Numerical simulations of a falling film on the inner surface of a rotating cylinder.

A flow in which a thin film falls due to gravity on the inner surface of a vertical, rotating cylinder is investigated. This is performed using two-dimensional (2D) and 3D direct numerical simulations, with a volume-of-fluid approach to treat the interface. The problem is parameterized by the Reynolds, Froude, Weber, and Ekman numbers. The variation of the Ekman number (Ek), defined to be proportional to the rotational speed of the cylinder, has a strong effect on the flow characteristics. Simulations are conducted over a wide range of Ek values (0≤Ek≤484) in order to provide detailed insight into how this parameter influences the flow. Our results indicate that increasing Ek, which leads to a rise in the magnitude of centrifugal forces, produces a stabilizing effect, suppressing wave formation. Key flow features, such as the transition from a 2D to a more complex 3D wave regime, are influenced significantly by this stabilization and are investigated in detail. Furthermore, the imposed rotation results in distinct flow characteristics such as the development of angled waves, which arise due to the combination of gravitationally and centrifugally driven motion in the axial and azimuthal directions, respectively. We also use a weighted residuals integral boundary layer method to determine a boundary in the space of Reynolds and Ekman numbers that represents a threshold beyond which waves have recirculation regions.

Fluid-solid phase transition of n-alkane mixtures: Coarse-grained molecular dynamics simulations and diffusion-ordered spectroscopy nuclear magnetic resonance.

Wax appearance temperature (WAT), defined as the temperature at which the first solid paraffin crystal appears in a crude oil, is one of the key flow assurance indicators in the oil industry. Although there are several commonly-used experimental techniques to determine WAT, none provides unambiguous molecular-level information to characterize the phase transition between the homogeneous fluid and the underlying solid phase. Molecular Dynamics (MD) simulations employing the statistical associating fluid theory (SAFT) force field are used to interrogate the incipient solidification states of models for long-chain alkanes cooled from a melt to an arrested state. We monitor the phase change of pure long chain n-alkanes: tetracosane (C24H50) and triacontane (C30H62), and an 8-component surrogate n-alkane mixture (C12-C33) built upon the compositional information of a waxy crude. Comparison to Diffusion Ordered Spectroscopy Nuclear Magnetic Resonance (DOSY NMR) results allows the assessment of the limitations of the coarse-grained models proposed. We show that upon approach to freezing, the heavier components restrict their motion first while the lighter ones retain their mobility and help fluidize the mixture. We further demonstrate that upon sub-cooling of long n-alkane fluids and mixtures, a discontinuity arises in the slope of the self-diffusion coefficient with decreasing temperature, which can be employed as a marker for the appearance of an arrested state commensurate with conventional WAT measurements.

Dynamics and universal scaling law in geometrically-controlled sessile drop evaporation.

The evaporation of a liquid drop on a solid substrate is a remarkably common phenomenon. Yet, the complexity of the underlying mechanisms has constrained previous studies to spherically symmetric configurations. Here we investigate well-defined, non-spherical evaporating drops of pure liquids and binary mixtures. We deduce a universal scaling law for the evaporation rate valid for any shape and demonstrate that more curved regions lead to preferential localized depositions in particle-laden drops. Furthermore, geometry induces well-defined flow structures within the drop that change according to the driving mechanism. In the case of binary mixtures, geometry dictates the spatial segregation of the more volatile component as it is depleted. Our results suggest that the drop geometry can be exploited to prescribe the particle deposition and evaporative dynamics of pure drops and the mixing characteristics of multicomponent drops, which may be of interest to a wide range of industrial and scientific applications.

A Langevin model for fluctuating contact angle behaviour parametrised using molecular dynamics.

Molecular dynamics simulations are employed to develop a theoretical model to predict the fluid-solid contact angle as a function of wall-sliding speed incorporating thermal fluctuations. A liquid bridge between counter-sliding walls is studied, with liquid-vapour interface-tracking, to explore the impact of wall-sliding speed on contact angle. The behaviour of the macroscopic contact angle varies linearly over a range of capillary numbers beyond which the liquid bridge pinches off, a behaviour supported by experimental results. Nonetheless, the liquid bridge provides an ideal test case to study molecular scale thermal fluctuations, which are shown to be well described by Gaussian distributions. A Langevin model for contact angle is parametrised to incorporate the mean, fluctuation and auto-correlations over a range of sliding speeds and temperatures. The resulting equations can be used as a proxy for the fully-detailed molecular dynamics simulation allowing them to be integrated within a continuum-scale solver.

The effect of adsorption kinetics on the rate of surfactant-enhanced spreading.

A comparison of the kinetics of spreading of aqueous solutions of two different surfactants on an identical substrate and their short time adsorption kinetics at the water/air interface has shown that the surfactant which adsorbs slower provides a higher spreading rate. This observation indicates that Marangoni flow should be an important part of the spreading mechanism enabling surfactant solutions to spread much faster than pure liquids with comparable viscosities and surface tensions.

Nonequilibrium hysteresis and Wien effect water dissociation at a bipolar membrane.

As in electrochemical cyclic voltammetry, time-periodic reverse voltage bias across a bipolar membrane is shown to exhibit hysteresis due to transient effects. This is due to the incomplete depletion of mobile ions, at the junction between the membranes, within two adjoining polarized layers; the layer thickness depends on the applied voltage and the surface charge densities. Experiments show that the hysteresis consists of an Ohmic linear rise in the total current with respect to the voltage, followed by a decay of the current. A limiting current is established for a long period when all the mobile ions are depleted from the polarized layer. If the resulting high field within the two polarized layers is sufficiently large, water dissociation occurs to produce proton and hydroxyl traveling wave fronts which contribute to another large jump in the current. We use numerical simulation and asymptotic analysis to interpret the experimental results and to estimate the amplitude of the transient hysteresis and the water-dissociation current.

Dynamics of a climbing surfactant-laden film--I: base-state flow.

The dynamics of a surfactant-laden film climbing up an inclined plane is investigated through a two-dimensional (2-D), nonlinear evolution equation for the interface coupled to convective-diffusion equations for the surfactant, derived using lubrication theory. One-dimensional (1-D) solutions, representing the base-state flow, are investigated for constant flux and constant volume configurations; these flows are parameterised by capillarity, gravity, convection-diffusion ratios (represented by Péclét numbers at the surface and bulk), a solubility parameter, sorption kinetics constants, the number of surfactant monomers in a micelle, and the nonlinearity of the surfactant equation of state. In both configurations studied, a front develops spreading up the substrate against the direction of gravity whereby the leading edge of the front follows a power-law as a function of time. The effect of system parameters on the base-state flow is explored through an extensive parametric study, while the stability of the above-mentioned system to spanwise perturbations is the focus of Part II.

Dynamics of a climbing surfactant-laden film II: stability.

The linear and nonlinear stability of a spreading film of constant flux and a drop of constant volume, discussed in [1], are examined here. A linear stability analysis (LSA) is carried out to investigate the stability to spanwise perturbations, by linearisation of the two-dimensional (2-D) evolution equations derived in [1] for the film thickness and surfactant concentration fields. The latter correspond to convective-diffusion equations for the surfactant, existing in the form of monomers (present at the free surface and in the bulk) and micelles (present in the bulk). The results of the LSA indicate that the thinning region, present upstream of the leading front in the constant flux case, and the leading ridge in the constant volume case, are unstable to spanwise perturbations. Numerical simulations of the 2-D system of equations demonstrate that the above-mentioned regions exhibit finger formation; the effect of selected system parameters on the fingering patterns is discussed.

Breakup of an electrified, perfectly conducting, viscous thread in an AC field.

We study the axisymmetric breakup and satellite formation of slender jets surrounded by a concentrically placed cylindrical electrode and subjected to time-dependent AC electric fields. The jet is assumed to be a perfectly conducting viscous fluid and surrounded by a dielectric inviscid gas. We use the long-wave approximation to derive coupled evolution equations for the interface position and the axial velocity component, which accounts for electrostatic forcing. The electrostatic force in this case is large and competes with capillary forces near the rupture point, causing the interface to oscillate and the satellite to have shapes that are distinct from the DC case. In particular, our results indicate that it may be possible to use the AC field to control the number of satellites accompanying breakup as well as their size.

Surfactant-enhanced rapid spreading of drops on solid surfaces.

We study the surfactant-enhanced spreading of drops on the surfaces of solid substrates. This work is performed in connection with the unique ability of aqueous trisiloxane solutions to wet highly hydrophobic substrates effectively, which has been studied for nearly two decades. We couple a lubrication model to advection-diffusion equations for surfactant transport. We allow for micelle formation and breakup in the bulk and adsorptive flux at both the gas-liquid and liquid-solid interfaces and use appropriate equations of state to model variations in surface tension and wettability. Our numerical results show the effect of basal adsorption, kinetic rates, and the availability of surfactant on the deformation of the droplet and its spreading rate. We demonstrate that this rate is maximized for intermediate rates of basal adsorption and the total mass of surfactant.

Pinning, retraction, and terracing of evaporating droplets containing nanoparticles.

We consider the dynamics of a slender, evaporating droplet containing nanoparticles. We use lubrication theory to derive a coupled system of equations that govern the film thickness and the concentration of nanoparticles. These equations account for capillarity, Marangoni stresses, evaporation, and disjoining pressure; the nanoparticle-induced structural component of the disjoining pressure is also considered. Contact line singularities are avoided through the adsorption of ultrathin films wherein evaporation is suppressed by the disjoining pressure; a similar approach has recently been used by Ajaev [J. Fluid Mech. 2005, 528, 279-296] who has built on the previous work of Moosman and Homsy [J. Colloid Interface Sci. 1980, 73, 212-223]. The results of our numerical simulations indicate that, depending on the value of system parameters, the droplet exhibits a variety of different behaviours, which include spreading, evaporation-driven retraction, contact line pinning, and "terrace" formation.

Interfacial dynamics in pressure-driven two-layer laminar channel flow with high viscosity ratios.

The large-scale dynamics of an interface separating two immiscible fluids in a channel is studied in the case of large viscosity contrasts. A long-wave analysis in conjunction with the Kármán-Polhausen method to approximate the velocity profile in the less viscous fluid is used to derive a single equation for the interface. This equation accounts for the presence of interfacial stress, capillarity, and viscous retardation as well as inertia in the less viscous fluid layer where the flow is considered to be quasistatic; the equation is shown to reduce to a Benney-type equation and the Kuramoto-Sivashinskiy equation in the relevant limits. The solutions of this equation are parametrized by an initial thickness ratio h0 and a dimensionless parameter S , which measures the relative significance of inertial to capillary forces. A parametric continuation technique is employed, which reveals that nonuniqueness of periodic solutions is possible in certain regions of (h0,S) space. Transient numerical simulations are also reported, whose results demonstrate good agreement with the bifurcation structure obtained from the parametric continuation results.

On autophobing in surfactant-driven thin films.

Recent experiments (Afsar-Siddiqui, A. B.; Luckham, P. F.; Matar, O. K. Langmuir 2004, 20, 7575-7582) on the spreading of aqueous droplets containing cationic surfactants over thin aqueous films supported by negatively charged substrates demonstrated trends in the spreading behavior with either increasing surfactant concentration or increasing film thickness. Although the substrate is initially hydrophilic and the droplet spreads, surfactant adsorption at the substrate renders it hydrophobic leading to droplet retraction. We generate a model here using lubrication theory that allows the effect of the surfactant on the wettability to be taken into account. Our numerical results show that due to basal adsorption of surfactant at the interface, the initially hydrophilic solid substrate is rendered hydrophobic. This then drives droplet retraction and dewetting, which is in agreement with the experimentally observed trends.

Drop manipulation and surgery using electric fields.

We study the dynamics of a slender drop sandwiched between two electrodes using lubrication theory. A coupled system of evolution equations for the film thickness and interfacial charge density is derived and simplified for the case of a highly conducting fluid. The contact line singularity is relieved by postulating the existence of a wetting precursor film, which is stabilised by intermolecular forces. We examine the motion of the drop as a function of system parameters: the electrode separation, beta, an electric capillary number, C, and a spatio-temporally varying bottom electrode potential. The possibility of drop manipulation and surgery, which include drop spreading, translation, splitting and recombination, is demonstrated using appropriate tuning of the properties of the bottom potential; these results could have potential implications for drop manipulation schemes in various microfluidic applications. For relatively small beta and/or large C values, the drop assumes cone-like structures as it approaches the top electrode; the latter stages of this approach are found to be self-similar and a power-law exponent has been extracted for this case.

Falling films on flexible inclines.

The nonlinear stability and dynamic behavior of falling fluid films is studied for flow over a flexible substrate. We use asymptotic methods to deduce governing equations valid in various limits. Long-wave theory is used to derive Benney-like coupled equations for the film thickness and substrate deflection. Weakly nonlinear equations are then derived from these equations that, in the limit of large wall damping and/or large wall tension, reduce to the Kuramoto-Sivashinsky equation. These models break down when inertia becomes more significant, so we also use a long-wave approximation in conjunction with integral theory to derive three strongly coupled nonlinear evolution equations for the film thickness, substrate deflection, and film volumetric flow rate valid at higher Reynolds numbers. These equations, accounting for inertia, capillary, viscous, wall tension, and damping effects, are solved over a wide range of parameters. Our results suggest that decreasing wall damping and/or wall tension can promote the development of chaos in the weakly nonlinear regime and lead to severe substrate deformations in the strongly nonlinear regime; these can give rise to situations in which the free surface and underlying substrate come into contact in finite time.

Dynamic spreading of droplets containing nanoparticles.

Recent experiments and models for the spreading of liquids laden with nanoparticles have demonstrated particle layering at the three-phase contact line; this is associated with the structural component of the disjoining pressure. Effects driven by structural disjoining pressures occur on scales longer than the diameter of a particle, below which other disjoining pressure components such as van der Waals and electrostatic forces are dominant. Motivated by these experimental observations, we investigate the dynamic spreading of a droplet laden with nanoparticles in the presence of structural disjoining pressure effects. We use lubrication theory to derive evolution equations for the interfacial location and the concentration of particles. These equations account for the presence of the structural component of the disjoining pressure for film thicknesses exceeding the diameter of a nanoparticle; below such thicknesses, van der Waals forces are assumed to be operative. The resulting evolution equations, for the particle motion and free surface position, are solved allowing for the viscosity to vary as a function of nanoparticle concentration. The results of our numerical simulations demonstrate qualitative agreement with experimental observations of a "step" emerging from the contact line. The results are also relevant to a wide range of other phenomena involving layering, or terraced spreading of nanodroplets, or stepwise thinning of micellar thin films.

Collapse of a bubble in an electric field.

A low-Mach-number analysis is presented of the collapse of a bubble in an electric field, which is assumed to be homogeneous, but may be unsteady. Ellipsoidal shape deformations are accounted for in the analysis, but are assumed to be small. It is shown that the presence of an electric field leads to additional terms in a modified Rayleigh-Plesset equation. This differential equation for the bubble radius and a corresponding equation for ellipsoidal shape deformations have been integrated numerically. The results indicate that a bubble can be made to collapse by instantaneously switching on an electric field. Also, nonharmonic volumetric oscillations are observed for time-dependent electric fields of sufficiently large amplitude. It is shown that the rate of a collapse driven by external pressure variations due, for instance, to acoustic forcing can be accelerated.

On the dynamics of liquid lenses.

The spreading of a lens of one liquid on the surface of another liquid is examined. Lubrication theory is used to derive a coupled system of equations for the air-liquid and liquid-liquid interfaces. In the case of highly viscous lenses, extensional stresses are promoted and an additional equation for the lens velocity is derived. The potential singularity at the three-phase line is relieved by a microscopic precursor layer of the spreading fluid assumed to be present ahead of the macroscopic lens. This layer is stabilised via the inclusion of disjoining pressure effects in the lens. The results of our full parametric study show that, for weak gravitational forces, the shape of the lens at equilibrium depends solely on the surface tension ratio for sufficiently deep substrate thicknesses. For thin substrates, the underlying liquid film deforms severely near the point of deposition exhibiting flattening and dimpling.

Collisions of liquid coated solid spherical particles in a viscous fluid.

An analytical description is presented for the head-on collision of two spherical rigid particles that are coated with a thin layer of one liquid and immersed in another. Lubrication theory is used to resolve the spatio-temporal evolution of the coating surfaces, in conjunction with the fluid flow in the gap region between the particles. The analysis is carried out up to the point where the gap region has almost completely been drained; intermolecular forces are neglected. The effects of particle inertia, the ratio of particle radii, surface tension, and the viscosity ratio of the coating and carrier fluids are studied; these are parameterised by St, beta, Ca and m, respectively. The results of the present work elucidate the effect of the above-mentioned factors on the conditions under which particles rebound (assumed to occur if the distance between the particles becomes very short while the relative velocity does not vanish) or stick. In particular, summarizing flowmaps show that the likelihood of particles rebounding increases with increasing St and decreasing beta, Ca and m. On the other hand, it is shown that the force on approaching particles depends on all of these parameters in a non-monotonic manner.

A note on the coating of an inclined plane in the presence of soluble surfactant.

We consider the flow of a thin liquid film coating an inclined plane in the presence of a soluble surfactant. A two-dimensional three-equation model is derived using lubrication theory in the rapid diffusion limit and then used to investigate the stability of the fluid height and the surfactant surface and bulk concentrations. We present solutions for an insoluble surfactant system, which are then contrasted with those obtained for a system containing a soluble surfactant; both transient growth and fully nonlinear two-dimensional simulation results are discussed. Our results indicate that the characteristics of the fingering phenomena which accompany the flow are altered by the effects of solubility. In particular, we find that these effects de-stabilise the system further over an intermediate range of surfactant solubility.