Schematic of free-surface film flow over a corrugated rigid substrate, together with defining geometric parameters.
Sketch of the four measurables used to quantify resulting flow behavior. The solid streamline and open circle mark the separatrix and eddy center positions under general flow conditions, while the dashed streamline and cross give their corresponding positions for Stokes flow. The quantities and therefore refer to the difference in the position of the eddy center and the eddy depth between the general and Stokes flow cases.
Flow structure under Stokes flow conditions: ; ; (a) , (b) , and (c) . In the latter the free-surface contour is out of shot. Comparison with the experiments of Wierschem et al. (Ref. 24) and overlaid CV (left) and FE streamline predictions (right).
Onset and development of kinematic eddies for three different values of for a substrate inclined at . For both FE and CV solutions are compared with experimental measurements taken from Ref. 24 where two different oils (BC5000cs and B1000) were used. Note that, for the case the eddy is persistent.
Effect of increasing inertia on local flow structure; the associated free-surface location is out of shot: FE solutions, compared in part to existing experimental results (Ref. 24) for BC5000cs and B200 oils with , , and ; (a) (corresponds to ); (b) (corresponds to ); (c) (corresponds to ); (d) (corresponds to ).
Effect of increasing inertia on the horizontal shift of the eddy center, (lower graph), and corresponding change in eddy depth (upper graph). Other flow parameters are , , and . The inlaid plots (A)–(E) show the separatrix and eddy center position at the values of indicated by the corresponding black circles on the graphs. In plots (B)–(E), the separatrix is shown as a solid line, and the eddy center as an open circle. The Stokes flow separatrix and eddy center position from A are included in each case as a dashed line and “” symbol, respectively.
Closeup streamline plots of the effect of on the local flow structure for fixed : ; [(a)–(c)]; [(d)–(f)]. Associated flow structure for (a) , (b) , (c) , (d) , (e) , and (f) . The free-surface location is out of shot.
Effect of on the horizontal shift of the eddy center, (lower graph), and on eddy depth, (upper graph): ; . The three curves in the bottom left corner of the upper plot correspond to changes in depth of the second eddy, which appears at sufficiently small wavelengths. The closeup streamline plots inlaid illustrate the local flow structure for parameter values corresponding to the points shown as the black circles. The inlaid experimentally observed flow is that of Fig. 5(d). The open circle marks the transition from a kinematically to an inertially induced eddy.
Predicted streamline closeups of the disappearance of a kinematically induced eddy by decreasing the corrugation steepness and the creation of an inertially induced eddy at fixed geometry. In (a) , , and (corresponding to ). In (b) , , and (corresponding to ). In (c) and (d), and are as in (b), but is increased: (c) and (d) . In all cases and the free-surface contour is out of shot.
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