Blood rheology is completely determined by its major corpuscles which are erythrocytes,or red blood cells (RBCs). That is why understanding and correct mathematical descriptionof RBCs behavior in blood is a critical step in modelling the blood dynamics. Variousphenomena provided by RBCs such as aggregation, deformation, shear-induced diffusion andnon-uniform radial distribution affect the passage of blood through the vessels. Hence,they have to be taken into account while modelling the blood dynamics. Other importantblood corpuscles are platelets, which are crucial for blood clotting. RBCs strongly affectthe platelet transport in blood expelling them to the vessel walls and increasing theirdispersion, which has to be considered in models of clotting. In this article we give abrief review of basic modern approaches in mathematical description of these phenomena,discuss their applicability to real flow conditions and propose further pathways fordeveloping the theory of blood flow.

We propose a mixed formulation for non-isothermal Oldroyd–Stokes problem where the bothextra stress and the heat flux’s vector are considered. Based on such a formulation, adual mixed finite element is constructed and analyzed. This finite element method enablesus to obtain precise approximations of the dual variable which are, for the non-isothermalfluid flow problems, the viscous and polymeric components of the extra-stress tensor, aswell as the heat flux. Furthermore, it has properties analogous to the finite volumemethods, namely, the local conservation of the momentum and the mass.

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.

Caused by the problem of unilateral contact during vibrations of satellite solar arrays, the aim of this paper is to better understand such a phenomenon. Therefore, it is studied here a simplified model composed by a beam moving between rigid obstacles. Our purpose is to describe and compare some families of fully discretized approximations and their properties, in the case of non-penetration Signorini's conditions. For this, starting from the works of Dumont and Paoli, we adapt to our beam model the singular dynamic method introduced by Renard. A particular emphasis is given in the use of a restitution coefficient in the impact law. Finally, various numerical results are presented and energy conservation capabilities of the schemes are investigated.

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.

A growing body of literature testifies to the importance of quantitative reasoning skillsin the 21st-century biology curriculum, and to the learning benefits associated withactive pedagogies. The process of modeling a biological system provides an approach thatintegrates mathematical skills and higher-order thinking with existing course contentknowledge. We describe a general strategy for teaching model-building in an introductorybiology course, using the example of a model of an infectious disease outbreak.Preliminary assessment data suggest that working through the formal process of modelconstruction may help students develop their scientific reasoning and communicationskills.

Infectious diseases ranging from the common cold to cholera affect our society physically, emotionally, ecologically, and economically. Yet despite their importance and impact, there remains a lack of effective teaching materials for epidemiology and disease ecology in K-12, undergraduate, and graduate curricula [2]. To address this deficit, we’ve developed a classroom lesson with three instructional goals: (1) Familiarize students on basic concepts of infectious disease ecology; (2) Introduce students to a classic compartmental model and its applications in epidemiology; (3) Demonstrate the application and importance of mathematical modeling as a tool in biology. The instructional strategy uses a game-based mathematical manipulative designed to engage students in the concepts of infectious disease spread. It has the potential to be modified for target audiences ranging from Kindergarten to professional schools in science, public health, policy, medical, and veterinarian programs. In addition, we’ve provided variations of the activity to enhance the transfer of fundamental concepts covered in the initial lesson to more complex concepts associated with vaccination and waning immunity. While 10 variations are presented here, the true number of directions in which the game might extend will only be limited by the imagination of its students [6].

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.

An analysis of all possible icosahedral viral capsids is proposed. It takes into accountthe diversity of coat proteins and their positioning in elementary pentagonal andhexagonal configurations, leading to definite capsid size. We show that theself-organization of observed capsids during their production implies a definitecomposition and configuration of elementary building blocks. The exact number of differentprotein dimers is related to the size of a given capsid, labeled by itsT-number. Simple rules determining these numbers for each value ofT are deduced and certain consequences concerning the probabilities ofmutations and evolution of capsid viruses are discussed.

The newly developed unifying discontinuous formulation named the correction procedure viareconstruction (CPR) for conservation laws is extended to solve the Navier-Stokesequations for 3D mixed grids. In the current development, tetrahedrons and triangularprisms are considered. The CPR method can unify several popular high order methodsincluding the discontinuous Galerkin and the spectral volume methods into a more efficientdifferential form. By selecting the solution points to coincide with the flux points,solution reconstruction can be completely avoided. Accuracy studies confirmed that theoptimal order of accuracy can be achieved with the method. Several benchmark test casesare computed by solving the Euler and compressible Navier-Stokes equations to demonstrateits performance.

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.

Over the last three decades Computational Fluid Dynamics (CFD) has gradually joined thewind tunnel and flight test as a primary flow analysis tool for aerodynamic designers. CFDhas had its most favorable impact on the aerodynamic design of the high-speed cruiseconfiguration of a transport. This success has raised expectations among aerodynamiciststhat the applicability of CFD can be extended to the full flight envelope. However, thecomplex nature of the flows and geometries involved places substantially increased demandson the solution methodology and resources required. Currently most simulations involveReynolds-Averaged Navier-Stokes (RANS) codes although Large Eddy Simulation (LES) andDetached Eddy Suimulation (DES) codes are occasionally used for component analysis ortheoretical studies. Despite simplified underlying assumptions, current RANS turbulencemodels have been spectacularly successful for analyzing attached, transonic flows. Whetheror not these same models are applicable to complex flows with smooth surface separation isan open question. A prerequisite for answering this question is absolute confidence thatthe CFD codes employed reliably solve the continuous equations involved. Too often,failure to agree with experiment is mistakenly ascribed to the turbulence model ratherthan inadequate numerics. Grid convergence in three dimensions is rarely achieved. Evenresidual convergence on a given grid is often inadequate. This paper discusses issuesinvolved in residual and especially grid convergence.

A finite volume method for the simulation of compressible aerodynamic flows is described.Stabilisation and shock capturing is achieved by the use of an HLLC consistent numericalflux function, with acoustic wave improvement. The method is implemented on anunstructured hybrid mesh in three dimensions. A solution of higher order accuracy isobtained by reconstruction, using an iteratively corrected least squares process, and by anew limiting procedure. The numerical performance of the complete approach is demonstratedby considering its application to the simulation of steady turbulent transonic flow overan ONERA M6 wing and to a steady inviscid supersonic flow over a modern military aircraftconfiguration.

This paper is devoted to solving of dynamic problems in biomechanics that requiredetailed study of fast processes. Numerical method of characteristics is used to model thetemporal development of the processes with high accuracy.

The evolution of a force-free granular gas with a constant restitution coefficient isstudied by means of granular hydrodynamics. We numerically solve the hydrodynamicequations and analyze the mechanisms of cluster formation. According to our findings, thepresently accepted mode-enslaving mechanism may not be responsible for the latterphenomenon. On the contrary, we observe that the cluster formation is mainly driven byshock-waves, which spontaneously originate and develop in the system. This agrees with apreviously suggested mechanism of formation of density singularities in one-dimensionalgranular gases.

This paper proposes a linear discrete-time scheme for general nonlinear cross-diffusion systems. The scheme can be regarded as an extension of a linear scheme based on the nonlinear Chernoff formula for the degenerate parabolic equations, which proposed by Berger et al. [RAIRO Anal. Numer. 13 (1979) 297–312]. We analyze stability and convergence of the linear scheme. To this end, we apply the theory of reaction-diffusion system approximation. After discretizing the scheme in space, we obtain a versatile, easy to implement and efficient numerical scheme for the cross-diffusion systems. Numerical experiments are carried out to demonstrate the effectiveness of the proposed scheme.

For flows with strong periodic content, time-spectral methods can be used to obtaintime-accurate solutions at substantially reduced cost compared to traditionaltime-implicit methods which operate directly in the time domain. However, these methodsare only applicable in the presence of fully periodic flows, which represents a severerestriction for many aerospace engineering problems. This paper presents an extension ofthe time-spectral approach for problems that include a slow transient in addition tostrong periodic behavior, suitable for applications such as transient turbofan simulationor maneuvering rotorcraft calculations. The formulation is based on a collocation methodwhich makes use of a combination of spectral and polynomial basis functions and results inthe requirement of solving coupled time instances within a period, similar to the timespectral approach, although multiple successive periods must be solved to capture thetransient behavior.

The implementation allows for two levels of parallelism, one in the spatial dimension,and another in the time-spectral dimension, and is implemented in a modular fashion whichminimizes the modifications required to an existing steady-state solver. For dynamicallydeforming mesh cases, a formulation which preserves discrete conservation as determined bythe Geometric Conservation Law is derived and implemented. A fully implicit approach whichtakes into account the coupling between the various time instances is implemented andshown to preserve the baseline steady-state multigrid convergence rate as the number oftime instances is increased. Accuracy and efficiency are demonstrated for periodic andnon-periodic problems by comparing the performance of the method with a traditionaltime-stepping approach using a simple two-dimensional pitching airfoil problem, athree-dimensional pitching wing problem, and a more realistic transitioning rotor problem.

We present the fiber-spring elastic model of the arterial wall with atheroscleroticplaque composed of a lipid pool and a fibrous cap. This model allows us to reproducepressure to cross-sectional area relationship along the diseased vessel which is used inthe network model of global blood circulation. Atherosclerosis attacks a region ofsystemic arterial network. Our approach allows us to examine the impact of the diseasedregion onto global haemodynamics.