While being computationally advantageous, solution of equation sets based on the long-wave approximations are limited to exploring free-surface features only, they reveal nothing about the internal flow structures within the film.
Continuous film flow over step-up and step-down topographies have become classi- cal problems where the associated inner flow structure induced is other than purely unidirectional. Eddies are observed to exist, an example of which is shown in Figure 1.6; here, flow structures formed under Stokes flow conditions can be seen from ex- periments and complementary numerical solutions for increasing film thickness on an undulating plane inclined at 45o- as the film thickness increases the emergence of an eddy can be observed, a further increase induces an increase in magnitude of the size of the separated flow region. The potential existence of eddies is important
as flow separation, between the bulk flow coating the surface and eddies trapped within a topographical feature, can strongly influence the flow and thus the asso- ciated rates of chemical reactions, heat (Scholle, Haas, Aksel, Thompson, Hewson and Gaskell, 2009) and mass transfer (Wierschem and Aksel, 2004).
FIGURE 1.6: An example of eddy formation in flow under Stokes flow conditions in the topography of an undulating substrate, with amplitude of the wavy topography de- fined as a = 2π/5 and inclined at 45o to the horizontal, for increasing film thickness; (a) h = 16π/25, (b) h = 18π/25 and (c) h = 24π/5. The left hand side shows experimental observations of Wierschem and Aksel (2003) (with complex variable numerical solutions overlaid) and the right column showing corresponding finite element numerical results of
Scholle et al. (2008).
Flow separation is found for a variety of substrate undulations, depending on the level of inertia and topography shape and steepness. Taneda (1979) visualised the flow of silicone oil over square and triangular shaped topographies demonstrating the effect the topographies had on the formation of eddies including separation of eddies with increasing trench topography length; such results have been observed in full three-dimensional computations see, for example, Veremieiev (2011). Zhao and Cerro (1992) found eddies to exist even under laminar flow conditions for a
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large range of film thickness; Wierschem et al. (2003) found that the generation of eddies at very low Reynolds numbers depended on surface tension, film thickness and substrate waviness. Wierschem and Aksel (2004) observed eddies created at low Reynolds numbers were then modified by increasing inertia, and similar obser- vations have been reported in Scholle et al. (2004, 2006).
The subtle interplay between geometric and inertial effects on the formation of lo- cal flow structures in the corrugations of a wavy substrate was revealed by Scholle et al.(2008). The effects can be measured by considering local (based on topogra- phy length scale) and global Reynolds numbers; two types of eddies were observed, those induced kinematically and those induced by inertia. The manipulation of ed- dies has also been investigated for shear-driven flow over a corrugated substrate (Scholle, Haas, Aksel, Wilson, Thompson and Gaskell, 2009); it was found that, as well as geometric and inertial effects seen in free-surface flow, the mean plate separation (the average distance between the top moving plate and bottom corru- gated plate) also influences the associated flow structures. Should no eddies be present in the flow at a certain mean separation, by decreasing the mean separation an eddy could be induced - this is the opposite to free-surface flow where increasing the Nusselt film thickness would induce an eddy where previously there was none (Scholle et al., 2008).
Waves are generated on the surface of a film coating an undulating substrate when inertia becomes an important factor due to instability. Wierschem and Aksel (2003) found that the critical Reynolds number for the instability was higher than on a flat inclined plate. The generation of surface waves and the resonance phenomena, the amplification of the free-surface and film thickness amplitude, seen in the case of flow over topography, has been investigated extensively; see for example Bonto- zoglou and Papapolymerou (1997), Malamataris and Bontozoglou (1999), Bonto- zoglou (2000), Vlachogiannis and Bontozoglou (2002) and Heining et al. (2009). Linear resonance is investigated within the context of corrugations whose depth is much smaller than its wavelength and the film thickness Wierschem et al. (2008)
whereas non-linear resonance effects are influential when this assumption no longer holds (Heining et al., 2009). As inertia is increased the free-surface of the flow is further disturbed, the magnitude of the disturbances have been found to depend on the depth of the valleys comprising the substrate (Argyriadi et al., 2006). Wier- schem and Aksel (2004) found that material transport between eddies was induced by disturbances to the free-surface resulting in the formation of surface waves; they observed the motion of the separatrix created by the local changes in film thickness induced a lobe mechanism of material transfer between the re-circulating flow and the bulk flow above.
Mass transfer and mixing is an important aspect within fluid flows in general in an engineering context. The enhancement or suppression of transport rates from re- circulation zones is important in the cleansing of rough surfaces (Tighe and Mid- dleman, 1985), the mechanisms involved in pitting corrosion (Frankel, 1998) and transport to cells in perfusion bioreactors (Horner et al., 1998). Investigations into mixing and transport enhancement in open cavities are limited; Jana and Ottino (1992) briefly looked at how oscillating the motion of an impinging jet above a cavity can induce removal of material, Howes and Shardlow (1997) pulsed the inlet flow to clean out a number of cavities in a channel. By placing obstacles upstream of an open cavity, oscillations can be induced in the flow to enhance transport (Gar- rison and Rogers, 1994). Horner et al. (2002) considered oscillating the shape and speed of a moving wall, driving flow past an open square cavity to increase the rate of fluid transport between the bulk and the cavity flow; the authors noted that changing the frequency and amplitude of the forcing induced a turnstile lobe mech- anism of transport. The dependence of the increased flux of fluid transport reaches a maximum at a critical value of the frequency of the forcing, after which the the flux decreases. In the case where the amplitude of the wall forcing is increased there is no limiting criteria, the flux is always increased.
The turnstile lobe mechanism is the same phenomenon as observed in free-surface flow over an undulating substrate (Wierschem and Aksel, 2004). Transport in the
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bulk flow over a wavy substrate can be enhanced by optimising the film thickness for an increase in mean transport velocity, Scholle et al. (2006); for small to moder- ate waviness they found a reduction in the mean transport velocity, when the onset of eddies is observed the reduction in material transport is partially compensated for by these inner flow structures which act like a fluid roller bearing promoting mate- rial transport. Heining et al. (2012) investigated film flow over undulating surfaces that were both partially- and fully-submerged focusing on how laminar mixing is affected by the topography.
FIGURE1.7: Experimental images of mixing between two counter-rotating rollers via lobe
formation, Wilson et al. (2006); the speed of one of the rollers is varied inducing a turnstile lobe mechanism of fluid transport creating the mixing pattern that can be observed.
Mixing and transport enhancement is important in many different thin film based applications, for instance in roll coating. Stirring and transport enhancement of fluid entrapped in the nip between two counter-rotating rollers was shown to increase when increasing the speed ratio of the two rollers by Wilson et al. (2006), one rotating at fixed speed, the other varying. As above, they noted that transport and mixing were described by a turnstile lobe mechanism induced by the tangling of the invariant stable and unstable manifolds associated with the separation boundary, see Figure 1.7. A complication with this method of transport enhancement was the disturbance induced at the free-surface - the percentage change in film thickness at the free-surface was found to be approximately a quarter of the percentage change of roll speed although this could be counter-acted by modulating the speed of both rollers.