Linear Stability of Thin Liquid Film on Solid Surface Under Effect of Apolar and Electrostatic Forces

The stability of thin liquid film on solid surface under apolar and electrostatic forces is investigated. The flow was represented by a two-dimensional Navier-Stokes equation coupled with continuity equation and associated boundary conditions. The film is modeled as a two-dimensional Newtonian liquid of constant density, ρ and viscosity, μ flowing on a horizontal plane. The liquid film of mean thickness, h0 is assumed to be thin enough to neglect the gravitational effect and bounded above by a passive gas and laterally extends to infinity (two-dimensional model). The body force term in the Navier-Stokes equation is modified by the inclusion of excess intermolecular interactions (apolar and electrostatic forces) between fluid film and the solid surface owing to apolar and electrostatic forces. The modified Navier-Stokes equation with associated boundary conditions is solved under long wave approximation method to obtain a nonlinear equation of evolution of the film interface. The results indicated that the behavior of the growth rate (Figure 2(a) and 2(b)) is a wave reducing in size and gradually disappearing altogether, whereas the value of growth rate starts at negative values and then increases gradually until it reaches the highest point, after which it decrease gradually until it disappears with increase in film thickness to reach stabilization which equals zero. This situation occurs when the film thickness is ≤ 30 nm, after the thickness of the film equals to 30 nm, it is noted that there is no effect on growth rate, so it can be interpreted that the effect of apolar and electrostatic forces occurs only when the thickness of thin film ≤ 30 nm. Thus, it can be noted that any increase in the value of the wavenumber leads to a decrease in the value of growth rate. Therefore, it can be established that the relationship between wavenumber and growth rate is proportional