arXiv subjects

Alexandra Tzella

Publications and source records attributed to Alexandra Tzella.

8 recordsLinked to original sources

arXiv2025

The role of mutants in the spatio-temporal progression of inflammatory bowel disease: three classes of permanent form travelling waves

Despite its high prevalence and impact on the lives of those affected, a complete understanding of the cause of inflammatory bowel disease (IBD) is lacking. In this paper, we investigate a novel mechanism which proposes that mutant epithelial cells are significant to the progression of IBD since they promote inflammation and are resistant to death. We develop a simple model encapsulating the propagation of mutant epithelial cells and immune cells which results from interactions with the intestinal barrier and bacteria. Motivated by the slow growth of mutant epithelial cells, and relatively slow response rate of the adaptive immune system, we combine geometric singular perturbation theory with matched asymptotic expansions to determine the one-dimensional slow invariant manifold that characterises the leading order dynamics at all times beyond a passive initial adjustment phase. The dynamics on this manifold are controlled by a bifurcation parameter, $\sigma$, which depends on the ratio of growth to decay rates of all components except mutants and determines three distinct classes of permanent-form travelling waves that describe the propagation of mutant epithelial and immune cells. These are obtained from scalar reaction-diffusion equations with the reaction being (i) a bistable nonlinearity with a cut-off, (ii) a cubic Fisher nonlinearity and (iii) a KPP or Fisher nonlinearity. Our results suggest that mutant epithelial cells are critical to the progression of IBD. However, their effect on the speed of progression is subdominant.

math.AP↗

arXiv2020

Diffusion in arrays of obstacles: beyond homogenisation

We revisit the classical problem of diffusion of a scalar (or heat) released in a two-dimensional medium with an embedded periodic array of impermeable obstacles such as perforations. Homogenisation theory provides a coarse-grained description of the scalar at large times and predicts that it diffuses with a certain effective diffusivity, so the concentration is approximately Gaussian. We improve on this by developing a large-deviation approximation which also captures the non-Gaussian tails of the concentration through a rate function obtained by solving a family of eigenvalue problems. We focus on cylindrical obstacles and on the dense limit, when the obstacles occupy a large area fraction and non-Gaussianity is most marked. We derive an asymptotic approximation for the rate function in this limit, valid uniformly over a wide range of distances. We use finite-element implementations to solve the eigenvalue problems yielding the rate function for arbitrary obstacle area fractions and an elliptic boundary-value problem arising in the asymptotics calculation. Comparison between numerical results and asymptotic predictions confirm the validity of the latter.

cs.CE↗

arXiv2018

Chemical front propagation in periodic flows: FKPP vs G

We investigate the influence of steady periodic flows on the propagation of chemical fronts in an infinite channel domain. We focus on the sharp front arising in Fisher--Kolmogorov--Petrovskii--Piskunov (FKPP) type models in the limit of small molecular diffusivity and fast reaction (large Péclet and Damköhler numbers, $\mathrm{Pe}$ and $\mathrm{Da}$) and on its heuristic approximation by the G equation. We introduce a variational formulation that expresses the two front speeds in terms of periodic trajectories minimizing the time of travel across the period of the flow, under a constraint that differs between the FKPP and G equations. This formulation shows that the FKPP front speed is greater than or equal to the G equation front speed. We study the two front speeds for a class of cellular vortex flows used in experiments. Using a numerical implementation of the variational formulation, we show that the differences between the two front speeds are modest for a broad range of parameters. However, large differences appear when a strong mean flow opposes front propagation; in particular, we identify a range of parameters for which FKPP fronts can propagate against the flow while G fronts cannot. We verify our computations against closed-form expressions derived for $\mathrm{Da}\ll \mathrm{Pe}$ and for $\mathrm{Da}\gg \mathrm{Pe}$.

physics.flu-dyn↗

arXiv2016

Dispersion in rectangular networks: effective diffusivity and large-deviation rate function

The dispersion of a diffusive scalar in a fluid flowing through a network has many applications including to biological flows, porous media, water supply and urban pollution. Motivated by this, we develop a large-deviation theory that predicts the evolution of the concentration of a scalar released in a rectangular network in the limit of large time $t \gg 1$. This theory provides an approximation for the concentration that remains valid for large distances from the centre of mass, specifically for distances up to $O(t)$ and thus much beyond the $O(t^{1/2})$ range where a standard Gaussian approximation holds. A byproduct of the approach is a closed-form expression for the effective diffusivity tensor that governs this Gaussian approximation. Monte Carlo simulations of Brownian particles confirm the large-deviation results and demonstrate their effectiveness in describing the scalar distribution when $t$ is only moderately large.

physics.flu-dyn↗

arXiv2015

FKPP fronts in cellular flows: the large-Péclet regime

We investigate the propagation of chemical fronts arising in Fisher--Kolmogorov--Petrovskii--Piskunov (FKPP) type models in the presence of a steady cellular flow. In the long-time limit, a steadily propagating pulsating front is established. Its speed, on which we focus, can be obtained by solving an eigenvalue problem closely related to large-deviation theory. We employ asymptotic methods to solve this eigenvalue problem in the limit of small molecular diffusivity (large Péclet number, $\text{Pe} \gg 1$) and arbitrary reaction rate (arbitrary Damköhler number $\text{Da}$). We identify three regimes corresponding to the distinguished limits $\text{Da} = O(\text{Pe}^{-1})$, $\text{Da}=O\left((\log \text{Pe})^{-1}\right)$ and $\text{Da} = O(\text{Pe})$ and, in each regime, obtain the front speed in terms of a different non-trivial function of the relevant combination of $\text{Pe}$ and $\text{Da}$. Closed-form expressions for the speed, characterised by power-law and logarithmic dependences on $\text{Da}$ and $\text{Pe}$ and valid in intermediate regimes, are deduced as limiting cases. Taken together, our asymptotic results provide a complete description of the complex dependence of the front speed on $\text{Da}$ for $\text{Pe} \gg 1$. They are confirmed by numerical solutions of the eigenvalue problem determining the front speed, and illustrated by a number of numerical simulations of the advection--diffusion--reaction equation.

physics.flu-dyn↗

arXiv2014

Front propagation in cellular flows for fast reaction and small diffusivity

We investigate the influence of fluid flows on the propagation of chemical fronts arising in FKPP type models. We develop an asymptotic theory for the front speed in a cellular flow in the limit of small molecular diffusivity and fast reaction, i.e., large Péclet ($Pe$) and Damköhler ($Da$) numbers. The front speed is expressed in terms of a periodic path -- an instanton -- that minimizes a certain functional. This leads to an efficient procedure to calculate the front speed, and to closed-form expressions for $(\log Pe)^{-1}\ll Da\ll Pe$ and for $Da\gg Pe$. Our theoretical predictions are compared with (i) numerical solutions of an eigenvalue problem and (ii) simulations of the advection--diffusion--reaction equation.

physics.flu-dyn↗

arXiv2011

Bounding the scalar dissipation scale for mixing flows in the presence of sources

We investigate the mixing properties of scalars stirred by spatially smooth, divergence-free flows and maintained by a steady source-sink distribution. We focus on the spatial variation of the scalar field, described by the {\it dissipation wavenumber}, $k_d$, that we define as a function of the mean variance of the scalar and its gradient. We derive a set of upper bounds that for large Péclet number ($\Pe\gg1$) yield four distinct regimes for the scaling behavior of $k_d$, one of which corresponds to the Batchelor regime. The transition between these regimes is controlled by the value of $\Pe$ and the ratio $ρ=\ell_u/\ell_s$, where $\ell_u$ and $\ell_s$ are respectively, the characteristic lengthscales of the velocity and source fields. A fifth regime is revealed by homogenization theory. These regimes reflect the balance between different processes: scalar injection, molecular diffusion, stirring and bulk transport from the sources to the sinks. We verify the relevance of these bounds by numerical simulations for a {two-dimensional, chaotically mixing} example flow and discuss their relation to previous bounds. Finally, we note some implications for three dimensional turbulent flows.

physics.flu-dyn↗

arXiv2009

The role of a delay time on the spatial structure of chaotically advected reactive scalars

The stationary-state spatial structure of reacting scalar fields, chaotically advected by a two-dimensional large-scale flow, is examined for the case for which the reaction equations contain delay terms. Previous theoretical investigations have shown that, in the absence of delay terms and in a regime where diffusion can be neglected (large Péclet number), the emergent spatial structures are filamental and characterized by a single scaling regime with a Hölder exponent that depends on the rate of convergence of the reactive processes and the strength of the stirring measured by the average stretching rate. In the presence of delay terms, we show that for sufficiently small scales all interacting fields should share the same spatial structure, as found in the absence of delay terms. Depending on the strength of the stirring and the magnitude of the delay time, two further scaling regimes that are unique to the delay system may appear at intermediate length scales. An expression for the transition length scale dividing small-scale and intermediate-scale regimes is obtained and the scaling behavior of the scalar field is explained. The theoretical results are illustrated by numerical calculations for two types of reaction models, both based on delay differential equations, coupled to a two-dimensional chaotic advection flow. The first corresponds to a single reactive scalar and the second to a nonlinear biological model that includes nutrients, phytoplankton and zooplankton. As in the no-delay case, the presence of asymmetrical couplings among the biological species results in a non-generic scaling behavior.

physics.flu-dyn↗