Interfacial flow

Interfacial flow refers to fluid transport that is governed by the presence of an interface between two immiscible fluids. Below you will find examples of our research in the area of interfacial flow.

Incompressible surfactants on the move

Even relatively small amounts of surfactants, molecules that attach to the interface between two fluids, can behave nearly incompressible at the interface. For liquid flowing over an elongated rectangular gas-filled cavity embedded in a planar wall, this incompressibility may render the interface immobile as long as the interface remains planar. By adjusting the gas-pressure in the cavity, the gas-liquid interface can be deflected above or below the planar wall. We find that in this case liquid flowing over the cavity sets the interface in motion, inducing a recirculating flow pattern at the interface.

Baier, Hardt, Shear flow over a surface containing a groove covered by an incompressible surfactant phase, J. Fluid Mech., 949, A34 (2022),

Tuning the wavelength of electrically induced waves

When a time-varying electric field acts on the interface between two immiscible liquids, characteristic waves form at the interface. A similar thing happens when two superposed liquids are vibrated using a shaker. These interface waves form in a self-organization process, and up to now, it was unknown how to tune their wavelength. We have found a way to exactly do that, as we show in a recently published paper: We superpose the time-varying field that drives the waves with a constant field. The results nicely agree with the predictions of a theoretical model we have developed.

S. Dehe, M. Hartmann, A. Bandopadhyay, and S. Hardt, Controlling the electrostatic Faraday instability using superposed electric fields, Phys. Rev. Fluids 7 (2022), L082002

Electrically-induced Faraday waves

It is well-known that wave patterns (so-called Faraday waves) emerge on a vibrated liquid film. Not only mechanical forces can induce such wave patterns, but also electric forces, for example when keeping the liquid in a parallel-plate capacitor to which an AC voltage is applied. Remarkably, electrically induced Faraday waves have hardly been studied experimentally, although electric forcing offers options to control the wave patterns not available with mechanical forcing. We have studied the wave patterns of the electrically induced Faraday instability for the first time and compared them to theoretical predictions accounting for viscosity effects, with the result that there is a good agreement between experiments and theory.

S. Dehe, M. Hartmann, A. Bandopadhyay, and S. Hardt, The spatial structure of electrostatically forced Faraday waves, Journal of Fluid Mechanics 939 (2022), A6.

From Taylor cones to surface dimples

It is well known that fluid interfaces disintegrate under sufficiently strong electric fields, leading to electrohydrodynamic tip streaming, often in form of a Taylor cone. However, when we studied the influence of a local electric field on an oil-water interface, an alternative deformation mode emerged: Here, the interface is pushed away from the electrode, and additional cone structures form at the rim of the dimple. Using experimental and numerical methods, we show that small droplets inside the oil phase play a crucial role in the dimple formation. They induce a background flow, which in turn perturbs the interface.

Sebastian Dehe and Steffen Hardt, Deformation modes of an oil-water interface under a local electric field: From Taylor cones to surface dimples, Phys. Rev. Fluids 6 (2021), 123702.

Partial droplet coalescence – where does it end?

When an electric field derives a water droplet to an interface between oil and water, under certain circumstances partial coalescence is observed. This means that instead of merging with the water volume, the droplet bounces back, whereupon it looses a part of its volume. The product droplet can again be driven to the oil-water interface, and the whole process can be repeated, resulting in sucessively smaller droplets. We have performed corresponding experiments with a large number of successive partial-coalescence events and could produce droplets with diameters of about 400 nm. The fact that we could not observe any smaller droplets is probably due to the optical resolution limits of our microscope. Would this process still work for droplet diameters of a few nanometers?

M. Shojaeian and S. Hardt, Manipulation of single sub-femtolitre droplets via partial coalescence in a direct-current electric field, Flow 1 (2021), E12.

Fluid interfaces can become solid-like

As a stream encounters an obstacle, floating debris sometimes piles up in front of the obstacle, forming a dense layer on the water surface. At a microscopic scale, surfactants, molecules attached to the interface between two fluids, can suffer the same fate, immobilizing the interface upstream of an obstacle. ‘Super-hydrophobic’ surfaces with embedded gas pockets have been suggested for reducing drag on objects moving through water. With this in mind, we investigated the impact that surfactants have on this mechanism and found that parts of the gas-water interface become ‘solidified’ by surfactant pile-up.

T. Baier and S. Hardt, Influence of insoluble surfactants on shear flow over a surface in Cassie state at large Péclet numbers, Journal of Fluid Mechanics 907 (2020), A3.

Also fluid interfaces can be viscous

Usually, viscosity is a quantity associated with the bulk of a fluid. If, however, small particles or surfactants are adsorbed to a gas-liquid or liquid-liquid interface, the interface itself becomes viscous. In that case, the viscosity coefficients represent the dissipation due to the presence of particles or surfactants in an average manner. We have recently derived analytical expressions for the viscosity coefficients of a particle-laden fluid interface in the limit of low particle concentrations. These results could significantly simplify simulations of two-phase flows with particle-laden interfaces.

M. Eigenbrod and S. Hardt, The effective shear and dilatational viscosities of a particle-laden interface in the dilute limit, Journal of Fluid Mechanics 903 (2020), A26.

“Tears of wine” in capillary tubes

“Tears of wine” is a phenomenon that can be observed in wine glasses: Driven by Marangoni stresses, a liquid film crawls up along the glass wall. Now imaging shrinking the diameter of the wine glass to one millimeter. In that case, ring-like structures are formed from the film. These liquid rings finally collapse and form plugs that are propelled along the capillary tube by evaporation. We have studied this phenomenon based on experiments and theory.

Reference: C. Lv, S. N. Varanakkottu, and S. Hardt, Liquid plug formation from heated binary mixtures in capillary tubes, Journal of Fluid Mechanics 889 (2020), A15. DOI: 10.1017/jfm.2020.80

Species transfer using small droplets as carriers

Transferring chemical species or nanoparticles in a controlled way becomes challenging when very small species amounts are considered. We have found a solution to this problem, where a droplet with a diameter of the order of 5 µm serves as carrier for the cargo. The method is sketched in the schematic on the left. The droplet reciprocates between two aqueous reservoirs under the influence of a DC electric field. Upon touching the reservoir, the droplet reverses its charge and its direction of motion. We have studied the transfer of chemical species between the two reservoirs mediated by the droplet.

Reference: M. Shojaeian and S. Hardt, Mass transfer via femtoliter droplets in ping-pong mode, Physical Review Applied 13 (2020), 014015. DOI: 10.1103/PhysRevApplied.13.014015

Non-contact electrostatic manipulation of droplets on liquid-infused surfaces

Liquid-infused surfaces have unique liquid-repelling properties. In comparison with conventional superhydrophobic surfaces, liquid-infused surfaces provide stable wetting states and pronounced self-healing properties. We have established the manipulation of highly mobile droplets sitting on silicone-oil infused surfaces by exploiting a non-uniform electric field between a grounded substrate and a non-touching pin electrode placed above it. Droplets are attracted towards the pin electrode, and translational velocities can exceed 1 cm/s.

Reference: N. Sinn, M. T. Schür, and S. Hardt, No-contact electrostatic manipulation of droplets on liquid-infused surfaces: Experiments and numerical simulations, Applied Physics Letters 114 (2019), 213704. DOI: 10.1063/1.5091836

On-demand production of femtoliter droplets in microchannels

We have developed a method to produce monodisperse droplets inside microchannels with volumes in the femtoliter range on demand. The method utilizes pulsed electric fields deforming the interface between an aqueous and an oil phase and pinching off droplets. It can be applied even to solutions with a zero-shear rate viscosity more than ten thousand-fold higher than that of water. The droplets may serve as biological reaction compartments.

Reference: M. Shojaeian, F. X. Lehr, H. U. Göringer, and S. Hardt, On-demand production of femtoliter drops in microchannels and their use as biological reaction compartments, Analytical Chemistry 91 (2019) 3484-3491. DOI: 10.1021/acs.analchem.8b05063

Controlling the trajectories of nano/micro particles using light-actuated Marangoni flow

The ability to manipulate small objects and to produce patterns on the nano- and microscale is of great importance, both with respect to fundamentals and technological applications. The manipulation of particles with diameters of the order of 100 nm or below is a challenge because of their Brownian motion but also because of the scaling behavior of methods such as optical trapping. We have developed a method enabling the trapping and manipulation of nano- and microparticles based on interfacial flows controlled by visible light. The inherent advantages of this method are the linear scaling of the trapping force with the particle diameter and the fact that the force is less dependent on particle properties than in the case of conventional methods.

Reference: C. Lv, S. N. Varanakkottu, T. Baier, and S. Hardt, Controlling the trajectories of nano/micro particles using light-actuated Marangoni flow, Nano Letters 18 (2018), 6924−6930. DOI: 10.1021/acs.nanolett.8b02814

Electrophoresis of surface particles

The electrophoresis of particles immersed in a liquid is a well-studied phenomenon. However, what happens when a particle attached to a liquid surface translates along that surface driven by an electric field has been largely unknown. We have computed the electrophoretic mobility of a particle at the interface between two fluids with large viscosity contrast. For thin Debye layers, the Smoluchowki mobility is recovered. Generally, the mobility depends on the contact angle between the fluids and the particle. We have also calculated the interfacial deformation caused by the Debye layer around the particle.

Reference: M. Eigenbrod, F. Bihler, and S. Hardt, Electrokinetics of a particle attached to a fluid interface: Electrophoretic mobility and interfacial deformation, Physical Review Fluids 3 (2018), 103701. DOI: 10.1103/PhysRevFluids.3.103701

Stability of holes in liquid films

If you spill liquid over a surface, it forms a film with a thickness of the order of its capillary length. If the surface is bounded and you did not spill enough liquid, a hole will form. The stability of the hole depends on its size. There is a threshold size below which it collapses. For a water film, the collapse is dominated by inertia. We studied the stability and collapse of holes in liquid films in detail and compared the experimental results with mathematical models and simulations.

Reference: C. Lv, M. Eigenbrod, and S. Hardt, Stability and collapse of holes in liquid layers, Journal of Fluid Mechanics 855 (2018), 1130-1155. DOI:10.1017/jfm.2018.680

Instabilities in microchannel gas-liquid flows

In a parallel-plates microchannel where the walls are coated with liquid films, separated by a gas layer in the middle, the gas flow triggers characteristic instabilities of the films. Interestingly, the instability patterns become synchronized between the two films, i.e. the upper film “knows” what the lower film does. For example, in many cases the mirror-symmetric mode (with respect to x2, see the figure) dominates. We have performed a detailed analysis of the coupled instability modes.

Reference: M. Vécsei, M. Dietzel, and S. Hardt, Interfacial instability of liquid films coating the walls of a parallel-plate channel and sheared by a gas flow, Microfluidics and Nanofluidics 22 (2018), 91. DOI: 10.1007/s1040 4-018-2111-z

Individual droplet size control in a microfluidic T-junction droplet generator

Electric fields can be used to control the size of individual aqueous droplets produced at a microfluidic T-junction. Under a sufficiently strong electric field, the droplet diameter is about half of the diameter obtained without electric field. Using electric field pulses with a duration of the order of the inverse droplet production frequency, the size control of individual droplets produced in a continuous stream becomes possible. Based on that, arbitrary droplet size sequences can be encoded.

Reference: M. Shojaeian and S. Hardt, Fast electric control of the droplet size in a microfluidic T-junction droplet generator, Applied Physics Letters 112 (2018) 194102. DOI: 10.1063/1.5025874

Excitation of fluid interfaces with AC electric fields

An interface between a dielectric and a conducting fluid can be excited using an AC electric field. When the electric field strength exceeds a threshold voltage, patterns with a characteristic wavelength appear at the fluid interface. Different from the Faraday instability, a related case where the interface becomes unstable due to a periodic acceleration, the response of the system is very complex when excited by a superposition of different frequencies. We have analyzed the instability modes using Floquet theory. Corresponding instabilities may be utilized for pattern formation based on self-organization.

Reference: A. Bandopadhyay and S. Hardt, Stability of horizontal viscous fluid layers in a vertical arbitrary time periodic electric field, Physics of Fluids 29 (2017) 124101. DOI: 10.1063/1.4999429

Stokes drag on a sphere translating along a fluid-fluid interface

Presumably the most well-known achievement of George Gabriel Stokes is his formula for the drag force experienced by a sphere translating in an unbounded fluid at low Reynolds numbers. However, in some scenarios small particles are attached to an interface between two immiscible fluids, constrained to tangential motion relative to the interface. We have analyzed this situation and derived a generalization of Stokes’ law for large viscosity contrast and small capillary number. The image at the left is a sketch of the corresponding streamlines.

Reference: A. Dörr, S. Hardt, H. Masoud and H. A. Stone, Drag and diffusion coefficients of a spherical particle attached to a fluid interface, Journal of Fluid Mechanics 790 (2016), 607–618.

Structuring thin liquid films utilizing the Bénard-Marangoni instability

If a system of two superposed liquid films is exposed to a temperature gradient normal to the films, the upper layer exhibits convection cells which deform the lower thin film in an identical pattern. This effect can be seen in the figure above, which displays the hexagonal convection cells and the humps of the thin film below. The principle could find applications in fabricating regular, highly ordered surface structures.

Reference: I.Nejati, M. Dietzel and S. Hardt, Conjugated liquid layers driven by the short-wavelength Bénard-Marangoni instability: experiment and numerical simulation, J. Fluid Mech., 783 (2015), 46-71.