A long-wave model for film flow inside a tube with slip

Abstract

A long-wave asymptotic model is developed for the flow of an axisymmetric viscous film lining the interior of a tube for the case where slip occurs at the tube wall. Both the case of a falling film with a passive air core and that of a film driven up the tube by pressure-driven airflow are considered. The impact of slip on the net liquid volume flux is discussed, and linear stability analysis of the evolution equation is conducted to identify the impact of slip on the phase speed and growth rates of disturbances in each case. The presence of slip enhances the growth rates, though its impact on phase speed depends on the film thickness and the strength of the core airflow. For some parameter combinations, slip can modify the phase speed without altering the base flow. The nonlinear evolution of the free surface is then studied numerically. For falling films, increasing the slip length reduces the critical thickness required for plug formation to occur. Families of travelling wave solutions are found via continuation and are used to derive a simple formula for the dependence of this critical thickness on the slip length; this formula is shown to hold for small slip length. For air-driven films, the topology of streamlines in the film can be altered by slip at the wall; if the slip length is large enough, it can prevent regions of recirculation from forming at the wave crest.