Continuum mechanics (e.g., hydrodynamics, elasticity theory) is based on the assumption that a small set of fields provides a closed description on large space and time scales. Conditions governing the choice for these fields are discussed in the context of granular fluids and multi-component fluids. In the first case, the relevance of temperature or energy as a hydrodynamic field is justified. For mixtures, the use of a total temperature and single flow velocity is compared with the use of multiple species temperatures and velocities.

An overview of recent results pertaining to the hydrodynamic description (both Newtonian and non-Newtonian) of granular gases described by the Boltzmann equation for inelastic Maxwell models is presented. The use of this mathematical model allows us to get exact results for different problems. First, the Navier–Stokes constitutive equations with explicit expressions for the corresponding transport coefficients are derived by applying the Chapman–Enskog method to inelastic gases. Second, the non-Newtonian rheological properties in the uniform shear flow (USF) are obtained in the steady state as well as in the transient unsteady regime. Next, an exact solution for a special class of Couette flows characterized by a uniform heat flux is worked out. This solution shares the same rheological properties as the USF and, additionally, two generalized transport coefficients associated with the heat flux vector can be identified. Finally, the problem of small spatial perturbations of the USF is analyzed with a Chapman–Enskog-like method and generalized (tensorial) transport coefficients are obtained.

In this paper we build and analyze networks using the statistical and programming environment R and the igraph package. We investigate random, small-world, and scale-free networks and test a standard problem of connectivity on a random graph. We then develop a method to study how vaccination can alter the structure of a disease transmission network. We also discuss a variety of other uses for networks in biology.

Peristaltic pumping of fluid is a fundamental method of transport in many biological processes. In some instances, particles of appreciable size are transported along with the fluid, such as ovum transport in the oviduct or kidney stones in the ureter. In some of these biological settings, the fluid may be viscoelastic. In such a case, a nonlinear constitutive equation to describe the evolution of the viscoelastic contribution to the stress tensor must be included in the governing equations. Here we use an immersed boundary framework to study peristaltic transport of a macroscopic solid particle in a viscoelastic fluid governed by a Navier-Stokes/Oldroyd-B model. Numerical simulations of peristaltic pumping as a function of Weissenberg number are presented. We examine the spatial and temporal evolution of the polymer stress field, and also find that the viscoelasticity of the fluid does hamper the overall transport of the particle in the direction of the wave.

We present a novel, cell-local shock detector for use with discontinuous Galerkin (DG) methods. The output of this detector is a reliably scaled, element-wise smoothness estimate which is suited as a control input to a shock capture mechanism. Using an artificial viscosity in the latter role, we obtain a DG scheme for the numerical solution of nonlinear systems of conservation laws. Building on work by Persson and Peraire, we thoroughly justify the detector’s design and analyze its performance on a number of benchmark problems. We further explain the scaling and smoothing steps necessary to turn the output of the detector into a local, artificial viscosity. We close by providing an extensive array of numerical tests of the detector in use.

Plant growth occurs due to cell proliferation in the meristem. We model the case of apical meristem specific for branch growth and the case of basal meristem specific for bulbous plants and grass. In the case of apical growth, our model allows us to describe the variety of plant forms and lifetimes, endogenous rhythms and apical domination. In the case of basal growth, the spatial structure, which corresponds to the appearance of leaves, results from dissipative instability of the homogeneous in space solution. We study nonlinear dynamics and wave propagation of the corresponding reaction-diffusion systems. Bifurcation of periodic at infinity waves is investigated numerically.

We construct interfacial solitary structures (spots) generated by a bistable chemical reaction or a non-equilibrium phase transition in a surfactant film. The structures are stabilized by Marangoni flow that prevents the spread of a state with a higher surface tension when it is dynamically favorable. In a system without surfactant mass conservation, a unique radius of a solitary spot exists within a certain range of values of the Marangoni number and of the deviation of chemical potential from the Maxvell construction, but multiple spots attract and coalesce. In a conservative system, there is a range of stable spot sizes, but solitary spots may exist only in a limited parametric range, beyond which multiple spots nucleate. Repeated coalescence and nucleation leads to chaotic dynamics of spots observed computationally in Ref. [1].

A mathematical model of infiltrative tumour growth is investigated taking into account transitions between two possible states of malignant cells: proliferation and migration. These transitions are considered to depend on oxygen level in a threshold manner where high oxygen concentration allows cell proliferation, while concentration below a certain critical value induces cell migration. The infiltrative tumour spreading rate dependence on model parameters is obtained. It is shown that the tumour growth rate depends on the tissue oxygen level in a threshold manner.

There is a growing interest in high-order finite and spectral/hp element methods using continuous and discontinuous Galerkin formulations. In this paper we investigate the effect of h- and p-type refinement on the relationship between runtime performance and solution accuracy. The broad spectrum of possible domain discretisations makes establishing a performance-optimal selection non-trivial. Through comparing the runtime of different implementations for evaluating operators over the space of discretisations with a desired solution tolerance, we demonstrate how the optimal discretisation and operator implementation may be selected for a specified problem. Furthermore, this demonstrates the need for codes to support both low- and high-order discretisations.

We determine the steady-state structures that result from liquid-liquid demixing in a free surface film of binary liquid on a solid substrate. The considered model corresponds to the static limit of the diffuse interface theory describing the phase separation process for a binary liquid (model-H), when supplemented by boundary conditions at the free surface and taking the influence of the solid substrate into account. The resulting variational problem is numerically solved employing a Finite Element Method on an adaptive grid. The developed numerical scheme allows us to obtain the coupled steady-state film thickness profile and the concentration profile inside the film. As an example we determine steady state profiles for a reflection-symmetric two-dimensional droplet for various surface tensions of the film and various preferential attraction strength of one component to the substrate. We discuss the relation of the results of the present diffuse interface theory to the sharp interface limit and determine the effective interface tension of the diffuse interface by several means.

We consider a dense granular shear flow in a two-dimensional system. Granular systems (composed of a large number of macroscopic particles) are far from equilibrium due to inelastic collisions between particles: an external driving is needed to maintain the motion of particles. Theoretical description of driven granular media is especially challenging for dense granular flows. This paper focuses on a gravity-driven dense granular Poiseuille flow in a channel. A special focus here is on the intriguing phenomenon of fluid-solid coexistence: a solid plug in the center of the system, surrounded by fluid layers. To find and analyze various flow regimes, a multi-scale approach is taken. On macro scale, granular hydrodynamics is employed. On micro scale, event-driven molecular dynamics simulations are performed. The entire phase diagram of parameters is explored, in order to determine which flow regime occurs in various regions in the parameter space.

The paper is devoted to the method of computer simulation of protein interactions taking part in photosynthetic electron transport reactions. Using this method we have studied kinetic characteristics of protein-protein complex formation for four pairs of proteins involved in photosynthesis at a variety of ionic strength values. Computer simulations describe non-monotonic dependences of complex formation rates on the ionic strength as the result of long-range electrostatic interactions. Calculations confirm that the decrease in the association second order rate constant at low values of the ionic strength is caused by the protein pairs spending more time in “wrong” orientations which do not satisfy the docking conditions and so do not form the final complex capable of the electron transfer.

We propose a novel approach to introducing hypothesis testing into the biology curriculum. Instead of telling students the hypothesis and what kind of data to collect followed by a rigid recipe of testing the hypothesis with a given test statistic, we ask students to develop a hypothesis and a mathematical model that describes the null hypothesis. Simulation of the model under the null hypothesis allows students to compare their experimental data to what they would expect under the null hypothesis, thus leading to a much more intuitive understanding of hypothesis testing. This approach has been tested both in the classroom and in faculty workshops, and we provide some suggestions for implementations based on our experiences.

This paper deals with the evolution Fokker-Planck-Smoluchowski configurational probability diffusion equation for the FENE dumbbell model in dilute polymer solutions. We prove the exponential convergence in time of the solution of this equation to a corresponding steady-state solution, for arbitrary velocity gradients.

The development of functional-structural plant models has opened interesting perspectives for a better understanding of plant growth as well as for potential applications in breeding or decision aid in farm management. Parameterization of such models is however a difficult issue due to the complexity of the involved biological processes and the interactions between these processes. The estimation of parameters from experimental data by inverse methods is thus a crucial step. This paper presents some results and discussions as first steps towards the construction of a general framework for the parametric estimation of functional-structural plant models. A general family of models of Carbon allocation formalized as dynamic systems serves as the basis for our study. An adaptation of the 2-stage Aitken estimator to this family of model is introduced as well as its numerical implementation, and applied in two different situations: first a morphogenetic model of sugar beet growth with simple plant structure, multi-stage and detailed observations, and second a tree growth model characterized by sparse observations and strong interactions between functioning and organogenesis. The proposed estimation method appears robust, easy to adapt to a wide variety of models, and generally provides a satisfactory goodness-of-fit. However, it does not allow a proper evaluation of estimation uncertainty. Finally some perspectives opened by the theory of hidden models are discussed.

A number of exciting new laboratory techniques have been developed using the Watson-Crick complementarity properties of DNA strands to achieve the self-assembly of graphical complexes. For all of these methods, an essential step in building the self-assembling nanostructure is designing the component molecular building blocks. These design strategy problems fall naturally into the realm of graph theory. We describe graph theoretical formalism for various construction methods, and then suggest several graph theory exercises to introduce this application into a standard undergraduate graph theory class. This application provides a natural framework for motivating central concepts such as degree sequence, Eulerian graphs, Fleury’s algorithm, trees, graph genus, paths, cycles, etc. There are many open questions associated with these applications which are accessible to students and offer the possibility of exciting undergraduate research experiences in applied graph theory.

Blood clotting system (BCS) modelling is an important issue with a plenty of applications in medicine and biophysics. The BCS main function is to form a localized clot at the site of injury preventing blood loss. Mutual influence of fibrin clot consisting mainly of fibrin polymer gel and blood flow is an important factor for BCS to function properly. The process of fibrin polymer mesh formation has not adequately been described by current mathematical models. That is why it is not possible to define the borders of growing clot and model its interaction with a blood flow. This paper main goal is to propose physically well-founded mathematical model of fibrin polymerization and gelation. The proposed model defines the total length of fibrin polymer fibers in the unit volume, determines a position of the border between gel and liquid and allows to evaluate the permeability of growing gel. Without significant structural changes the proposed model could be modified to include the blood shear rate influence on the fibrin polymerization and gelation.

Theoretical studies and numerical experiments suggest that unstructured high-order methods can provide solutions to otherwise intractable fluid flow problems within complex geometries. However, it remains the case that existing high-order schemes are generally less robust and more complex to implement than their low-order counterparts. These issues, in conjunction with difficulties generating high-order meshes, have limited the adoption of high-order techniques in both academia (where the use of low-order schemes remains widespread) and industry (where the use of low-order schemes is ubiquitous). In this short review, issues that have hitherto prevented the use of high-order methods amongst a non-specialist community are identified, and current efforts to overcome these issues are discussed. Attention is focused on four areas, namely the generation of unstructured high-order meshes, the development of simple and efficient time integration schemes, th e development of robust and accurate shock capturing algorithms, and finally the development of high-order methods that are intuitive and simple to implement. With regards to this final area, particular attention is focused on the recently proposed flux reconstruction approach, which allows various well known high-order schemes (such as nodal discontinuous Galerkin methods and spectral difference methods) to be cast within a single unifying framework. It should be noted that due to the experience of the authors the review is directed somewhat towards aerodynamic applications and compressible flow. However, many of the discussions have a wider applicability. Moreover, the tone of the review is intended to be generally accessible, such that an extended scientific community can gain insight into factors currently pacing the adoption of unstructured high-order methods.

Spatiotemporal patterns near a codimension-2 Turing-Hopf point of the one-dimensional superdiffusive Brusselator model are analyzed. The superdiffusive Brusselator model differs from its regular counterpart in that the Laplacian operator of the regular model is replaced by ∂α/∂|ξ|α, 1 < α < 2, an integro-differential operator that reflects the nonlocal behavior of superdiffusion. The order of the operator, α, is a measure of the rate of superdiffusion, which, in general, can be different for each of the two components. A weakly nonlinear analysis is used to derive two coupled amplitude equations describing the slow time evolution of the Turing and Hopf modes. We seek special solutions of the amplitude equations, namely a pure Turing solution, a pure Hopf solution, and a mixed mode solution, and analyze their stability to long-wave perturbations. We find that the stability criteria of all three solutions depend greatly on the rates of superdiffusion of the two components. In addition, the stability properties of the solutions to the anomalous diffusion model are different from those of the regular diffusion model. Numerical computations in a large spatial domain, using Fourier spectral methods in space and second order Runge-Kutta in time are used to confirm the analysis and also to find solutions not predicted by the weakly nonlinear analysis, in the fully nonlinear regime. Specifically, we find a large number of steady state patterns consisting of a localized region or regions of stationary stripes in a background of time periodic cellular motion, as well as patterns with a localized region or regions of time periodic cells in a background of stationary stripes. Each such pattern lies on a branch of such solutions, is stable and corresponds to a different initial condition. The patterns correspond to the phenomenon of pinning of the front between the stripes and the time periodic cellular motion. While in some cases it is difficult to isolate the effect of the diffusion exponents, we find characteristics in the spatiotemporal patterns for anomalous diffusion that we have not found for regular (Fickian) diffusion.

The study of the fluctuations in the steady state of a heated granular system is reviewed. A Boltzmann-Langevin description can be built requiring consistency with the equations for the one- and two-particle correlation functions. From the Boltzmann-Langevin equation, Langevin equations for the total energy and the transverse velocity field are derived. The existence of a fluctuation-dissipation relation for the transverse velocity field is also studied.