Materials adsorbed to the surface of a fluid -- for instance, crude oil, biogenic slicks, or industrial/medical surfactants -- will move in response to surface waves. Due to the difficulty of non-invasive measurement of the spatial distribution of a molecular monolayer, little is known about the dynamics that couple the surface waves and the evolving density field. Here, we report measurements of the spatiotemporal dynamics of the density field of an insoluble surfactant driven by gravity-capillary waves in a shallow cylindrical container. Standing Faraday waves and traveling waves generated by the meniscus are superimposed to create a non-trivial surfactant density field. We measure both the height field of the surface using moiré-imaging, and the density field of the surfactant via the fluorescence of NBD-tagged phosphatidylcholine, a lipid. Through phase-averaging stroboscopically-acquired images of the density field, we determine that the surfactant accumulates on the leading edge of the traveling meniscus waves and in the troughs of the standing Faraday waves. We fit the spatiotemporal variations in the two fields using an ansatz consisting of a superposition of Bessel functions, and report measurements of the wavenumbers and energy damping factors associated with the meniscus and Faraday waves, as well as the spatial and temporal phase shifts between them. While these measurements are largely consistent for both types of waves and both fields, it is notable that the damping factors for height and surfactant in the meniscus waves do not agree, suggesting the presence of longitudinal waves.
The effect of surfactants on the tail and film dynamics of elongated gas bubbles propagating through circular capillary tubes is investigated by means of an extensive three-dimensional numerical study using a hybrid front-tracking/level-set method. The focus is on the visco-inertial regime, which occurs when the Reynolds number of the flow is much larger than unity. Under these conditions, `clean' bubbles exhibit interface undulations in the proximity of the tail \cite{Magnini_prf_2017}, with an amplitude that increases with the Reynolds number. We perform a systematic analysis of the impact o…
We derive a general integrodifferential equation for the transient behaviour of small-amplitude capillary waves on the planar surface of a viscous fluid in the presence of the Marangoni effect. The equation is solved for an insoluble surfactant solution in concentration below the critical micelle concentration (cmc) undergoing convective-diffusive surface transport. The special case of a diffusion-driven surfactant is considered near the the critical damping wavelength. The Marangoni effect is shown to contribute to the overall damping mechanism and a first-order term correction to the critica…
Flexible rings and rectangle structures floating at the surface of water are prone to deflect under the action of surface pressure induced by the addition of surfactant molecules on the bath. While the frames of rectangles bend inward or outward for any surface pressure difference, circles are only deformed by compression beyond a critical buckling load. However, compressed frames also undergo a secondary buckling instability leading to a rhoboidal shape. Following the pioneering works of \cite{Hu} and \cite{Zell}, we describe both experimentally and theoretically the different elasto-capillar…
We present a parametric study of the unsteady phenomena associated with the flow of elongated gas bubbles travelling through liquid-filled square capillaries under high Weber number conditions. These conditions consistently induce the formation of a re-entrant jet at the back of the bubble that commonly gives way to a deep liquid cavity. Subsequent steps include pinch-off events in the cavity to generate one or multiple encapsulated drops which may coalesce, in conjunction with the bursting of the bubble-liquid interface by either the cavity or the drops. Some of these interfacial instabilitie…
Recent experimental studies on the instability appearance of capillary jets have revealed the capabilities of linear spatiotemporal instability analysis to predict the parametrical map where steady jetting or dripping takes place. In this work, we present an extensive analytical, numerical and experimental analysis of confined capillary jets extending previous studies. We propose an extended, accurate analytic model in the limit of low Reynolds flows, and introduce a numerical scheme to predict the system response when the liquid inertia is not negligible. Theoretical predictions show a remark…
We study spreading of a droplet, with insoluble surfactant covering its capillary surface, on a textured substrate. In this process, the surfactant-dependent surface tension dominates the behaviors of the whole dynamics, particularly the moving contact lines. This allows us to derive the full dynamics of the droplets laid by the insoluble surfactant: (i) the moving contact lines, (ii) the evolution of the capillary surface, (iii) the surfactant dynamics on this moving surface with a boundary condition at the contact lines and (iv) the incompressible viscous fluids inside the droplet. Our deriv…
In this work we present a non-standard model for microbial enhanced oil recovery including the oil-water interfacial area. Including the interfacial area in the model, we eliminate the hysteresis in the capillary pressure relationship. One of the characteristics that a surfactant should have, it is biological production at the oil-water interface. Therefore, we consider the production rate of surfactants not only as a function of the nutrient concentration, but also the interfacial area. To solve the model equations, we use an efficient and robust linearization scheme that considers a linear a…
We study the effect of insoluble surfactants on the wave dynamics of vertically-falling liquid films. We use three-dimensional numerical simulations and employ a hybrid interface-tracking/level-set method, taking into account Marangoni stresses induced by gradients of interfacial surfactant concentration. Our numerical predictions for the evolution of the surfactant-free, three-dimensional wave topology are validated against the experimental work of Park & Nosoko (2003). The addition of surfactants is found to influence significantly the development of horseshoe-shaped waves. At low Marangoni …
We use both theory and experiment to study the response of {\it partially wetting} films of water and surfactant solutions to a propagating MHz vibration in the solid substrate in the form of a Rayleigh surface acoustic wave (SAW). The SAW invokes a drift of mass in the liquid film, which is associated with the Schlichting boundary layer flow (also known as the Schlichting streaming). We study thin films that are governed by a balance between the drift and capillary stress alone. We demonstrate weak capillary contributions, such as for silicon oil films, support dynamic wetting and lead to the…
We present three-dimensional numerical simulations, employing the well-established lattice Boltzmann method, and investigate similarities and differences between surfactants and nanoparticles as additives at a fluid-fluid interface. We report on their respective effects on the surface tension of such an interface. Next, we subject a fluid droplet to shear and explore the deformation properties of the droplet, its inclination angle relative to the shear flow, the dynamics of the particles at the interface, and the possibility of breakup. Particles are seen not to affect the surface tension of t…
Experiments over the last 50 years have suggested a tentative correlation between the surface (shear) viscosity and the stability of a foam or emulsion. We examine this link theoretically using small-amplitude capillary waves in the presence of a surfactant solution of dilute concentrations where the associated Marangoni and surface viscosity effects are modelled via the Boussinesq-Scriven formulation. The resulting integro-differential initial value problem is solved analytically and surface viscosity is found to contribute an overall damping effect on the amplitude of the capillary wave with…
In this paper we present experimental and numerical studies of the electrohydrodynamic stretching of a sub-millimetre-sized salt water drop, immersed in oil with added non-ionic surfactant, and subjected to a suddenly applied electric field of magnitude approaching 1 kV/mm. By varying the drop size, electric field strength and surfactant concentration we cover the whole range of electric capillary numbers ($Ca_E$) from 0 up to the limit of drop disintegration. The results are compared with the analytical result by Taylor (1964) which predicts the asymptotic deformation as a function of $Ca_E$.…
The sedimentation of a surfactant-laden deformable viscous drop acted upon by an electric field is considered theoretically. The convection of surfactants in conjunction with the the combined effect electrohydrodynamic flow and sedimentation leads to a locally varying surface tension, which subsequently alters the drop dynamics via the interplay of Marangoni, Maxwell and hydrodynamic stresses. Assuming small capillary number and small electric Reynolds number, we employ a regular perturbation technique to solve the coupled system of governing equations. It is shown that when a leaky dielectric…
For more than two decades, a single model for the spreading of a surfactant-driven thin liquid film has dominated the applied mathematics literature on the subject. Recently, through the use of fluorescently-tagged lipids, it has become possible to make direct, quantitative comparisons between experiments and models. These comparisons have revealed two important discrepancies between simulations and experiments: the spatial distribution of the surfactant layer, and the timescale over which spreading occurs. In this paper, we present numerical simulations that demonstrate the impact of the part…
The linear spatiotemporal stability properties of axisymmetric laminar capillary jets with fully developed initial velocity profiles are studied for large values of both the Reynolds number, $Re=Q/(π\,a\,ν)$, and the Froude number, $Fr=Q^2/(π^2\,g\,a^5)$, where $a$ is the injector radius, $Q$ the volume flow rate, $ν$ its kinematic viscosity, and $g$ the gravitational acceleration. The downstream development of the basic flow and its stability are addressed with an approximate formulation that takes advantage of the jet slenderness. The base flow is seen to depend on two parameters, namely a S…
Marangoni propulsion is a form of locomotion wherein an asymmetric release of surfactant by a body located at the surface of a liquid leads to its directed motion. We present in this paper a mathematical model for Marangoni propulsion in the viscous regime. We consider the case of a thin rigid circular disk placed at the surface of a viscous fluid and whose perimeter has a prescribed concentration of an insoluble surfactant, to which the rest of its surface is impenetrable. Assuming a linearized equation of state between surface tension and surfactant concentration, we derive analytically the …
We analyze both theoretically and experimentally the breakup of a pendant water droplet loaded with Sodium Dodecyl Sulfate (SDS). The free surface minimum radius measured in the experiments is compared with that obtained from a numerical solution of the full Navier-Stokes equations for different values of the shear and dilatational surface viscosities. This comparison shows the small but measurable effect of the surface viscous stresses on the system dynamics for sufficiently small spatiotemporal distances from the breakup point, and allows to establish upper bounds for the values of the shear…
We analyse the effect of surface viscoelasticity on the temporal stability of a free cylindrical liquid jet coated with insoluble surfactant, extending the results of Timmermans & Lister (J. Fluid Mech., vol. 459, 2002, pp. 289-306). Our development requires, in particular, deriving the correct expressions for the normal and tangential stress boundary conditions at a general axisymmetric interface when surface viscosity is modelled with the Boussinesq-Scriven constitutive equation. These stress conditions are applied to obtain a new dispersion relation for the liquid thread, which is solved to…
