Starting and steady quadrupolar flow

Planar flow induced in a viscous fluid by a small cylinder oscillating in the direction normal to its axis is modeled theoretically and reproduced experimentally. In the model, a line force dipole (force doublet) was used as the source of motion. In an initially quiescent unbounded fluid this source produces zero net momentum and generates symmetrical quadrupolar flow consisting of two dipolar vorticity fronts propagating in opposite directions from the source. For starting flow at low Reynolds numbers, a second-order unsteady solution is obtained in terms of a power series of the Reynolds number, Re=Q/4 pi nu(2), where Q is the forcing amplitude and nu is the kinematic viscosity. This solution demonstrates that, as time t-->infinity, the flow in the vicinity of the source becomes steady and radial. To describe this steady asymptote, the Jeffery-Hamel nonlinear solution for radial flow is used. A particular solution is derived using the nondimensional intensity Re of the force dipole as a governing parameter. It is shown that the problem permits a similarity solution for all values of Re when a mass sink of prescribed intensity q=q(Re) is added to the flow. This steady asymptote is reproduced experimentally, using a vertical porous cylinder that oscillates horizontally in the shallow upper layer of a two-layer fluid and sucks fluid through its porous walls. (C) 1996 American Institute of Physics.