Slow viscous flow over rectangular cavities
Date
1971
Authors
Downes, Gaye Lenore
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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.
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