TY - JOUR

T1 - Prediction of two-dimensional dripping onset of a liquid film under an inclined plane

AU - Kofman, Nicolas

AU - Rohlfs, Wilko

AU - Gallaire, François

AU - Scheid, Benoit

AU - Ruyer-Quil, Christian

PY - 2018/7

Y1 - 2018/7

N2 - Falling films flowing down a vertical plane are unstable due to inertial effects. This instability, first studied by Kapitza in 1948, is always of convective nature. Films flowing underneath an inclined plane are additionally destabilized by gravity as in the classical Rayleigh–Taylor instability for an horizontal plane. In such a situation, the instability can be of absolute nature if the plane is sufficiently inclined from the vertical. Thus, the unstable falling film exhibits an absolute/convective transition, as it has been shown recently using first a lubrication equation, and then a set of equations which include inertia and second-order viscous terms, referred to as the Weighted Residual Integral Boundary Layer (WRIBL) model. Experimental observations suggest that the absolute/convective transition corresponds roughly to the limit of dripping. This study investigates numerically the dripping limit by comparing stationary waves of the WRIBL model to numerical solutions of the Navier–Stokes equations. It is shown that this limit can be characterized by two alternative arguments: i) a secondary instability of stationary solutions of the model, or a loss of stationary solutions if the full expression of the interface curvature is used, and ii) a critical value of the area enclosed in a wave period that provides a sufficient mass for dripping.

AB - Falling films flowing down a vertical plane are unstable due to inertial effects. This instability, first studied by Kapitza in 1948, is always of convective nature. Films flowing underneath an inclined plane are additionally destabilized by gravity as in the classical Rayleigh–Taylor instability for an horizontal plane. In such a situation, the instability can be of absolute nature if the plane is sufficiently inclined from the vertical. Thus, the unstable falling film exhibits an absolute/convective transition, as it has been shown recently using first a lubrication equation, and then a set of equations which include inertia and second-order viscous terms, referred to as the Weighted Residual Integral Boundary Layer (WRIBL) model. Experimental observations suggest that the absolute/convective transition corresponds roughly to the limit of dripping. This study investigates numerically the dripping limit by comparing stationary waves of the WRIBL model to numerical solutions of the Navier–Stokes equations. It is shown that this limit can be characterized by two alternative arguments: i) a secondary instability of stationary solutions of the model, or a loss of stationary solutions if the full expression of the interface curvature is used, and ii) a critical value of the area enclosed in a wave period that provides a sufficient mass for dripping.

U2 - 10.1016/j.ijmultiphaseflow.2018.02.007

DO - 10.1016/j.ijmultiphaseflow.2018.02.007

M3 - Article

VL - 104

SP - 286

EP - 293

JO - International journal of multiphase flow

JF - International journal of multiphase flow

SN - 0301-9322

ER -