A computer-aided method for accurately carrying out the Chapman-Enskog expansion of the Boltzmann equation, including its inelastic variant, is presented and employed to derive a hydrodynamic description of a dilute binary mixture of smooth inelastic spheres. Constitutive relations, formally valid for all physical values of the coefficients of restitution, are calculated by carrying out the pertinent Chapman-Enskog expansion to sufficient high orders in the Sonine polynomials to ensure numerical convergence. The resulting hydrodynamic description is applied to the analysis of a vertically vibrated binary mixture of particles (under gravity) differing only in their respective coefficients of restitution. It is shown that even with this “minor”difference the mixture partly segregates, its steady state exhibiting a sandwich-like configuration.

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

In this paper we build and analyze networks using the statistical and programmingenvironment R and the igraph package. We investigate random, small-world, and scale-freenetworks and test a standard problem of connectivity on a random graph. We then develop amethod 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.

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 smoothnessestimate which is suited as a control input to a shock capture mechanism. Using anartificial viscosity in the latter role, we obtain a DG scheme for the numerical solutionof nonlinear systems of conservation laws. Building on work by Persson and Peraire, wethoroughly justify the detector’s design and analyze its performance on a number ofbenchmark problems. We further explain the scaling and smoothing steps necessary to turnthe output of the detector into a local, artificial viscosity. We close by providing anextensive array of numerical tests of the detector in use.

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

We construct interfacial solitary structures (spots) generated by a bistable chemicalreaction or a non-equilibrium phase transition in a surfactant film. The structures arestabilized by Marangoni flow that prevents the spread of a state with a higher surfacetension when it is dynamically favorable. In a system without surfactant massconservation, a unique radius of a solitary spot exists within a certain range of valuesof the Marangoni number and of the deviation of chemical potential from the Maxvellconstruction, but multiple spots attract and coalesce. In a conservative system, there isa range of stable spot sizes, but solitary spots may exist only in a limited parametricrange, beyond which multiple spots nucleate. Repeated coalescence and nucleation leads tochaotic dynamics of spots observed computationally in Ref. [1].

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

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 elementmethods using continuous and discontinuous Galerkin formulations. In this paper weinvestigate the effect of h- and p-type refinement onthe relationship between runtime performance and solution accuracy. The broad spectrum ofpossible domain discretisations makes establishing a performance-optimal selectionnon-trivial. Through comparing the runtime of different implementations for evaluatingoperators over the space of discretisations with a desired solution tolerance, wedemonstrate how the optimal discretisation and operator implementation may be selected fora specified problem. Furthermore, this demonstrates the need for codes to support bothlow- and high-order discretisations.

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 toinelastic collisions between particles: an external driving is needed to maintain themotion of particles. Theoretical description of driven granular media is especiallychallenging for dense granular flows. This paper focuses on a gravity-driven densegranular Poiseuille flow in a channel. A special focus here is on the intriguingphenomenon 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-scaleapproach is taken. On macro scale, granular hydrodynamics is employed. On micro scale,event-driven molecular dynamics simulations are performed. The entire phase diagram ofparameters is explored, in order to determine which flow regime occurs in various regionsin the parameter space.

We determine the steady-state structures that result from liquid-liquid demixing in afree surface film of binary liquid on a solid substrate. The considered model correspondsto the static limit of the diffuse interface theory describing the phase separationprocess for a binary liquid (model-H), when supplemented by boundary conditions at thefree surface and taking the influence of the solid substrate into account. The resultingvariational problem is numerically solved employing a Finite Element Method on an adaptivegrid. The developed numerical scheme allows us to obtain the coupled steady-state filmthickness profile and the concentration profile inside the film. As an example wedetermine steady state profiles for a reflection-symmetric two-dimensional droplet forvarious surface tensions of the film and various preferential attraction strength of onecomponent to the substrate. We discuss the relation of the results of the present diffuseinterface theory to the sharp interface limit and determine the effective interfacetension of the diffuse interface by several means.

The paper is devoted to the method of computer simulation of protein interactions takingpart in photosynthetic electron transport reactions. Using this method we have studiedkinetic characteristics of protein-protein complex formation for four pairs of proteinsinvolved in photosynthesis at a variety of ionic strength values. Computer simulationsdescribe non-monotonic dependences of complex formation rates on the ionic strength as theresult of long-range electrostatic interactions. Calculations confirm that the decrease inthe association second order rate constant at low values of the ionic strength is causedby the protein pairs spending more time in “wrong” orientations which do not satisfy thedocking conditions and so do not form the final complex capable of the electrontransfer.

The development of functional-structural plant models has opened interesting perspectivesfor a better understanding of plant growth as well as for potential applications inbreeding or decision aid in farm management. Parameterization of such models is however adifficult issue due to the complexity of the involved biological processes and theinteractions between these processes. The estimation of parameters from experimental databy inverse methods is thus a crucial step. This paper presents some results anddiscussions as first steps towards the construction of a general framework for theparametric estimation of functional-structural plant models. A general family of models ofCarbon allocation formalized as dynamic systems serves as the basis for our study. Anadaptation of the 2-stage Aitken estimator to this family of model is introduced as wellas its numerical implementation, and applied in two different situations: first amorphogenetic model of sugar beet growth with simple plant structure, multi-stage anddetailed observations, and second a tree growth model characterized by sparse observationsand strong interactions between functioning and organogenesis. The proposed estimationmethod appears robust, easy to adapt to a wide variety of models, and generally provides asatisfactory goodness-of-fit. However, it does not allow a proper evaluation of estimationuncertainty. Finally some perspectives opened by the theory of hidden models arediscussed.

We propose a novel approach to introducing hypothesis testing into the biologycurriculum. Instead of telling students the hypothesis and what kind of data to collectfollowed by a rigid recipe of testing the hypothesis with a given test statistic, we askstudents to develop a hypothesis and a mathematical model that describes the nullhypothesis. Simulation of the model under the null hypothesis allows students to comparetheir experimental data to what they would expect under the null hypothesis, thus leadingto a much more intuitive understanding of hypothesis testing. This approach has beentested both in the classroom and in faculty workshops, and we provide some suggestions forimplementations based on our experiences.

This paper deals with the evolution Fokker-Planck-Smoluchowski configurationalprobability diffusion equation for the FENE dumbbell model in dilute polymer solutions. Weprove the exponential convergence in time of the solution of this equation to acorresponding steady-state solution, for arbitrary velocity gradients.

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-ordermethods can provide solutions to otherwise intractable fluid flow problems within complexgeometries. However, it remains the case that existing high-order schemes are generallyless robust and more complex to implement than their low-order counterparts. These issues,in conjunction with difficulties generating high-order meshes, have limited the adoptionof high-order techniques in both academia (where the use of low-order schemes remainswidespread) and industry (where the use of low-order schemes is ubiquitous). In this shortreview, issues that have hitherto prevented the use of high-order methods amongst anon-specialist community are identified, and current efforts to overcome these issues arediscussed. Attention is focused on four areas, namely the generation of unstructuredhigh-order meshes, the development of simple and efficient time integration schemes, th edevelopment of robust and accurate shock capturing algorithms, and finally the developmentof high-order methods that are intuitive and simple to implement. With regards to thisfinal area, particular attention is focused on the recently proposed flux reconstructionapproach, which allows various well known high-order schemes (such as nodal discontinuousGalerkin methods and spectral difference methods) to be cast within a single unifyingframework. It should be noted that due to the experience of the authors the review isdirected somewhat towards aerodynamic applications and compressible flow. However, many ofthe discussions have a wider applicability. Moreover, the tone of the review is intendedto be generally accessible, such that an extended scientific community can gain insightinto factors currently pacing the adoption of unstructured high-order methods.

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

Spatiotemporal patterns near a codimension-2 Turing-Hopf point of the one-dimensionalsuperdiffusive Brusselator model are analyzed. The superdiffusive Brusselator modeldiffers from its regular counterpart in that the Laplacian operator of the regular modelis replaced by ∂ α /∂|ξ|α , 1 < α< 2, an integro-differential operator that reflects the nonlocal behavior ofsuperdiffusion. The order of the operator, α, is a measure of the rate ofsuperdiffusion, which, in general, can be different for each of the two components. Aweakly nonlinear analysis is used to derive two coupled amplitude equations describing theslow time evolution of the Turing and Hopf modes. We seek special solutions of theamplitude equations, namely a pure Turing solution, a pure Hopf solution, and a mixed modesolution, and analyze their stability to long-wave perturbations. We find that thestability criteria of all three solutions depend greatly on the rates of superdiffusion ofthe two components. In addition, the stability properties of the solutions to theanomalous diffusion model are different from those of the regular diffusion model.Numerical computations in a large spatial domain, using Fourier spectral methods in spaceand second order Runge-Kutta in time are used to confirm the analysis and also to findsolutions 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 localizedregion 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 abackground of stationary stripes. Each such pattern lies on a branch of such solutions, isstable and corresponds to a different initial condition. The patterns correspond to thephenomenon of pinning of the front between the stripes and the time periodic cellularmotion. While in some cases it is difficult to isolate the effect of the diffusionexponents, we find characteristics in the spatiotemporal patterns for anomalous diffusionthat we have not found for regular (Fickian) diffusion.