Puiseux asymptotic expansions for convection-dominated transport problems in thin graph-like networks: strong boundary interactions

Abstract

This article completes the study of the influence of the intensity parameter $\alpha$ in the boundary condition $\epsilon {\partial }_{{\mathbf{\nu }}_{\epsilon }}{u}_{\epsilon }-u_{\epsilon }\overrightarrow{V_{\epsilon }}\cdot {\mathbf{\nu }}_{\epsilon }={\epsilon }^{\alpha }{\phi }_{\epsilon }$ given on the boundary of a thin three-dimensional graph-like network consisting of thin cylinders that are interconnected by small domains (nodes) with diameters of order $\mathcal{O}(\epsilon ).$ Inside of the thin network a time-dependent convection-diffusion equation with high Péclet number of order $\mathcal{O}({\epsilon }^{-1})$ is considered. The novelty of this article is the case of $\alpha <1,$ which indicates a strong intensity of physical processes on the boundary, described by the inhomogeneity ${\phi }_{\epsilon }$ (the cases $\alpha =1$ and $\alpha >1$ were previously studied by the same authors). A complete Puiseux asymptotic expansion is constructed for the solution $u_{\epsilon }$ as $\epsilon \to 0,$ i.e., when the diffusion coefficients are eliminated and the thin network shrinks into a graph. Furthermore, the corresponding uniform pointwise and energy estimates are proved, which provide an approximation of the solution with a given accuracy in terms of the parameter $\epsilon .$

22 pages, 3 figures

References in corpus (1)

• Asymptotic approximations for semilinear parabolic convection-dominated transport problems in thin graph-like networks

Cited by in corpus (1)

• Muskat-Leverett two-phase flow in thin cylindric porous media: Asymptotic approach