Unsteady Liquid Film Flow with a Prescribed Free-Surface Velocity

Abstract

A liquid film flow over a flat plate is investigated by prescribing the unsteady interface velocity. With this prescribed surface velocity, the governing Navier–Stokes (NS) equations are transformed into a similarity ordinary differential equation, which is solved numerically. The flow characteristics is controlled by an unsteadiness parameter S and the flow direction parameter Λ. The results show that solutions only exist for a certain range of the unsteadiness parameter, i.e., for Λ = −1 and for Λ = 1. In the solution domain, the dimensionless liquid film thickness β decreases with S for both the cases. The wall shear stress increases with the decrease of S for Λ = −1. However, for Λ = −1 the shear stress magnitude first decreases and then increases with the decrease of S. There are no zero crossing points for the velocity profiles for both the cases. The profiles of velocity stay either positive or negative all the time, except for the wall zero velocity. Consequently, the vertical velocity becomes a monotonic function. To maintain the prescribed velocity, mass transpiration is generally needed, but for the shrinking film case it is possible to have an impermeable wall. The results are also an exact solution to the full NS equations.