King et al. (2019) introduced a model for Fickian diffusion into core-shell geometry. The purpose of this model is to study diffusion of oxygen through protective shells encapsulating pancreatic Langerhan islets. These core-shells are of interest for the preparation of artificial pancreas to treat diabetes. In this paper we prove the existence of viable core-shell solutions for King’s model using a topological shooting method. The governing equations of the diffusion model can be reduced to a 2-dimensional non-autonomous first order ordinary differential equation. Solutions which correspond to viable core-shell diffusion are required to satisfy global constraints and boundary conditions in both the core and the encapsulating shell. These boundary conditions each give rise to one free parameter. We call solutions satisfying the core boundary condition core solutions. We identify two parameter spaces corresponding to core solution families. The viable core-shell solutions are on the boundary of these two core solution families. Using analytically obtained bounds we apply the intermediate value theorem to prove the existence of core-shell solutions. In addition, we obtain rigorous approximations for the boundary conditions of the viable diffusion core-shell solution.
|Title of host publication
|Nonlinear Dynamics of Discrete and Continuous Systems
|Andrei K. Abramian, Igor V. Andrianov, Valery A. Gaiko
|Springer International Publishing
|Number of pages
|Published - 2020
| Advanced Structured Materials