The transport of fluid from a recirculation region adjacent to a free surface is studied using a numerical method validated with experimental flow visualization. The flow is an example of a liquid film coating process, and consists of two counter-rotating rolls placed side-by-side and half-submerged in a bath of fluid. In the gap between the rolls a recirculation zone exists just below the free surface, around which the flow splits into two films. Fluid recirculating for long periods has been identified as a source of coating defects, so this paper considers a possible method of inducing stirring. The flow is modulated by driving one of the rolls through a Hooke's joint, which delivers a well-characterized periodic perturbation to the roll speed. In response to this speed modulation, the free surface undergoes a periodic change in position and shape which drives an exchange of fluid between the recirculation region and the surrounding flow. The amplitude of the free-surface motion is strongly dependent on modulation frequency.
The dynamics of the free surface preclude a quasi-steady approach, even in the small-frequency limit, and so a fully time-dependent analysis based on the finite element method is employed. Trigonometric temporal interpolation of the finite element data is used to make passive tracer advection calculations more efficient, and excellent agreement is seen between simulation and experiment. Computations of the stable and unstable invariant manifolds associated with periodic points on the free surface reveal that the exchange of fluid is governed by a self-intersecting turnstile mechanism, by which most fluid entrained during a modulation cycle is ejected later in the same cycle.
Transport over several cycles is explored by observation of the evacuation of passive tracers initially distributed uniformly in the recirculation zone. Results demonstrate the persistence of unmixed cores whose size is dependent on the modulation frequency. By considering the percentage of tracers remaining after a fixed number of cycles, contours in frequency–amplitude space show that for each modulation amplitude there is a frequency which produces the most effective transport, with up to 80 % of tracers removed by a modulation which produces only a 5 % change in film thickness. Finally it is shown how modulation of both rolls at slightly different phases can reduce the film thickness variation to about 1 % while maintaining the level of transport.