Air Entrainment in Dynamic Wetting: Knudsen Effects and the Influence of Ambient Air Pressure

Recent experiments on coating flows and liquid drop impact both demonstrate that wetting failures caused by air entrainment can be suppressed by reducing the ambient gas pressure. Here, it is shown that non-equilibrium effects in the gas can account for this behaviour, with ambient pressure reductions increasing the gas' mean free path and hence the Knudsen number $Kn$. These effects first manifest themselves through Maxwell slip at the gas' boundaries so that for sufficiently small $Kn$ they can be incorporated into a continuum model for dynamic wetting flows. The resulting mathematical model contains flow structures on the nano-, micro- and milli-metre scales and is implemented into a computational platform developed specifically for such multiscale phenomena. The coating flow geometry is used to show that for a fixed gas-liquid-solid system (a) the increased Maxwell slip at reduced pressures can substantially delay air entrainment, i.e. increase the `maximum speed of wetting', (b) unbounded maximum speeds are obtained as the pressure is reduced only when slip at the gas-liquid interface is allowed for and (c) the observed behaviour can be rationalised by studying the dynamics of the gas film in front of the moving contact line. A direct comparison to experimental results obtained in the dip-coating process shows that the model recovers most trends but does not accurately predict some of the high viscosity data at reduced pressures. This discrepancy occurs because the gas flow enters the `transition regime', so that more complex descriptions of its non-equilibrium nature are required. Finally, by collapsing onto a master curve experimental data obtained for drop impact in a reduced pressure gas, it is shown that the same physical mechanisms are also likely to govern splash suppression phenomena.Comment: Accepted for publication in the Journal of Fluid Mechanic

The Dynamics of Liquid Drops Coalescing in the Inertial Regime

We examine the dynamics of two coalescing liquid drops in the `inertial regime', where the effects of viscosity are negligible and the propagation of the bridge front connecting the drops can be considered as `local'. The solution fully computed in the framework of classical fluid-mechanics allows this regime to be identified and the accuracy of the approximating scaling laws proposed to describe the propagation of the bridge to be established. It is shown that the scaling law known for this regime has a very limited region of accuracy and, as a result, in describing experimental data it has frequently been applied outside its limits of applicability. The origin of the scaling law's shortcoming appears to be the fact that it accounts for the capillary pressure due only to the longitudinal curvature of the free surface as the driving force for the process. To address this deficiency, the scaling law is extended to account for both the longitudinal and azimuthal curvatures at the bridge front which, fortuitously, still results in an explicit analytic expression for the front's propagation speed. This new expression is then shown to offer an excellent approximation for both the fully-computed solution and for experimental data from a range of flow configurations for a remarkably large proportion of the coalescence process. The derived formula allows one to predict the speed at which drops coalesce for the duration of the inertial regime which should be useful for the analysis of experimental data.Comment: Accepted for publication in Physical Review

Kinetic effects in dynamic wetting

The maximum speed at which a liquid can wet a solid is limited by the need to displace gas lubrication films in front of the moving contact line. The characteristic height of these films is often comparable to the mean free path in the gas so that hydrodynamic models do not adequately describe the flow physics. This Letter develops a model which incorporates kinetic effects in the gas, via the Boltzmann equation, and can predict experimentally-observed increases in the maximum speed of wetting when (a) the liquid’s viscosity is varied, (b) the ambient gas pressure is reduced or (c) the meniscus is confined

Viscous flows in corner regions: Singularities and hidden eigensolutions

Numerical issues arising in computations of viscous flows in corners formed by a liquid-fluid free surface and a solid boundary are considered. It is shown that on the solid a Dirichlet boundary condition, which removes multivaluedness of velocity in the `moving contact-line problem' and gives rise to a logarithmic singularity of pressure, requires a certain modification of the standard finite-element method. This modification appears to be insufficient above a certain critical value of the corner angle where the numerical solution becomes mesh-dependent. As shown, this is due to an eigensolution, which exists for all angles and becomes dominant for the supercritical ones. A method of incorporating the eigensolution into the numerical method is described that makes numerical results mesh-independent again. Some implications of the unavoidable finiteness of the mesh size in practical applications of the finite element method in the context of the present problem are discussed.Comment: Submitted to the International Journal for Numerical Methods in Fluid

A Parametric Study of the Coalescence of Liquid Drops in a Viscous Gas

The coalescence of two liquid drops surrounded by a viscous gas is considered in the framework of the conventional model. The problem is solved numerically with particular attention to resolving the very initial stage of the process which only recently has become accessible both experimentally and computationally. A systematic study of the parameter space of practical interest allows the influence of the governing parameters in the system to be identified and the role of viscous gas to be determined. In particular, it is shown that the viscosity of the gas suppresses the formation of toroidal bubble predicted in some cases by early computations where the gas' dynamics was neglected. Focussing computations on the very initial stages of coalescence and considering the large parameter space allows us to examine the accuracy and limits of applicability of various `scaling laws' proposed for different `regimes' and, in doing so, reveal certain inconsistencies in recent works. A comparison to experimental data shows that the conventional model is able to reproduce many qualitative features of the initial stages of coalescence, such as a collapse of calculations onto a `master curve' but, quantitatively, overpredicts the observed speed of coalescence and there are no free parameters to improve the fit. Finally, a phase diagram of parameter space, differing from previously published ones, is used to illustrate the key findings.Comment: Accepted for publication in the Journal of Fluid Mechanic

Dynamics of liquid nano-threads : fluctuation-driven instability and rupture

The instability and rupture of nanoscale liquid threads is shown to strongly depend on thermal fluctuations. These fluctuations are naturally occurring within molecular dynamics (MD) simulations and can be incorporated via fluctuating hydrodynamics into a stochastic lubrication equation (SLE). A simple and robust numerical scheme is developed for the SLE that is validated against MD for both the initial (linear) instability and the nonlinear rupture process. Particular attention is paid to the rupture process and its statistics, where the `double-cone’ profile reported by Moseler & Landmann [Science, 2000, 289(5482): 1165-1169] is observed, as well as other distinct profile forms depending on the flow conditions. Comparison to the Eggers’ similarity solution [Physical Review Letters, 2002, 89(8): 084502], a power law of the minimum thread radius against time to rupture, shows agreement only at low surface tension; indicating that surface tension cannot generally be neglected when considering rupture dynamics

Capillary breakup of a liquid bridge : identifying regimes and transitions

Computations of the breakup of a liquid bridge are used to establish the limits of applicability of similarity solutions derived for different breakup regimes. These regimes are based on particular viscous-inertial balances, that is different limits of the Ohnesorge number Oh. To accurately establish the transitions between regimes, the minimum bridge radius is resolved through four orders of magnitude using a purpose-built multiscale finite element method. This allows us to construct a quantitative phase diagram for the breakup phenomenon which includes the appearance of a recently discovered low-Oh viscous regime. The method used to quantify the accuracy of the similarity solutions allows us to identify a number of previously unobserved features of the breakup, most notably an oscillatory convergence towards the viscous-inertial similarity solution. Finally, we discuss how the new findings open up a number of challenges for both theoretical and experimental analysis

The Formation of a Bubble from a Submerged Orifice

The formation of a single bubble from an orifice in a solid surface, submerged in an in- compressible, viscous Newtonian liquid, is simulated. The finite element method is used to capture the multiscale physics associated with the problem and to track the evolution of the free surface explicitly. The results are compared to a recent experimental analysis and then used to obtain the global characteristics of the process, the formation time and volume of the bubble, for a range of orifice radii; Ohnesorge numbers, which combine the material parameters of the liquid; and volumetric gas flow rates. These benchmark calculations, for the parameter space of interest, are then utilised to validate a selection of scaling laws found in the literature for two regimes of bubble formation, the regimes of low and high gas flow rates.Comment: Accepted for publication in the European Journal of Mechanics B/Fluid

A computational study of fluctuating viscoelastic forces on trapped interfaces in porous media

In immiscible liquid–liquid Newtonian flow through a porous medium, one phase often becomes trapped in corners or narrow regions by capillary forces as blobs (e.g. oil ganglia) deep within the matrix, whilst the second flow exhibits a steady and laminar flow (e.g. of water) that has negligible influence on the trapped liquid–liquid interfaces. However, recent microfluidic experiments have shown the situation radically changes when using a viscoelastic liquid, which is capable of exhibiting pore-scale unsteady flow that can deform such interfaces. Here, a computational model is developed which allows us to capture the forces that cause this behaviour and provide a framework for future investigations of this system. In this paper, the forces on trapped interfaces are investigated for the first time. Notably, when the viscoelastic flow becomes unsteady the forces on the trapped interfaces not only fluctuate but also become amplified, thus supporting experimental findings showing they can be used to free such interfaces. At Weissenberg values of 1.5 and above the fluctuations become important and the mean values of the forces on the interfaces decrease as the fluctuations grow

On the anomalous dynamics of capillary rise in porous media

The anomalous dynamics of capillary rise in a porous medium discovered experimentally more than a decade ago (Delker et al., Phys. Rev. Lett. 76 (1996) 2902) is described. The developed theory is based on considering the principal modes of motion of the menisci that collectively form the wetting front on the Darcy scale. These modes, which include (i) dynamic wetting mode, (ii) threshold mode and (iii) interface de-pinning process, are incorporated into the boundary conditions for the bulk equations formulated in the regular framework of continuum mechanics of porous media, thus allowing one to consider a general case of three-dimensional flows. The developed theory makes it possible to describe all regimes observed in the experiment, with the time spanning more than four orders of magnitude, and highlights the dominant physical mechanisms at different stages of the process