We present two-dimensional simulations of chemotactic self-propelled bacteria swimming ina viscous fluid. Self-propulsion is modelled by a couple of forces of same intensity andopposite direction applied on the rigid bacterial body and on an associated region in thefluid representing the flagellar bundle. The method for solving the fluid flow and themotion of the bacteria is based on a variational formulation written on the whole domain,strongly coupling the fluid and the rigid particle problems: rigid motion is enforced bypenalizing the strain rate tensor on the rigid domain, while incompressibility is treatedby duality. This model allows to achieve an accurate description of fluid motion andhydrodynamic interactions in moderate to concentrated active suspensions. A mesoscopicmodel is also used, in which the size of the bacteria is supposed to be much smaller thanthe elements of fluid: the perturbation of the fluid due to propulsion and motion of theswimmers is neglected, and the fluid is only subjected to the buoyant forcing induced bythe presence of the bacteria, which are denser than the fluid. Although this model doesnot accurately take into account hydrodynamic interactions, it is able to reproducecomplex collective dynamics observed in concentrated bacterial suspensions, such asbioconvection. From a mathematical point of view, both models lead to a minimizationproblem which is solved with a standard Finite Element Method. In order to ensurerobustness, a projection algorithm is used to deal with contacts between particles. Wealso reproduce chemotactic behaviour driven by oxygen: an advection-diffusion equation onthe oxygen concentration is solved in the fluid domain, with a source term accounting foroxygen consumption by the bacteria. The orientations of the individual bacteria aresubjected to random changes, with a frequency that depends on the surrounding oxygenconcentration, in order to favor the direction of the concentration gradient.

In this article a variational reduction method, how to handle the case of heterogenousdomains for the Transport equation, is presented. This method allows to get rid of therestrictions on the size of time steps due to the thin parts of the domain. In the thinpart of the domain, only a differential problem, with respect to the space variable, is tobe approximated numerically. Numerical results are presented with a simple example. Thevariational reduction method can be extended to thin domains multi-branching in 3dimensions, which is a work in progress.

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.

How does DNA, the molecule containing genetic information, change its three-dimensionalshape during the complex cellular processes of replication, recombination and repair? Thisis one of the core questions in molecular biology which cannot be answered without helpfrom mathematical modeling. Basic concepts of topology and geometry can be introduced inundergraduate teaching to help students understand counterintuitive complex structuraltransformations that occur in every living cell. Topoisomerases, a fascinating class ofenzymes involved in replication, recombination and repair, catalyze a change in DNAtopology through a series of highly coordinated mechanistic steps. Undergraduate biologyand mathematics students can visualize and explore the principles of topoisomerase actionby using easily available materials such as Velcro, ribbons, telephone cords, zippers andtubing. These simple toys can be used as powerful teaching tools to engage students inhands-on exploration with the goal of learning about both the mathematics and the biologyof DNA structure.

The interplay between dissipation and long-range repulsive/attractive forces inhomogeneous, dilute, mono-disperse particle systems is studied. Thepseudo-Liouville operator formalism, originallyintroduced for hard-sphere interactions, is modified such that it provides very goodpredictions for systems with weak long-range forces at low densities, with the ratio ofpotential to fluctuation kinetic energy as control parameter. By numerical simulations,the theoretical results are generalized with empirical, density dependentcorrection-functions up to moderate densities.

The main result of this study on dissipative cooling is an analytical prediction for the reduced cooling rate due torepulsive forces and for the increased rate due to attractive forces. In the latter case,surprisingly, for intermediate densities, similar cooling behavior is observed as insystems without long-range interactions. In the attractive case, in general, dissipationleads to inhomogeneities earlier and faster than in the repulsive case.

We present a dynamic geometric model of phyllotaxis based on two postulates, primordiaformation and meristem expansion. We find that Fibonacci, Lucas, bijugate and multijugateare all variations of the same unifying phenomenon and that the difference lies in thechanges in position of initial primordia. We explore the set of all initial positions andcolor-code its points depending on the phyllotactic pattern that arises.

An exploratory study is performed to investigate the use of a time-dependent discreteadjoint methodology for design optimization of a high-lift wing configuration augmentedwith an active flow control system. The location and blowing parameters associated with aseries of jet actuation orifices are used as design variables. In addition, a geometricparameterization scheme is developed to provide a compact set of design variablesdescribing the wing shape. The scaling of the implementation is studied using severalthousand processors and it is found that asynchronous file operations can greatly improvethe overall performance of the approach in such massively parallel environments. Threedesign examples are presented which seek to maximize the mean value of the liftcoefficient for the coupled system, and results demonstrate improvements as high as 27%relative to the lift obtained with non-optimized actuation. This lift gain is more thanthree times the incremental lift provided by the non-optimized actuation.

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.

We study molecular motor-induced microtubule self-organization in dilute and semi-dilutefilament solutions. In the dilute case, we use a probabilistic model of microtubuleinteraction via molecular motors to investigate microtubule bundle dynamics. Microtubulesare modeled as polar rods interacting through fully inelastic, binary collisions. Ourmodel indicates that initially disordered systems of interacting rods exhibit anorientational instability resulting in spontaneous ordering. We study the existence anddynamic interaction of microtubule bundles analytically and numerically. Our resultsreveal a long term attraction and coalescing of bundles indicating a clear coarsening inthe system; microtubule bundles concentrate into fewer orientations on a slow logarithmictime scale. In semi-dilute filament solutions, multiple motors can bind a filament toseveral others and, for a critical motor density, induce a transition to an ordered phasewith a nonzero mean orientation. Motors attach to a pair of filaments and walk along thepair bringing them into closer alignment. We develop a spatially homogenous, mean-fieldtheory that explicitly accounts for a force-dependent detachment rate of motors, which inturn affects the mean and the fluctuations of the net force acting on a filament. We showthat the transition to the oriented state can be both continuous and discontinuous whenthe force-dependent detachment of motors is important.

Waxy Crude Oils (WCO’s) are characterized by the presence of heavy paraffins insufficiently large concentrations. They exhibit quite complex thermodynamical andrheological behaviour and present the peculiar property of giving rise to the formation ofsegregated wax deposits, when temperature falls down the so called WAT, or Wax AppearanceTemperature. In extreme cases, segregated waxes may lead to pipeline occlusion due todeposition on cold walls. In this paper we review the mathematical models formulated todescribe: (i) wax cystallization or thawing in cooling/heating cycles; (ii) the mechanismsof mass transport in saturated non-isothermal solutions; (iii) the experimental deviceused to measure wax solubility and wax diffusivity; (iv) wax deposition in pipelinescarrying a warm, wax-saturated WCO through cold regions; (v) wax deposition accompanied bygelification during the cooling of a WCO under a thermal gradient.

Physics based simulation is widely seen as a way of increasing the information aboutaircraft designs earlier in their definition, thus helping with the avoidance ofunanticipated problems as the design is refined. This paper reports on an effort to assessthe automated use of computational fluid dynamics level aerodynamics for the developmentof tables for flight dynamics analysis at the conceptual stage. These tables are then usedto calculate handling qualities measures. The methodological questions addressed area)geometry and mesh treatment for automated analysis from a high level conceptual aircraftdescription and b) sampling and data fusion to allow the timely calculation of large datatables. The test case used to illustrate the approaches is based on a refined designpassenger jet wind tunnel model. This model is reduced to a conceptual description, andthe ability of this geometry to allow calculations relevant to the final design to bedrawn is then examined. Data tables are then generated and handling qualitiescalculated.

A computer aided method using symbolic computations that enables the calculation of thesource terms (Boltzmann) in Grad’s method of moments is presented. The method is extremelypowerful, easy to program and allows the derivation of balance equations to very highmoments (limited only by computer resources). For sake of demonstration the method isapplied to a simple case: the one-dimensional stationary granular gas under gravity. Themethod should find applications in the field of rarefied gases, as well. Questions ofconvergence, closure are beyond the scope of this article.

There are two mathematical models of elastic walls of healthy and atherosclerotic bloodvessels developed and studied. The models are included in a numerical model of globalblood circulation via recovery of the vessel wall state equation. The joint model allowsus to study the impact of arteries atherosclerotic disease of a set of arteries onregional haemodynamics.

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.

We study pattern-forming instabilities in reaction-advection-diffusion systems. Wedevelop an approach based on Lyapunov-Bloch exponents to figure out the impact of aspatially periodic mixing flow on the stability of a spatially homogeneous state. We dealwith the flows periodic in space that may have arbitrary time dependence. We propose adiscrete in time model, where reaction, advection, and diffusion act as successiveoperators, and show that a mixing advection can lead to a pattern-forming instability in atwo-component system where only one of the species is advected. Physically, this can beexplained as crossing a threshold of Turing instability due to effective increase of oneof the diffusion constants.

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.

In a simple FitzHugh-Nagumo neuronal model with one fast and two slow variables, asequence of period-doubling bifurcations for small-scale oscillations precedes thetransition into the spiking regime. For a wide range of values of the timescale separationparameter, this scenario is recovered numerically. Its relation to the singularlyperturbed integrable system is discussed.

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.

Diffusion preserves the positivity of concentrations, therefore, multicomponent diffusionshould be nonlinear if there exist non-diagonal terms. The vast variety of nonlinearmulticomponent diffusion equations should be ordered and special tools are needed toprovide the systematic construction of the nonlinear diffusion equations formulticomponent mixtures with significant interaction between components. We develop anapproach to nonlinear multicomponent diffusion based on the idea of the reaction mechanismborrowed from chemical kinetics.

Chemical kinetics gave rise to very seminal tools for the modeling of processes. This isthe stoichiometric algebra supplemented by the simple kinetic law. The results of thisinvention are now applied in many areas of science, from particle physics to sociology. Inour work we extend the area of applications onto nonlinear multicomponent diffusion.

We demonstrate, how the mechanism based approach to multicomponent diffusion can beincluded into the general thermodynamic framework, and prove the corresponding dissipationinequalities. To satisfy thermodynamic restrictions, the kinetic law of an elementaryprocess cannot have an arbitrary form. For the general kinetic law (the generalized MassAction Law), additional conditions are proved. The cell–jump formalism gives anintuitively clear representation of the elementary transport processes and, at the sametime, produces kinetic finite elements, a tool for numerical simulation.

We have developed a chemical kinetics simulation that can be used as both an educationaland research tool. The simulator is designed as an accessible, open-source project thatcan be run on a laptop with a student-friendly interface. The application can potentiallybe scaled to run in parallel for large simulations. The simulation has been successfullyused in a classroom setting for teaching basic electrochemical properties. We have shownthat this can be used for simulating fundamental molecular and chemical processes and evensimplified models of predator–prey interactions. By giving the simulated entities spatialextent in the lattice, the particles do not interpenetrate, and clusters of particles canspatially exclude one another. Our simulation demonstrates that spatial inhomogeneityleads to different results than those that are obtained by using standard ordinarydifferential equation models, as previously reported.