Volume 30, Issue 3

This note introduces the linear stability of a ferrofluid thick layer submitted to gravity and to a normal magnetic field. The compatibility condition is derived taking into account the Rayleigh–Taylor instability and the Marangoni one. The pure normal deformation is analytically solved and these results can be extended to explain the heat conducting case.

One of the effects of radio-frequency (RF) alternating magnetic fields generated in the vicinity of electrically conducting liquid components is the generation of a free surface shear flow. This effect is generally undesired, but could be used to improve convection control in some applications – such as the Czochralski or floating zone crystal growth processes for example. Since the low value of the magnetic skin depth most often leads to high numerical difficulties, we have developed a mathematical model for the electromagnetic field distribution in the conductors for axisymmetric configurations, which provides equivalent magnetic surface forces and heat flux. The choice of an appropriate inductor shape, and the use of several out-of-phase induction components, allows us to equalize the orders of magnitude of tangential electromagnetic and thermocapillary Marangoni forces. In such case, the electromagnetic field can be used to counteract the Marangoni effect and to control the flow. Numerical calculations of the convection induced by electromagnetic and thermocapillary forces are carried out in order to depict typical flow patterns obtained in an axisymmetric liquid bridge surrounded by inductors.

The utility of composite error-function velocity profiles for the modeling of gas–liquid shear layers is examined by comparing temporal stability results from such approximate error-function profiles with results from the exact velocity profiles based on the two-fluid boundary layer equations. The stability analysis is restricted to two-dimensional perturbations. The two-fluid boundary layer equations are solved numerically using a shooting method in each fluid layer. The composite error-function profile is constructed by matching displacement thicknesses with the exact solution. With given fluid properties, the displacement thicknesses depend on the asymptotic velocity ratio of the liquid and gas stream in the laboratory frame. For different sets of fluid properties, the maximum growth rates of the Kelvin–Helmholtz instability show good numerical agreement between the exact and the approximate velocity profiles, especially when the asymptotic velocities of the gas and liquid phase are close.

Thermocapillarity-induced flow and evolution of thin liquid films on horizontal structured heated walls are studied theoretically and numerically. The steady deformation of the liquid–gas interface and the vortex flow pattern within the film are predicted for the steady continuous films in the case of grooved walls. The effect of the thickness of the gas layer above the film on the film deformation is quantified. A linear stability analysis of the film is performed. The dynamics of film evolution toward rupture is simulated numerically in the framework of the long-wave theory. The effect of the wall topography on the rupture time and on the film topology at the moment just before rupture is studied. The long-wave model is verified by comparison with full-scale numerical simulations using the volume of fluid method.

This paper presents a numerical strategy for the simulation of phenomena disclosed by very recent experimentation within the framework of microscale studies devoted to the investigation of dissolution phenomena in liquid couples with a miscibility gap. Possible approaches to the problem are reviewed and discussed through a focused and critical comparison of similarities and differences, advantages, and intrinsic limitations. The proposed technique comes through major modification of existing volume tracking methods. Some effort is devoted to demonstrate its relevance when trying to capture the fluid-dynamic instabilities dependent on mechanisms at the liquid/liquid interface driven by non-equilibrium thermodynamic conditions.

The appearance and development of thermoconvective oscillatory flows are investigated experimentally with numerical support in a liquid bridge of aspect ratio Γ = 1.2 filled with 5- and 10-cSt silicone oils and subjected to a temperature difference Δ T . The experiments with high Prandtl number fluids were carried out under terrestrial conditions for a wide range of volumes of liquid bridges in open air and with shielding. For the experiments in open air, the temperatures of the gas and liquid close to the interface remain practically constant along most of the interface. Moving across the interface, i.e., from the liquid to the gas, a temperature ‘‘jump’’ is found with  0.4Δ T  ≈ 9  K over a distance of less than 1 mm for slender liquid bridges. This tendency remains valid for fat liquid bridges. This sharp temperature drop represents additional disturbances in destabilizing the system. The flow is steady and axisymmetric at suffciently low values of Δ T . The magnitude of Δ T at which the flow becomes oscillatory depends on the relative volume of the liquid and on the external conditions. To better understand the influence of the thermal conditions of the surroundings on transition to oscillations, the experiments were carried out under two different conditions: in an open-air environment and in shielded liquid bridges. In this way, different heat transfer possibilities at the free surface are realized. Complementary numerical calculations were also performed to support our explanations of the experimental observations.

We study experimentally and theoretically convective flows, which are induced in a horizontal liquid layer by a concentrated heat source: the split coherent beam of laser radiation. The layer surface is deformable. Depending on controlling factors, laboratory experiments demonstrate a variety of flow structures and surface configurations. The flow primarily has a single vortex pattern, but in a certain range of governing parameters, secondary nonstationary vortices are superimposed. Among the surface configurations there are the concave meniscus, the convex one, and the concavo-convex one. The numerical simulation is performed on a mathematical model, which involves nonlinear partial differential equations for two-dimensional amplitude functions associated with distributions of temperature, vorticity, and surface deformation. In the long-wave approximation, the model describes the contribution of thermocapillary convection to the heat transfer as well as the degree of the interface deformation. The proposed model generalizes the existing one by taking into account the heating inhomogeneity.

The oscillatory behaviour of the Rayleigh–Marangoni–Bénard convective instability (R-M-B instability) regarding two combinations of two-layer fluid systems has been investigated theoretically and numerically. For the two-layer system of Silicone oil (10cSt) over Fluorinert (FC70), both linear instability analysis and 2D numerical simulation show that the instability of the system depends strongly on the depth ratio H r  =  H 1 / H 2 of the two-layer liquid. The oscillatory regime at the onset of R-M-B convection enlarges with reducing Γ =  Ra  / Ma values. In the two-layer system of Silicone oil (2cSt) over water, it loses its stability and onsets to steady convection at first, then the steady convection bifurcates to oscillatory convection with increasing Rayleigh number Ra. This behaviour was found through numerical simulation above the onset of steady convection in the case of Γ = 2.9, ε = ( Ra  −  Ra c  ) /  Ra c  = 1.0, and H r  = 0.5. Our findings are different from the previous study of the Rayleigh–Bénard instability and show the strong effects of the thermocapillary force at the interface on the time-dependent oscillations at or after the onset of convection. We propose a secondary oscillatory instability mechanism to explain the experimental observation of Degen et al. [Phys. Rev. E, 57 (1998), 6647–6659].