Three-dimensional deformation analyses of the ultra-thin liquid film surface (linearized analyses for the steady state)

Long wave theory, which is the time evolution equation for the shape and deformation of thin liquid films and includes surface tension and surface forces such as van der Waals forces, was used to examine steady and three-dimensional deformations of ultra-thin but continuous liquid films. As liquid film deformations caused by gas pressures and shear stresses at the gas–liquid interface are usually very small, the linearized long wave equation, which is obtained for infinitesimal deformations, was employed to predict the steady-state liquid surface deformations produced by gas pressures and shear stresses. As the velocity of the solid increases and the liquid film thickness decreases, the deformation decreases and is nearly constant along solid running direction almost everywhere except at the applied position of the pressure and shearing stresses. The results obtained using the linearized equation agrees well with those obtained using the nonlinear equation and the calculation time is greatly reduced.