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

TL;DR

This paper analytically predicts the Saffman-Taylor finger width in tapered Hele-Shaw cells, revealing how gap gradients influence finger stability and matching experimental data.

Contribution

Introduces an analytical expression for finger width in tapered Hele-Shaw cells considering small depth gradients, extending previous parallel cell results.

Findings

Gap gradient significantly affects finger width and stability.

Derived expression matches experimental and stability analysis data.

Control over fingering instabilities is possible via gap gradient sign.

Abstract

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 \[ \Lambda - \frac{1}{2} \sim f(\alpha) Ca_m^{2/3} \quad \mbox{as} \quad Ca_m \rightarrow 0, \;\;\; \mbox{and} \;\;\; \lvert \alpha \rvert \ll 1.\] Here, $\Lambda$ is the dimensionless finger width, $Ca_m$ 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 $\Lambda$, 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.