Rigorous results in steady finger selection in viscous fingering (Q1566401)

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 \(\varepsilon\).