SOME LIMITATIONS OF 2-DIMENSIONAL UNBOUNDED STOKES-FLOW

The basic approach toward a theoretical understanding of slow viscous flows in two dimensions is through solutions of the biharmonic equation; however, when the fluid is unbounded some of the solutions corresponding to locally generated flows lead to paradoxical behavior (e.g., Jeffery [Proc. R. Soc. London Ser. A 101, 169 (1922)]). Here one case in detail (the source-sink flow in front of a circular cylinder) is described, and then whether such a flow is possible as either (a) the limit of a bounded flow in two dimensions as the radius of the outer boundary grows without bound, or (b) the limit of an unbounded three-dimensional flow as lengths in the third dimension grow without bound, or (c) the limit of an impulsively started two-dimensional unbounded flow as time grows without bound is considered. In each case it is found that the formal limit does exist, but that the error is only as small as O [ (1n μ)-1] when the appropriate parameter μ→∞. This indicates that such locally generated unbounded flows are not realistically attainable. © 1990 American Institute of Physics.