  We present a theoretical and numerical study on the (in)stability of the
interface between two immiscible liquids, i.e., viscous fingering, in angled
Hele-Shaw cells across a range of capillary numbers ($Ca$). We consider two
types of angled Hele-Shaw cells: diverging cells with a positive depth gradient
and converging cells with a negative depth gradient, and compare those against
parallel cells without a depth gradient. A modified linear stability analysis
is employed to derive an expression for the growth rate of perturbations on the
interface and for the critical capillary number ($Ca_c$) for such tapered
Hele-Shaw cells with small gap gradients. Based on this new expression for
$Ca_c$, a three-regime theory is formulated to describe the interface
(in)stability: (i) in Regime I, the growth rate is always negative, thus the
interface is stable; (ii) in Regime II, the growth rate remains zero (parallel
cells), changes from negative to positive (converging cells), or from positive
to negative (diverging cells), thus the interface (in)stability possibly
changes type at some location in the cell; (iii) in Regime III, the growth rate
is always positive, thus the interface is unstable. We conduct
three-dimensional direct numerical simulations of the full Navier--Stokes
equations, using a phase field method to enforce surface tension at the
interface, to verify the theory and explore the effect of depth gradient on the
interface (in)stability. We demonstrate that the depth gradient has only a
slight influence in Regime I, and its effect is most pronounced in Regime III.
Finally, we provide a critical discussion of the stability diagram derived from
theoretical considerations versus the one obtained from direct numerical
simulations.
