Session HD: Interfacial and Thin Flim Instabilities V

Flow down a fibre at moderate Reynolds number

We consider the axisymmetric flow of a film down the exterior of a rigid straight vertical cylinder. A system of evolution equations for the film thickness and volumetric flow rate that was first derived by Trifonov~(1992), has been re-formulated as an extension of the classical falling film problem. To inspire confidence in the predictions of this model, its linear stability characteristics versus those of the full Navier-Stokes equations are examined yielding very good agreement. Travelling wave solutions are determined and analysed in detail over a wide range of system parameters. The solutions resemble those associated with a film flowing down a plane for sufficiently small film thickness to fibre radius ratios, and beads when these ratios are relatively large. Transient computations are also performed for comparison with the travelling wave solutions and demonstrate the selection mechanism leading to the development of so-called `dominating' waves for comparison with experimental observations.   

Experimental and numerical study of film flows down fibers at moderate Reynolds numbers

We consider the stability and nonlinear dynamics of a falling liquid film down a fiber. At moderate flow rate, the primary instability involves two different mechanisms that reinforce each other: the Rayleigh--Plateau instability, promoted by the curvature of the fiber, and the hydrodynamic instability of a falling liquid film due to inertia. An experimental set-up have been built, that enables to impose a periodic forcing at inlet. Moreover, a reduced system of evolution equations for the flow rate $q$ and the the film thickness $h$ have been obtained trough averaging of the basic equations. This model includes all physical effects, especially viscous dispersion and inertia. Preliminary results show excellent agreement between numerical and experimental observations. The spatial response of the film to an inlet periodic forcing have been analysed, indicating that, for sufficiently large fiber radius, the film behaves as a selective noise amplifier. The characteristics of the travelling-wave solutions to our model compare well to the experimental data.   

Effect of wall topography on the wavy motion on falling liquid films

Falling liquid films are used in various technological processes, including cooling of electronic devices and other components, water desalination, and powder production. Walls with topography often improve the performance of such processes. We use the shadow method and the chromatic white light sensor (CHR) method to study the wavy motion on falling liquid films on tubes with smooth walls and on tubes with longitudinal grooves. The measurements are performed at different distances below the film distributor. We show that the wavy pattern substantially changes on walls with topography. The wave frequency, the wave propagation velocity and the area of the liquid-gas interface decrease on grooved walls. The effect of the wall topography on the falling films stability is studied theoretically and numerically. We show that the longitudinal grooves stabilize the films flowing in the continuous regime (when the wall topography is completely flooded) and lead to increase of the wave length of the fastest growing wave, as well as to the decrease of the wave propagation velocity. A good qualitative and quantitative agreement between the numerical predictions and the experimental results has been found.   

Rivulet patterns in heated falling films up to moderate Reynolds numbers.

Three-dimensional wave patterns on a film flowing down a uniformly heated wall are investigated. Combining a gradient expansion with a Galerkin method, a model of four evolution equations for the film thickness, the surface temperature and both the streamwise and spanwise flow rates is shown to be robust and accurate in describing the competition between hydrodynamic waves and thermocapillary effects in a large range parameters. For small Reynolds number, {\it i.e.} in the drag-gravity regime, regularly spaced rivulets are observed, aligned with the flow and fostering quasi-two-dimensional waves of larger amplitude and phase speed than those observed in isothermal conditions. For larger Reynolds number, {\it i.e.} in the drag-inertia regime, the picture is similar to the isothermal case and no rivulets are observed. The transition between these two situations shows complex cooperative behaviors between both hydrodynamic and thermocapillary modes. Additionally, this transition is found to be related to the variations of amplitude and speed of the spanwise independent solitary wave solutions to the model.   

Rupture of thin films with resonant substrate patterning

We study the stability and rupture of thin liquid films on patterned substrates. It is shown that striped patterning on a length scale comparable to that of the spinodal instability leads to a resonance effect and an imperfect bifurcation of equilibrium film shapes. Weakly nonlinear analysis gives predictions for film shapes, stability, growth rates, and rupture times, which are confirmed by numerical solution of the thin-film equation. Film behavior is qualitatively different in the resonant patterning regime, but with sufficiently large domains rupture occurs on a spinodal length scale regardless of patterning. Instabilities transverse to the patterning are examined and shown to behave similarly as disturbances to films on uniform substrates. We explain some previously reported effects in terms of the imperfect bifurcation.   

Unusual Result of Non-Linear Wave Interaction

Combination of different types of convective instabilities has been often observed in liquid layers and thin films. In large systems with cylindrical symmetry the investigation of the thermocapillary flows usually revealed the appearance of hydrothermal waves, either standing or travelling azimuthally. However, the increase of the liquid bridge aspect ratio (cylinder length) may cause the development of some unexpected flow patterns. The time-dependent thermocapillary convection has been studied numerically in a liquid bridge with aspect ratio \textit{height/radius =1.8} and Prandtl number \textit{Pr=14}. The animation of 3D numerical results shows that slightly above the threshold of instability a new scenario is realized instead an azimuthally travelling wave. At the mid-height of the liquid bridge, spots of the temperature disturbances travel counter-clockwise during about 2/3 of an oscillation period, and then they change the direction of motion. A detailed non-linear analysis has shown that this phenomenon is caused by the presence of an additional almost stationary (slowly varying in time) 3D structure. The superposition of both structures creates an illusion that the azimuthal direction of the disturbance propagation changes its sign during a certain part of the period.