  Viscous flows in a quasi-two-dimensional Hele-Shaw geometry can lead to an
interfacial instability when one fluid, of viscosity $\eta_{in}$ displaces
another of higher viscosity, $\eta_{out}$. Recent studies have shown that there
is a delay in the onset of fingering in miscible fluids as the viscosity ratio,
$\eta_{in}/\eta_{out}$, increases and approaches unity; the interface can
remain stable even though the displacing liquid is less viscous. This paper
shows that a delayed onset and stable pattern can occur in immiscible fluids as
well. However, there are two significant differences between the two cases.
First, in miscible fluids, stable patterns are obtained whenever
$\eta_{in}/\eta_{out} > 0.33$ while in immiscible fluids, the radius at which
the onset of fingering starts, $R_{onset}$, increases steadily until
$\eta_{in}/\eta_{out}=1$. A stable pattern is obtained only when the total size
of the plate used is smaller than $R_{onset}$. Second, once the delayed
fingering starts in immiscible fluids, the fingers grow faster than the central
circular region. In miscible fluids, there is a regime in which the fingers and
the central region grow in proportion to each other. These differences between
miscible and immiscible fingering are maintained even when we compare
immiscible fluids that have very low interfacial tension with miscible fluids
that have negligible diffusion.
