Slow viscous flow over rectangular cavities

Abstract

Analytic solutions of the equations of motion of slow viscous flow (R = 0) of an incompressible fluid over two-dimensional rectangular cavities are found for a variety of boundary conditions. The flows considered are generated (a) by the uniform translation of one wall of a cavity parallel to itself; (b) by an outer stream over a cavity; and (c) by an outer stream, when the cavity walls are moving. cavities. In each case, recirculating flows are generated in the deeper In (a) a sequence of eddies is found while in (b), for the deep cavities, a similar set of eddies results, separating from the main outer stream a small distance inside the cavity. In the shallow cavities of cases (b) or (c), the outer flow dips in and out, forming large corner eddies in the bottom of the cavity in the case of (b) . obtained. In the deep cavities of type (c), a single large eddy is Further, it is found that an infinite array of cavities whose boundaries are moving can generate a uniform velocity at a far distance. This last result has significance in determining the hydrodynamics of propulsion of miscroscopic organisms. The Reynolds number for such organisms are much less than one and the approximation that R = O, or, that the inertia forces are negligible, is legitimate. Taylor (1951), using the model of an infinite flexible sheet (inextensible), found that transverse sine waves of small amplitude moving through the sheet give rise to a propulsion of the sheet through the fluid in the direction opposite to the direction of propagation of the waves. There is some debate as to whether for large amplitudes the propulsion is in the direction of propagation. The results we obtain for large amplitude square waves provide some insight into this problem. Series solutions of the biharmonic equation were obtained by fitting some of the boundary conditions exactly. The coefficients of the series were then evaluated by fitting the remaining boundary conditions by the method of collocation. The results compare very well with those found by other authors using various numerical procedures of solution, for the cases (a) and (b). The cases (c) do not appear to have been attempted previously. Show more