Long-wavelength surface-tension-driven Bénard convection: experiment and theory

Abstract

Surface-tension-driven Bénard (Marangoni) convection in liquid layers heated from below can exhibit a long-wavelength primary instability that differs from the more familiar hexagonal instability associated with Bénard. This long-wavelength instability is predicted to be significant in microgravity and for thin liquid layers. The instability is studied experimentally in terrestrial gravity for silicone oil layers 0.007 to 0.027 cm thick on a conducting plate. For shallow liquid depths (<.017 cm for 0.102 cm2 s−1 viscosity liquid), the system evolves to a strongly deformed long-wavelength state which can take the form of a localized depression (‘dry spot’) or a localized elevation (‘high spot’), depending on the thickness and thermal conductivity of the gas layer above the liquid. For slightly thicker liquid depths (0.017–0.024 cm), the formation of a dry spot induces the formation of hexagons. For even thicker liquid depths (>0.024 cm), the system forms only the hexagonal convection cells. A two-layer nonlinear theory is developed to account properly for the effect of deformation on the interface temperature profile. Experimental results for the long-wavelength instability are compared to our two-layer theory and to a one-layer theory that accounts for the upper gas layer solely with a heat transfer coefficient. The two-layer model better describes the onset of instability and also predicts the formation of localized elevations, which the one-layer model does not predict. A weakly nonlinear analysis shows that the bifurcation is subcritical. Solving for steady states of the system shows that the subcritical pitchfork bifurcation curve never turns over to a stable branch. Numerical simulations also predict a subcritical instability and yield long-wavelength states that qualitatively agree with the experiments. The observations agree with the onset prediction of the two-layer model, except for very thin liquid layers; this deviation from theory may arise from small non-uniformities in the experiment. Theoretical analysis shows that a small non-uniformity in heating produces a large steady-state deformation (seen in the experiment) that becomes more pronounced with increasing temperature difference across the liquid. This steady-state deformation becomes unstable to the long-wavelength instability at a smaller temperature difference than that at which the undeformed state becomes unstable in the absence of non-uniformity.

Nonlinear evolution of nonuniformly heated falling liquid films

“…͑8͒ contains both mechanisms of thermocapillarity. Small perturbations of a uniform heating, i.e., for ␦ӷ1, were already studied by Van Hook et al 21 in the different context for a horizontal layer only ͑␤ϭ0͒. The difference between the horizontal and inclined heated layers is profound, since in the latter case the mean flow can prevent the inherent tendency of dry spot formation and allow steady-state deformations of much higher amplitude, arising from the application of a nonuniform heating.…”

“…Moreover, inclusion of the van der Waals forces in the analysis, for very thin film, either leads to spontaneous film rupture or prevents the occurrence of any dry spot on the microscopic scale, depending on the attractive or repulsive character of this force, hence on the nature of liquid and plate. Small perturbations of uniform heating and their effect on the dynamics of the film were also studied by Van Hook et al 21 for a horizontal layer (␤ϭ0). They showed that nonuniformity in heating produces a steady-state deformation for any temperature difference across the layer.…”

Validity domain of the Benney equation including the Marangoni effect for closed and open flows

“…The qualitative agreement between their results and those obtained from the long-wave approximation for horizontal layers (Oron 2000) indicates that the main features of the physical system are well captured by this approximation. Weakly nonlinear analysis done by VanHook et al (1997) has demonstrated the subcritical character of the pure long-wave Marangoni instability, i.e. for a horizontal film.…”

“…Now, we can raise the question, is this subcritical behavior physical with the Marangoni effect? Actually, for horizontal layer, VanHook et al (1997) found that the Marangoni instability is subcritical and predicts a blow-up of the solution. However, as pointed out by Kalliadasis et al (2003a), this is not a true singularity formation, as forces of nonhydrodynamic origin, namely van der Waals forces not included here, become increasingly important in the region of very thin films and will arrest this blow-up behaviour.…”

Thermocapillary effect on the dynamics of viscous beads on vertical fiber

“…This long-wavelength instability mode (the S mode) was first predicted and investigated theoretically by Scriven and Sternling [17] and Smith [18]. VanHook et al [19,20] investigated both experimentally and theoretically on the long-wave instability of a thin liquid layer heated from below or cooled from above. In the experiments, the long-wave instability mode takes the form of a localized depression ("dry spot") or a localized elevation ("high spot"), depending on the thickness and thermal conductivity of the gas layer above the liquid.…”