Capillary Ridge Research Articles

Coating flow of a liquid film with colloidal particles on a vertical fiber

A thin liquid film of colloidal suspension is considered which is spreading down on a vertical cylinder under the effect of gravity. A precursor film model is employed at the three-phase contact line to relieve the stress singularity. The curvature pressure leads to the formation of a capillary ridge at the contact-line. Bulk and surface colloids are assumed to be present in the liquid film. The interfacial pattern is governed by the film evolution equation, obtained by simplifying mass and momentum balance equations within the lubrication assumption, while rapid vertical diffusion is assumed for the advection–diffusion equation of the bulk concentration. Fluid viscosity and diffusivity are considered to be functions of the particle volume fraction. The effects of the bulk and surface colloids on the spreading film dynamics are systematically studied. The presence of bulk colloids leads to the thinning of the capillary ridge for smaller inlet bulk colloid concentrations. On the other hand, a larger inlet bulk concentration leads to the formation of a secondary advancing front behind the capillary ridge in the film profile. A reduced contact point velocity is observed with an increase in the upstream bulk concentration. Surface concentration is found to result in a hump-like structure behind the capillary ridge in the upstream direction due to solutal Marangoni stress. The Marangoni stress also hinders the surface-tension-driven spreading, resulting in a smaller contact line velocity. Reducing the Bond number results in unstable film profiles, which lead to a wave-like structure in the region with no bulk concentration gradients.

Thickness of Nanoscale Poly(Dimethylsiloxane) Layers Determines the Motion of Sliding Water Drops.

Layers of nanometer thick polydimethylsiloxane (PDMS) are applied as hydrophobic coatings because of their environmentally friendly and chemically inert properties. In applications such as heat exchangers or fog harvesting, low water drop friction on surfaces is required. While the onset of motion (static friction) has been studied, the knowledge of dynamic friction needs to be improved. To minimize drop friction, it is essential to understand which processes lead to energy dissipation and cause dynamic friction? Here, the dynamic friction of drops on PDMS brushes of different thicknesses is measured, covering the whole available velocity regime. The brush thickness L turns out to be a predictor for drop friction. 4-5nm thick PDMS brush shows the lowest dynamic friction. A certain minimal thickness is necessary to form homogeneous surfaces and reduce the attractive van der Waals interaction between water and the substrate. The increase in dynamic friction above L = 5nm is also attributed to the increasing viscoelastic dissipation of the capillary ridge formed at the contact line. The height of the ridge is related to the brush thickness. Fluorescence correlation spectroscopy and atomic force measurements support this interpretation. Sum-frequency generation further indicates a maximum order at the PDMS-water interface at intermediate thickness.

Computational study of a thin film flow over a topographical feature using phase-field lattice Boltzmann method

The study employs the phase-field lattice Boltzmann method (PFLBM) to explore the dynamics of a thin film flowing over a topographical feature such as a mound or a trench. The mesoscopic nature of PFLBM makes it a suitable technique for problems involving tracking the evolution of a liquid–air interface. PFLBM simulation results are validated with experimental and analytical results confirming the viability of the numerical approach for such problems. The effect of changing the topographical height, aspect ratio, viscosity ratio, and presence of multiple mounds on the film profiles are systematically analyzed. It is observed that a steady-state solution could not be obtained for large height topographical features. The transition from a steady-state interfacial pattern to an unsteady-steady state is found to depend on the width of the topography. Geometry-based condition is employed to deal with the contact points present in the film dynamics beyond rupture. For large contact angles, the unsteady cases result in film rupture and form a continuous array of droplets of equivalent dimensions at a periodic interval. Increasing the aspect ratio reduces the width of the capillary ridge formed above the topographical feature, while the viscosity ratio reduces the maximum height of the ridge. The shapes of the capillary ridges formed over multiple mounds in the flow direction are independent if the separation between the successive mounds is beyond a critical value. This critical value strongly depends on the capillary number and is independent of the dimensions of the mound.

Direct force measurement of microscopic droplets pulled along soft surfaces

When a droplet is placed on a soft surface, surface tension deforms the substrate, creating a capillary ridge. We study how the motion of the ridge dissipates energy in microscopic droplets. Using a micropipette based method, we are able to simultaneously image and measure forces on a microscopic droplet moving at a constant speed along a soft film supported on a rigid substrate. Changing the thickness of the thin film tunes the effective stiffness of the substrate. Thus we can control the ridge size without altering the surface chemistry. We find that the dissipation depends strongly on the film thickness, decreasing monotonically as effective stiffness increases. This monotonic trend is beyond the realm of small deformation theory, but can be explained with a simple scaling analysis.

Contact line motion on heated patterned surfaces

We develop a lubrication-type model of contact line motion on a surface patterned by an array of square patches of decreased wettability. The model accounts for evaporation of the liquid due to surface heating, as well as effects of surface tension and disjoining pressure. Stick-slip motion and strong interface deformations are found from the simulations. The height of the capillary ridge becomes spatially nonuniform when the stable contact line travels over the region of decreased wettability. Coupling between the surface pattern and intrinsic contact line instability is investigated. For the weak evaporation limit, the model predictions are compared to previous studies of contact line dynamics under isothermal conditions.

Colloidal exchange flow in ducts

Buoyancy-driven exchange flow, i.e. drainage of a heavy colloidal suspension by an immiscible light pure fluid in a vertical duct, is studied analytically. A lubrication model is developed for two-dimensional channel and axisymmetric pipe geometries. The outcome transport equations are solved numerically to capture the effect of important governing dimensionless parameters: Péclet number, precursor film thickness, Reynolds number, mixtures’ viscosity ratio, initial volume fraction of hard-sphere particles, particles concentration within the precursor film, and capillary number. Recent study of colloidal film flow over a flat surface reveals interface thinning over time due to diffusion impact on suspension viscosity. However, the current model suggests that there is no such thinning phenomenon observed for colloidal flows within confined geometry unless in the absence of surface tension. The extent of the mixed particle zone along the duct grows as the Péclet number is decreased. Precursor film thickness is found to decrease the capillary ridge height forming at heavy suspension front. Either an increase in the Reynolds number or a decrease in light-to-heavy-fluid viscosity ratio extends the interpenetration rate of mixtures. As the bulk volume fraction of particles is increased close to 50 %, interesting secondary fronts emerge within the flow that coincide with the range where the diffusion coefficient is minimized. The heavy suspension layer drains slightly faster when the gradient in particle concentration between the bulk and precursor film regions is maximum. Finally, the non-monotonic impact of surface tension on the extent of mixtures’ exchange zone is uncovered via simulations conducted at various capillary numbers.

Biphasic co-flow through a sudden expansion or contraction of a Hele-Shaw channel

We study the biphasic co-flow through an expansion (or contraction) of the shallow (Hele-Shaw) microchannel. Assuming gentle streamwise variation of the flow, we derive the 3rd-order nonlinear differential equation governing the steady-state shape of the interface separating the two fluids. The interfacial profiles obtained by integrating this nonlinear equation are further compared to numerical solution of the two-dimensional free-boundary problem and experimental results. The amplitude of the capillary ridge emerging upstream from a sudden expansion and leading to narrowing of the thread of the inner phase is substantial, however, not large enough to trigger instability and breakup.

Experimental study of a liquid film flowing over a perforation

AbstractPerforations are one of the recognized geometrical features that contribute to liquid redistribution in corrugated sheet packings. Our experimental study focuses on a simplified but relevant configuration: a thin liquid film flowing on either side of a vertical plate with a circular perforation. We focus on the curtain mode when the liquid fills the perforation. Confocal chromatic imaging reveals a capillary ridge upstream of the perforation, an inertial ridge downstream, and a varicose capillary wave standing on the liquid curtain. We show that the wavelength is selected such that the velocity of the wave both satisfies Taylor's dispersion relation and matches the curtain local speed. We examine the effect of perforation size, supply conditions, and liquid properties on the curtain transition. Lastly, we propose a simple model based on a momentum balance that describes the effect of these parameters on the Reynolds number at which curtain forms.

Modeling and simulation of capillary ridges on the free surface dynamics of third-grade fluid

Abstract Most of the viscoelastic fluids have deformation while flowing over a heated plate. A typical feature of a thin viscous or viscoelastic fluid is the formation of the capillary ridges over locally heated plates. The creation of such ridges in the thin-film surface can affect the smoothness of the coating. This work particularly concerned the flow of non-Newtonian third-grade fluid over an inclined heated plate and the formation of ridges. The conservation laws associated with free surface and wall boundary conditions model the two-dimensional fluid flow. The long wave approximation of the model results in an equation of evolution to explain the structure of free surfaces. The resulting equation is discretized implicitly using the finite volume method. The obtained results are discussed for different flow parameters that affect capillary ridge emergence on the free surface. Variation in the height of capillary ridges of third-grade fluid is compared with the second-grade fluid and Newtonian fluid flow. We observe, the ridge size gets smaller for the third-grade fluid compared to the Newtonian and the second-grade fluid. Our analysis investigates how the third-grade viscoelastic parameters affect the dynamics of the free surface and the size of the capillary ridge concerning temperature changes and other phenomena of interest.

Contact line instability of gravity driven thin films flowing down an inclined plane with wall slippage

Coating liquid films exist widely in nature and industrial processes, such as lava flows, falling-film evaporators, and microfabrication processes. In this paper, we aim to give a comprehensive study on contact line instability of thin film flows on a slippery substrate, which has different slippery lengths in the streamwise direction and the spanwise direction. It is reported here for thin film flows with contact lines, the streamwise slippery effect plays an opposite role compared with the spanwise slippery effect, different from the conclusion by Ding and Wong that the wall slippage always promoted the instability of the thin film flow [Ding and Wong, Int. J. Heat Mass Tran. 90, 2015]. The evolution equation of the thin film flow is derived by the lubrication theory, and traveling wave solutions are obtained, showing that with a larger streamwise slippery length, the traveling wave speed is promoted and a higher capillary ridge is induced. Linear stability analysis (LSA) and energy analysis are carried out to elucidate the mechanism of the flow instability. Results indicate that for thin film flows with dynamic contact lines, the slippery effect in the streamwise and spanwise direction plays different roles in the flow instability. The wall slippage in the streamwise direction suppresses the wave height of traveling waves and impedes the instability of the flow, while the wall slippage in the spanwise direction promotes the instability with a smaller cut off wavenumber. Besides, numerical simulations for thin film flows are performed, and the flow phenomenon agrees well with the prediction by the LSA. Our findings offer insight into the influence of the wall slippage on the dynamics of thin films coating on biomaterials and compliant substrates.

Formation of the Capillary Ridge on the Free Surface Dynamics of Second-Grade Fluid Over an Inclined Locally Heated Plate

Abstract The formation of capillary ridges is the typical features of thin viscous or viscoelastic fluids over a locally heated plate. This ridge leads to the nonuniformity in the thin film coating. In this work, the formation of capillary ridges on the free surface of thin second-grade non-Newtonian fluid flowing over an inclined heated plate is discussed. The flow is modelled by two-dimensional laws of conservation of mass, momentum, and energy with corresponding boundary conditions at the plate and the free surface. An evolution equation for the description of the liquid thin film height is derived from the two-dimensional balance equations using the long-wave approximation. The resulting nonlinear dynamic equation is discretised implicitly on a uniform grid using the finite volume method. The obtained results on the capillary ridge in the free surface are discussed for the different flow parameters. It is noted that the capillary ridge height is higher for the second-grade viscoelastic fluid in comparison to the Newtonian one. This study can be a starting point to investigate the influence of second-grade viscoelastic parameter on the free surface instability and other phenomena of interest.

Fingering instability on curved substrates: optimal initial film and substrate perturbations

We investigate the stability of a thin Newtonian fluid spreading on a horizontal cylinder under the action of gravity. The capillary ridge forming at the advancing front is known to be unstable with respect to spanwise perturbations, resulting in the formation of fingers. In contrast to the classic case of a flow over an inclined plane, the gravity components along a cylindrical substrate vary in space and the draining flow is time-dependent, making a modal stability analysis inappropriate. A linear optimal transient growth analysis is instead performed to find the optimal spanwise wavenumber. We not only consider the optimal perturbations of the initial film thickness, as commonly done in the literature, but also the optimal topographical perturbations of the substrate, which are of significant practical relevance. We found that, in both cases, the optimal gains are obtained when the perturbation structures are the least affected by the time horizon. The optimal spanwise wavenumber is found to be dependent on the front location, due to the dependence of the characteristic length of the capillary ridge on its polar location.

Instability of pressure-driven gas–liquid two-layer channel flows in two and three dimensions

We study unstable waves in gas–liquid two-layer channel flows driven by a pressure gradient, under stable stratification, not assumed to be set in motion impulsively. The basis of the study is direct numerical simulation (DNS) of the two-phase Navier–Stokes equations in two and three dimensions for moderately large Reynolds numbers, accompanied by a theoretical description of the dynamics in the linear regime (Orr–Sommerfeld–Squire equations). The results are compared and contrasted across a range of density ratios $r=\unicode[STIX]{x1D70C}_{liquid}/\unicode[STIX]{x1D70C}_{gas}$. Linear theory indicates that the growth rate of small-amplitude interfacial disturbances generally decreases with increasing $r$; at the same time, the cutoff wavenumbers in both streamwise and spanwise directions increase, leading to an ever-increasing range of unstable wavenumbers, albeit with diminished growth rates. The analysis also demonstrates that the most dangerous mode is two-dimensional in all cases considered. The results of a comparison between the DNS and linear theory demonstrate a consistency between the two approaches: as such, the route to a three-dimensional flow pattern is direct in these cases, i.e. through the strong influence of the linear instability. We also characterize the nonlinear behaviour of the system, and we establish that the disturbance vorticity field in two-dimensional systems is consistent with a mechanism proposed previously by Hinch (J. Fluid Mech., vol. 144, 1984, p. 463) for weakly inertial flows. A flow-pattern map constructed from two-dimensional numerical simulations is used to describe the various flow regimes observed as a function of density ratio, Reynolds number and Weber number. Corresponding simulations in three dimensions confirm that the flow-pattern map can be used to infer the fate of the interface there also, and show strong three-dimensionality in cases that exhibit violent behaviour in two dimensions, or otherwise the development of behaviour that is nearly two-dimensional behaviour possibly with the formation of a capillary ridge. The three-dimensional vorticity field is also analysed, thereby demonstrating how streamwise vorticity arises from the growth of otherwise two-dimensional modes.

Thermocapillary migration of droplets under molecular and gravitational forces

We study the thermocapillary migration of two-dimensional droplets of partially wetting liquids on a non-uniformly heated surface. The effect of a non-zero contact angle is imposed through a disjoining–conjoining pressure term. The numerical results for two different molecular interactions are compared: on the one hand, London–van der Waals and ionic–electrostatics molecular interactions that account for polar liquids; on the other hand, long- and short-range molecular forces that model molecular interactions of non-polar fluids. In addition, the effect of gravity on the velocity of the drop is analysed. We find that for small contact angles, the long-time dynamics is independent of the molecular potential, and the footprint of the droplet increases with the square root of time. For intermediate contact angles we observe that polar droplets are more likely to break up into smaller volumes than non-polar ones. A linear stability analysis allows us to predict the number of droplets after breakup occurs. In this regime, the effect of gravity is stabilizing: it reduces the growth rates of the unstable modes and increases the shortest unstable wavelength. When breakup is not observed, the droplet moves steadily with a profile that consists in a capillary ridge followed by a film of constant thickness, for which we find power law dependencies with the cross-sectional area of the droplet, the contact angle and the temperature gradients. For large contact angles, non-polar liquids move faster than polar ones, and the velocity is proportional to the Marangoni stress. We find power law dependencies for the velocity for the different regimes of flow. The numerical results allow us to shed light on experimental facts such as the origin of the elongation of droplets and the existence of saturation velocity.

Effect of an insoluble surfactant on the dynamics of a thin liquid film flowing over a non-uniformly heated substrate.

The stability analysis of a gravity-driven thin liquid film with an insoluble surfactant flowing over a surface with embedded, regularly spaced heaters is investigated. At the leading edge of a heater, the presence of a temperature gradient induces an opposing Marangoni stress at the interface leading to the formation of a capillary ridge. This ridge has been shown to be susceptible to thermocapillary (oscillating in the flow direction) and rivulet (spanwise periodic pattern) instabilities. The presence of an insoluble surfactant is shown to have a stabilizing effect on this system. The governing equations for the evolution of the film thickness and surfactant concentration are obtained within the lubrication approximation. The coupled two-dimensional base solutions for the film thickness and surfactant concentration show that there is no significant change in the height of the capillary ridge at the subsequent heaters downstream. The height of the capillary ridge is reduced by the presence of the surfactant. For very small Peclet number, the presence of multiple heaters has almost no significant effect on the film stability as compared to a single heater and similar trends are observed between the two configurations in the presence of the surfactant as for the case of a clean interface. However, for large Peclet number, the effect was observed on both types of instabilities for certain heater configurations. The Biot number is shown to have a strong effect on the stability results wherein the dominant mode of instability is altered (from rivulet to thermocapillary instability) for a passive or no surfactant case with increase in the Biot number. For an active surfactant thermocapillary instability is found to remain the dominant mode of instability for all the values of the Biot number. It is shown that increasing the number of heaters beyond a couple does not further affect the stability results.

Buoyant displacement flow of immiscible fluids in inclined ducts: A theoretical approach

We study the buoyant displacement flow of two immiscible Newtonian fluids in an inclined duct (two-dimensional channel) theoretically. The fluids may have different viscosities. The displacing fluid is denser than the displaced one, i.e., a density-unstable configuration. For simplicity, the fluids are assumed to behave as neutrally wetting in the vicinity of duct walls. The small diameter-to-length ratio of the duct considered (δ≪1) has been used as the perturbation parameter in developing a lubrication model (negligible inertia). Appropriate Navier-slip conditions have been applied at the walls to overcome contact-line problem singularity. The lubrication model developed has then been numerically solved using a robust total variation diminishing finite difference scheme. Completely different flow patterns have been observed compared to the miscible limit. Fluids immiscibility is found to cause a capillary ridge in the vicinity of the displacing front, which diminishes as the surface tension is increased. For small values of surface tension parameter, the fluids immiscibility is found to decelerate the advancement of interpenetrating heavy and light layers. More efficient displacement (less fingering within the displacing layer) has been observed at small density differences and when the displacing fluid is more viscous than the displaced one. The limit of zero imposed velocity corresponding to the exchange flow has further been considered in the lubrication model. An interesting jump in the interface height occurs close to the vicinity of the gate region due to the immiscibility, which has been similarly reported in other recent computational works. Detailed mathematical notes on the similarity solution of the flow at long times are moreover provided. Investigating the short-time dynamics of the flow reveals the dominance of diffusive surface tension effects over buoyancy.

Free Surface Thin Film Flow of a Sisko’s Fluid over a Surface Topography

The flow of a thin film down an inclined surface over topography is considered for the case of liquids with Sisko’s model viscosity. For the first time lubrication theory is used to reduce the governing equations to a non-linear evolution equation for a current of a Sisko’s model non-Newtonian fluid on an inclined plane under the action of gravity and the viscous stresses. This model is solved numerically using an efficient Full Approximation Storage (FAS) multigrid algorithm. Free surface results are plotted and carefully examined near the topography for different values of power-law index np, viscosity parameter m, the aspect ratio A and for different inclination angle of the plane with the horizontal. Number of complications and additional physical effects are discussed that enrich real situations. It is observed that the flows into narrow trenches develop a capillary ridge just in front of the upstream edge of a trench followed by a small trough. For relatively small width trenches, the free surface is almost everywhere flat as the dimensional width of the trench is much smaller than the capillary length scale. In this region, surface tension dominates the solution and acts so as to stretch a membrane across the trench leading to smaller height deviations. The ridge originates from the topographic forcing which works to force fluid upstream immediately prior to the trench before helping to accelerate it over. The upstream forcing slows down the fluid locally and increases the layer thickness.

Interfacial stability and self-similar rupture of evaporating liquid layers under vapor recoil

We investigate interfacial stability of an evaporating viscous liquid layer above/below a horizontal heated substrate in the framework of a long-wave model that accounts for surface tension, positive/negative gravity, and evaporation effects of mass loss and vapor recoil. With the time-dependent linear stability analysis, it is found that the interface instability is enhanced by vapor recoil with time using an effective growth rate. The destabilizing mechanism of vapor thrust competes with the stabilizing surface tension, and the effects of the latter are not asymptotically negligible near rupture, reflected by a rescaled effective interfacial pressure. A two-dimensional nonlinear evolution is investigated for the quasi-equilibrium evaporating layers with different evaporative conditions for Rayleigh–Taylor unstable and sessile layers. For weak mass loss and strong vapor recoil, the well-defined capillary ridges emerge around a deepening narrow valley with increasing wavelength under a positive gravity, while, on the basis of initial condition, main and secondary droplets are either coalesced partially or separated by a sharp dry-out point under a negative gravity. The rupture location depends strongly on the characteristics of a given initial condition, except for the random perturbation. For both the cases, an increase in the modified evaporation number tends to reduce the rupture time tr and droplet thickness remarkably. Similarity analysis along with numerical strategy is presented for the final stage of touch-down dynamics, determined by a physical balance between the vapor recoil and capillary force. The evaporation-driven rupture with a significant vapor recoil and negligible mass loss is shown to contain a countably infinite number of similarity solutions whose horizontal and vertical length scales behave as (tr − t)1/2 and (tr − t)1/3. The first similarity solution represents a stable single-point rupture.

Inertialess multilayer film flow with surfactant: Stability and traveling waves

Multilayer film flow down an inclined plane in the presence of an insoluble surfactant is investigated with particular emphasis on determining flow stability and investigating the possibility of traveling-wave solutions. The investigation is conducted for two or three layers under conditions of Stokes flow and, separately, on the basis of a long-wave assumption. A normal mode linear stability analysis for Stokes flow shows that adding surfactant to one of the film surfaces can destabilize an otherwise stable flow configuration. For the long-wave system, periodic traveling-wave branches are detected and traced, revealing solutions with pulselike solitary waves on each film surface traveling in phase with each other, traveling waves with capillary ridge structures, and solutions with two of the film surfaces almost in contact. Time-periodic traveling-wave solutions are also found. The stability of the traveling waves is determined by solving initial-value problems and by computing eigenvalue spectra. Boundary element simulations for Stokes flow confirm the existence of traveling waves outside the long-wave regime.

Steady viscoelastic film flow over 2D Topography: II. The effect of capillarity, inertia and substrate geometry

We examine the two-dimensional, steady flow of a viscoelastic film under the action of gravity over a substrate with periodic topographical features. We account for the rheology of the viscoelastic material using the exponential Phan–Thien and Tanner (PTT) constitutive model. The conservation equations are solved via the mixed finite element method combined with a quasi-elliptic grid generation scheme, while the viscoelastic stresses are discretized using the EVSS-G/SUPG method. Our scheme allows the computation of accurate steady-state solutions up to high values of Deborah, Reynolds and capillary numbers. We perform a thorough parametric analysis to investigate the effect of the elastic, capillary and inertia forces on the flow characteristics. Our results indicate that surface tension and elasticity affect the film closer to the location with abrupt changes of the substrate topography; the sizes of the capillary ridge before a step down and of the depression before a step up are increased and move upstream as fluid elasticity or interfacial tension increase. It is shown that under creeping flow conditions the length scale of the capillary ridge increases with De following a power law of ¼, which can also be predicted by simple scaling arguments. Inertia has a more global effect on the film affecting larger portions of it, while in its presence the length scale of the capillary features is not affected significantly by the material elasticity. Moreover, it is shown that similarly to the case of Newtonian liquids, high inertia causes the formation of a ridge just after the step up. We also explore the effect of the geometrical characteristics of the substrate as well as its inclination angle and it is shown that the interface shape becomes more deformed as the topography appears wider, deeper or it approaches the vertical plane.