We examine the flow in a horizontal Hele-Shaw cell in which the
unidirectional flow at infinity is required to stream around a vertical
the gap between the two (horizontal) plates of the cell. A combination
asymptotic expansions and numerical methods is employed to elucidate the
of the boundary layer near the surface of the cylinder. The two length
scales of the
problem are the gap, h, and the length of the body, l;
it is assumed that h/l<<1. The
characteristic Reynolds number based on l is O(1).
The length scales associated with
the boundary layer and the classical Hele-Shaw flow pattern are
O(h) and O(l), respectively.
It is found that the boundary layer contains streamwise vorticity. This
generated at the three no-slip surfaces (the two plates and the cylinder
wall) as a result
of the cross-flow induced by the streamwise acceleration/deceleration
of the flow
around the curved cylinder. The strength of the secondary flow, hence the
streamwise vorticity, is proportional to changes in body curvature. The
validity of the
classical Hele-Shaw flow is examined systematically, and higher-order corrections
worked out. This results in a displacement thickness that is roughly 30%
of the gap.
In other words, the lowest-order correction to the classical Hele-Shaw
flow may be
obtained by requiring the outer flow (on the scale O(l))
to satisfy the no-penetration
boundary condition on a displaced cylinder surface. The boundary layer
‘corner’ vortices at the intersections of the horizontal
plates and the vertical cylinder surface.