On the selection of Saffman-Taylor fingers in a tapered Hele-Shaw cell

We present an analytical study for predicting the finger width of the Saffman-Taylor finger in a tapered Hele-Shaw cell. We consider a rectilinear geometry with a constant depth gradient and apply analytical techniques of singular perturbation analysis and WKB approximation to derive an expression for the finger selection mechanism for such tapered Hele-Shaw cells with small depth gradients. We establish $$\mathrm{\Lambda }-\frac{1}{2}\sim f(\alpha )C{a}_m^{2\mathrm{/}3}asC{a}_m\to 0,and\mid \alpha \mid \ll 1.$$ Here, $\mathrm{\Lambda }$ is the dimensionless finger width, $C{a}_m$ denotes the modified Capillary parameter, and $f(\alpha )$ is a linear function of the gap gradient $\alpha$, such that $f(\alpha =0)=1$ recovering the results of parallel Hele-Shaw cell (Hong and Langer \cite{hong1986analytic}, Combescot \emph{et al.} \cite{Combescot1986}, Shraiman \cite{shraiman1986velocity}). Our findings indicate that the Hele-Shaw cell gap gradient plays a crucial role in determining $\mathrm{\Lambda }$, allowing for control over fingering instabilities such that the single-finger steady state can be stabilised or destabilised depending on the sign of the gradient, compared to the standard Hele-Shaw cell. The theoretical estimates reveal excellent agreement with experimental finger-width data and predictions from linear stability analyses.