air interface and derive asymptotic approximations suitable to describe the behaviour of a thin film of fluid above an array of electrodes. In this case, the problem is reduced to a single nonlinear partial differential equation describing the film height, coupled to the electrostatic problem via suitable numerical solution or via an asymptotic formula for electrostatic forcing. Comparison against numerical simulations of the full problem shows that the reduced models successfully capture key features of the film dynamics in appropriate regimes; all three approaches are shown to reproduce experimental results