A Investigation of a Viscous Coating Flow

The gravity driven flow of a viscous fluid coating an inclined dry solid substrate was investigated as a prototype coating flow problem. This flow is observed to be unstable in the sense that a pool of liquid with an initially straight contact line will not advance uniformly across the substrate. Instead, rivulets form. This flow was investigated by a combination of numerical and experimental methods. A numerical method based on boundary integral techniques was developed to solve the free boundary problem. Numerical results for the two-dimensional base-state revealed that the free surface of the fluid bulges upward near the contact line. The magnitude of the bulge depends on the inclination angle of the plane, the contact angle of the air-liquid interface, and the relative importance of viscous and surface tension forces. By interrogation of the stresses, the bulge was found to be a manifestation of the kinematics of the flow and in turn was thought to be instrumental in the mechanism of the instability. An experimental technique was developed that used light extinction measurements and digital imaging to resolve the morphology of the free surface as it evolved in time. With this technique, the existence of the bulge was confirmed and the magnitude was measured. The experimentally measured interface shape was found to be in good agreement with the predictions of the numerical simulations. Furthermore, the bulge was always observed in the unstable flows. Data were taken on the growth of the instability. The dispersion relation, i.e., the growth rate-wavenumber relation, for the linear regime of the instability was measured experimentally for the first time. The results were in qualitative agreement with existing linear stability theory. The long time growth rate was also measured and found to be equally well represented by exponential or power-law growth laws. The main conclusions of this work were: (1) that a bulge which forms near the advancing contact line as a result of flow stresses and kinematics is instrumental in the mechanism of the instability and (2) that the instability is described qualitatively by linear stability theory.