Singular limit for reactive diffusive transport through an array of thin\n channels in case of critical diffusivity

2020/03/30 by Markus Gahn, Gahn, Markus, Maria Neuss‐Radu +1 · 2 citations
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Nonlinear Partial Differential Equations #Numerical methods in inverse problems

paper · pdf · doi:10.48550/arxiv.2003.13310

openalex publication_date 2020/03/30 · openalex created_date 2022/07/26 · openalex updated_date 2026/07/28

Abstract

We consider a nonlinear reaction--diffusion equation in a domain consisting\nof two bulk regions connected via small channels periodically distributed\nwithin a thin layer. The height and the thickness of the channels are of order\n\ε, and the equation inside the layer depends on the parameter\n\ε. We consider the critical scaling of the diffusion coefficients in\nthe channels and nonlinear Neumann-boundary condition on the channels' lateral\nboundaries. We derive effective models in the limit \ε \→ 0 , when the\nchannel-domain is replaced by an interface \Σ between the two\nbulk-domains. Due to the critical size of the diffusion coefficients, we obtain\njumps for the solution and its normal fluxes across \Σ, involving the\nsolutions of local cell problems on the reference channel in every point of the\ninterface \Σ.\n

Cited by

• Effective transmission through an interface with evolving microstructure
• Homogenized limits of Stokes flow and advective transport in thin perforated domains

Related

• Effective transmission through an interface with evolving microstructure
• Effective interface laws for fluid flow and solute transport through thin reactive porous layers
• Fick's las selects the Neumann boundary condition
• Correctors and error estimates for reaction-diffusion processes through thin heterogeneous layers in case of homogenized equations with interface diffusion
• Reaction-diffusion equation on thin porous media