arXiv subjects

Sarah L. Waters

Publications and source records attributed to Sarah L. Waters.

14 recordsLinked to original sources

arXiv2026

Modelling flow-driven pore closure of weakening poroelastic media

Poroelastic materials, such as polymer tissue scaffolds, porous rocks, and hydrogels, can weaken due to interactions between the solid skeleton and chemical species in the interstitial fluid. We develop a mathematical model for a poroelastic material to provide fundamental mechanistic insight into how weakening the material can affect the time-varying mechanics of the system. Our model couples large-deformation poroelasticity with an advection-diffusion equation for the solute. Furthermore, we introduce a decay equation for the material stiffness, whose rate of decay depends on the solute concentration. In this way, we describe a three-way coupling between poroelastic deformation, weakening of the skeleton and transport of solute through the material. We exploit numerical and analytical techniques to reveal the flow-driven uniaxial compression of a weakening poroelastic material and determine parameter regimes for which weakening the material facilitates pore closure at the downstream boundary. We identify parameter regimes in which (1) a steady state is attained without pore closure, (2) pore closure occurs at a finite time or (3) the pores close instantaneously; we uncover case (2) through the introduction of weakening into the system. We provide insights into the relationship between the differing behaviours and the separation between the timescales of the system. For systems with slow weakening, we derive a leading-order approximation for the time of pore closure, treating the ratio of the timescales of poroelastic relaxation and weakening as a small parameter, and investigate the accuracy of this approximation and the new behaviours that arise when these timescales become comparable.

physics.flu-dyn↗

arXiv2026

Reliable mechanistic operator recovery with biologically-informed neural networks: principles for architecture and optimisation design

Many biological processes are governed by complex dynamical mechanisms that remain incompletely understood despite increasing volumes of experimental data. Biologically-informed neural networks (BINNs) seek to address this challenge by embedding mechanistic differential equations into neural network training, enabling interpretable constitutive operators to be recovered directly from sparse and noisy observations. However, reliable operator recovery depends sensitively on network architecture, optimisation strategy, and data informativeness. Here, we present a systematic empirical study of how these factors influence mechanistic inference using BINNs applied to canonical one-dimensional advection-diffusion-reaction partial differential equation models. Across a suite of benchmark problems, we investigate how network expressivity, learning rate, loss weighting, and batch size influence optimisation behaviour and operator recovery. We show that successful mechanistic inference depends on balancing competing objectives rather than maximising any single aspect of the model or optimisation. Moderately expressive architectures outperform overly complex networks, intermediate learning rates improve optimisation stability, balanced data and PDE losses are essential for accurate operator recovery, and intermediate batch sizes provide the best compromise between computational efficiency and reproducibility. We further identify practical diagnostics for recognising common failure modes, including over-fitting, unstable optimisation, and poor mechanistic recovery when the ground truth is unavailable. Together, these findings provide evidence-based guidelines for deploying BINNs as credible tools for biological model discovery.

q-bio.QM↗

arXiv2026

Uniaxial poroelastic tendon model with crimped fibre recruitment

Fibre recruitment plays an important role in tendon and other biological soft tissue mechanics. Due to their large water content, a popular modelling approach for tendons is poroelasticity. Within this framework some tendon studies have included fibres, though none have included crimped fibre recruitment. We present a one dimensional poroelastic model in which the solid skeleton is composed of a soft neo-Hookean background matrix and crimped fibrils which do not bear load (FIB model). As the tissue is stretched, fibrils are straightened and contribute to load bearing. The fibre-reinforced tissue is compared to a tissue with a purely neo-Hookean (NH) skeleton in response to a uniaxial constant applied load (loading) and release of the load (unloading). The system dynamics are governed by a diffusion equation where the diffusion coefficient depends on stiffness. Within tendon parameter ranges, the FIB model is softer than the NH model, and so approaches steady state more slowly during loading. The presence of crimped fibrils allows the tendon to stretch further without excessively straining the fibrils or the NCM, providing a natural protection mechanism for the tendon's structural components to load, in agreement with experiments. During unloading, the FIB model is much slower to relax as the tissue softens due to fibril re-crimping. This asymmetry in loading and unloading manifests as a hysteresis loop in the stress-strain curve averaged over the tendon. The hysteresis is reduced with increasing applied load. The inclusion of fibrils allows for clearer biological interpretation and potential comparison to data. While the stress law employed in this study is bespoke for the application at hand by accounting for crimp and fibril recruitment, other fibril constitutive laws can readily be considered and incorporated into this framework.

physics.bio-ph↗

arXiv2025

The influence of cell phenotype on collective cell invasion into the extracellular matrix

Understanding the interactions between cells and the extracellular matrix (ECM) during collective cell invasion is crucial for advancements in tissue engineering, cancer therapies, and regenerative medicine. This study focuses on the roles of contact guidance and ECM remodelling in directing cell behaviour, with a particular emphasis on exploring how differences in cell phenotype impact collective cell invasion. We present a computationally tractable two-dimensional hybrid model of collective cell migration within the ECM, where cells are modelled as individual entities and collagen fibres as a continuous tensorial field. Our model incorporates random motility, contact guidance, cell-cell adhesion, volume filling, and the dynamic remodelling of collagen fibres through cellular secretion and degradation. Through a comprehensive parameter sweep, we provide valuable insights into how differences in the cell phenotype, in terms of the ability of the cell to migrate, secrete, degrade, and respond to contact guidance cues from the ECM, impacts the characteristics of collective cell invasion.

q-bio.CB↗

arXiv2025

Cyclic loading of a heterogeneous non-linear poroelastic material

Cyclic loading is a common feature in poroelastic systems, the material response depending non-trivially on the exact form of boundary conditions, pore structure, and mechanical properties. The situation becomes more complex when heterogeneity is introduced in the properties of the poroelastic material, yet heterogeneity too is common in physical poroelastic structures. In this paper, we analyse the behaviour of a soft porous material in response to a uniaxial cyclic stress or displacement, with a focus on understanding how this response is affected by continuous heterogeneity in the stiffness or permeability. Our work is motivated by observed altered material properties of the diseased tendon, but the framework we develop and analyse is generically applicable. We construct a one-dimensional non-linear poroelastic model, assuming Darcy flow through the pores of the solid skeleton which we assume has neo-Hookean elasticity. The system is driven by an applied uniaxial cyclic stress or a uniaxial cyclic displacement at one boundary. Heterogeneity in the stiffness or permeability profile is imposed via a Gaussian bump function. By exploring a range of loading frequencies together with magnitudes and locations of heterogeneity, we characterise the effect of heterogeneity on the response of the material, and show that the response of the system to an applied stress is qualitatively distinct from the response to an applied displacement. Our analysis of this simple model provides a foundation for understanding how heterogeneity affects the poroelastic response to cyclic loading.

cond-mat.soft↗

arXiv2024

Stokes flows in a two-dimensional bifurcation

The flow network model is an established approach to approximate pressure-flow relationships in a bifurcating network, and has been widely used in many contexts. Existing models typically assume unidirectional flow and exploit Poiseuille's law, and thus neglect the impact of bifurcation geometry and finite-sized objects on the flow. We determine the impact of bifurcation geometry and objects by computing Stokes flows in a two-dimensional (2D) bifurcation using the LARS (Lightning-AAA Rational Stokes) algorithm, a novel mesh-free algorithm for solving 2D Stokes flow problems utilising an applied complex analysis approach based on rational approximation of the Goursat functions. We compute the flow conductances of bifurcations with different channel widths, bifurcation angles, curved boundary geometries, and fixed circular objects. We quantify the difference between the computed conductances and their Poiseuille's law approximations to demonstrate the importance of incorporating detailed bifurcation geometry into existing flow network models. We parameterise the flow conductances of 2D bifurcation as functions of the dimensionless parameters of bifurcation geometry and a fixed object using a machine learning approach, which is simple to use and provides more accurate approximations than Poiseuille's law. Finally, the details of the 2D Stokes flows in bifurcations are presented.

physics.flu-dyn↗

arXiv2023

Computation of 2D Stokes flows via lightning and AAA rational approximation

Low Reynolds number fluid flows are governed by the Stokes equations. In two dimensions, Stokes flows can be described by two analytic functions, known as Goursat functions. Brubeck and Trefethen (2022) recently introduced a lightning Stokes solver that uses rational functions to approximate the Goursat functions in polygonal domains. In this paper, we present the "LARS" algorithm (Lightning-AAA Rational Stokes) for computing 2D Stokes flows in domains with smooth boundaries and multiply-connected domains using lightning and AAA rational approximation (Nakatsukasa et al., 2018). After validating our solver against known analytical solutions, we solve a variety of 2D Stokes flow problems with physical and engineering applications. Using these examples, we show rational approximation can now be used to compute 2D Stokes flows in general domains. The computations take less than a second and give solutions with at least 6-digit accuracy.

math.NA↗

arXiv2023

A continuum model for the elongation and orientation of Von Willebrand Factor with applications in arterial flow

The blood protein Von Willebrand Factor (VWF) is critical in facilitating arterial thrombosis. At pathologically high shear rates the protein unfolds and binds to the arterial wall, enabling the rapid deposition of platelets from the blood. We present a novel continuum model for VWF dynamics in flow based on a modified viscoelastic fluid model that incorporates a single constitutive relation to describe the propensity of VWF to unfold as a function of the scalar shear rate. Using experimental data of VWF unfolding in pure shear flow, we fix the parameters for VWF's unfolding propensity and the maximum VWF length, so that the protein is half unfolded at a shear rate of approximately 5,000 s$^{-1}$. We then use the theoretical model to predict VWF's behaviour in two complex flows where experimental data is challenging to obtain: pure elongational flow and stenotic arterial flow. In pure elongational flow, our model predicts that VWF is 50% unfolded at approximately 2,000 s$^{-1}$, matching the established hypothesis that VWF unfolds at lower shear rates in elongational flow than in shear flow. We demonstrate the sensitivity of this elongational flow prediction to the value of maximum VWF length used in the model, which varies significantly across experimental studies, predicting that VWF can unfold between 600 - 3,200 s$^{-1}$ depending on the selected value. Finally, we examine VWF dynamics in a range of idealised arterial stenoses, predicting the relative extension of VWF in elongational flow structures in the centre of the artery compared to high-shear regions near the arterial walls.

physics.bio-ph↗

arXiv2022

The electric double layer at the interface between a polyelectrolyte gel and salt bath

The electric double layer (EDL) that forms at the interface between a polyelectrolyte gel and a salt bath is studied using asymptotic and numerical methods. Specifically, matched asymptotic expansions, based on the smallness of the Debye length relative to the typical gel dimensions, are used to construct solutions of the governing equations and derive electroneutral models with consistent jump conditions across the gel-bath interface. A general approach for solving the equations of incompressible nonlinear elasticity in a curved boundary layer is developed and used to resolve the gel mechanics in the EDL. A critical feature of the model is that it accounts for phase separation within the gel, which gives rise to diffuse interfaces with a characteristic thickness described by the Kuhn length. We show that the solutions of the electroneutral model can only be asymptotically matched to the solutions in the EDL, in general, when the Kuhn length greatly exceeds the Debye length. Conversely, if the Debye length is similar to or larger than the Kuhn length, then the entire gel can self-organise into periodic, electrically charged domains via phase separation. The breakdown of electroneutrality demonstrates that the commonly invoked electroneutral assumption must be used with caution, as it generally only applies when the Debye length is much smaller than the Kuhn length.

cond-mat.soft↗

arXiv2021

A kinetic model of a polyelectrolyte gel undergoing phase separation

In this study we use non-equilibrium thermodynamics to systematically derive a phase-field model of a polyelectrolyte gel coupled to a hydrodynamic model for a salt solution surrounding the gel. The governing equations for the gel account for the free energy of the internal interfaces which form upon phase separation, the nonlinear elasticity of the polyelectrolyte network, and multi-component diffusive transport following a Stefan--Maxwell approach. The time-dependent model describes the evolution of the gel across multiple time and spatial scales and so is able to capture the large-scale solvent flux and the emergence of long-time pattern formation in the system. We explore the model for the case of a constrained gel undergoing uni-axial deformations. Numerical simulations show that rapid changes in the gel volume occur once the volume phase transition sets in, as well as the triggering of spinodal decomposition that leads to strong inhomogeneities in the lateral stresses, potentially leading to experimentally visible patterns.

cond-mat.soft↗

arXiv2021

The dynamics of a collapsing polyelectrolyte gel

We analyse the dynamics of different routes to collapse of a constrained polyelectrolyte gel in contact with an ionic bath. The evolution of the gel is described by a model that incorporates non-linear elasticity, Stefan-Maxwell diffusion and interfacial gradient free energy to account for phase separation of the gel. A bifurcation analysis of the homogeneous equilibrium states reveals three solution branches at low ion concentrations in the bath, giving way to only one above a critical ion concentration. We present numerical solutions that capture both the spatial heterogeneity and the multiple time-scales involved in the process of collapse. These solutions are complemented by two analytical studies. Firstly, a phase-plane analysis that reveals the existence of a depletion front for the transition from the highly swollen to the new collapsed equilibrium state. This depletion front is initiated after the fast ionic diffusion has set the initial condition for this time regime. Secondly, we perform a linear stability analysis about the homogeneous states that show that for a range of ion concentrations in the bath, spinodal decomposition of the swollen state gives rise to localized solvent-rich(poor) and, due to the electro-neutrality condition, ion-poor(rich) phases that coarsen on the route to collapse. This dynamics of a collapsing polyelectrolyte gel has not been described before.

cond-mat.soft↗

arXiv2018

Bifurcations and dynamics emergent from lattice and continuum models of bioactive porous media

We study dynamics emergent from a two-dimensional reaction--diffusion process modelled via a finite lattice dynamical system, as well as an analogous PDE system, involving spatially nonlocal interactions. These models govern the evolution of cells in a bioactive porous medium, with evolution of the local cell density depending on a coupled quasi--static fluid flow problem. We demonstrate differences emergent from the choice of a discrete lattice or a continuum for the spatial domain of such a process. We find long--time oscillations and steady states in cell density in both lattice and continuum models, but that the continuum model only exhibits solutions with vertical symmetry, independent of initial data, whereas the finite lattice admits asymmetric oscillations and steady states arising from symmetry-breaking bifurcations. We conjecture that it is the structure of the finite lattice which allows for more complicated asymmetric dynamics. Our analysis suggests that the origin of both types of oscillations is a nonlocal reaction-diffusion mechanism mediated by quasi-static fluid flow.

q-bio.TO↗

arXiv2018

Lattice and Continuum Modelling of a Bioactive Porous Tissue Scaffold

A contemporary procedure to grow artificial tissue is to seed cells onto a porous biomaterial scaffold and culture it within a perfusion bioreactor to facilitate the transport of nutrients to growing cells. Typical models of cell growth for tissue engineering applications make use of spatially homogeneous or spatially continuous equations to model cell growth, flow of culture medium, nutrient transport, and their interactions. The network structure of the physical porous scaffold is often incorporated through parameters in these models, either phenomenologically or through techniques like mathematical homogenization. We derive a model on a square grid lattice to demonstrate the importance of explicitly modelling the network structure of the porous scaffold, and compare results from this model with those from a modified continuum model from the literature. We capture two-way coupling between cell growth and fluid flow by allowing cells to block pores, and by allowing the shear stress of the fluid to affect cell growth and death. We explore a range of parameters for both models, and demonstrate quantitative and qualitative differences between predictions from each of these approaches, including spatial pattern formation and local oscillations in cell density present only in the lattice model. These differences suggest that for some parameter regimes, corresponding to specific cell types and scaffold geometries, the lattice model gives qualitatively different model predictions than typical continuum models. Our results inform model selection for bioactive porous tissue scaffolds, aiding in the development of successful tissue engineering experiments and eventually clinically successful technologies.

q-bio.TO↗

arXiv2016

On the boundary layer structure near a highly permeable porous interface

The method of matched asymptotic expansions is used to study the canonical problem of steady laminar flow through a narrow two-dimensional channel blocked by a tight-fitting finite-length highly permeable porous obstacle. We investigate the behaviour of the local flow close to the interface between the single-phase and porous regions (governed by the incompressible Navier--Stokes and Darcy flow equations, respectively). We solve for the flow in these inner regions in the limits of low and high Reynolds number, facilitating an understanding of the nature of the transition from Poiseuille to plug to Poiseuille flow in each of these limits. Significant analytic progress is made in the high-Reynolds-number limit, and we explore in detail the rich boundary layer structure that occurs. We consider the three-dimensional generalization to unsteady laminar flow through and around a tight-fitting highly permeable cylindrical porous obstacle within a Hele-Shaw cell. For the high-Reynolds-number limit, we give the coupling conditions and interfacial stress in terms of the outer flow variables, allowing information from a nonlinear three-dimensional problem to be obtained by solving a linear two-dimensional problem. Finally, we illustrate the utility of our analysis by considering the specific example of time-dependent forced far-field flow in a Hele-Shaw cell containing a porous cylinder with a circular cross-section. We determine the internal stress within the porous obstacle, which is key for tissue engineering applications, and the interfacial stress on the boundary of the porous obstacle, which has applications to biofilm erosion. In the high-Reynolds-number limit, we demonstrate that the fluid inertia can result in the cylinder experiencing a time-independent net force, even when the far-field forcing is periodic with zero mean.

physics.flu-dyn↗