Asymptotic expansion for convection-dominated transport in a thin graph-like junction

TL;DR

This paper develops a rigorous asymptotic expansion for convection-dominated transport in thin graph-like junctions, deriving limit hyperbolic problems with novel gluing conditions as the thin dimensions vanish.

Contribution

It introduces a comprehensive asymptotic analysis for a convection-diffusion problem in thin junctions, including new gluing conditions and limit equations on a graph structure.

Findings

Asymptotic expansion constructed with uniform estimates.

Limit problems are hyperbolic equations on graph branches.

New gluing conditions generalize Kirchhoff transmission conditions.

Abstract

We consider for a small parameter $\varepsilon >0$ a parabolic convection-diffusion problem with P\'eclet number of order $\mathcal{O}(\varepsilon^{-1})$ in a three-dimensional graph-like junction consisting of thin curvilinear cylinders with radii of order $\mathcal{O}(\varepsilon)$ connected through a domain (node) of diameter $\mathcal{O}(\varepsilon).$ Inhomogeneous Neumann type boundary conditions, that involve convective and diffusive fluxes, are prescribed both on the lateral surfaces of the thin cylinders and the boundary of the node. The asymptotic behaviour of the solution is studied as $\varepsilon \to 0,$ i.e., when the diffusion coefficients are eliminated and the thin junction is shrunk into a three-part graph connected in a single vertex. A rigorous procedure for the construction of the complete asymptotic expansion of the solution is developed and the corresponding energetic and uniform pointwise estimates are proven. Depending on the directions of the limit convective fluxes, the corresponding limit problems $(\varepsilon = 0)$ are derived in the form of first-order hyperbolic differential equations on the one-dimensional branches with novel gluing conditions at the vertex. These generalize the classical Kirchhoff transmission conditions and might require the solution of a three-dimensional cell-like problem associated with the vertex to account for the local geometric inhomogeneity of the node and the physical processes in the node. The asymptotic ansatz consists of three parts, namely, the regular part, node-layer part, and boundary-layer one. Their coefficients are classical solutions to mixed-dimensional limit problems. The existence and other properties of those solutions are analyzed.