Lubrication Theory Of Gas-liquid Interfaces In Dymamic Wetting

The evolution of gas-liquid interfaces on a substrate is encountered in a variety of environmental and technological fields.In a typical dip coating process,the hydrodynamic mechanisms of the interface and contact line remain challenging,due to the stress singularity of moving contact lines and the geometric singularity of interface.Lubrication theory,supplemented with contact line models,is one of the significant theoretical approaches to solve interface problems,but remains as a challenge when generalized into complicated cases.In this thesis,we investigate the morphologies of gas-liquid interface and the behaviour of contact lines in a few specific problems by theoretical analysis and numerical calculation.The main results are briefly given as follows:(1)The morphology of two-dimensional steady liquid films is studied by generalized lubrication theory,based on assumption of wedge flow.This generalized lubrication theory is established by decoupling the effect of plate motion and flux in liquid,and can reduce to classical lubrication equation where slope of interface is small and Snoeijer(2006b)’s lubrication equation with a moving contact line.The results are verified by considering the liquid film above or below an inclined plate in complete wetting situation,and are consistent with those of asymptotic expansion theory(Wilson,1982).Two specific cases,dimple and capillary shock,are discussed in detail,showing significant improvement compared with classical theory.In particular,the threshold for wetting transition is predicted and the overturning phenomenon is discussed.(2)We investigate the behaviour of a two-phase flow in a Couette geometry,in which one plate is stationary and a gas film is entrained over the moving plate.We report an asymptotic theory of the selection of gas film thickness,based on lubrication approximations.It is found that the gas film thickness relies primarily on the curvature of the meniscus,which represents a balance between the capillary force and gravity.The influence of the plate speed follows the classical 2/3 power law analogous to liquid deposition,with a mild modification from the viscosity ratio.The asymptotic predictions agree well with the exact solutions of the lubrication equation.(3)The conical structure of the rear of a sliding drop on the plate is studied theoretically.A new contact line model is introduced to derive the self-similar lubrication theory on this structure.The cross-section of interfacial profile is well approximated by a parabola,leading to the relation between the cone angles,which agrees better with the exact results compared with the relations from Limat&Stone(2004)and Snoeijer et al.(2005).The speed-dependence of the cone angles is examined via numerical calculation of the lubrication equation,showing that two distinct stages exist with the increase of capillary number:slowly varying stage and typical cone stage.We discover the threshold for this conical structure,beyond which the teardrop or rivulet is pulled out,and the corresponding apparent angle doesn’t disappear.In particular,The speed of contact line has a non-monotonic feature with the increasing of plate speed.(4)We investigated the gas films with saw-tooth contact lines,observed in dip coating when wetting failure occurs.We present a hydrodynamic model for the morphology of cornered gas films,which is assumed to have a conical geometry and can be described by a self-similar solution.It is found that the opening angles from top and side views are well connected and both depends on the capillary number and viscosity ratio,which is different from the case of receding contact lines.By coupling the cone solution to a contact-line model,we found the cone geometry disappears when the capillary number is beyond a threshold,giving rise to the ejection of gas bubbles.