An Effective Model for Biofilm Growth Made by Chemotactical Bacteria in Evolving Porous Media

Schulz, Raphael; Knabner, Peter

doi:10.1137/16m108817x

Cited by 18 publications

(37 citation statements)

References 33 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Similar to e.g. [21,25,26,30,32] the upscaled model was derived formally in the current paper. For a rigorous derivation the well-posedness of the pore-scale model and corrector estimates (e.g.…”

Section: Discussionmentioning

confidence: 99%

“…Due to the fact that the coupled system at the pore-scale is a rather complex free boundary problem (in particular it contains elliptic, parabolic and hyperbolic equations), the question about well-posedness for this and related models, e.g. [21,25,26,30,32], is still open. It is an important challenge to close this gap in the future.…”

Section: Discussionmentioning

confidence: 99%

“…An extension of formal two‐scale asymptotic expansion to a level‐set framework to capture the evolving solid–liquid interface was introduced in []. Beside others, this method was recently applied to locally‐periodic porous media, drug delivery systems and biofilm growth . In [] the main ideas for modeling the pore‐scale process are basically adopted from [], such that we strongly refer to this work for more details.…”

Section: Introductionmentioning

confidence: 99%

“…Upscaled diffusion–precipitation equations with effective coefficients coupled with a level‐set equation are considered in []. An effective model describing biofilm growth in porous media is already derived and analyzed in [], but neglects chemotactical effects completely, which were involved in [].…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

Biofilm modeling in evolving porous media with Beavers‐Joseph condition

Schulz

2019

Z Angew Math Mech

Self Cite

View full text Add to dashboard Cite

In this article, we derive an effective model for biofilm growth in a saturated porous medium. Several experimental investigations found out that biomass is in general very heterogeneous and contains in particular channels filled with fluid. These channels facilitate the transport of nutritive substances within the biofilm by advection. Referring to this, we model the biofilm itself at the pore‐scale as a porous medium. Hence the starting point at the pore‐scale is a Darcy‐Stokes system, where the Beavers‐Joseph boundary condition at the corresponding interface is proposed. By formal periodic homogenization we derive an averaged model describing the process via Darcy's law and upscaled transport equations with effective coefficients given by the evolving microstructure. Based on the assumption of uniform evolve of the underlying pore geometry, solvability in a weak sense global in time or at least up to a possible clogging phenomenon is stated.

show abstract

Section: Discussionmentioning

confidence: 99%

Section: Discussionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Biofilm modeling in evolving porous media with Beavers‐Joseph condition

Schulz

2019

Z Angew Math Mech

Self Cite

View full text Add to dashboard Cite

show abstract

“…Firstly, we treat the second integral in (114). We want to prove that By taking into account the assumptions on the reaction terms given by (21)- (24) and the fact that M 1 = M 2 , the left-hand side of (115) can be rewritten as…”

Section: Proof Of Theoremmentioning

confidence: 99%

Analysis of a Degenerate and Singular Volume-Filling Cross-Diffusion System Modeling Biofilm Growth

Daus¹,

Milišić²,

Zamponi³

2019

SIAM J. Math. Anal.

View full text Add to dashboard Cite

We analyze the mathematical properties of a multi-species biofilm crossdiffusion model together with very general reaction terms and mixed Dirichlet-Neumann boundary conditions on a bounded domain. This model belongs to the class of volumefilling type cross-diffusion systems which exhibit a porous medium-type degeneracy when the total biomass vanishes as well as a superdiffusion-type singularity when the biomass reaches its maximum cell capacity, which make the analysis extremely challenging. The equations also admit a very interesting non-standard entropy structure. We prove the existence of global-in-time weak solutions, study the asymptotic behavior and the uniqueness of the solutions, and complement the analysis by numerical simulations that illustrate the theoretically obtained results.

show abstract