The Poiseuille flow (centreline velocity
$U_c$
) of a fluid (kinematic viscosity
$\nu$
) past a circular cylinder (radius
$R$
) in a Hele-Shaw cell (height
$2h$
) is traditionally characterised by a Stokes flow (
$\varLambda =(U_cR/\nu )(h/R)^2 \ll 1$
) through a thin gap (
$\epsilon =h/R \ll 1$
). In this work we use asymptotic methods and direct numerical simulations to explore the parameter space
$\varLambda$
–
$\epsilon$
when these conditions are not met. Starting with the Navier–Stokes equations and increasing
$\varLambda$
(which corresponds to increasing inertial effects), four successive regimes are identified, namely the linear regime, nonlinear regimes I and II in the boundary layer (the ‘ inner’ region) and a nonlinear regime III in both the inner and outer region. Flow phenomena are studied with extensive comparisons made between reduced calculations, direct numerical simulations and previous analytical work. For
$\epsilon =0.01$
, the limiting condition for a steady flow as
$\varLambda$
is increased is the instability of the Poiseuille flow. However, for larger
$\epsilon$
, this limit is at a much higher
$\varLambda$
, resulting in a laminar separation bubble, of size
${O}(h)$
, forming for a certain range of
$\epsilon$
at the back of the cylinder, where the azimuthal location was dependent on
$\epsilon$
. As
$\epsilon$
is increased to approximately 0.5, the secondary flow becomes increasingly confined adjacent to the sidewalls. The results of the analysis and numerical simulations are summarised in a plot of the parameter space
$\varLambda$
–
$\epsilon$
.
