Thomas Bickel

We investigate the thermocapillary flow in a thin liquid film which is subjected to local heating, in the presence of insoluble surfactants. While surfactant molecules are first advected from warmer to cooler regions, the resulting concentration gradient drives a solutal counterflow in the opposite direction. This competition is theoretically addressed within the lubrication approximation. Assuming small deviations with respect to the mean surfactant concentration, we derive the time evolution equation governing the shape of the interface. Our study reveals that both interfacial deformations and velocities are progressively suppressed as the solutal Marangoni number increases. Our versatile model, adaptable to a range of experimental setups, offers a quantitative tool for understanding the effect of surfactants in thermocapillary-driven systems.

We investigate the thermocapillary flow in a thin liquid film which is subjected to local heating, in the presence of insoluble surfactants. While surfactant molecules are first advected from warmer to cooler regions, the resulting concentration gradient drives a solutal counterflow in the opposite direction. This competition is theoretically addressed within the lubrication approximation. Assuming small deviations with respect to the mean surfactant concentration, we derive the time evolution equation governing the shape of the interface. Our study reveals that both interfacial deformations and velocities are progressively suppressed as the solutal Marangoni number increases. Our versatile model, adaptable to a range of experimental setups, offers a quantitative tool for understanding the effect of surfactants in thermocapillary-driven systems.

Thermocapillary convection is particularly effective for the control of thin liquid film topography or for the actuation of microparticles at the liquid-air interface. Experiments with water are challenging, however, as its interface is prone to contamination by surface-active impurities that profoundly alter its hydrodynamic response. Despite numerous reports highlighting the hindrance of thermocapillary flows, quantitative information on interface contamination is definitely lacking. We therefore introduce a general framework in order to account for the presence of a low concentration of insoluble surfactants at the water-air interface. Focusing on the low-compressibility limit, we identify the inverse Marangoni number as the appropriate small parameter in the modeling. The transport equations can then be linearized without any further assumption regarding the other dimensionless groups. It is shown in particular that the surfactant concentration adapts to the thermocapillary forcing in order to cancel the interfacial viscous stress. Several experimentally relevant situations are discussed with emphasis on physical observables. Besides the excellent agreement between theory and experiment, our predictions provide a quantitative assessment of the impurity concentration.

The unsteady hydrodynamic drag exerted on an oscillating sphere near a planar wall is addressed experimentally, theoretically and numerically. The experiments are performed by using colloidal-probe atomic force microscopy in thermal noise mode. The resonance frequencies and quality factors are extracted from the measurement of the power spectrum density of the probe oscillation for a broad range of gap distances and Womersley numbers. The shift in the resonance frequency of the colloidal probe as the probe goes close to a solid wall infers the wall-induced variations of the effective mass of the probe. Interestingly, a crossover from a positive to a negative shift is observed as the Womersley number increases. In order to rationalize the results, the confined unsteady Stokes equation is solved numerically using a finite-element method, as well as asymptotic calculations. The in-phase and out-of-phase terms of the hydrodynamic drag acting on the sphere are obtained and agree well with the experimental results. All together, the experimental, theoretical and numerical results show that the hydrodynamic force felt by an immersed sphere oscillating near a wall is highly dependent on the Womersley number.

When surface-active molecules are released at a liquid interface, their spreading dynamics is controlled by Marangoni flows. Though such Marangoni spreading was investigated in different limits, exact solutions remain very few. Here we consider the spreading of an insoluble surfactant along the interface of a deep fluid layer. For two-dimensional Stokes flows, it was recently shown that the nonlinear transport problem can be exactly mapped to a complex Burgers equation [D. Crowdy, SIAM J. Appl. Math. 81, 2526 (2021)]. We first present a very simple derivation of this equation. We then provide fully explicit solutions and find that varying the initial surfactant distribution—pulse, hole, or periodic—results in distinct spreading behaviors. By obtaining the fundamental solution, we also discuss the influence of surface diffusion. We identify situations where spreading can be described as an effective diffusion process but observe that this approximation is not generally valid. Finally, the case of a three-dimensional flow with axial symmetry is briefly considered. Our findings should provide reference solutions for Marangoni spreading that may be tested experimentally with fluorescent or photoswitchable surfactants.

Extremely small amounts of surface-active contaminants are known to drastically modify the hydrodynamic response of the water-air interface. Surfactant concentrations as low as a few thousand molecules per square micron are sufficient to eventually induce complete stiffening. In order to probe the shear response of a water-air interface, we design a radial flow experiment that consists in an upward water jet directed to the interface. We observe that the standard no-slip effect is often circumvented by an azimuthal instability with the occurrence of a vortex pair. Supported by numerical simulations, we highlight that the instability occurs in the (inertia-less) Stokes regime and is driven by surfactant advection by the flow. The latter mechanism is suggested as a general feature in a wide variety of reported and yet unexplained observations.

We report on the effect of intermolecular forces on the fluctuations of supported liquid films. Using an optically induced thermal gradient, we form nanometer-thin films of wetting liquids on glass substrates, where van der Waals forces are balanced by thermocapillary forces. We show that the fluctuation dynamics of the film interface is strongly modified by intermolecular forces at lower frequencies. Data spanning three frequency decades are in excellent agreement with theoretical predictions accounting for van der Waals forces. Our results emphasize the relevance of intermolecular forces on thermal fluctuations when fluids are confined at the nanoscale.

We consider the creeping flow of a Newtonian fluid in a hemispherical region. In a domain with spherical or nearly spherical geometry, the solution of the Stokes equation can be expressed as a series of spherical harmonics. However, the original Lamb solution is not complete when the flow is restricted to a semi-infinite space. The general solution in hemispherical geometry is then constructed explicitly. As an application, we discuss the solutions of Marangoni flows due to a local source at the liquid–air interface.