Local and global instabilities of flow in a flexible-walled channel

We consider laminar high-Reynolds-number flow through a long finite-length planar channel, where a segment of one wall is replaced by a massless membrane held under longitudinal tension. The flow is driven by a fixed pressure difference across the channel and is described using an integral form of the unsteady boundary-layer equations. The basic flow state, for which the channel has uniform width, exhibits static and oscillatory global instabilities, having distinct modal forms. In contrast, the corresponding local problem (neglecting boundary conditions associated with the rigid parts of the system) is found to be convectively, but not absolutely, unstable to small-amplitude disturbances in the absence of wall damping. We show how amplification of the primary global oscillatory instability can arise entirely from wave reflections with the rigid parts of the system, involving interacting travelling wave flutter and static-divergence modes that are convectively stable; alteration of the mean flow by oscillations makes the onset of this primary instability subcritical. We also show how distinct mechanisms of energy transfer differentiate the primary global mode from other modes of oscillatory instability

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.Comment: 30 pages, 21 figure

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.Comment: 38 pages, 16 figures. This version includes a much-expanded introduction, and a new section on nonlinear diffusion in addition to polish throughou

Is the Donnan effect sufficient to explain swelling in brain tissue slices?

Brain tissue swelling is a dangerous consequence of traumatic injury and is associated with raised intracranial pressure and restricted blood flow. We consider the mechanical effects that drive swelling of brain tissue slices in an ionic solution bath, motivated by recent experimental results that showed that the volume change of tissue slices depends on the ionic concentration of the bathing solution. This result was attributed to the presence of large charged molecules that induce ion concentration gradients to ensure electroneutrality (the Donnan effect), leading to osmotic pressures and water accumulation. We use a mathematical triphasic model for soft tissue to characterize the underlying processes that could lead to the volume changes observed experimentally. We suggest that swelling is caused by an osmotic pressure increase driven by both non-permeating solutes released by necrotic cells, in addition to the Donnan effect. Both effects are necessary to explain the dependence of the tissue slice volume on the ionic bath concentration that was observed experimentally

Wrinkling, creasing, and folding in fiber-reinforced soft tissues

Many biological tissues develop elaborate folds during growth and development. The onset of this folding is often understood in relation to the creasing and wrinkling of a thin elastic layer that grows whilst attached to a large elastic foundation. In reality, many biological tissues are reinforced by fibres and so are intrinsically anisotropic. However, the correlation between the fiber directions and the pattern formed during growth is not well understood. Here, we consider the stability of a two-layer tissue composed of a thin hyperelastic strip adhered to an elastic half-space in which are embedded elastic fibers. The combined object is subject to a uniform compression and, at a critical value of this compression, buckles out of the plane — it wrinkles. We characterize the wrinkle wavelength at onset as a function of the fiber orientation both computationally and analytically and show that the onset of surface instability can be either promoted or inhibited as the fiber stiffness increases, depending on the fibre angle. However, we find that the structure of the resulting folds is approximately independent of the fiber orientation. We also explore numerically the formation of large creases in fiber-reinforced tissue in the post-buckling regime

Stokes flows in a 2D bifurcation

The flow network model is an established approach to approximate pressure-flow relationships in a network, which has been widely used in many contexts. However, little is known about the impact of bifurcation geometry on such approximations, so the existing models mostly rely on unidirectional flow assumption and Poiseuille's law, and thus neglect the flow details at each bifurcation. In this work, we address these limitations by computing Stokes flows in a 2D bifurcation using LARS (Lightning-AAA Rational Stokes), 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. Using our 2D bifurcation model, we show that the fluxes in two child branches depend on not only pressures and widths of inlet and outlet branches, as most previous studies have assumed, but also detailed bifurcation geometries (e.g. bifurcation angle), which were not considered in previous studies. The 2D Stokes flow simulations allow us to represent the relationship between pressures and fluxes of a bifurcation using an updated flow network, which considers the bifurcation geometry and can be easily incorporated into previous flow network approaches. The errors in the flow conductance of a channel in a bifurcation approximated using Poiseuille's law can be greater than 16%, when the centreline length is twice the inlet channel width and the bifurcation geometry is highly asymmetric. In addition, we present details of 2D Stokes flow features, such as flow separation in a bifurcation and flows around fixed objects at different locations, which previous flow network models cannot capture. These findings suggest the importance of incorporating detailed flow modelling techniques alongside existing flow network approaches when solving complex flow problems

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 a solver for computing 2D Stokes flows in domains with smooth boundaries and multiply-connected domains using lightning and AAA rational approximation (Nakatsukasa et al., 2018). This leads to a new rational approximation algorithm "LARS" that is suitable for computing many bounded 2D Stokes flow problems. After validating our solver against known analytical solutions, we solve a variety of 2D Stokes flow problems with physical and engineering applications. The computations take less than a second and give solutions with at least 6-digit accuracy

Mathematical modelling of tissue-engineering angiogenesis

We present a mathematical model for the vascularisation of a porous scaffold following implantation in vivo. The model is given as a set of coupled non-linear ordinary differential equations (ODEs) which describe the evolution in time of the amounts of the different tissue constituents inside the scaffold. Bifurcation analyses reveal how the extent of scaffold vascularisation changes as a function of the parameter values. For example, it is shown how the loss of seeded cells arising from slow infiltration of vascular tissue can be overcome using a prevascularisation strategy consisting of seeding the scaffold with vascular cells. Using certain assumptions it is shown how the system can be simplified to one which is partially tractable and for which some analysis is given. Limited comparison is also given of the model solutions with experimental data from the chick chorioallantoic membrane (CAM) assay

Spinodal decomposition and collapse of a polyelectrolyte gel

The collapse of a polyelectrolyte gel in a (monovalent) salt solution is analysed using a new model that includes interfacial gradient energy to account for phase separation in the gel, finite elasticity and multicomponent transport. We carry out a linear stability analysis to determine the stable and unstable spatially homogeneous equilibrium states and how they phase separate into localized regions that eventually coarsen to a new stable state. We then investigate the problem of a collapsing gel as a response to increasing the salt concentration in the bath. A phase space analysis reveals that the collapse is obtained by a front moving through the gel that eventually ends in a new stable equilibrium. For some parameter ranges, these two routes to gel shrinking occur together