On a viscous two-fluid channel flow including evaporation
The author considers the steady Navier-Stokes-Stephan problem \((v\cdot \nabla )v-\nu \nabla ^{2}v+\frac{1}{\eta }\nabla p=ge_{g}\), \(\nabla \cdot v=0 \), \((v\cdot \nabla )\theta -\lambda \nabla ^{2}\theta =0\) posed in the 2D semi-infinite domain \(\Omega =\Omega _{1}\cup \Omega _{2}\) partly separated by an interface \(\Sigma _{1}\). On the other boundary \(\Sigma _{0}\) of \( \Omega _{1}\), the boundary conditions \(v=R\) and \(\theta =\theta _{0}=0\) are imposed. On the boundaries \(\Sigma _{k}\), \(k=1,2\), of \(\Omega _{2}\), the boundary conditions \(v=0\) and \(\theta =\theta _{k}\) are imposed. On the free interface \(\Gamma \) between the two fluids flows (after \(\Sigma _{1}\)) the conditions \([\theta ]=0=[v]\), \(v\cdot n\mid _{\Gamma ^{-}}=[\lambda \frac{ \partial \theta }{\partial n}]\mid _{\Gamma }\), \([\tau \cdot S(v)n]\mid _{\Gamma }=0\), \(\frac{d}{dx_{1}}\frac{\varphi ^{\prime }(x_{1})}{\sqrt{ 1+\varphi ^{\prime }(x_{1})^{2}}}=\frac{1}{\sigma (\theta )}[-p+n\cdot S(v)n]\mid _{\Gamma }\) and \(\int_{\delta _{k}(t)}v\cdot ndx_{2}=F_{k}\) are imposed. Here \(F_{k}\) is the flux, \(\delta _{k}(t)\) is a cross-section of \( \Omega _{k}\) and \(S\)\ is the frictional tensor \(S_{jk}=v\eta (\partial _{k}v_{j}-\partial _{j}v_{k})\). The main purpose of the paper is to prove the existence of a unique solution \((v,p,\theta ,\varphi )\) in weighted H ölder spaces to this problem under smallness hypotheses on the data, among others. The author gives a representation of this solution and he proves regularity properties of this solution. For the proof, the author refers to his thesis and to the paper he previously wrote [Z. Angew. Math. Mech. 72, No. 7, 251--268 (1992; Zbl 0770.76067)].
scientific article; zbMATH DE number 6892342
scientific article