On the selection of Saffman-Taylor fingers in a tapered Hele-Shaw cell
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arXiv
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| Natura: | Preprint |
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2026
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| _version_ | 1866914466316156928 |
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| author | Ghosh, Dipa Pramanik, Satyajit |
| author_facet | Ghosh, Dipa Pramanik, Satyajit |
| contents | 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 \[ Λ- \frac{1}{2} \sim f(α) Ca_m^{2/3} \quad \mbox{as} \quad Ca_m \rightarrow 0, \;\;\; \mbox{and} \;\;\; \lvert α\rvert \ll 1.\] Here, $Λ$ is the dimensionless finger width, $Ca_m$ denotes the modified Capillary parameter, and $f(α)$ is a linear function of the gap gradient $α$, such that $f(α= 0) = 1$ 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 $Λ$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_10407 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the selection of Saffman-Taylor fingers in a tapered Hele-Shaw cell Ghosh, Dipa Pramanik, Satyajit Soft Condensed Matter Fluid Dynamics 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 \[ Λ- \frac{1}{2} \sim f(α) Ca_m^{2/3} \quad \mbox{as} \quad Ca_m \rightarrow 0, \;\;\; \mbox{and} \;\;\; \lvert α\rvert \ll 1.\] Here, $Λ$ is the dimensionless finger width, $Ca_m$ denotes the modified Capillary parameter, and $f(α)$ is a linear function of the gap gradient $α$, such that $f(α= 0) = 1$ 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 $Λ$, 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. |
| title | On the selection of Saffman-Taylor fingers in a tapered Hele-Shaw cell |
| topic | Soft Condensed Matter Fluid Dynamics |
| url | https://arxiv.org/abs/2604.10407 |