STOKES-FLOW IN CLOSED, RECTANGULAR DOMAINS

Three practically relevant, Stokes flows in closed, rectangular caviti es are considered. The first involves a solid-walled cavity where flow is driven by the motion of either one or both of its horizontal bound ing walls; the other two have an upper free surface and are driven eit her by the motion of vertical side walls or by a horizontally-moving l ower wall. Each problem is formulated as a biharmonic boundary value p roblem (bvp) for the streamfunction. The relative merits of two differ ent coefficient determination methods for the corresponding analytical solutions are assessed and, in addition, each solution is compared wi th its numerical counterpart obtained using a finite element formulati on of the governing equations. It is shown that, provided the number N of terms in each solution is sufficiently large, they are in extremel y good agreement and, similarly, they compare well with work from othe r published theoretical and experimental studies. Streamlines are pres ented, over a wide range of operating parameters, for the geometries c ontaining an upper free surface. For the how generated by two moving v ertical side walls two how transformation mechanisms are identified. F or cavities with small and decreasing width to depth (aspect) ratios, there is a sequence of critical aspect ratios at which flow bifurcatio ns arise with a centre becoming a saddle point and vice versa, whereas for large aspect ratios increasing the ratio further leads to eddy gr owth from the lower wall, resulting in a regular sequence of separatri ces along the cavity width. In the case of flow generated by a horizon tally-moving lower wall the streamlines are simpler and exhibit the re gular array of separatrices reported previously for flow in a solid-wa lled cavity with a single moving wall. (C) 1998 Elsevier Science Inc. All rights reserved.