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Personne# Shahab Eghbali

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Edouard Boujo, Yves-Marie François Ducimetière, Shahab Eghbali, François Gallaire

We study numerically and theoretically the gravity-driven flow of a viscous liquid film coating the inner side of a horizontal cylindrical tube and surrounding a shear-free dynamically inert gaseous core. The liquid-gas interface is prone to the Rayleigh-Plateau and Rayleigh-Taylor instabilities. Here we focus on the limit of low and intermediate Bond numbers, Bo, where the capillary and gravitational forces are comparable and the Rayleigh-Taylor instability is known to be suppressed. We first study the evolution of the axially invariant draining flow, initiating from a uniform film thickness until reaching a quasistatic regime as the bubble approaches the upper tube wall. We then investigate the flow's linear stability within two frameworks: frozen time-frame (quasisteady) stability analysis and transient growth analysis. We explore the effect of the surface tension (Bo) and inertia (measured by the Ohnesorge number, Oh) on the flow and its stability. The linear stability analysis suggests that the interface deformation at large Bo results in the suppression of the Rayleigh-Plateau instability in the asymptotic long-time limit. Furthermore, the transient growth analysis suggests that the initial flow evolution does not lead to any considerable additional amplification of initial interface perturbations, a posteriori rationalizing the quasisteady assumption. The present study yields a satisfactory prediction of the stabilization threshold found experimentally by Duclaux et al. [J. Fluid Mech. 556, 217 (2006)].

This thesis is dedicated to the analysis of a subclass of interfacial flows, columnlike free-interface flows, from two view angles: (i) the symmetry breaking under geometry-induced or external forces, (ii) their stability against infinitesimal disturbances. We employ the domain perturbation method to address three flow types by means of linear stability analysis. First, we examine the flow down an eccentric vertical fibre: the non-axisymmetric base flow brings interfacial shear into play to deform the capillary-driven Rayleigh-Plateau modes. A large enough eccentricity destabilises extra whirl modes despite the surface energy barrier to coil the interface. The linear analysis concludes according with our experiments that the combination of a thin fibre (with respect to the liquid column), a large Bond number ($Bo$, that compares gravitational forces with surface tension), and large eccentricity leads to the destabilisation and dominance of the whirl mode. We secondly study numerically and theoretically the draining liquid film coating the inside of a horizontal tube at moderate $Bo$. The buoyancy-driven rising interface deforms as $Bo$ increases, and a large enough deformation can suppress the Rayleigh-Plateau instability at large times. The linear analysis seconds pre-existing experiments in the literature, showing that the critical stabilising $Bo$ increases with the average film thickness, irrespective of finite inertia and transient growth of the perturbations. Thirdly, we explore the draining film down a horizontal cylinder, a configuration suitable for the co-existence of the Rayleigh-Plateau and Rayleigh-Taylor instabilities. The base flow either reaches a quasi-static pendant equilibrium or keeps falling until a two-dimensional rupture occurs. Nonlinear simulations suggest that the critical $Bo$ to maintain a pendant curtain scales inversely with mean film thickness. The resulting quasi-static state is linearly unstable and the collective action of capillary and gravitational effects can form two distinct patterns: (i) pearls enveloping the cylinder when surface tension dominates, (ii) vertical fingers underneath the cylinder when gravity dominates. The most linearly amplified mode will either form an array of pendant drops or result in a three-dimensional rupture, a threshold found unaffected by the transient growth of the perturbations. Lastly, we inspect numerically an electrified liquid jet falling vertically from a nozzle by coupling the flow and electric field equations. When electrical forces dominate surface tension (at large electric Bond number), the interface smoothly transitions to a conical meniscus at the nozzle tip emitting a fine jet downstream. This is due to the tangential electrical stress at the interface that folds the streamlines in the vicinity of the nozzle tip. Further raising the electrical Bond number reinforces the thinning, increases the cone half-angle, and sets in a recirculating cell at the nozzle tip to conserve the flow rate.

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We study a gravity-driven viscous flow coating a vertical cylindrical fibre. The destabilisation of a draining liquid column into a downward moving train of beads has been linked to the conjunction of the Rayleigh-Plateau and Kapitza instabilities in the limit of small Bond numbers Bo. Here, we focus on quasi-inertialess flows (large Ohnesorge number Oh) and conduct a linear stability analysis on a unidirectional flow along a rigid eccentric fibre for intermediate to large Bo. We show the existence of two unstable modes, namely pearl and whirl modes. The pearl mode depicts asymmetric beads, similar to that of the Rayleigh-Plateau instability, whereas a single helix forms along the axis in the whirl mode instability. The geometric and hydrodynamic thresholds of the whirl mode instability are investigated, and phase diagrams showing the transition thresholds between different regimes are presented. Additionally, an energy analysis is carried out to elucidate the whirl formation mechanism. This analysis reveals that despite the unfavourable capillary energy cost, the asymmetric interface shear distribution, caused by the fibre eccentricity, has the potential to sustain a whirling interface. In general, small fibre radius and large eccentricity tend to foster the whirl mode instability, while reducing Bo tends to favour the dominance of the pearl mode instability. Finally, we compare the predictions of our model with the results of some illustrative experiments, using highly viscous silicone oils flowing down fibres. Whirling structures are observed for the first time, and the measured wavenumbers match our stability analysis prediction.