86 research outputs found
Homogenization of some degenerate pseudoparabolic variational inequalities
Multiscale analysis of a degenerate pseudoparabolic variational inequality,
modelling the two-phase flow with dynamical capillary pressure in a perforated
domain, is the main topic of this work. Regularisation and penalty operator
methods are applied to show the existence of a solution of the nonlinear
degenerate pseudoparabolic variational inequality defined in a domain with
microscopic perforations, as well as to derive a priori estimates for solutions
of the microscopic problem. The main challenge is the derivation of a priori
estimates for solutions of the variational inequality, uniformly with respect
to the regularisation parameter and to the small parameter defining the scale
of the microstructure. The method of two-scale convergence is used to derive
the corresponding macroscopic obstacle problem
Homogenization of degenerate cross-diffusion systems
Two-scale homogenization limits of parabolic cross-diffusion systems in a
heterogeneous medium with no-flux boundary conditions are proved. The
heterogeneity of the medium is reflected in the diffusion coefficients or by
the perforated domain. The diffusion matrix is of degenerate type and may be
neither symmetric nor positive semi-definite, but the diffusion system is
assumed to satisfy an entropy structure. Uniform estimates are derived from the
entropy production inequality. New estimates on the equicontinuity with respect
to the time variable ensure the strong convergence of a sequence of solutions
to the microscopic problems defined in perforated domains
Derivation of a macroscopic model for transport of strongly sorbed solutes in the soil using homogenization theory
In this paper we derive a model for the diffusion of strongly sorbed solutes in soil taking into account diffusion within both the soil fluid phase and the soil particles. The model takes into account the effect of solutes being bound to soil particle surfaces by a reversible nonlinear reaction. Effective macroscale equations for the solute movement in the soil are derived using homogenization theory. In particular, we use the unfolding method to prove the convergence of nonlinear reaction terms in our system. We use the final, homogenized model to estimate the effect of solute dynamics within soil particles on plant phosphate uptake by comparing our double-porosity model to the more commonly used single-porosity model. We find that there are significant qualitative and quantitative differences in the predictions of the models. This highlights the need for careful experimental and theoretical treatment of plant-soil interaction when trying to understand solute losses from the soil
Homogenization of a system of elastic and reaction-diffusion equations modelling plant cell wall biomechanics
In this paper we present a derivation and multiscale analysis of a
mathematical model for plant cell wall biomechanics that takes into account
both the microscopic structure of a cell wall coming from the cellulose
microfibrils and the chemical reactions between the cell wall's constituents.
Particular attention is paid to the role of pectin and the impact of
calcium-pectin cross-linking chemistry on the mechanical properties of the cell
wall. We prove the existence and uniqueness of the strongly coupled microscopic
problem consisting of the equations of linear elasticity and a system of
reaction-diffusion and ordinary differential equations. Using homogenization
techniques (two-scale convergence and periodic unfolding methods) we derive a
macroscopic model for plant cell wall biomechanics
Homogenization approach to water transport in plant tissues with periodic microstructures
Water flow in plant tissues takes place in two different physical domains
separated by semipermeable membranes: cell insides and cell walls. The assembly
of all cell insides and cell walls are termed symplast and apoplast,
respectively. Water transport is pressure driven in both, where osmosis plays
an essential role in membrane crossing. In this paper, a microscopic model of
water flow and transport of an osmotically active solute in a plant tissue is
considered. The model is posed on the scale of a single cell and the tissue is
assumed to be composed of periodically distributed cells. The flow in the
symplast can be regarded as a viscous Stokes flow, while Darcy's law applies in
the porous apoplast. Transmission conditions at the interface (semipermeable
membrane) are obtained by balancing the mass fluxes through the interface and
by describing the protein mediated transport as a surface reaction. Applying
homogenization techniques, macroscopic equations for water and solute transport
in a plant tissue are derived. The macroscopic problem is given by a Darcy law
with a force term proportional to the difference in concentrations of the
osmotically active solute in the symplast and apoplast; i.e. the flow is also
driven by the local concentration difference and its direction can be different
than the one prescribed by the pressure gradient.Comment: 31 page
Multiscale analysis and simulation of a signalling process with surface diffusion
We present and analyze a model for cell signaling processes in biological tissues. The model includes diffusion and nonlinear reactions on the cell surfaces and both inter- and intracellular signaling. Using techniques from the theory of two-scale convergence as well the unfolding method, we show convergence of the solutions to the model to solutions of a two-scale macroscopic problem. We also present a two-scale bulk-surface finite element method for the approximation of the macroscopic model. We report on some benchmarking results as well as numerical simulations in a biologically relevant regime that illustrate the influence of cell-scale heterogeneities on macroscopic concentrations
Mathematical Modelling of Auxin Transport in Plant Tissues:Flux Meets Signalling and Growth
Plant hormone auxin has critical roles in plant growth, dependent on its
heterogeneous distribution in plant tissues. Exactly how auxin transport and
developmental processes such as growth coordinate to achieve the precise
patterns of auxin observed experimentally is not well understood. Here we use
mathematical modelling to examine the interplay between auxin dynamics and
growth and their contribution to formation of patterns in auxin distribution in
plant tissues. Mathematical models describing the auxin-related signalling
pathway, PIN and AUX1 dynamics, auxin transport, and cell growth in plant
tissues are derived. A key assumption of our models is the regulation of PIN
proteins by the auxin-responsive ARF-Aux/IAA signalling pathway, with
upregulation of PIN biosynthesis by ARFs. Models are analysed and solved
numerically to examine the long-time behaviour and auxin distribution. Changes
in auxin-related signalling processes are shown to be able to trigger
transition between passage and spot type patterns in auxin distribution. The
model was also shown to be able to generate isolated cells with oscillatory
dynamics in levels of components of the auxin signalling pathway which could
explain oscillations in levels of ARF targets that have been observed
experimentally. Cell growth was shown to have influence on PIN polarisation and
determination of auxin distribution patterns. Numerical simulation results
indicate that auxin-related signalling processes can explain the different
patterns in auxin distributions observed in plant tissues, whereas the
interplay between auxin transport and growth can explain the `reverse-fountain'
pattern in auxin distribution observed at plant root tips
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