  We consider viscous steady streaming induced by oscillatory flow past a
cylinder between two plates, where the cylinder's axis is normal to the plates.
While this phenomenon was first studied in the 1930s, it has received renewed
interest recently for possible applications in particle manipulations and
non-Newtonian flows. The flow is driven at the ends of the channel by the
boundary condition which is a series solution of the oscillating flow problem
in a rectangular channel in the absence of a cylinder. We use a combination of
Fourier series and an asymptotic expansion to study the confinement effects for
steady-streaming. The Fourier series in time naturally simplifies to a finite
series. In contrast, it is necessary to truncate the Fourier series in z, which
is in the direction of the axis of the cylinder, to solve numerically. The
successive equations for the Fourier coefficients resulting from the asymptotic
expansion are then solved numerically using finite element methods. We use our
model to evaluate how steady streaming depends on the domain width and distance
from the cylinder to the outer walls, including the possible breaking of the
four-fold symmetry due to the domain shape. We utilize the tangential
steady-streaming velocity along the radial chord at an angle of pi/4 to analyze
our solutions over an extensive range of oscillating frequencies and multiple
levels in the z-direction. Finally, higher-order solutions are computed and an
asymptotic correction to steady streaming is included.
