Rigorous results in steady finger selection in viscous fingering
type: Item
instance of: scholarly article
title: Rigorous results in steady finger selection in viscous fingering
zbMATH Open document ID: 1042.76020
author: Xuming Xie
author: Saleh Tanveer
published in: Archive for Rational Mechanics and Analysis
publication date: 2003-06-02T00:00:00Z
full work available at URL: http://rave.ohiolink.edu/etdc/view
review text: The authors prove the existence of steadily translating finger solutions in a Hele-Shaw cell for small but non-zero surface tension \(\varepsilon^2\). Assuming that the relative finger width \(\lambda\) lies in the interval \([{1\over 2}, \lambda_m]\) (with \(\lambda-{1\over 2}\) small but independent of \(\varepsilon\)), the existence of an analytic, symmetric finger solution is characterized by the fact that the Stokes constant \(\widetilde S(\beta_0)\) for a certain nonlinear ODE vanishes. In this case, the authors' main result states that if \(\widetilde S'(\beta_0)\neq 0\), then there exists an analytic function \(\beta(\varepsilon^{{2\over 3}})\) with \(\lim_{\varepsilon\to 0} \beta(\varepsilon^{{2\over 3}})= \beta_0\), such that a solution with the above-mentioned properties exists when the parameters \(\varepsilon\) and \(\lambda\) are related by \({2\lambda- 1\over 1-\lambda}= \varepsilon^{{4\over 3}}\beta(\varepsilon^{{2\over 3}})\). Furthermore, if \(a= {2\lambda- 1\over \varepsilon^{{4\over 3}}(1-\lambda)}\) and \(\widetilde S(a)\neq 0\), then no steady finger solution can exist for small \(\varepsilon\).
reviewed by: Klaus Deckelnick
Mathematics Subject Classification ID: 76D27
Mathematics Subject Classification ID: 35Q35
Mathematics Subject Classification ID: 76D45
zbMATH DE Number: 1922448
zbMATH Keywords: Hele-Shaw flow
zbMATH Keywords: Stokes constant
MaRDI profile type: Publication
OpenAlex ID: W2029086555
DOI: 10.1007/S00205-002-0235-4
description: scientific article; zbMATH DE number 1922448