Kerstin Eckert

Spin-up of a liquid metal flow driven by a rotating magnetic field in a finite cylinder: A numerical and an analytical study

This paper presents a combined numerical and analytical study of the impulsive axisymmetric spin-up from rest of an isothermal liquid metal in a closed cylinder. The motion of the liquid is caused by the action of a low-frequency, low-induction rotating magnetic field, whose magnetic Taylor number is in the range (0.01–0.9) Tacr3D with Tacr3D given by Grants and Gerbeth [“Linear three-dimensional instability of a magnetically driven rotating flow,” J. Fluid Mech. 463, 229 (2002)]. The computations were performed for cylindrical containers of aspect ratios (diameter/height) R equal to 0.5, 1, and 2. The numerical simulations are compared with the predictions of an analytical model, valid for small Ekman numbers E extending a former work by Ungarish [“The spin-up of liquid metal driven by a rotating magnetic field,” J. Fluid Mech. 347, 105 (1997)]. The first phase of the motion from rest is an initial adjustment: the inviscid fluid begins to rotate due to the externally forced azimuthal acceleration, and co...

Chemical pattern formation driven by a neutralization reaction. I. Mechanism and basic features

We study the chemohydrodynamic pattern formation during interfacial mass transfer accompanied by a neutralization reaction. The system, which is placed in a Hele-Shaw cell, is a configuration of two immiscible liquid phases in contact along a plane interface. In the upper, organic layer a carboxylic acid is dissolved, the concentration of which is far beyond the equilibrium partition ratio. Interfacial acid transfer initiates the neutralization with an organic base dissolved in the lower, aqueous layer. Focus is on the exploration of a novel instability consisting of a regular cellular structure penetrating into the aqueous bulk solution. By several complementary experimental methods, including shadowgraph visualization with different magnifications, particle image velocimetry, differential interferometry, and detailed measurements of relevant material properties, the driving mechanism of the instability is identified. Synthesis of the experimental results suggests that lateral differences in buoyancy are...

Convective mixing induced by acid-base reactions.

When two miscible solutions, each containing a reactive species, are put in contact in the gravity field, local variations in the density due to the reaction can induce convective motion and mixing. We characterize here both experimentally and theoretically such buoyancy-driven instabilities induced by the neutralization of a strong acid by a strong base in aqueous solutions. The diverse patterns obtained are shown to depend on the type of reactants used and on their relative concentrations. They have their origin in a combination of classical hydrodynamic instabilities including differential diffusion of the solutes involved while temperature effects only play a marginal role.

Pattern formation and mass transfer under stationary solutal Marangoni instability

According to the seminal theory by Sternling and Scriven, solutal Marangoni convection during mass transfer of surface-active solutes may occur as either oscillatory or stationary instability. With strong support of Manuel G. Velarde, a combined initiative of experimental works, in particular to mention those of Linde, Wierschem and coworkers, and theory has enabled a classification of dominant wave types of the oscillatory mode and their interactions. In this way a rather comprehensive understanding of the nonlinear evolution of the oscillatory instability could be achieved. A comparably advanced state-of-the-art with respect to the stationary counterpart seemed to be out of reach a short time ago. Recent developments on both the numerical and experimental side, in combination with assessing an extensive number of older experiments, now allow one to draw a more unified picture. By reviewing these works, we show that three main building blocks exist during the nonlinear evolution: roll cells, relaxation oscillations and relaxation oscillations waves. What is frequently called interfacial turbulence results from the interaction between these partly coexisting basic patterns which may additionally occur in different hierarchy levels. The second focus of this review lies on the practical importance of such convection patterns concerning their influence on mass transfer characteristics. Particular attention is paid here to the interaction between Marangoni and buoyancy effects which frequently complicates the pattern formation even more. To shed more light on these dependencies, new simulations regarding the limiting case of stabilizing density stratification and vanishing buoyancy are incorporated.

Dynamics of Single Hydrogen Bubbles at a Platinum Microelectrode.

Bubble dynamics, including the formation, growth, and detachment, of single H2 bubbles was studied at a platinum microelectrode during the electrolysis of 1 M H2SO4 electrolyte. The bubbles were visualized through a microscope by a high-speed camera. Electrochemical measurements were conducted in parallel to measure the transient current. The periodic current oscillations, resulting from the periodic formation and detachment of single bubbles, allow the bubble lifetime and size to be predicted from the transient current. A comparison of the bubble volume calculated from the current and from the recorded bubble image shows a gas evolution efficiency increasing continuously with the growth of the bubble until it reaches 100%. Two different substrates, glass and epoxy, were used to embed the Pt wire. While nearly no difference was found with respect to the growth law for the bubble radius, the contact angle differs strongly for the two types of cell. Data provided for the contact point evolution further complete the image of single hydrogen bubble growth. Finally, the velocity field driven by the detached bubble was measured by means of PIV, and the effects of the convection on the subsequent bubble were evaluated.

Multiscale structures in solutal Marangoni convection: Three-dimensional simulations and supporting experiments

Transient solutal Marangoni convection in a closed two-layer system is studied by a combination of numerical simulations and supplementary validation experiments. The initially quiescent, equally sized liquid layers are the phases of a cyclohexanol/water mixture. Butanol is additionally dissolved in the upper organic layer. Its diffusion across the interface is sensitive to the Marangoni instability. Complex convective patterns emerge that develop a hierarchical cellular structure in the course of the mass transfer. Our highly resolved simulations based on a pseudospectral method are the first to successfully reproduce the multiscale flow observed in the experiments. We solve the three-dimensional Navier-Stokes-Boussinesq equations with an undeformable interface, which is modeled using the linear Henry relation for the partition of the weakly surface-active butanol. Length scales in the concentration and velocity fields associated with the small and large-scale cells agree well with our experimental data from shadowgraph images. Moreover, the simulations provide detailed information on the local properties of the flow by which the evolution of the patterns and their vertical structure are analyzed. Apart from relatively weak influences due to buoyancy, the evolution of the convective structures is self-similar between different initial butanol concentrations when length and time are appropriately rescaled.

Chemo-Marangoni convection driven by an interfacial reaction: Pattern formation and kinetics

A combined study devoted to chemo-Marangoni convection and the underlying kinetics is presented for a biphasic system in which surfactants are produced in situ by an interfacial reaction. The pattern formation studied in a Hele-Shaw cell in both microgravity and terrestrial environments initially shows an ensemble of chemo-Marangoni cells along a nearly planar interface. Soon, a crossover occurs to periodic large-scale interfacial deformations which coexist with the Marangoni cells. This crossover can be correlated with the autocatalytic nature of the interfacial reaction identified in the kinetic studies. The drastic increase in the product concentration is associated with an enhanced aggregate-assisted transfer after the critical micellar concentration is approached. In this context, it was possible to conclusively explain the changes in the periodicity of the interfacial deformations depending on the reactant concentration ratio.

Enrichment of Paramagnetic Ions from Homogeneous Solutions in Inhomogeneous Magnetic Fields.

Applying interferometry to an aqueous solution of paramagnetic manganese ions, subjected to an inhomogeneous magnetic field, we observe an unexpected but highly reproducible change in the refractive index. This change occurs in the top layer of the solution, closest to the magnet. The shape of the layer is in accord with the spatial distribution of the largest component of the magnetic field gradient force. It turns out that this layer is heavier than the underlying solution because it undergoes a Rayleigh-Taylor instability upon removal of the magnet. The very good agreement between the magnitudes of buoyancy, associated with this layer, and the field gradient force at steady state provides conclusive evidence that the layer formation results from an enrichment of paramagnetic manganese ions in regions of high magnetic field gradient.

Convective Instability in a Liquid-Liquid System Due to Complexation with a Crown Ether

Periodic convective instability has been observed in a biphasic system during the complexation reaction of alkali picrate and dicyclohexano-18-crown-6 which undergoes mass transfer from the hexane phase into the aqueous phase. The convection was visualized by means of precipitated crystals that are formed in both phases by the complexation reaction. The fluid motion was observed with an optical microscope and further analyzed with the particle image velocimetry (PIV) technique. The partition at the extraction of cesium into the organic phase was followed by means of the radioactive isotope (137)Cs. The type of the hydrodynamic instability is governed by the alkali metal expressed via its stability constants for the complex formed. More stable complexes trigger a higher precipitation, thereby favoring a Raleigh-Taylor instability. Complexes with a lower stability constant induce Marangoni cells which show a pulsating character in a cubic container. Depending on the confinement of the experiment cell the fluid motion can also follow a back-and-forth movement. Possible mechanisms for the occurring oscillations are discussed.

A + B → C reaction fronts in Hele-Shaw cells under modulated gravitational acceleration.

The dynamics of A + B → C reaction fronts is studied under modulated gravitational acceleration by means of a combination of parabolic flight experiments and numerical simulations. During modulated gravity the front position undergoes periodic modulation with an accelerated front propagation under hyper-gravity together with a slowing down under low gravity. The underlying reason for this is an amplification and a decay, respectively, of the buoyancy-driven double vortex associated with the front propagation under standard gravitational acceleration, as explained by reaction-diffusion-convection simulations of convection around an A + B → C front. Deeper insights into the correlation between grey-value changes in the experimental shadowgraph images and characteristic changes in the concentration profiles are obtained by a numerical simulation of the imaging process.