Marcel Hennenberg

Mass transfer, marangoni effect, and instability of interfacial longitudinal waves: I. Diffusional exchanges

Abstract A general formalism is developed to study interfacial instability of two immiscible incompressible fluids. Mass diffusion fluxes across the interface are the determining step. The surface mass balance equation depends upon the surface diffusion and convection and on the net flux. Discussion is restricted to longitudinal perturbations. Using the concept of surface elasticity, necessary and sufficient instability conditions for oscillating and non oscillating regimes are given for long wavelengths. The obtained criteria are extensions of the Sternling and Scriven ones.

Mass transfer, marangoni effect, and instability of interfacial longitudinal waves. II. Diffusional exchanges and adsorption-desorption processes

Abstract A general formalism is developed to study interfacial convective instability of two immiscible incompressible fluids. As an extension of the previous development (see part I) mass transfer occurs through diffusional fluxes and through adsorption-desorption processes. A linear stability analysis is performed and restricted to pure longitudinal perturbations for long wavelengths. Only oscillatory regimes are considered. We derive the stability criteria which are related to the surface elasticity. Its explicit formulation is linked to the steady diffusional fluxes, to the adsorption-desorption barrier, to the diffusion coefficients, and to the kinematic viscosities.

Electric impedance of cellulose acetate membranes and a composite membrane at different salt concentrations

Impedance measurements have been performed on dense and asymmetric cellulose acetate membranes cast in our laboratory. The frequency range was from 1 mHz to 65 kHz. Different concentrations of NaCl in water at 25°C were used. Two different potentiostats and different types of measurement cells were used and compared. The repeatability of the time constants was within ±7% except at extreme dilutions. A hybrid RC model is proposed, where the resistance (R) is calculated from the Donnan distribution of the ions and the ratios of membrane diffusion coefficients found in earlier work. The diffusion constant of Na+ is determined as a new parameter, taking the fixed charge of the membrane from previous data. DNA+ is of the order of 10−8–10−9 cm2/s. At low NaCl concentrations the diffusion coefficient of Na+ decreases, probably due to electrostatic binding to the unshielded glucuronic acid groups. The capacity (C) exhibited a marked increase with concentration, most drastically seen in the case of asymmetric membranes. This feature is explained using a theory of Trukhan combined with the integration method of Bruggemann. The effect arises from the dynamically created double layer generated by the applied field, when ions are confined in spherical alveoles. Alveolar sizes of about 70 ± 30 A are determined for the dense membranes. For the asymmetric we determine 2000 ± 1000 A, but the alveoles here seem to form a (fractal?) hierarchy of sizes. This is reflected in the lower value of the Cole-Cole α-parameter, especially at high NaCl concentrations. Heat curing of an asymmetric membrane only influenced the resistance, indicating a tightening of the pores in the skin layer. In contrast to the Trukhan-Bruggemann theory, the Maxwell-Wagner-Sillars theory is not able to account for the observed large variations in the membrane capacitances with concentration. A commercial composite membrane for seawater desalination (Fluid Systems) shows two relaxations corresponding to a skin layer and a support layer. The capacitance of the polysulfone support layer is determined by the Trukhan-Bruggemann theory, too.

Onset of oscillatory interfacial instability and wave motions in Bénard layers

Publisher Summary This chapter describes the onset of oscillatory interfacial instability and wave motions in Benard layers. An account of the basic equations and approximations needed to study Benard convection with heat or mass transfer and Marangoni stresses is presented. According to Pearsons theory, a liquid layer open to passive air is unstable to a well-defined short-wave planform of steady cellular convection for a critical value of the Marangoni number when the heating is from the liquid side. It is shown that that oscillatory instability is possible for gradients of the opposite sense if due account is taken of the dynamics of both the upper and lower phases as for the case of an interface between two liquids with transport from either side. It is found that when the Rayleigh numbers of two layers are very different from one another, the onset of instability in one of the layers drives the other, and hence the appearance of two counter-rotating cells. It is observed that when both Rayleigh numbers approach a common value, for about the same critical wavenumbers, the situation is more complex. The nonlinear waves and dissipative solitons are also elaborated.

Transverse and longitudinal waves at the air-liquid interface in the presence of an adsorption barrier

Abstract Analytical and computer results are provided here for the onset of oscillatory convective motions induced at the free surface of a liquid open to air, when there is transfer of a surfactant from one to the other phase. Special attention is paid to the role of a potential barrier, the interfacial deformability of the surface, and the competition between sorption and solute diffusion processes.

Chemical and hydrodynamical analysis of stability of a spherical interface

Abstract The hydrodynamical and chemical stability of deformation of the interface of a spherical drop suspended in an infinite amount of another immiscible liquid is investigated by the methods of linear, hydrodynamical stability theory. The two bulk fluids are homogeneous and continuous throughout. A general determinantal dispersion relation is evaluated between the complex frequency of the perturbation and the number characterising the surface harmonic normal mode of perturbation in the case of an arbitrary number of fluctuating and reacting species on the interface. The coupling between chemical reactions, surface diffusion, and hydrodynamics is effected by the interfacial through the equation of state of the interface and by convection motions on the interface. Surface shear and dilatational viscosity are taken into account assuming the surface fluid to be Newtonian. The stability of a stationary state of the interfacial chemical reaction with the bulk fluids in hydrodynamical rest with regard to small perturbations in the surface concentrations and in the velocity of the fluid is then studied. The fluxes from the bulk fluids to the interface remain constant, or otherwise they are perturbed proportional to the fluctuations in the surface concentrations. The case of one fluctuating species and small drop radii is treated in detail. The necessary condition for the system to be unstable is that the surface chemical reaction is unstable itself. In addition, the coefficient of autocatalysis of the surface reactions has to exceed a threshold value composed by the quenching effects of surface diffusion and of the bulk and surface viscosities. In cases with more than one fluctuating species there exist possibilities for the total system to be unstable even for stable surface reactions. The present theory is an extension of the theory of oscillations of a viscous drop due to capillary forces. It is thought to be an introduction to the study of “kicking drops” and motile events connected with the deformation of the biological cell membrane.

Thermocapillary convection in a multilayer system

The Marangoni–Benard instability for a symmetrical three‐layer system is examined theoretically. Linear stability analysis and nonlinear numerical simulations show that the ratio of the heat diffusivities determines the nature of the instability. Monotonic disturbances exist only when this parameter is far enough from one, the motion being driven by one interface. When the heat diffusivity ratio is close to one, oscillatory convection is observed. This is explained on a physical base: the oscillation rests on the coupling of both interfaces, which creates a flip–flop mechanism leading to a double inversion of the vortices rotation during one period of oscillation.

Deformational instability of a plane interface with perpendicular linear and exponential concentration gradients

Abstract A linear, hydrodynamical stability analysis is carried out for the deformation of an originally plane interface between two immiscible liquid phases with perpendicular linear or exponential concentration gradients of a third component. The results are compared with observations on the ethylene glycol-ethyl acetate-acetic acid system studied by Orell and Westwater. The normal mode of maximum instability is well in accordance with the dimensions of the convection cells reported by these authors. Two critical wavelengths of perturbation are found in the case of an exponential profile, in contrast to the single critical wavelength found for a linear profile. This fact is qualitatively explained by means of an exergy release/excess dissipation principle pertinent to the appearance of hydrodynamical dissipative structures.

Investigation of thermocapillary convection in a three-liquid-layer system

We present the first experimental results on Marangoni-Benard instability in a symmetrical three-layer system. A pure thermocapillary phenomenon has been observed by performing the experiment in a microgravity environment where buoyancy forces can be neglected. This configuration enables the hydrodynamic stability of two identical liquid-liquid interfaces subjected to a normal gradient of temperature to be studied. The flow is driven by one interface only and obeys the criterion based on the heat diffusivity ratio proposed by Scriven & Sternling and Smith. The measured critical temperature difference for the onset of convection is compared to the value obtained from two-dimensional numerical simulations. The results of the simulations are in reasonable agreement with the velocimetry and the thermal experimental data for moderate supercriticality. Numerically and experimentally, the convective pattern exhibits a transition between different convective regimes for similar temperature gradients. Their common detailed features are discussed

Deformational instability of a plane interface with transfer of matter. Part 1: Non-oscillatory critical states with a linear concentration profile

General laws of conservation of mass and momentum are formulated for a moving and arbitrarily deformed interface in local equilibrium with the adjacent, immiscible bulk liquids in which a third component is distributed. The equations are linearised for the case of small deformations from a plane interface.The stability of a plane interface with a perpendicular, linear concentration gradient with respect to deformation is investigated by means of linear, hydrodynamic stability theory. A general dispersion relation between the complex time constant of perturbation ω and the wavenumber k is obtained. An explicit solution for the exchange of stability boundary is found. For spontaneous, interfacial deformation to occur, the diffusion has to be directed from the liquid with the smallest value for the diffusion coefficient of the third component to the liquid with the greatest. The direction of the gravitational field has no importance in the linear theory. All wavenumbers below a certain critical number kcr will be unstable for a fixed difference of slopes of the concentration profiles. On the other hand, when k is fixed, the difference of slopes has to exceed an instability threshold which increases with the bulk and surface viscosities and the diffusion coefficients, and decreases with increasing value of the coupling coefficient between the surface pressure and the surface concentration of the third component.