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Scaling effects and homogenization of reaction-diffusion problems with nonlinear drift
Karlstad University, Faculty of Health, Science and Technology (starting 2013), Department of Mathematics and Computer Science (from 2013).ORCID iD: 0000-0001-5168-0841
2024 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

We study the periodic homogenization of reaction-diffusion problems with nonlinear drift describing the transport of interacting particles in composite materials. The microscopic model is derived as the hydrodynamic limit of a totally asymmetric simple exclusion process for a population of interacting particles crossing a domain with obstacles. We are particularly interested in exploring how the scalings of the drift affect the structure of the upscaled model.

We first look into a situation when the interacting particles cross a thin layer that has a periodic microstructure. To understand the effective transmission condition, we perform homogenization together with the dimension reduction of the aforementioned reaction-diffusion-drift problem with variable scalings.

One particular physically interesting scaling that we look at separately is when the drift is very large compared to both the diffusion and reaction rate. In this case, we consider the overall process taking place in an unbounded porous media. Since we have the presence of a large nonlinear drift in the microscopic problem, we first upscale the model using the formal asymptotic expansions with drift. Then, with the help of two-scale convergence with drift, we rigorously derive the homogenization limit for a similar microscopic problem with a nonlinear Robin-type boundary condition. Additionally, we show the strong convergence of the corrector function. 

In the large drift case, the resulting upscaled equation is a nonlinear reaction-dispersion equation that is strongly coupled with a system of nonlinear elliptic cell problems. We study the solvability of a similar strongly coupled two-scale system with nonlinear dispersion by constructing an iterative scheme. Finally, we illustrate the behavior of the solution using the iterative scheme.

Abstract [en]

We study the homogenization of reaction-diffusion problems with nonlinear drift. The microscopic model is derived as the hydrodynamic limit of a totally asymmetric simple exclusion process of interacting particles. We first look into a situation when the interacting particles cross a thin composite layer. To understand the effective transmission condition, we perform the homogenization and dimension reduction of the model with variable scalings. One physically interesting scaling that we look at separately is when the drift is large. In this case, we consider the overall process taking place in an unbounded porous media. We first upscale the model using the asymptotic expansions with drift. Then, using two-scale convergence with drift, we rigorously derive the homogenization limit for a similar microscopic problem with a nonlinear boundary condition. Additionally, we show the strong convergence of the corrector function. In the large drift case, the resulting upscaled model is a nonlinear reaction-dispersion equation strongly coupled with a system of nonlinear elliptic cell problems. We study the solvability of a similar strongly coupled two-scale system with nonlinear dispersion by constructing an iterative scheme. Finally, we illustrate the behavior of the solution using the iterative scheme.

Place, publisher, year, edition, pages
Karlstads universitet, 2024. , p. 24
Series
Karlstad University Studies, ISSN 1403-8099 ; 2024:7
Keywords [en]
homogenization, asymptotic expansion with drift, two-scale convergence with drift, effective transmission condition, dimension reduction, two-scale system, nonlinear dispersion
National Category
Mathematics
Research subject
Mathematics
Identifiers
URN: urn:nbn:se:kau:diva-98720DOI: 10.59217/fjww2863ISBN: 978-91-7867-440-4 (print)ISBN: 978-91-7867-441-1 (electronic)OAI: oai:DiVA.org:kau-98720DiVA, id: diva2:1841801
Public defence
2024-04-18, Sjöström lecture hall, 1B309, Karlstads universitet, Karlstad, 13:15 (English)
Opponent
Supervisors
Available from: 2024-03-28 Created: 2024-02-29 Last updated: 2024-03-28Bibliographically approved
List of papers
1. Scaling effects on the periodic homogenization  of a reaction-diffusion-convection problem posed in homogeneous domains connected by a thin composite layer
Open this publication in new window or tab >>Scaling effects on the periodic homogenization  of a reaction-diffusion-convection problem posed in homogeneous domains connected by a thin composite layer
2022 (English)In: Quarterly of Applied Mathematics, ISSN 0033-569X, E-ISSN 1552-4485, Vol. 80, p. 157-200Article in journal (Refereed) Published
Abstract [en]

We study the question of periodic homogenization of a variably scaled reaction-diffusion problem with non-linear drift posed for a domain crossed by a flat composite thin layer. The structure of the non-linearity in the drift was obtained in earlier works as hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) for a population of interacting particles crossing a domain with obstacle.

Using energy-type estimates as well as concepts like thin-layer convergence and two-scale convergence, we derive the homogenized evolution equation and the corresponding effective model parameters for a regularized problem. Special attention is paid to the derivation of the effective transmission conditions across the separating limit interface in essentially two different situations: (i) finitely thin layer and (ii) infinitely thin layer.

This study should be seen as a preliminary step needed for the investigation of averaging fast non-linear drifts across material interfaces—a topic with direct applications in the design of thin composite materials meant to be impenetrable to high-velocity impacts.

Place, publisher, year, edition, pages
American Mathematical Society (AMS), 2022
Keywords
non-linear drift, thin-layer convergence, periodic homogenization, reaction-diffusion problem
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-86396 (URN)10.1090/qam/1607 (DOI)000752021500006 ()2-s2.0-85123893480 (Scopus ID)
Projects
Homogenization and dimension reduction of thin heterogeneous layers
Funder
Swedish Research Council, 2018-03648
Available from: 2021-10-30 Created: 2021-10-30 Last updated: 2024-02-29Bibliographically approved
2. Upscaling of a reaction-diffusion-convection problem with exploding non-linear drift
Open this publication in new window or tab >>Upscaling of a reaction-diffusion-convection problem with exploding non-linear drift
2022 (English)In: Quarterly of Applied Mathematics, ISSN 0033-569X, E-ISSN 1552-4485, Vol. 80, no 4, p. 641-667Article in journal (Refereed) Published
Abstract [en]

We study a reaction-diffusion-convection problem with non-linear drift posed in a domain with periodically arranged obstacles. The non-linearity in the drift is linked to the hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) governing a population of interacting particles crossing a domain with obstacle. Because of the imposed large drift scaling, this non-linearity is expected to explode in the limit of a vanishing scaling parameter. As main working techniques, we employ twoscale formal homogenization asymptotics with drift to derive the corresponding upscaled model equations as well as the structure of the effective transport tensors. Finally, we use Schauder???s fixed point theorem as well as monotonicity arguments to study the weak solvability of the upscaled model posed in an unbounded domain. This study wants to contribute with theoretical understanding needed when designing thin composite materials that are resistant to high velocity impacts.

Place, publisher, year, edition, pages
American Mathematical Society (AMS), 2022
National Category
Computational Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-89438 (URN)10.1090/qam/1622 (DOI)000804247500001 ()
Projects
Homogenization and dimension reduction of thin heterogeneous layers
Funder
Swedish Research Council, 2018-03648
Available from: 2022-04-09 Created: 2022-04-09 Last updated: 2024-02-29Bibliographically approved
3. Homogenization of a reaction-diffusion problem with large nonlinear drift and Robin boundary data
Open this publication in new window or tab >>Homogenization of a reaction-diffusion problem with large nonlinear drift and Robin boundary data
2024 (English)In: Quarterly of Applied Mathematics, ISSN 0033-569X, E-ISSN 1552-4485Article in journal (Refereed) Epub ahead of print
Abstract [en]

We study the periodic homogenization of a reaction-diffusion problem with large nonlinear drift and Robin boundary condition posed in an unbounded perforated domain. The nonlinear problem is associated with the hydrodynamic limit of a totally asymmetric simple exclusion process (TASEP) governing a population of interacting particles crossing a domain with obstacle. We are interested in deriving rigorously the upscaled model equations and the corresponding effective coefficients for the case when the microscopic dynamics are linked to a particular choice of characteristic length and time scales that lead to an exploding nonlinear drift. The main mathematical difficulty lies in proving the two-scale compactness and strong convergence results needed for the passage to the homogenization limit. To cope with the situation, we use the concept of two-scale compactness with drift, which is similar to the more classical two-scale compactness result but it is defined now in moving coordinates. We provide as well a strong convergence result for the corrector function, starting this way the search for the order of the convergence rate of the homogenization process for our target nonlinear drift problem.

Place, publisher, year, edition, pages
American Mathematical Society (AMS), 2024
Keywords
Homogenization, reaction-diffusion equations with large nonlinear drift, two-scale convergence with drift, strong convergence in moving coordinates, effective dispersion tensors for reactive flow in porous media
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-97991 (URN)10.1090/qam/1687 (DOI)001162782500001 ()
Funder
Swedish Research Council, 2018-03648
Available from: 2024-01-14 Created: 2024-01-14 Last updated: 2024-03-08Bibliographically approved
4. Strongly Coupled Two-scale System with Nonlinear Dispersion: Weak Solvability and Numerical Simulation
Open this publication in new window or tab >>Strongly Coupled Two-scale System with Nonlinear Dispersion: Weak Solvability and Numerical Simulation
Show others...
(English)Manuscript (preprint) (Other academic)
Abstract [en]

We investigate a two-scale system featuring an upscaled parabolic dispersion-reaction equation intimately linked to a family of elliptic cell problems.    The system is strongly coupled through a dispersion tensor, which depends on the solutions to the cell problems, and via the cell problems themselves, where the solution of the parabolic problem interacts nonlinearly with the drift term.        This particular mathematical structure is motivated by a rigorously derived upscaled reaction-diffusion-convection model that describes the evolution of a population of interacting particles pushed by a large drift through an array of periodically placed obstacles (i.e., through a regular porous medium). 

    We prove the existence of weak solutions to our system by means of an iterative scheme, where particular care is needed to ensure the uniform positivity of the dispersion tensor.      Additionally, we use finite element-based approximations for the same iteration scheme to perform multiple simulation studies. Finally, we highlight how the choice of micro-geometry (building the regular porous medium) and of the nonlinear drift coupling affects the macroscopic dispersion of particles.

Keywords
Two-scale system, Nonlinear dispersion, Weak solutions, Iterative scheme, Simulation.
National Category
Mathematics
Research subject
Mathematics
Identifiers
urn:nbn:se:kau:diva-98725 (URN)
Available from: 2024-02-29 Created: 2024-02-29 Last updated: 2024-02-29

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Raveendran, Vishnu

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